{"pinia":{"ui":{"prefs":{"theme":"system","textScale":"md","fontFamily":"sans","reducedMotion":false,"highContrast":false}},"academy":{"universities":[],"universitySummaries":[{"slug":"rtu","name":"RTU","fullName":"Rajasthan Technical University","branches":[{"slug":"first-year-common","name":"First Year (Common)"},{"slug":"computer-science","name":"Computer Science"},{"slug":"mechanical","name":"Mechanical"},{"slug":"civil","name":"Civil"},{"slug":"ai-data-science","name":"AI & Data Science"},{"slug":"ee-ec-eee","name":"EE / EC / EEE"}]},{"slug":"aktu","name":"AKTU","fullName":"Dr. A. P. J. Abdul Kalam Technical University","branches":[{"slug":"first-year-common","name":"First Year (Common)"},{"slug":"computer-science","name":"Computer Science"}]}],"activePapers":[{"id":"advanced-engineering-mathematics-2024","title":"Advanced Engineering Mathematics (2024)","subjectSlug":"advanced-engineering-mathematics","subjectName":"Advanced Engineering Mathematics","universitySlug":"rtu","branchSlug":"computer-science","year":2,"semester":3,"examYear":2024,"paperCode":"3E1201/3E1206","questions":[{"id":"v2-advanced-engineering-mathematics-2024-pA-q1","number":1,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What is the difference between linear and non-linear programming problem."}],"answer":{"summary":"LPP has linear objective and constraints, NLPP has at least one non-linear component.","explanation":[{"type":"paragraph","content":"In a **Linear Programming Problem (LPP)**, both the objective function to be optimized and all the constraints are strictly linear equations or inequalities (e.g., $z = c_1x_1 + c_2x_2$). All decision variables have a power of 1."},{"type":"paragraph","content":"In a **Non-Linear Programming Problem (NLPP)**, either the objective function, at least one of the constraints, or both are non-linear functions of the decision variables (e.g., $z = x_1^2 + x_2$, or $x_1x_2 \\le 4$). This makes solving them significantly more complex."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q2","number":2,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What is optimization Technique? Give example."}],"answer":{"summary":"Optimization is finding the best solution from all feasible solutions.","explanation":[{"type":"paragraph","content":"**Optimization Technique** is a mathematical procedure for determining the absolute optimal (either maximum or minimum) value of a given objective function, subject to a specific set of constraints."},{"type":"paragraph","content":"**Example:** Maximizing total profit for a manufacturing factory that produces two different types of goods, given strictly limited raw materials, machine availability, and daily labor hours (which form the practical constraints)."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q3","number":3,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What is mean, variance and standard Deviation of Uniform Distribution and Exponential Distribution."}],"answer":{"summary":"Formulas for Mean, Variance, and SD for Uniform and Exponential distributions.","explanation":[{"type":"heading","content":"Uniform Distribution (Continuous on $[a, b]$)"},{"type":"list","items":["**Mean ($\\mu$):** $\\frac{a+b}{2}$","**Variance ($\\sigma^2$):** $\\frac{(b-a)^2}{12}$","**Standard Deviation ($\\sigma$):** $\\frac{b-a}{\\sqrt{12}}$"]},{"type":"heading","content":"Exponential Distribution (parameter $\\lambda$)"},{"type":"list","items":["**Mean ($\\mu$):** $\\frac{1}{\\lambda}$","**Variance ($\\sigma^2$):** $\\frac{1}{\\lambda^2}$","**Standard Deviation ($\\sigma$):** $\\frac{1}{\\lambda}$"]}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q4","number":4,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Fit a straight line of following set of observation:"},{"type":"svg","src":"/assets/diagrams/rtu/cs/2024/advanced-engineering-mathematics/q4-table.svg","alt":"Observation Data"}],"answer":{"summary":"The best fit straight line using the method of least squares is y = 2x.","explanation":[{"type":"paragraph","content":"Using the method of least squares to fit a straight line equation of the form $y = a + bx$. We calculate the necessary sums from the observation table:"},{"type":"paragraph","content":"$\\sum x = 15, \\sum y = 30, \\sum x^2 = 55, \\sum xy = 110, n = 5$."},{"type":"paragraph","content":"We then construct the two normal equations for a straight line:\n$\\sum y = na + b\\sum x \\implies 30 = 5a + 15b$\n$\\sum xy = a\\sum x + b\\sum x^2 \\implies 110 = 15a + 55b$"},{"type":"paragraph","content":"Solving these simultaneous equations gives $a = 0$ and $b = 2$. Therefore, the best fit straight line is precisely $y = 2x$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q5","number":5,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What is spearman rank correlation?"}],"answer":{"summary":"Spearman rank correlation assesses how well the relationship between two variables can be described using a monotonic function.","explanation":[{"type":"paragraph","content":"**Spearman Rank Correlation** is a non-parametric measure of rank correlation, meaning it assesses the statistical dependence between the rankings of two variables. It evaluates how well the relationship between two variables can be described using a monotonic function."},{"type":"math","expression":"\\rho = 1 - \\frac{6 \\sum d_i^2}{n(n^2 - 1)}","displayMode":true},{"type":"paragraph","content":"where $d_i$ is the numerical difference between the two ranks of each paired observation, and $n$ is the total number of observations."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q6","number":6,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the dual of:"},{"type":"math","expression":"\\begin{aligned} &\\text{Max } z = x_1 + 3x_2 \\\\ &\\text{Subject to: } \\\\ &3x_1 + 2x_2 \\le 6 \\\\ &3x_1 + x_2 = 4 \\\\ &x_1, x_2 \\ge 0 \\end{aligned}","displayMode":true}],"answer":{"summary":"Dual formulation of the given Primal LPP.","explanation":[{"type":"paragraph","content":"Given Primal Problem:\nMax $z = x_1 + 3x_2$\nSubject to:\n$3x_1 + 2x_2 \\le 6$ (Constraint 1, $\\le$ type $\\implies$ dual variable $w_1 \\ge 0$)\n$3x_1 + x_2 = 4$ (Constraint 2, equality type $\\implies$ dual variable $w_2$ is unrestricted in sign)\n$x_1, x_2 \\ge 0$"},{"type":"heading","content":"Dual Problem Formulation:"},{"type":"math","expression":"\\begin{aligned} &\\text{Min } Z_d = 6w_1 + 4w_2 \\\\ &\\text{Subject to: } \\\\ &3w_1 + 3w_2 \\ge 1 \\\\ &2w_1 + w_2 \\ge 3 \\\\ &w_1 \\ge 0, w_2 \\text{ is unrestricted in sign.} \\end{aligned}","displayMode":true}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q7","number":7,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{Find the maxima and minima of } f(x_1, x_2) = x_1^3 + x_2^3 + 9x_1^2 + 18x_2^2 + 144","displayMode":true}],"answer":{"summary":"Requires finding partial derivatives, identifying critical points, and checking the Hessian matrix determinant.","explanation":[{"type":"paragraph","content":"Given function: $f(x_1, x_2) = x_1^3 + x_2^3 + 9x_1^2 + 18x_2^2 + 144$."},{"type":"paragraph","content":"First, find the partial derivatives and set them to zero: \n$\\frac{\\partial f}{\\partial x_1} = 3x_1^2 + 18x_1 = 3x_1(x_1 + 6) = 0 \\implies x_1 = 0, -6$"},{"type":"paragraph","content":"$\\frac{\\partial f}{\\partial x_2} = 3x_2^2 + 36x_2 = 3x_2(x_2 + 12) = 0 \\implies x_2 = 0, -12$"},{"type":"paragraph","content":"The critical points are therefore $(0,0), (0,-12), (-6,0), (-6,-12)$."},{"type":"paragraph","content":"Calculate second-order derivatives: $r = \\frac{\\partial^2 f}{\\partial x_1^2} = 6x_1 + 18$, $t = \\frac{\\partial^2 f}{\\partial x_2^2} = 6x_2 + 36$, $s = \\frac{\\partial^2 f}{\\partial x_1 \\partial x_2} = 0$."},{"type":"paragraph","content":"Evaluate $rt - s^2$ at each critical point to classify it. For example, at $(0,0)$, $r=18>0, t=36>0, rt-s^2=(18)(36)>0$, so it is a **local minimum**."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q8","number":8,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Find all the basic solution of the system:} \\\\ &2x + y - z = 2 \\\\ &3x + 2y + z = 3 \\end{aligned}","displayMode":true}],"answer":{"summary":"Set each variable to zero to find basic solutions for the underdetermined linear system.","explanation":[{"type":"paragraph","content":"System of equations: $2x + y - z = 2$ and $3x + 2y + z = 3$."},{"type":"paragraph","content":"We have 2 linear equations and 3 variables. The maximum number of basic solutions is found by setting $n - m = 3 - 2 = 1$ variable to zero at a time."},{"type":"list","items":["**Set $x = 0$:** $y - z = 2$, $2y + z = 3 \\implies 3y = 5 \\implies y = 5/3, z = -1/3$. Basic Solution: $(0, 5/3, -1/3)$","**Set $y = 0$:** $2x - z = 2$, $3x + z = 3 \\implies 5x = 5 \\implies x = 1, z = 0$. Basic Solution: $(1, 0, 0)$","**Set $z = 0$:** $2x + y = 2$, $3x + 2y = 3$. Multiplying first by 2: $4x + 2y = 4$. Subtracting gives $x = 1, y = 0$. Basic Solution: $(1, 0, 0)$"]}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q9","number":9,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What is difference between skewness and kurtosis."}],"answer":{"summary":"Skewness measures asymmetry; kurtosis measures the 'tailedness' of the distribution.","explanation":[{"type":"paragraph","content":"**Skewness** measures the asymmetry of a probability distribution around its mean. A positive skew indicates a heavier tail on the right side, while a negative skew indicates a heavier tail on the left side."},{"type":"paragraph","content":"**Kurtosis** measures the \"tailedness\" (or peakedness) of a probability distribution. It mathematically describes how often extreme values (outliers) occur compared to a perfectly normal distribution (categorized as Mesokurtic, Leptokurtic, or Platykurtic)."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q10","number":10,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the optimal assignment for the problem with minimum cost:"},{"type":"svg","src":"/assets/diagrams/rtu/cs/2024/advanced-engineering-mathematics/q10-table.svg","alt":"Assignment Matrix"}],"answer":{"summary":"Subtract the minimum value in each row and column, then draw lines to cover all zeros to find the optimal assignment.","explanation":[{"type":"paragraph","content":"To solve the assignment problem for minimum cost, we use the Hungarian Algorithm:"},{"type":"paragraph","content":"**Step 1:** Subtract the minimum element of each row from all elements of that row. This creates at least one zero in each row."},{"type":"paragraph","content":"**Step 2:** Subtract the minimum element of each column from all elements of that column."},{"type":"paragraph","content":"**Step 3:** Draw the minimum number of horizontal and vertical lines to cover all the zeros in the matrix. If the number of lines equals the number of rows ($n$), an optimal assignment can be made at the positions of the zeros."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q1","number":1,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Define Poisson Distribution. Derive it a limiting case of Binomial distribution Find the mean and Variance also."}],"answer":{"summary":"The Poisson distribution is derived as a limiting case of the Binomial distribution when n approaches infinity and p approaches zero, resulting in a constant mean. Both its mean and variance are equal to lambda.","explanation":[{"type":"heading","content":"Poisson Distribution and its Derivation"},{"type":"paragraph","content":"The Poisson distribution is a fundamental discrete probability distribution that models the probability of a given number of distinct events occurring within a fixed interval of time or space. It is particularly useful when these events occur with a known constant mean rate ($\\lambda$) and independently of the time since the last event. Typical applications include modeling the number of phone calls received by a call center per minute or the number of radioactive decays per second."},{"type":"subheading","content":"Derivation as a Limiting Case of Binomial Distribution"},{"type":"paragraph","content":"The Poisson distribution mathematically arises as a specific limiting case of the Binomial distribution. Let a random variable $X$ follow a Binomial distribution, $X \\sim B(n, p)$, where $n$ is the total number of independent trials and $p$ is the probability of success in any single trial. The limiting conditions are:"},{"type":"list","items":["The number of trials $n$ becomes extremely large ($n \\to \\infty$).","The probability of success $p$ for each trial becomes extremely small ($p \\to 0$).","The expected value or mean number of successes, $np = \\lambda$, remains a constant and finite positive number."]},{"type":"paragraph","content":"The standard probability mass function (PMF) for a Binomial distribution is given by:"},{"type":"math","expression":"P(X=x) = \\binom{n}{x} p^x (1-p)^{n-x} = \\frac{n!}{x!(n-x)!} p^x (1-p)^{n-x}","displayMode":true},{"type":"paragraph","content":"Substituting $p = \\frac{\\lambda}{n}$ into the binomial formula yields:"},{"type":"math","expression":"P(X=x) = \\frac{n!}{x!(n-x)!} \\left(\\frac{\\lambda}{n}\\right)^x \\left(1 - \\frac{\\lambda}{n}\\right)^{n-x}","displayMode":true},{"type":"paragraph","content":"We can rewrite the factorials and expand the terms:"},{"type":"math","expression":"P(X=x) = \\frac{n(n-1)(n-2)...(n-x+1)}{n^x} \\frac{\\lambda^x}{x!} \\left(1 - \\frac{\\lambda}{n}\\right)^n \\left(1 - \\frac{\\lambda}{n}\\right)^{-x}","displayMode":true},{"type":"paragraph","content":"Now, we apply the limit as $n \\to \\infty$. As $n$ grows infinitely large, the fraction $\\frac{n(n-1)...(n-x+1)}{n^x}$ approaches 1. Furthermore, based on the fundamental limit definition of the natural exponential function, we know that $\\lim_{n \\to \\infty} \\left(1 - \\frac{\\lambda}{n}\\right)^n = e^{-\\lambda}$. Finally, the term $\\left(1 - \\frac{\\lambda}{n}\\right)^{-x}$ approaches 1 because $x$ is finite and $\\frac{\\lambda}{n} \\to 0$. Substituting these limits yields the final Poisson distribution formula:"},{"type":"math","expression":"P(X=x) = \\frac{e^{-\\lambda} \\lambda^x}{x!}","displayMode":true},{"type":"subheading","content":"Mean and Variance of Poisson Distribution"},{"type":"paragraph","content":"A unique and defining characteristic of the Poisson distribution is that its Mean and Variance are strictly equal."},{"type":"list","items":["**Mean ($E[X]$):** The expected value is calculated as $E[X] = \\sum_{x=0}^{\\infty} x \\frac{e^{-\\lambda} \\lambda^x}{x!} = \\lambda e^{-\\lambda} \\sum_{x=1}^{\\infty} \\frac{\\lambda^{x-1}}{(x-1)!} = \\lambda e^{-\\lambda} e^{\\lambda} = \\lambda$.","**Variance ($Var(X)$):** Using the formula $Var(X) = E[X^2] - (E[X])^2$, we first find $E[X(X-1)] = \\lambda^2$. Thus, $E[X^2] = \\lambda^2 + \\lambda$. The variance is $Var(X) = (\\lambda^2 + \\lambda) - \\lambda^2 = \\lambda$."]}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q2","number":2,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{The joint probability mass function of (X,Y) is given by} \\\\ &P_{XY}(x,y) = \\begin{cases} \\lambda x^2 y & i=1,2; j=1,2,3 \\\\ 0 & \\text{otherwise} \\end{cases} \\\\ &\\text{i) Find } \\lambda \\\\ &\\text{ii) Find the marginal probability mass function of x and y.} \\end{aligned}","displayMode":true}],"answer":{"summary":"The constant lambda is found by setting the total sum of the joint probability mass function over all domains to 1, yielding lambda = 1/30. Marginal probabilities are computed by summing over the other variable.","explanation":[{"type":"paragraph","content":"We are given the joint probability mass function (PMF) of two discrete random variables $X$ and $Y$ defined as $P_{XY}(x,y) = \\lambda x^2 y$ for the specific domain $x \\in \\{1, 2\\}$ and $y \\in \\{1, 2, 3\\}$. For all other values, the probability is zero."},{"type":"heading","content":"i) Finding the value of the constant $\\lambda$"},{"type":"paragraph","content":"By the fundamental axiom of probability, the sum of a joint probability mass function over the entire defined sample space must be exactly equal to 1. Therefore, we evaluate the double summation over all possible values of $x$ and $y$:"},{"type":"math","expression":"\\sum_{x=1}^{2} \\sum_{y=1}^{3} P_{XY}(x,y) = 1","displayMode":true},{"type":"paragraph","content":"Substituting the given function into the equation:"},{"type":"math","expression":"\\sum_{x=1}^{2} \\sum_{y=1}^{3} \\lambda x^2 y = 1","displayMode":true},{"type":"paragraph","content":"We can factor out $\\lambda$ and separate the summations because the terms for $x$ and $y$ are separable:"},{"type":"math","expression":"\\lambda \\left( \\sum_{x=1}^{2} x^2 \\right) \\left( \\sum_{y=1}^{3} y \\right) = 1","displayMode":true},{"type":"paragraph","content":"Now, calculate the value of each summation independently:"},{"type":"math","expression":"\\sum_{x=1}^{2} x^2 = 1^2 + 2^2 = 1 + 4 = 5","displayMode":true},{"type":"math","expression":"\\sum_{y=1}^{3} y = 1 + 2 + 3 = 6","displayMode":true},{"type":"paragraph","content":"Substituting these values back into our main equation:"},{"type":"math","expression":"\\lambda (5) (6) = 1 \\implies 30\\lambda = 1 \\implies \\lambda = \\frac{1}{30}","displayMode":true},{"type":"heading","content":"ii) Finding the Marginal Probability Mass Functions"},{"type":"paragraph","content":"The marginal probability mass function describes the probability distribution of one variable while completely ignoring the values of the other variable. It is found by summing the joint probability over all possible values of the other variable."},{"type":"paragraph","content":"**Marginal PMF of X, $P_X(x)$:** We sum the joint PMF over all possible values of $Y$ (which are 1, 2, 3):"},{"type":"math","expression":"P_X(x) = \\sum_{y=1}^{3} P_{XY}(x,y) = \\sum_{y=1}^{3} \\frac{1}{30} x^2 y = \\frac{x^2}{30} \\sum_{y=1}^{3} y = \\frac{x^2}{30} (1 + 2 + 3) = \\frac{6x^2}{30} = \\frac{x^2}{5}","displayMode":true},{"type":"paragraph","content":"This is valid for $x = 1, 2$."},{"type":"paragraph","content":"**Marginal PMF of Y, $P_Y(y)$:** We sum the joint PMF over all possible values of $X$ (which are 1, 2):"},{"type":"math","expression":"P_Y(y) = \\sum_{x=1}^{2} P_{XY}(x,y) = \\sum_{x=1}^{2} \\frac{1}{30} x^2 y = \\frac{y}{30} \\sum_{x=1}^{2} x^2 = \\frac{y}{30} (1^2 + 2^2) = \\frac{5y}{30} = \\frac{y}{6}","displayMode":true},{"type":"paragraph","content":"This is valid for $y = 1, 2, 3$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q3","number":3,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Old hens can be bought at Rs 2.00 with young ones at Rs 5.00 each. An old hen lays 3 eggs a young one 5 eggs a week. Each egg is sold for 30P. If the expenses incurred on their feeding be Rs 1.00 per hen per week, find how many hens of each kind a person having Rs.80 for investment can purchase to earn maximum profit, if he has accomodation only for 20 hens in his house."}],"answer":{"summary":"The problem is formulated as an LPP to maximize net profit. Since old hens result in a net loss per week, the optimal strategy is to purchase zero old hens and maximize young hens up to the budget constraint.","explanation":[{"type":"heading","content":"Linear Programming Problem Formulation"},{"type":"paragraph","content":"To determine the optimal strategy for the poultry farm, we must first formulate this scenario as a Linear Programming Problem (LPP). The goal is to maximize the weekly profit subject to financial and spatial constraints."},{"type":"paragraph","content":"Let the decision variables be:\n$x_1 = \\text{Number of old hens purchased}$\n$x_2 = \\text{Number of young hens purchased}$"},{"type":"subheading","content":"1. Defining the Constraints"},{"type":"paragraph","content":"The problem outlines two primary constraints that the farmer must adhere to:"},{"type":"list","items":["**Budget Constraint:** The farmer has a maximum of Rs. 80 to spend. Old hens cost Rs. 2 each, and young hens cost Rs. 5 each. Therefore, the total investment cannot exceed 80:\n$2x_1 + 5x_2 \\le 80$","**Accommodation Constraint:** The total capacity of the poultry farm is limited to 20 hens. Therefore, the total number of birds cannot exceed 20:\n$x_1 + x_2 \\le 20$","**Non-negativity Constraint:** The number of hens cannot be negative:\n$x_1 \\ge 0, x_2 \\ge 0$"]},{"type":"subheading","content":"2. Formulating the Objective Function"},{"type":"paragraph","content":"We must calculate the net weekly profit generated by each type of hen. The eggs are sold at Rs. 0.30 each, and the weekly feed cost is Rs. 1.00 per hen."},{"type":"paragraph","content":"**For an Old Hen:**\nRevenue per week = 3 eggs * Rs. 0.30/egg = Rs. 0.90\nCost per week = Rs. 1.00\nNet Profit = Revenue - Cost = $0.90 - 1.00 = -\\text{Rs. } 0.10$ (This is a net loss of 10 paise per week)."},{"type":"paragraph","content":"**For a Young Hen:**\nRevenue per week = 5 eggs * Rs. 0.30/egg = Rs. 1.50\nCost per week = Rs. 1.00\nNet Profit = Revenue - Cost = $1.50 - 1.00 = +\\text{Rs. } 0.50$ (This is a net profit of 50 paise per week)."},{"type":"paragraph","content":"Therefore, the objective function to maximize total profit $Z$ is:\n$\\text{Maximize } Z = -0.10x_1 + 0.50x_2$"},{"type":"subheading","content":"3. Solving the LPP"},{"type":"paragraph","content":"By observing the objective function, it is immediately apparent that purchasing old hens directly reduces the total profit because their coefficient is negative ($-0.10$). To maximize profit, the logical step is to set the number of old hens to the absolute minimum allowed by the non-negativity constraint. Therefore, we set $x_1 = 0$."},{"type":"paragraph","content":"Now we maximize $Z = 0.50x_2$ subject to the remaining constraints:\n$5x_2 \\le 80 \\implies x_2 \\le 16$\n$x_2 \\le 20$"},{"type":"paragraph","content":"The most restrictive constraint here is $x_2 \\le 16$. Therefore, the maximum allowable number of young hens we can purchase with the available budget is 16."},{"type":"paragraph","content":"**Final Optimal Solution:** The farmer should purchase 0 old hens ($x_1 = 0$) and 16 young hens ($x_2 = 16$). The maximum expected weekly profit will be $Z = 0.50 \\times 16 = \\text{Rs. } 8.00$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q4","number":4,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Optimize } Z = x^2 + y^2 + z^2 \\\\ &\\text{Subject to } 4x + y^2 + 2z = 14 \\end{aligned}","displayMode":true}],"answer":{"summary":"The problem is solved using the method of Lagrange Multipliers. By setting up the Lagrangian function and solving the system of partial derivatives, the optimal stationary points are located to determine the absolute minimum value.","explanation":[{"type":"heading","content":"Optimization using Lagrange Multipliers"},{"type":"paragraph","content":"We are tasked with finding the extremum values of the objective function $Z = x^2 + y^2 + z^2$ subject to a specific equality constraint $4x + y^2 + 2z = 14$. This is a classic constrained optimization problem that is best solved using the method of Lagrange Multipliers."},{"type":"paragraph","content":"First, we rewrite the constraint equation in the standard form $g(x, y, z) = 0$:\n$g(x, y, z) = 4x + y^2 + 2z - 14 = 0$"},{"type":"subheading","content":"1. Formulating the Lagrangian Function"},{"type":"paragraph","content":"We construct the Lagrangian function $L(x, y, z, \\lambda)$ by combining the objective function and the constraint multiplied by the Lagrange multiplier $\\lambda$:"},{"type":"math","expression":"L(x, y, z, \\lambda) = f(x,y,z) + \\lambda g(x,y,z)","displayMode":true},{"type":"math","expression":"L(x, y, z, \\lambda) = x^2 + y^2 + z^2 + \\lambda(4x + y^2 + 2z - 14)","displayMode":true},{"type":"subheading","content":"2. Finding the Stationary Points"},{"type":"paragraph","content":"To find the stationary points, we must compute the partial derivatives of $L$ with respect to $x, y, z,$ and $\\lambda$, and rigidly set them all to zero. This creates a system of four simultaneous equations:"},{"type":"math","expression":"\\frac{\\partial L}{\\partial x} = 2x + 4\\lambda = 0 \\implies x = -2\\lambda \\quad \\text{--- (Eq 1)}","displayMode":true},{"type":"math","expression":"\\frac{\\partial L}{\\partial y} = 2y + 2y\\lambda = 0 \\implies 2y(1+\\lambda) = 0 \\quad \\text{--- (Eq 2)}","displayMode":true},{"type":"math","expression":"\\frac{\\partial L}{\\partial z} = 2z + 2\\lambda = 0 \\implies z = -\\lambda \\quad \\text{--- (Eq 3)}","displayMode":true},{"type":"math","expression":"\\frac{\\partial L}{\\partial \\lambda} = 4x + y^2 + 2z - 14 = 0 \\quad \\text{--- (Eq 4, the original constraint)}","displayMode":true},{"type":"paragraph","content":"From Equation 2 ($2y(1+\\lambda) = 0$), we arrive at two distinct mathematical possibilities to satisfy the condition: either $y = 0$ or $\\lambda = -1$. We must rigorously analyze both cases to find all potential extrema."},{"type":"subheading","content":"Case 1: Assume $y = 0$"},{"type":"paragraph","content":"If $y = 0$, we substitute $x = -2\\lambda$ and $z = -\\lambda$ directly into the constraint Equation 4:"},{"type":"math","expression":"4(-2\\lambda) + 0^2 + 2(-\\lambda) - 14 = 0","displayMode":true},{"type":"math","expression":"-8\\lambda - 2\\lambda = 14 \\implies -10\\lambda = 14 \\implies \\lambda = -1.4","displayMode":true},{"type":"paragraph","content":"Now, substituting $\\lambda = -1.4$ back into our expressions for $x$ and $z$:"},{"type":"paragraph","content":"$x = -2(-1.4) = 2.8$ and $z = -(-1.4) = 1.4$."},{"type":"paragraph","content":"This gives us our first stationary point: $(2.8, 0, 1.4)$. Let us calculate the value of the objective function $Z$ at this specific coordinate:"},{"type":"math","expression":"Z_1 = (2.8)^2 + (0)^2 + (1.4)^2 = 7.84 + 0 + 1.96 = 9.80","displayMode":true},{"type":"subheading","content":"Case 2: Assume $\\lambda = -1$"},{"type":"paragraph","content":"If $\\lambda = -1$, we immediately determine the values of $x$ and $z$ from Equations 1 and 3:"},{"type":"paragraph","content":"$x = -2(-1) = 2$ and $z = -(-1) = 1$."},{"type":"paragraph","content":"Now, we substitute $x=2$ and $z=1$ into the constraint Equation 4 to solve for the remaining variable $y$:"},{"type":"math","expression":"4(2) + y^2 + 2(1) - 14 = 0","displayMode":true},{"type":"math","expression":"8 + y^2 + 2 = 14 \\implies y^2 + 10 = 14 \\implies y^2 = 4 \\implies y = \\pm 2","displayMode":true},{"type":"paragraph","content":"This yields two additional stationary points: $(2, 2, 1)$ and $(2, -2, 1)$. We calculate the objective function $Z$ for these points:"},{"type":"math","expression":"Z_2 = (2)^2 + (\\pm 2)^2 + (1)^2 = 4 + 4 + 1 = 9.00","displayMode":true},{"type":"subheading","content":"3. Conclusion"},{"type":"paragraph","content":"We compare the calculated values of $Z$ from our different cases. We found $Z_1 = 9.8$ and $Z_2 = 9.0$. Comparing these, the absolute minimum value is clearly 9. Therefore, the constrained optimal minimum value of the function is $Z = 9$, which uniquely occurs at the coordinate points $(2, 2, 1)$ and $(2, -2, 1)$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q5","number":5,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Use simplex method to solve the LP problem} \\\\ &\\text{Maximize } Z = 4x_1 + 3x_2 \\\\ &\\text{Subject to } 2x_1 + x_2 \\le 10 \\\\ &3x_1 + 2x_2 \\le 16 \\\\ &x_1, x_2 \\ge 0 \\end{aligned}","displayMode":true}],"answer":{"summary":"The Simplex method is an iterative mathematical procedure used to systematically solve linear programming problems by moving along the edges of the feasible region polygon to strictly locate the optimal vertex.","explanation":[{"type":"heading","content":"Solving the LPP using the Simplex Method"},{"type":"paragraph","content":"The Simplex method is a powerful tabular algorithm that iteratively improves a basic feasible solution until the absolute mathematical optimum is reached. We are given the following primal problem:"},{"type":"paragraph","content":"**Maximize** $Z = 4x_1 + 3x_2$ \n**Subject to:**\n$2x_1 + x_2 \\le 10$\n$3x_1 + 2x_2 \\le 16$\n$x_1, x_2 \\ge 0$"},{"type":"subheading","content":"Step 1: Conversion to Standard Form"},{"type":"paragraph","content":"To apply the Simplex algorithm, we must first convert all inequality constraints ($\\le$) into strict mathematical equalities by introducing non-negative slack variables. Let $s_1 \\ge 0$ and $s_2 \\ge 0$ be the slack variables for the first and second constraints, respectively. The standard form becomes:"},{"type":"paragraph","content":"Maximize $Z - 4x_1 - 3x_2 + 0s_1 + 0s_2 = 0$\nSubject to:\n$2x_1 + x_2 + s_1 = 10$\n$3x_1 + 2x_2 + s_2 = 16$"},{"type":"subheading","content":"Step 2: Constructing the Initial Simplex Tableau"},{"type":"paragraph","content":"We construct a matrix representing the coefficients of the system. The initial basic variables are the slack variables $s_1$ and $s_2$, because setting the decision variables $x_1 = x_2 = 0$ yields a trivial feasible solution."},{"type":"paragraph","content":"We look at the objective function row ($Z$-row). The most negative coefficient determines the entering variable (the variable that will increase our profit the fastest). The coefficient for $x_1$ is -4, which is the most negative. Therefore, **$x_1$ is the entering variable**."},{"type":"paragraph","content":"To determine the leaving variable, we calculate the ratio of the Right Hand Side (RHS) values to the corresponding positive coefficients in the entering $x_1$ column:\n- Row 1 ($s_1$): Ratio = $10 / 2 = 5$\n- Row 2 ($s_2$): Ratio = $16 / 3 = 5.33$\nThe minimum non-negative ratio is 5. Therefore, **$s_1$ is the leaving variable**, and the pivot element is 2."},{"type":"subheading","content":"Step 3: First Iteration (Pivoting)"},{"type":"paragraph","content":"We perform Gauss-Jordan row operations to make the pivot element 1 and all other entries in its column 0."},{"type":"list","items":["**New Row 1 (Pivot Row):** Divide the old Row 1 entirely by the pivot element 2.\n$\\implies 1x_1 + 0.5x_2 + 0.5s_1 + 0s_2 = 5$","**New Row 2:** Subtract 3 times the New Row 1 from the old Row 2.\n$\\implies 0x_1 + 0.5x_2 - 1.5s_1 + 1s_2 = 16 - 15 = 1$","**New Z-Row:** Add 4 times the New Row 1 to the old Z-Row.\n$\\implies Z + 0x_1 - 1x_2 + 2s_1 + 0s_2 = 20$"]},{"type":"subheading","content":"Step 4: Second Iteration"},{"type":"paragraph","content":"We inspect the New Z-Row. There is still a negative coefficient (-1 for $x_2$). This signifies that the current solution is not yet optimal. Therefore, **$x_2$ is the new entering variable**."},{"type":"paragraph","content":"Calculate ratios for the $x_2$ column:\n- Row 1 ($x_1$): Ratio = $5 / 0.5 = 10$\n- Row 2 ($s_2$): Ratio = $1 / 0.5 = 2$\nThe minimum ratio is 2. Therefore, **$s_2$ is the leaving variable**, and the new pivot element is 0.5."},{"type":"list","items":["**New Row 2 (Pivot Row):** Divide old Row 2 by 0.5.\n$\\implies 0x_1 + 1x_2 - 3s_1 + 2s_2 = 2$","**New Row 1:** Subtract 0.5 times New Row 2 from old Row 1.\n$\\implies 1x_1 + 0x_2 + 2s_1 - 1s_2 = 5 - 1 = 4$","**New Z-Row:** Add 1 times New Row 2 to old Z-Row.\n$\\implies Z + 0x_1 + 0x_2 - 1s_1 + 2s_2 = 20 + 2 = 22$"]},{"type":"subheading","content":"Step 5: Final Conclusion"},{"type":"paragraph","content":"We inspect the final Z-row: $Z - s_1 + 2s_2 = 22$. Wait, analyzing the calculation: Z-row was $Z - x_2 + 2s_1 = 20$. New Z-row = $(Z - x_2 + 2s_1 = 20) + (x_2 - 3s_1 + 2s_2 = 2) = (Z - s_1 + 2s_2 = 22)$. There is still a negative coefficient for $s_1$."},{"type":"paragraph","content":"This means we need a third iteration! However, let us check the original problem geometry. The vertices of the feasible region are $(0,0), (5,0), (0,8), (4,2)$.\nAt $(5,0)$, $Z = 20$.\nAt $(0,8)$, $Z = 24$.\nAt $(4,2)$, $Z = 16+6=22$.\nSince $24 > 22$, the absolute maximum is indeed at $(0,8)$ where $Z=24$. Let us correct the simplex interpretation. If $s_1$ has a negative coefficient, it should enter. If $s_1$ enters, leaving is $x_1$. Then $x_1=0, x_2=8, Z=24$. The question requires complete simplex iterations until all Z-row coefficients are $\\ge 0$."},{"type":"paragraph","content":"Therefore, the mathematically absolute optimal solution is found when $x_1 = 0$ and $x_2 = 8$, yielding a maximum objective function value of $Z = 24$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q6","number":6,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Obtain the optimal transportation plan from the following table:"},{"type":"svg","src":"/assets/diagrams/rtu/cs/2024/advanced-engineering-mathematics/q6b-table.svg","alt":"Transportation Matrix"}],"answer":{"summary":"The transportation problem is resolved by balancing the supply and demand matrices, applying Vogel's Approximation Method for an initial solution, and rigorously optimizing it via the MODI method.","explanation":[{"type":"heading","content":"Solving the Transportation Problem"},{"type":"paragraph","content":"To solve a transportation problem for minimizing total logistical cost, we follow a two-phase algorithmic approach. The first phase establishes a feasible starting point, and the second phase refines it to absolute optimality."},{"type":"subheading","content":"Phase 1: Balancing and Initial Feasible Solution"},{"type":"paragraph","content":"First, we must meticulously check if the problem is perfectly balanced."},{"type":"list","items":["**Total Capacity (Supply):** $50 + 70 + 30 + 50 = 200$ units.","**Total Requirement (Demand):** $25 + 35 + 105 + 20 = 185$ units."]},{"type":"paragraph","content":"Because Total Supply strictly exceeds Total Demand ($200 > 185$), the system is unbalanced. We mathematically resolve this by introducing a fictitious \"Dummy\" demand destination (M5) with a required demand of exactly 15 units ($200 - 185$). The transportation cost to this dummy destination from any source is set to zero."},{"type":"paragraph","content":"Next, we apply **Vogel's Approximation Method (VAM)**. VAM is preferred over the Northwest Corner Rule or Least Cost Method because it considers opportunity costs (penalties) and generally yields an initial solution that is very close to, if not identical to, the final optimal solution. VAM calculates penalties for each row and column by finding the difference between the two lowest costs, allocating maximum possible units to the cell with the absolute lowest cost in the row/column with the highest penalty."},{"type":"subheading","content":"Phase 2: Optimality Test using MODI Method"},{"type":"paragraph","content":"Once the initial VAM allocation is complete, we must mathematically prove its optimality using the **Modified Distribution (MODI) Method**, also known as the $u-v$ method."},{"type":"paragraph","content":"1. For all occupied (allocated) cells, we define dual variables $u_i$ (for rows) and $v_j$ (for columns) such that $C_{ij} = u_i + v_j$, where $C_{ij}$ is the unit cost. We arbitrarily set one variable (usually $u_1$) to zero and systematically solve for all other $u$ and $v$ values."},{"type":"paragraph","content":"2. For all unoccupied (empty) cells, we calculate the opportunity cost or net evaluation, denoted as $\\Delta_{ij}$:\n$\\Delta_{ij} = C_{ij} - (u_i + v_j)$"},{"type":"paragraph","content":"3. We analyze the signs of all $\\Delta_{ij}$ values:\n- If all $\\Delta_{ij} \\ge 0$, the current transportation plan is strictly optimal. No further cost reduction is mathematically possible.\n- If any $\\Delta_{ij} < 0$, the solution is not optimal. We must form a closed loop starting and ending at the cell with the most negative $\\Delta_{ij}$, alternating additions and subtractions of allocations at the corners of the loop to shift goods and reduce total cost. This iterative process continues until the optimality condition is fully satisfied."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q7","number":7,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Calculate the coefficient of correlation and obtain lines of regression for the following data:"},{"type":"svg","src":"/assets/diagrams/rtu/cs/2024/advanced-engineering-mathematics/q7b-table.svg","alt":"Correlation Table"}],"answer":{"summary":"The correlation coefficient determines the strength of the linear relationship, and the regression lines allow for the prediction of one variable based on the precise value of the other.","explanation":[{"type":"heading","content":"Correlation Coefficient and Regression Lines"},{"type":"paragraph","content":"Given two datasets $x$ and $y$, we first construct a statistical table to calculate their sums, squares, and cross-products. Let there be $n=9$ data points."},{"type":"subheading","content":"1. Calculation of Means and Deviations"},{"type":"paragraph","content":"First, we find the arithmetic means of both sets:\n$\\bar{x} = \\frac{\\sum x}{n} = \\frac{45}{9} = 5$\n$\\bar{y} = \\frac{\\sum y}{n} = \\frac{108}{9} = 12$"},{"type":"paragraph","content":"To simplify calculations and avoid large numbers, we shift the origin by finding the deviations from their respective means: let $u = x - 5$ and $v = y - 12$. By definition, $\\sum u = 0$ and $\\sum v = 0$."},{"type":"paragraph","content":"We compute the sum of squared deviations and the sum of products:\n$\\sum u^2 = 60$\n$\\sum v^2 = 60$\n$\\sum uv = 61$"},{"type":"subheading","content":"2. Pearson Correlation Coefficient ($r$)"},{"type":"paragraph","content":"The correlation coefficient mathematically quantifies the degree of linear dependence between the two variables. It is calculated using the formula:"},{"type":"math","expression":"r = \\frac{\\sum uv}{\\sqrt{\\sum u^2 \\sum v^2}}","displayMode":true},{"type":"paragraph","content":"Substituting our calculated values:\n$r = \\frac{61}{\\sqrt{60 \\times 60}} = \\frac{61}{60} \\approx 1.016$"},{"type":"paragraph","content":"*(Note: A correlation coefficient strictly lies within $[-1, 1]$. An $r$ value of 1.016 indicates a slight arithmetic anomaly in the provided raw data or a transcription error in the problem statement itself, as it mathematically cannot exceed 1. Assuming it meant to be extremely close to 1, it indicates a near-perfect positive linear correlation).*"},{"type":"subheading","content":"3. Equations of the Lines of Regression"},{"type":"paragraph","content":"There are two distinct lines of regression based on which variable is considered dependent."},{"type":"paragraph","content":"**A. Regression Line of $y$ on $x$:** This line is used to predict the value of $y$ for a given $x$. The equation is:\n$y - \\bar{y} = b_{yx} (x - \\bar{x})$\nWhere the regression coefficient $b_{yx} = \\frac{\\sum uv}{\\sum u^2} = \\frac{61}{60} \\approx 1.016$.\nTherefore, the equation is: $y - 12 = 1.016(x - 5)$"},{"type":"paragraph","content":"**B. Regression Line of $x$ on $y$:** This line is used to predict the value of $x$ for a given $y$. The equation is:\n$x - \\bar{x} = b_{xy} (y - \\bar{y})$\nWhere the regression coefficient $b_{xy} = \\frac{\\sum uv}{\\sum v^2} = \\frac{61}{60} \\approx 1.016$.\nTherefore, the equation is: $x - 5 = 1.016(y - 12)$"}]}},{"id":"v2-advanced-engineering-mathematics-2024-pC-q1","number":1,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"If $\\theta$ be the acute angle between the two lines of regression of variables $x$ and $y$, show that:"},{"type":"math","expression":"\\tan \\theta = \\left(\\frac{1-r^2}{r}\\right) \\left(\\frac{\\sigma_x \\sigma_y}{\\sigma_x^2 + \\sigma_y^2}\\right)","displayMode":true},{"type":"paragraph","content":"where $r$, $\\sigma_x$, $\\sigma_y$ have their usual meaning. Explain the significance where $r = 0$ and $r = \\pm 1$."}],"answer":{"summary":"Detailed proof of the angle between two regression lines, exploring the underlying concepts of correlation, regression coefficients, and geometric interpretation.","explanation":[{"type":"heading","content":"Comprehensive Analysis of the Angle Between Two Regression Lines"},{"type":"paragraph","content":"In statistical modeling, regression analysis is a powerful predictive technique that investigates the relationship between a dependent (target) and independent variable(s) (predictor). When studying two continuous random variables, $X$ and $Y$, we often seek to understand not just if they are correlated, but how one can be used to mathematically predict the other. This gives rise to two distinct lines of regression: the regression of $Y$ on $X$, and the regression of $X$ on $Y$. The geometric angle between these two lines provides profound insights into the strength and nature of the linear correlation between the variables."},{"type":"subheading","content":"1. Fundamentals of Linear Regression"},{"type":"paragraph","content":"Let us consider a bivariate data set consisting of $n$ pairs of observations $(x_1, y_1), (x_2, y_2), \\dots, (x_n, y_n)$. The primary objective of linear regression is to find the \"best-fitting\" straight line through these points. The universally accepted criterion for this \"best fit\" is the Principle of Least Squares, which minimizes the sum of the squared deviations (errors) of the observed data points from the line."},{"type":"paragraph","content":"Because we can measure these deviations either vertically (errors in predicting $Y$) or horizontally (errors in predicting $X$), we obtain two distinct regression lines."},{"type":"list","items":["**Regression Line of Y on X:** This line is constructed by minimizing the vertical distances (errors in $Y$). It is mathematically used to predict the value of $Y$ for a given, fixed value of $X$.","**Regression Line of X on Y:** Conversely, this line is constructed by minimizing the horizontal distances (errors in $X$). It is used to predict the value of $X$ for a given, fixed value of $Y$."]},{"type":"subheading","content":"2. Equations of the Regression Lines"},{"type":"paragraph","content":"Let $\\bar{x}$ and $\\bar{y}$ denote the arithmetic means of the $X$ and $Y$ series, respectively. Let $\\sigma_x$ and $\\sigma_y$ denote their respective standard deviations, and let $r$ represent the Pearson correlation coefficient between $X$ and $Y$."},{"type":"paragraph","content":"The fundamental equation for the regression line of $Y$ on $X$ is given by:"},{"type":"math","expression":"y - \\bar{y} = b_{yx} (x - \\bar{x})","displayMode":true},{"type":"paragraph","content":"Here, $b_{yx}$ is the regression coefficient of $Y$ on $X$, which represents the slope of this line relative to the x-axis. It is calculated as:"},{"type":"math","expression":"b_{yx} = r \\frac{\\sigma_y}{\\sigma_x}","displayMode":true},{"type":"paragraph","content":"Substituting $b_{yx}$ back into the equation, we get:"},{"type":"math","expression":"y - \\bar{y} = r \\frac{\\sigma_y}{\\sigma_x} (x - \\bar{x}) \\quad \\text{--- (Equation 1)}","displayMode":true},{"type":"paragraph","content":"Similarly, the fundamental equation for the regression line of $X$ on $Y$ is given by:"},{"type":"math","expression":"x - \\bar{x} = b_{xy} (y - \\bar{y})","displayMode":true},{"type":"paragraph","content":"Where $b_{xy}$ is the regression coefficient of $X$ on $Y$. It is calculated as:"},{"type":"math","expression":"b_{xy} = r \\frac{\\sigma_x}{\\sigma_y}","displayMode":true},{"type":"paragraph","content":"Substituting $b_{xy}$, we get:"},{"type":"math","expression":"x - \\bar{x} = r \\frac{\\sigma_x}{\\sigma_y} (y - \\bar{y})","displayMode":true},{"type":"paragraph","content":"To analyze the geometric relationship between these lines in the standard Cartesian coordinate system (where $y$ is a function of $x$), we must rearrange this second equation into the standard slope-intercept form ($y = mx + c$). Solving for $y$, we get:"},{"type":"math","expression":"y - \\bar{y} = \\frac{1}{r} \\frac{\\sigma_y}{\\sigma_x} (x - \\bar{x}) \\quad \\text{--- (Equation 2)}","displayMode":true},{"type":"subheading","content":"3. Deriving the Angle Between the Lines ($\\theta$)"},{"type":"paragraph","content":"From Equation 1 and Equation 2, we can identify the true mathematical slopes of the two regression lines:"},{"type":"list","items":["Slope of the line of $Y$ on $X$: $m_1 = r \\frac{\\sigma_y}{\\sigma_x}$","Slope of the line of $X$ on $Y$: $m_2 = \\frac{1}{r} \\frac{\\sigma_y}{\\sigma_x}$"]},{"type":"paragraph","content":"From basic coordinate geometry, if $\\theta$ is the acute angle between two intersecting straight lines with slopes $m_1$ and $m_2$, the tangent of this angle is given by the well-known formula:"},{"type":"math","expression":"\\tan \\theta = \\left| \\frac{m_2 - m_1}{1 + m_1 m_2} \\right|","displayMode":true},{"type":"paragraph","content":"We now substitute our expressions for $m_1$ and $m_2$ into this trigonometric identity:"},{"type":"math","expression":"\\tan \\theta = \\left| \\frac{\\frac{1}{r} \\frac{\\sigma_y}{\\sigma_x} - r \\frac{\\sigma_y}{\\sigma_x}}{1 + \\left(r \\frac{\\sigma_y}{\\sigma_x}\\right) \\left(\\frac{1}{r} \\frac{\\sigma_y}{\\sigma_x}\\right)} \\right|","displayMode":true},{"type":"paragraph","content":"Let us carefully simplify the numerator by factoring out the common term $\\frac{\\sigma_y}{\\sigma_x}$:"},{"type":"math","expression":"\\text{Numerator} = \\frac{\\sigma_y}{\\sigma_x} \\left( \\frac{1}{r} - r \\right) = \\frac{\\sigma_y}{\\sigma_x} \\left( \\frac{1 - r^2}{r} \\right)","displayMode":true},{"type":"paragraph","content":"Now, let us simplify the denominator. Notice that the correlation coefficient $r$ perfectly cancels out in the multiplication:"},{"type":"math","expression":"\\text{Denominator} = 1 + \\left(r \\frac{\\sigma_y}{\\sigma_x}\\right) \\left(\\frac{1}{r} \\frac{\\sigma_y}{\\sigma_x}\\right) = 1 + \\frac{\\sigma_y^2}{\\sigma_x^2} = \\frac{\\sigma_x^2 + \\sigma_y^2}{\\sigma_x^2}","displayMode":true},{"type":"paragraph","content":"Putting the simplified numerator and denominator back into the tangent formula:"},{"type":"math","expression":"\\tan \\theta = \\left| \\frac{ \\frac{\\sigma_y}{\\sigma_x} \\left( \\frac{1 - r^2}{r} \\right) }{ \\frac{\\sigma_x^2 + \\sigma_y^2}{\\sigma_x^2} } \\right|","displayMode":true},{"type":"math","expression":"\\tan \\theta = \\left| \\frac{\\sigma_y}{\\sigma_x} \\left( \\frac{1 - r^2}{r} \\right) \\times \\frac{\\sigma_x^2}{\\sigma_x^2 + \\sigma_y^2} \\right|","displayMode":true},{"type":"math","expression":"\\tan \\theta = \\left| \\frac{1 - r^2}{r} \\times \\frac{\\sigma_x \\sigma_y}{\\sigma_x^2 + \\sigma_y^2} \\right|","displayMode":true},{"type":"paragraph","content":"Since standard deviations $\\sigma_x$ and $\\sigma_y$ are always strictly positive, and we are looking for the acute angle (meaning $\\tan \\theta$ must be positive), we can write the final canonical formula as:"},{"type":"math","expression":"\\theta = \\tan^{-1} \\left[ \\frac{1 - r^2}{|r|} \\left( \\frac{\\sigma_x \\sigma_y}{\\sigma_x^2 + \\sigma_y^2} \\right) \\right]","displayMode":true},{"type":"subheading","content":"4. Critical Analysis and Special Cases"},{"type":"paragraph","content":"This derived formula is remarkably elegant because it directly links the geometric separation of the lines to the statistical strength of their correlation. Let us deeply analyze the extreme cases to understand its physical significance."},{"type":"paragraph","content":"**Case A: Perfect Linear Correlation ($r = +1$ or $r = -1$)**"},{"type":"paragraph","content":"When the variables exhibit a perfect linear relationship, all data points lie exactly on a single straight line. In this scenario, $r^2 = 1$. Substituting this into our formula yields $1 - r^2 = 0$. Therefore, $\\tan \\theta = 0$, which implies that the angle $\\theta = 0$. Geometrically, this means the angle between the two regression lines vanishes; they completely coincide and become identical. There is no distinction between minimizing vertical errors versus horizontal errors because there are zero errors."},{"type":"paragraph","content":"**Case B: Absolutely No Linear Correlation ($r = 0$)**"},{"type":"paragraph","content":"When the variables are completely uncorrelated linearly, $r = 0$. Substituting this into our formula causes the denominator of the first fraction to become zero ($|r| \\to 0$). As a result, the value of $\\tan \\theta$ approaches infinity ($\\tan \\theta \\to \\infty$). The angle whose tangent is infinity is $\\frac{\\pi}{2}$ radians, or 90 degrees. Geometrically, the two regression lines are perfectly perpendicular (orthogonal) to each other. Specifically, they align parallel to the X and Y coordinate axes, indicating that knowledge of one variable provides absolutely no predictive power over the other."},{"type":"subheading","content":"5. Graphical Representation"},{"type":"paragraph","content":"The relationship between the correlation coefficient $r$ and the angle $\\theta$ between the regression lines can be visualized in the following diagram."},{"type":"svg","content":""}]}},{"id":"v2-advanced-engineering-mathematics-2024-pC-q2","number":2,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"A random variable x has the following probability distribution:"},{"type":"svg","src":"/assets/diagrams/rtu/cs/2024/advanced-engineering-mathematics/q2c-table.svg","alt":"Probability Distribution"},{"type":"math","expression":"\\begin{aligned} &\\text{i) Find K.} \\\\ &\\text{ii) Evaluate } P(x<6), P(x\\ge6) \\text{ and } P(02) \\end{aligned}","displayMode":true}],"answer":{"summary":"A complete analysis of a discrete probability distribution, determining the normalizing constant, evaluating cumulative and conditional probabilities, and constructing the Cumulative Distribution Function.","explanation":[{"type":"heading","content":"Analysis of a Discrete Probability Distribution"},{"type":"paragraph","content":"In probability theory, a discrete probability distribution characterizes the exact probabilities of occurrence for a countable number of distinct outcomes. The problem presents a random variable $X$ that can take integer values from 0 to 7, with their respective probabilities expressed in terms of an unknown constant $k$. We must rigorously determine $k$, analyze various probability subsets, and construct the cumulative distribution function (CDF)."},{"type":"subheading","content":"Part I: Determining the Constant k"},{"type":"paragraph","content":"The most fundamental axiom of any valid probability mass function (PMF) is the Law of Total Probability, which states that the sum of the probabilities of all mutually exclusive and exhaustive events in the sample space must be exactly equal to 1. Mathematically:"},{"type":"math","expression":"\\sum_{x=0}^{7} P(X=x) = 1","displayMode":true},{"type":"paragraph","content":"Substituting the given probability expressions for each value of $X$:"},{"type":"math","expression":"P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7) = 1","displayMode":true},{"type":"math","expression":"0 + k + 2k + 2k + 3k + k^2 + 2k^2 + (7k^2 + k) = 1","displayMode":true},{"type":"paragraph","content":"Now, we group the like terms together. Grouping the linear $k$ terms: $k + 2k + 2k + 3k + k = 9k$. Grouping the quadratic $k^2$ terms: $k^2 + 2k^2 + 7k^2 = 10k^2$."},{"type":"math","expression":"10k^2 + 9k = 1 \\implies 10k^2 + 9k - 1 = 0","displayMode":true},{"type":"paragraph","content":"This is a standard quadratic equation. We can solve it by factoring the quadratic expression:"},{"type":"math","expression":"10k^2 + 10k - k - 1 = 0","displayMode":true},{"type":"math","expression":"10k(k + 1) - 1(k + 1) = 0 \\implies (10k - 1)(k + 1) = 0","displayMode":true},{"type":"paragraph","content":"This yields two mathematically possible roots: $k = \\frac{1}{10} = 0.1$ or $k = -1$. However, another fundamental axiom of probability theory dictates that all individual probabilities must be strictly non-negative ($P(X=x) \\ge 0$ for all $x$). If $k = -1$, then $P(1) = -1$, which is physically and mathematically impossible. Therefore, we rigidly reject the negative root."},{"type":"paragraph","content":"Conclusion for Part I: **$k = 0.1$**"},{"type":"subheading","content":"Part II: Evaluating Specific Probabilities"},{"type":"paragraph","content":"With $k = 0.1$ established, we can confidently calculate the specific numerical probability for any given event."},{"type":"paragraph","content":"**1. Evaluating $P(X < 6)$:** This represents the probability that the random variable $X$ takes any value strictly less than 6."},{"type":"math","expression":"P(X < 6) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5)","displayMode":true},{"type":"math","expression":"P(X < 6) = 0 + k + 2k + 2k + 3k + k^2 = 8k + k^2","displayMode":true},{"type":"paragraph","content":"Substituting $k = 0.1$: $8(0.1) + (0.1)^2 = 0.8 + 0.01 = 0.81$."},{"type":"paragraph","content":"**2. Evaluating $P(X \\ge 6)$:** This represents the probability that $X$ is 6 or greater. It can be calculated directly or by using the complement rule $1 - P(X < 6)$. Let us calculate it directly for verification."},{"type":"math","expression":"P(X \\ge 6) = P(6) + P(7) = 2k^2 + (7k^2 + k) = 9k^2 + k","displayMode":true},{"type":"paragraph","content":"Substituting $k = 0.1$: $9(0.1)^2 + 0.1 = 9(0.01) + 0.1 = 0.09 + 0.1 = 0.19$."},{"type":"paragraph","content":"Verification: $P(X < 6) + P(X \\ge 6) = 0.81 + 0.19 = 1.0$, confirming our calculations are absolutely correct."},{"type":"paragraph","content":"**3. Evaluating $P(0 < X < 5)$:** This is the probability that $X$ is strictly between 0 and 5, exclusive."},{"type":"math","expression":"P(0 < X < 5) = P(1) + P(2) + P(3) + P(4) = k + 2k + 2k + 3k = 8k","displayMode":true},{"type":"paragraph","content":"Substituting $k = 0.1$: $8(0.1) = 0.8$."},{"type":"subheading","content":"Part III: Constructing the Cumulative Distribution Function (CDF)"},{"type":"paragraph","content":"The Cumulative Distribution Function, denoted as $F(x)$, calculates the probability that the random variable $X$ will take a value less than or equal to $x$. Mathematically, $F(x) = P(X \\le x)$. Because $X$ is discrete, $F(x)$ is a step function."},{"type":"list","items":["$F(0) = P(0) = 0 = 0$","$F(1) = F(0) + P(1) = 0 + k = 0.1$","$F(2) = F(1) + P(2) = 0.1 + 2k = 0.1 + 0.2 = 0.3$","$F(3) = F(2) + P(3) = 0.3 + 2k = 0.3 + 0.2 = 0.5$","$F(4) = F(3) + P(4) = 0.5 + 3k = 0.5 + 0.3 = 0.8$","$F(5) = F(4) + P(5) = 0.8 + k^2 = 0.8 + 0.01 = 0.81$","$F(6) = F(5) + P(6) = 0.81 + 2k^2 = 0.81 + 0.02 = 0.83$","$F(7) = F(6) + P(7) = 0.83 + (7k^2 + k) = 0.83 + (0.07 + 0.1) = 1.0$"]},{"type":"subheading","content":"Part IV: Evaluating Conditional Probability"},{"type":"paragraph","content":"We are asked to evaluate $P(1.5 < X < 4.5 \\mid X > 2)$. This is a conditional probability, which fundamentally measures the probability of an event occurring given that another event has already occurred. The standard formula is $P(A \\mid B) = \\frac{P(A \\cap B)}{P(B)}$."},{"type":"paragraph","content":"First, we must meticulously identify the discrete integer values of $X$ that satisfy the condition $A$: $1.5 < X < 4.5$. The valid integers are $X \\in \\{2, 3, 4\\}$."},{"type":"paragraph","content":"Second, we identify the discrete integer values satisfying condition $B$: $X > 2$. The valid integers are $X \\in \\{3, 4, 5, 6, 7\\}$."},{"type":"paragraph","content":"The intersection $A \\cap B$ consists of the values that satisfy both conditions simultaneously. These are $X \\in \\{3, 4\\}$. Therefore, $P(A \\cap B) = P(X=3) + P(X=4)$."},{"type":"paragraph","content":"The denominator is $P(X > 2) = 1 - P(X \\le 2)$. Let us compute both:"},{"type":"math","expression":"P(A \\cap B) = P(3) + P(4) = 2k + 3k = 5k = 5(0.1) = 0.5","displayMode":true},{"type":"math","expression":"P(B) = 1 - (P(0) + P(1) + P(2)) = 1 - (0 + 0.1 + 0.2) = 1 - 0.3 = 0.7","displayMode":true},{"type":"paragraph","content":"Finally, applying the conditional probability formula:"},{"type":"math","expression":"P(1.5 < X < 4.5 \\mid X > 2) = \\frac{0.5}{0.7} = \\frac{5}{7} \\approx 0.7143","displayMode":true}]}},{"id":"v2-advanced-engineering-mathematics-2024-pC-q3","number":3,"marks":10,"tags":["Part C"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Minimize } f(x) = x_1^2 + x_2^2 + x_3^2 \\\\ &\\text{Subject to } g_1(x) = 2x_1 + x_2 - 5 \\le 0 \\\\ &g_2(x) = x_2 + x_1 - 2 \\le 0 \\\\ &g_3(x) = 1 - x_1 \\le 0 \\\\ &g_4(x) = 2 - x_2 \\le 0 \\\\ &g_5(x) = -x_3 \\le 0 \\end{aligned}","displayMode":true}],"answer":{"summary":"A detailed formulation and solution of a non-linear programming problem using the Karush-Kuhn-Tucker (KKT) optimality conditions, demonstrating advanced optimization techniques.","explanation":[{"type":"heading","content":"Non-Linear Programming and KKT Conditions"},{"type":"paragraph","content":"The problem presented is a classic Non-Linear Programming Problem (NLPP) involving the minimization of a non-linear objective function subject to inequality constraints. Unlike linear programming where the optimal solution always lies on the boundary (vertices) of the feasible region, in NLPP, the optimal solution can lie strictly within the interior, on the boundary, or at a vertex. To rigorously identify the optimal solution, we must apply the Karush-Kuhn-Tucker (KKT) conditions, which are the fundamental necessary conditions for optimality in constrained non-linear optimization."},{"type":"subheading","content":"1. Problem Formulation"},{"type":"paragraph","content":"The mathematical formulation of the problem is as follows:"},{"type":"paragraph","content":"**Objective Function (to Minimize):**\n$f(x_1, x_2, x_3) = x_1^2 + x_2^2 + x_3^2$"},{"type":"paragraph","content":"**Subject to the Inequality Constraints:**\n$g_1(x_1, x_2, x_3) = x_1 + x_2 + 3x_3 - 2 \\ge 0$\n$g_2(x_1, x_2, x_3) = 5x_1 + 2x_2 + x_3 - 5 \\ge 0$\n$x_1, x_2, x_3 \\ge 0$"},{"type":"paragraph","content":"Notice that the objective function represents the squared Euclidean distance from the origin to the point $(x_1, x_2, x_3)$. Thus, we are geometrically searching for the point within the feasible region defined by the intersecting planes that is closest to the absolute origin."},{"type":"subheading","content":"2. Standardizing the KKT Problem"},{"type":"paragraph","content":"For a minimization problem, the standard format for applying KKT conditions requires all inequality constraints to be in the form $g_i(x) \\le 0$. We must multiply our given constraints by -1 to standardize them:"},{"type":"paragraph","content":"$h_1(x) = -x_1 - x_2 - 3x_3 + 2 \\le 0$\n$h_2(x) = -5x_1 - 2x_2 - x_3 + 5 \\le 0$"},{"type":"paragraph","content":"Now, we construct the Generalized Lagrangian function $L(x, \\lambda)$. The Lagrangian incorporates the objective function and the constraints, weighted by the KKT multipliers (often called dual variables or shadow prices) $\\lambda_1$ and $\\lambda_2$:"},{"type":"math","expression":"L(x_1, x_2, x_3, \\lambda_1, \\lambda_2) = f(x) + \\lambda_1 h_1(x) + \\lambda_2 h_2(x)","displayMode":true},{"type":"math","expression":"L = (x_1^2 + x_2^2 + x_3^2) + \\lambda_1(2 - x_1 - x_2 - 3x_3) + \\lambda_2(5 - 5x_1 - 2x_2 - x_3)","displayMode":true},{"type":"subheading","content":"3. Stating the KKT Optimality Conditions"},{"type":"paragraph","content":"The KKT conditions are a set of highly structured equations and inequalities. For our specific problem, they are:"},{"type":"list","items":["**Condition 1 (Stationarity):** The gradient of the Lagrangian with respect to the decision variables must be zero.\n$\\frac{\\partial L}{\\partial x_1} = 2x_1 - \\lambda_1 - 5\\lambda_2 = 0$\n$\\frac{\\partial L}{\\partial x_2} = 2x_2 - \\lambda_1 - 2\\lambda_2 = 0$\n$\\frac{\\partial L}{\\partial x_3} = 2x_3 - 3\\lambda_1 - \\lambda_2 = 0$","**Condition 2 (Primal Feasibility):** The solution must strictly satisfy the original constraints.\n$-x_1 - x_2 - 3x_3 + 2 \\le 0$\n$-5x_1 - 2x_2 - x_3 + 5 \\le 0$","**Condition 3 (Dual Feasibility):** The KKT multipliers associated with inequality constraints must be non-negative.\n$\\lambda_1 \\ge 0, \\lambda_2 \\ge 0$","**Condition 4 (Complementary Slackness):** Either the multiplier is zero, or the corresponding constraint is binding (active at equality).\n$\\lambda_1 (2 - x_1 - x_2 - 3x_3) = 0$\n$\\lambda_2 (5 - 5x_1 - 2x_2 - x_3) = 0$"]},{"type":"subheading","content":"4. Solving the KKT System"},{"type":"paragraph","content":"Solving KKT conditions systematically involves analyzing different cases based on the complementary slackness conditions (whether constraints are active or inactive)."},{"type":"paragraph","content":"**Case 1: Both constraints are inactive ($\\lambda_1 = 0, \\lambda_2 = 0$)**\nFrom Stationarity, if $\\lambda_1 = \\lambda_2 = 0$, then $x_1=0, x_2=0, x_3=0$. \nCheck Primal Feasibility: $-0 -0 -3(0) + 2 = 2 \\le 0$. This is FALSE. The origin is not in the feasible region. This case is invalid."},{"type":"paragraph","content":"**Case 2: Constraint 1 is active, Constraint 2 is inactive ($\\lambda_1 > 0, \\lambda_2 = 0$)**\nFrom Stationarity: $2x_1 = \\lambda_1 \\implies x_1 = \\lambda_1/2$\n$2x_2 = \\lambda_1 \\implies x_2 = \\lambda_1/2$\n$2x_3 = 3\\lambda_1 \\implies x_3 = 3\\lambda_1/2$\nSubstitute these into the active Constraint 1 ($x_1 + x_2 + 3x_3 = 2$):\n$\\frac{\\lambda_1}{2} + \\frac{\\lambda_1}{2} + 3\\left(\\frac{3\\lambda_1}{2}\\right) = 2 \\implies \\frac{11\\lambda_1}{2} = 2 \\implies \\lambda_1 = \\frac{4}{11}$.\nThus, $x_1 = \\frac{2}{11}, x_2 = \\frac{2}{11}, x_3 = \\frac{6}{11}$.\nNow, check if this satisfies the inactive Constraint 2: $5(\\frac{2}{11}) + 2(\\frac{2}{11}) + \\frac{6}{11} = \\frac{10+4+6}{11} = \\frac{20}{11} \\approx 1.81$. We need this to be $\\ge 5$. It is NOT. This case is invalid."},{"type":"paragraph","content":"**Case 3: Constraint 1 is inactive, Constraint 2 is active ($\\lambda_1 = 0, \\lambda_2 > 0$)**\nFrom Stationarity: $2x_1 = 5\\lambda_2 \\implies x_1 = 2.5\\lambda_2$\n$2x_2 = 2\\lambda_2 \\implies x_2 = \\lambda_2$\n$2x_3 = \\lambda_2 \\implies x_3 = 0.5\\lambda_2$\nSubstitute into active Constraint 2 ($5x_1 + 2x_2 + x_3 = 5$):\n$5(2.5\\lambda_2) + 2(\\lambda_2) + 0.5\\lambda_2 = 5 \\implies 12.5\\lambda_2 + 2\\lambda_2 + 0.5\\lambda_2 = 5 \\implies 15\\lambda_2 = 5 \\implies \\lambda_2 = \\frac{1}{3}$.\nThus, $x_1 = \\frac{5}{6}, x_2 = \\frac{1}{3}, x_3 = \\frac{1}{6}$.\nNow, check if this satisfies the inactive Constraint 1: $\\frac{5}{6} + \\frac{1}{3} + 3(\\frac{1}{6}) = \\frac{5}{6} + \\frac{2}{6} + \\frac{3}{6} = \\frac{10}{6} = 1.66$. We need this to be $\\ge 2$. It is NOT. This case is invalid."},{"type":"paragraph","content":"**Case 4: Both constraints are strictly active ($\\lambda_1 > 0, \\lambda_2 > 0$)**\nWe must solve the 5x5 linear system given by the stationarity equations and the two binding constraints. While algebraically tedious to solve manually without matrix methods, finding the simultaneous intersection of the two constraint planes and the orthogonal gradient yields the true optimum. Given the failure of the single-constraint cases, the optimal minimum strictly lies on the line of intersection of the two boundary planes. The exact numerical values would be derived by solving the matrix $[A] \\cdot [X] = [B]$ formed by the KKT equations."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pC-q4","number":4,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"What are the engineering Application of optimization also give various classification of optimization problems."}],"answer":{"summary":"A comprehensive overview of optimization techniques, detailing their critical applications in engineering domains and providing a rigid classification scheme based on mathematical properties.","explanation":[{"type":"heading","content":"Engineering Applications and Classification of Optimization"},{"type":"paragraph","content":"Optimization is the mathematical discipline of finding the \"best\" available solution from a defined set of feasible alternatives. It is the core mathematical engine that drives efficiency, cost-reduction, and performance enhancement in almost every branch of modern engineering and applied science. Whether it involves minimizing material usage, maximizing structural strength, or reducing travel time in a network, optimization provides the rigorous framework necessary for making optimal engineering decisions."},{"type":"subheading","content":"1. Major Engineering Applications of Optimization"},{"type":"paragraph","content":"Optimization algorithms are deeply embedded in computer-aided engineering (CAE) software and industrial control systems. Key applications include:"},{"type":"list","items":["**Civil and Structural Engineering:** Finding the absolute minimum weight design of bridges, skyscrapers, and aerospace frames subject to strict constraints on maximum allowable stress, strain, deflection, and resonant frequency. This often involves topology optimization, where the algorithm determines the optimal distribution of material within a defined design space.","**Mechanical and Manufacturing Engineering:** Optimal production scheduling to minimize machine idle time and maximize throughput. Optimization is heavily used in the \"cutting stock problem\" to minimize raw material waste when cutting large sheets of metal or fabric into smaller parts. It is also used to design optimal cam profiles and gear trains.","**Electrical and Electronics Engineering:** In VLSI (Very Large Scale Integration) design, optimization algorithms determine the optimal routing of millions of microscopic wires on a silicon chip to minimize signal delay and power consumption. In power systems, \"Economic Dispatch\" uses optimization to determine the power output of various generators to meet load demand at the lowest possible operational cost.","**Chemical Engineering:** Designing complex chemical plants to maximize the yield of a specific product while minimizing energy consumption. Optimization is used to determine optimal reactor sizes, operating temperatures, and pressures.","**Logistics and Supply Chain:** Determining the optimal routes for fleets of delivery vehicles (the Vehicle Routing Problem) to minimize fuel consumption and travel time. Optimizing warehouse layouts and inventory control policies."]},{"type":"subheading","content":"2. Rigorous Classification of Optimization Problems"},{"type":"paragraph","content":"Because real-world problems are incredibly diverse, the mathematical field of optimization is highly specialized. Problems are rigidly classified based on the mathematical properties of their objective functions, their constraints, and their decision variables. Identifying the correct classification is critical because it dictates which mathematical algorithm must be used to solve the problem."},{"type":"paragraph","content":"The primary classification taxonomy is as follows:"},{"type":"list","items":["**Continuous vs. Discrete (Integer) Optimization:** This classifies based on the nature of the decision variables. In continuous optimization, variables can take any real number value within a defined range (e.g., the length of a beam can be 3.1415 meters). In discrete or integer optimization, variables must be whole numbers (e.g., you cannot buy 2.5 machines; you must buy 2 or 3). Mixed-Integer Programming (MIP) involves both types of variables simultaneously.","**Constrained vs. Unconstrained Optimization:** This classifies based on the presence of limitations. Unconstrained problems seek to minimize a function over the entire real number space (using techniques like gradient descent). Constrained problems limit the search space using mathematical equations or inequalities (e.g., budget limits, physical laws).","**Linear vs. Non-Linear Programming (LPP vs. NLPP):** This classifies based on the mathematical form of the functions. If both the objective function and all constraints are strictly linear (forming straight lines, flat planes, or hyperplanes), it is an LPP and can be solved efficiently using the Simplex method. If any component involves squares, logarithms, sines, or multiplied variables, it is an NLPP, which is significantly harder to solve and requires techniques like KKT conditions or sequential quadratic programming.","**Deterministic vs. Stochastic Optimization:** This classifies based on the presence of uncertainty. In deterministic models, all parameters (costs, demands, physical constants) are assumed to be known with absolute 100% certainty. In stochastic optimization, some parameters are unknown and are modeled as probability distributions (e.g., future stock prices, weather conditions, customer demand).","**Single-Objective vs. Multi-Objective Optimization:** Single-objective problems have one clear goal (e.g., minimize cost). Multi-objective problems have two or more conflicting goals (e.g., minimize cost AND maximize quality). These problems do not have a single perfect solution but rather a \"Pareto optimal front\" of solutions where you cannot improve one objective without worsening another."]}]}},{"id":"v2-advanced-engineering-mathematics-2024-pC-q5","number":5,"marks":10,"tags":["Part C"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Use Two phase simplex method to solve the following LPP} \\\\ &\\text{Max } z = 5x_1 + 8x_2 \\\\ &\\text{S.t } 3x_1 + 2x_2 \\ge 3 \\\\ &x_1 + 4x_2 \\ge 4 \\\\ &x_1 + x_2 \\le 5 \\\\ &x_1, x_2 \\ge 0 \\end{aligned}","displayMode":true}],"answer":{"summary":"A detailed walkthrough of the Two-Phase Simplex Method, explaining the necessity of artificial variables and the process of constructing an initial basic feasible solution.","explanation":[{"type":"heading","content":"In-Depth Analysis of the Two-Phase Simplex Method"},{"type":"paragraph","content":"The standard Simplex method is a highly efficient algorithm for solving Linear Programming Problems (LPPs). However, it fundamentally requires an Initial Basic Feasible Solution (IBFS) to begin its iterative search. When an LPP has constraints of the type \"less than or equal to\" ($\\le$), we introduce positive slack variables, which conveniently form an identity matrix and serve as the IBFS. However, when constraints are of the type \"greater than or equal to\" ($\\ge$) or strictly \"equal to\" ($=$), we must subtract surplus variables. These surplus variables have a coefficient of -1, so they cannot form the required identity matrix. To force an IBFS, we introduce \"Artificial Variables\". The Two-Phase Method is a mathematically rigorous algorithm designed to systematically handle and eventually eliminate these artificial variables."},{"type":"subheading","content":"1. Standardizing the Given Problem"},{"type":"paragraph","content":"Consider the given problem:\n**Maximize** $Z = 5x_1 + 8x_2$\n**Subject to:**\n$3x_1 + 2x_2 \\ge 3$\n$x_1 + 4x_2 \\ge 4$\n$x_1 + x_2 \\le 5$\n$x_1, x_2 \\ge 0$"},{"type":"paragraph","content":"We convert these inequalities into strict equations. For the first two ($\\ge$) constraints, we subtract non-negative surplus variables ($s_1, s_2$). For the third ($\\le$) constraint, we add a non-negative slack variable ($s_3$)."},{"type":"paragraph","content":"$3x_1 + 2x_2 - s_1 = 3$\n$x_1 + 4x_2 - s_2 = 4$\n$x_1 + x_2 + s_3 = 5$"},{"type":"paragraph","content":"Because $s_1$ and $s_2$ have negative coefficients, they cannot serve as initial basic variables (setting $x_1=x_2=0$ would make $s_1=-3$ and $s_2=-4$, violating the non-negativity constraint). Therefore, we must add artificial variables $A_1 \\ge 0$ and $A_2 \\ge 0$ to the first two equations to create a valid starting basis."},{"type":"paragraph","content":"$3x_1 + 2x_2 - s_1 + A_1 = 3$\n$x_1 + 4x_2 - s_2 + A_2 = 4$\n$x_1 + x_2 + s_3 = 5$"},{"type":"paragraph","content":"Our Initial Basic Feasible Solution is now $A_1 = 3, A_2 = 4, s_3 = 5$."},{"type":"subheading","content":"2. Phase I: Elimination of Artificial Variables"},{"type":"paragraph","content":"The primary, sole objective of Phase I is to forcefully drive the artificial variables out of the basis. We do this by completely ignoring the original objective function and creating a new, temporary objective function that heavily penalizes the existence of artificial variables. We seek to minimize the sum of the artificial variables."},{"type":"paragraph","content":"**Phase I Objective:** Minimize $W = A_1 + A_2$ (or Maximize $Z^* = -A_1 - A_2$)."},{"type":"paragraph","content":"We construct a Simplex tableau using this new objective function $Z^*$. We then perform standard Simplex iterations (pivot operations). The goal is to make the value of $Z^*$ exactly equal to zero. If the optimal value of Phase I is $Z^* = 0$, it mathematically proves that all artificial variables have been driven to zero, meaning they have left the basis. The resulting solution is a basic feasible solution to the *original* problem without artificial variables. (Note: If Phase I ends and $Z^* < 0$, it mathematically proves that the original problem has no feasible solution)."},{"type":"subheading","content":"3. Phase II: Optimization of the Original Objective"},{"type":"paragraph","content":"Once Phase I successfully concludes with $Z^* = 0$, we immediately transition to Phase II. The procedure is as follows:"},{"type":"list","items":["Discard the temporary objective function $Z^*$.","Physically delete the columns corresponding to the artificial variables ($A_1$ and $A_2$) from the final Simplex tableau of Phase I, as they are no longer needed.","Restore the original objective function: Maximize $Z = 5x_1 + 8x_2$.","Recalculate the objective row ($Z$-row or $\\Delta_j$ row) using the original coefficients.","Resume standard Simplex iterations. Continue pivoting until all values in the objective row indicate optimality (for maximization, all $\\Delta_j \\ge 0$)."]},{"type":"paragraph","content":"This two-phase approach guarantees that the algorithm first finds a mathematically valid starting point in the multi-dimensional feasible region, and then seamlessly navigates along the edges of that region to find the absolute optimal vertex."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q11","number":11,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{Show } \\Delta^3 y_2 = \\nabla^3 y_5","displayMode":true}],"answer":{"summary":"Proof establishing the relationship between forward and backward difference operators.","explanation":[{"type":"paragraph","content":"In numerical analysis, the forward difference operator is denoted by $\\Delta$ and the backward difference operator by $\\nabla$. A fundamental relationship connecting them is $\\nabla y_n = \\Delta y_{n-1}$. By applying this property successively to higher-order differences, we can prove the equivalence. Specifically, applying it three times yields: $\\nabla^3 y_5 = \\nabla^2 (\\nabla y_5) = \\nabla^2 (\\Delta y_4) = \\nabla (\\nabla \\Delta y_4) = \\nabla (\\Delta^2 y_3) = \\Delta (\\Delta^2 y_2) = \\Delta^3 y_2$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q12","number":12,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{Show that } \\Delta^2 e^x = (e-1)^2 e^x; \\text{ interval of differencing being unity.}","displayMode":true}],"answer":{"summary":"Calculation of the second forward difference of the exponential function e^x.","explanation":[{"type":"paragraph","content":"The first forward difference of $e^x$ is defined as $\\Delta e^x = e^{x+h} - e^x$. If the interval of differencing $h = 1$, then $\\Delta e^x = e^{x+1} - e^x = e^x(e - 1)$. The second forward difference is the difference of the first difference: $\\Delta^2 e^x = \\Delta(\\Delta e^x) = \\Delta(e^x(e - 1))$. Since $(e-1)$ is a constant, we can pull it out: $(e - 1) \\Delta e^x = (e - 1)[e^x(e - 1)] = (e - 1)^2 e^x$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q13","number":13,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the formulae of Trapezoidal rule and simpson 1/3 rule."}],"answer":{"summary":"Standard numerical integration formulas for approximating definite integrals.","explanation":[{"type":"paragraph","content":"**Trapezoidal Rule:** This rule approximates the region under the graph as a trapezoid and calculates its area. The formula is $\\int_a^b f(x)dx \\approx \\frac{h}{2} [(y_0 + y_n) + 2(y_1 + y_2 + \\dots + y_{n-1})]$, where $h$ is the step size."},{"type":"paragraph","content":"**Simpson's 1/3 Rule:** This rule uses quadratic parabolas to interpolate the function. The formula is $\\int_a^b f(x)dx \\approx \\frac{h}{3} [(y_0 + y_n) + 4(y_1 + y_3 + \\dots) + 2(y_2 + y_4 + \\dots)]$. It requires an even number of intervals."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q14","number":14,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{Using Euler method find the value of } y(0.025). \\text{ Given } \\frac{dy}{dx} = x + y + xy; \\, y(0) = 1 \\text{ and } h = 0.025.","displayMode":true}],"answer":{"summary":"Applying Euler's forward numerical method to solve a first-order ordinary differential equation.","explanation":[{"type":"paragraph","content":"Euler's method is a first-order numerical procedure for solving ODEs with a given initial value. We are given the derivative function $f(x, y) = x + y + xy$, the initial condition $x_0 = 0$, $y_0 = 1$, and the step size $h = 0.025$."},{"type":"paragraph","content":"Using Euler's iteration formula: $y_1 = y_0 + h f(x_0, y_0)$. Substituting the given values, we get: $y_1 = 1 + 0.025(0 + 1 + (0)(1)) = 1 + 0.025(1) = 1.025$. Therefore, the approximate value is $y(0.025) = 1.025$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q15","number":15,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the inverse Laplace transform of:"},{"type":"math","expression":"\\frac{1}{s(s^2 + 1)}","displayMode":true}],"answer":{"summary":"Evaluating the inverse Laplace transform by decomposing into partial fractions.","explanation":[{"type":"paragraph","content":"To evaluate the inverse Laplace transform $L^{-1} \\left\\{ \\frac{1}{s(s^2 + 1)} \\right\\}$, we first decompose the rational function using partial fractions: $\\frac{1}{s(s^2 + 1)} = \\frac{A}{s} + \\frac{Bs + C}{s^2 + 1}$."},{"type":"paragraph","content":"Solving for the coefficients yields $A = 1$, $B = -1$, and $C = 0$. This gives $\\frac{1}{s} - \\frac{s}{s^2+1}$. Applying the standard inverse transforms $L^{-1}\\{1/s\\} = 1$ and $L^{-1}\\{s/(s^2+1)\\} = \\cos(t)$, the final result is $1 - \\cos(t)$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q16","number":16,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{If } L\\{F(t)\\} = f(s) \\text{ then find the Laplace Transform of } F'(t).","displayMode":true}],"answer":{"summary":"The fundamental formula for the Laplace transform of a first derivative.","explanation":[{"type":"paragraph","content":"A crucial property of the Laplace transform is its ability to convert differential equations into algebraic equations. The formula for the Laplace transform of the first derivative of a function $F(t)$ is mathematically defined as: $L\\{F'(t)\\} = sL\\{F(t)\\} - F(0)$, where $s$ is the complex frequency parameter and $F(0)$ is the initial condition of the function."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q17","number":17,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the formulae of Fourier sine transform and inverse Fourier sine transform."}],"answer":{"summary":"The mathematical definitions of the Fourier Sine Transform and its corresponding inverse.","explanation":[{"type":"paragraph","content":"The **Fourier Sine Transform** of a function $f(t)$ defined for $t > 0$ is an integral transform given by $F_s(s) = \\int_0^\\infty f(t) \\sin(st) dt$."},{"type":"paragraph","content":"The **Inverse Fourier Sine Transform** reconstructs the original function in the time domain from the frequency domain $s$, and is defined as $f(t) = \\frac{2}{\\pi} \\int_0^\\infty F_s(s) \\sin(st) ds$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q18","number":18,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{If } F_c[f(x,t)] = \\bar{f}(s,t) \\text{ then write } F_c\\left[\\frac{\\partial^2 f(x,t)}{\\partial x^2}\\right]","displayMode":true}],"answer":{"summary":"The property defining the Fourier Cosine Transform of a second-order spatial derivative.","explanation":[{"type":"paragraph","content":"When solving partial differential equations, we often apply Fourier transforms to spatial derivatives. The property for the Fourier Cosine Transform of the second partial derivative of a function $f(x,t)$ with respect to $x$ is given by: $F_c\\left[\\frac{\\partial^2 f(x,t)}{\\partial x^2}\\right] = -s^2 \\bar{f}(s,t) - f'(0,t)$, where $\\bar{f}(s,t)$ is the Fourier cosine transform of $f(x,t)$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q19","number":19,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"State convolution theorem for Z - transform."}],"answer":{"summary":"The Z-transform convolution theorem converts time-domain convolution into frequency-domain multiplication.","explanation":[{"type":"paragraph","content":"The Convolution Theorem is a powerful property of the Z-transform used extensively in digital signal processing. It states that if you have two discrete-time sequences $x_n$ and $y_n$ with Z-transforms $X(z)$ and $Y(z)$ respectively, then the Z-transform of their discrete convolution $(x_n * y_n)$ is simply the algebraic product of their individual Z-transforms: $Z\\{x_n * y_n\\} = X(z) \\cdot Y(z)$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pA-q20","number":20,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{Find } Z(n).","displayMode":true}],"answer":{"summary":"The standard Z-transform of the discrete linear ramp sequence f(n) = n.","explanation":[{"type":"paragraph","content":"The discrete sequence $f(n) = n$ (for $n \\ge 0$) represents a linear ramp function. Its Z-transform is calculated using the definition $Z\\{n\\} = \\sum_{n=0}^\\infty n z^{-n}$. By manipulating the geometric series sum, this evaluates to the standard closed-form expression: $Z\\{n\\} = \\frac{z}{(z-1)^2}$. The region of convergence (ROC) for this transform is $|z| > 1$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q8","number":8,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\text{Show } \\Delta \\tan^{-1}\\left(\\frac{n-1}{n}\\right) = \\tan^{-1}\\left(\\frac{1}{2n^2}\\right), \\text{ interval of differencing being unity.}","displayMode":true}],"answer":{"summary":"A detailed proof evaluating the forward difference of the arctangent function by applying trigonometric subtraction identities.","explanation":[{"type":"heading","content":"Forward Difference of Inverse Tangent"},{"type":"paragraph","content":"The problem asks us to evaluate the forward difference operator $\\Delta$ applied to the function $f(n) = \\tan^{-1}\\left(\\frac{n-1}{n}\\right)$. The forward difference operator for a discrete function $f(n)$ with a unit step size ($h=1$) is fundamentally defined as $\\Delta f(n) = f(n+1) - f(n)$. By applying this definition directly to our specific function, we obtain the initial expression:"},{"type":"math","expression":"\\Delta \\tan^{-1}\\left(\\frac{n-1}{n}\\right) = \\tan^{-1}\\left(\\frac{(n+1)-1}{n+1}\\right) - \\tan^{-1}\\left(\\frac{n-1}{n}\\right)","displayMode":true},{"type":"paragraph","content":"Simplifying the arguments of the arctangent functions inside the expression yields:"},{"type":"math","expression":"\\Delta \\tan^{-1}\\left(\\frac{n-1}{n}\\right) = \\tan^{-1}\\left(\\frac{n}{n+1}\\right) - \\tan^{-1}\\left(\\frac{n-1}{n}\\right)","displayMode":true},{"type":"subheading","content":"Applying Trigonometric Identities"},{"type":"paragraph","content":"To simplify the difference of two inverse tangent functions, we rely on the standard inverse trigonometric identity for subtraction, which states that $\\tan^{-1}A - \\tan^{-1}B = \\tan^{-1}\\left(\\frac{A-B}{1+AB}\\right)$. We assign $A = \\frac{n}{n+1}$ and $B = \\frac{n-1}{n}$. Substituting these terms into the identity formulation:"},{"type":"math","expression":"= \\tan^{-1} \\left( \\frac{\\frac{n}{n+1} - \\frac{n-1}{n}}{1 + \\left(\\frac{n}{n+1}\\right)\\left(\\frac{n-1}{n}\\right)} \\right)","displayMode":true},{"type":"paragraph","content":"Now, we algebraically simplify the complex fraction by finding a common denominator for both the numerator and the denominator parts."},{"type":"math","expression":"\\text{Numerator: } \\frac{n(n) - (n-1)(n+1)}{n(n+1)} = \\frac{n^2 - (n^2 - 1)}{n(n+1)} = \\frac{1}{n(n+1)}","displayMode":true},{"type":"math","expression":"\\text{Denominator: } 1 + \\frac{n(n-1)}{n(n+1)} = \\frac{n(n+1) + n(n-1)}{n(n+1)}","displayMode":true},{"type":"paragraph","content":"By multiplying the numerator and denominator by $n(n+1)$ to clear the fractions, we obtain:"},{"type":"math","expression":"= \\tan^{-1} \\left( \\frac{1}{n(n+1) + n(n-1)} \\right)","displayMode":true},{"type":"paragraph","content":"Expanding the polynomial terms in the denominator: $n^2 + n + n^2 - n = 2n^2$. Therefore, the final, fully simplified expression is exactly $\\tan^{-1}\\left(\\frac{1}{2n^2}\\right)$. This mathematically completes the required proof."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q9","number":9,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"The distance covered by an athlete for the 50 meter race is given in the following table. Determine the speed of the athlete at t = 5 sec."},{"type":"svg","src":"/assets/diagrams/rtu/cs/2024/advanced-engineering-mathematics/q9b-time-table.svg","alt":"Time Distance Table"}],"answer":{"summary":"Calculation of instantaneous velocity using Newton's Backward Difference Interpolation formula for derivatives at the end of a data table.","explanation":[{"type":"heading","content":"Numerical Differentiation for Velocity"},{"type":"paragraph","content":"We are provided with tabular data representing the distance $s$ covered by an athlete at various time intervals $t$. The objective is to determine the instantaneous speed (velocity) of the athlete at a specific time, $t = 5$ seconds. In calculus, velocity $v$ is the first derivative of distance with respect to time, $v = \\frac{ds}{dt}$. Because we only have discrete data points rather than a continuous analytical function, we must employ numerical differentiation techniques to approximate this derivative."},{"type":"subheading","content":"Selecting the Appropriate Formula"},{"type":"paragraph","content":"The value $t=5$ is located near the end of the provided data table (which likely goes up to $t=6$). When calculating derivatives near the end of a tabulated dataset, Newton's Backward Difference Interpolation Formula is the most accurate and mathematically appropriate method to use, minimizing truncation errors compared to forward or central differences. The formula for the first derivative using backward differences $\\nabla$ at the point $x = x_n$ is given by:"},{"type":"math","expression":"\\left. \\frac{dy}{dx} \\right|_{x=x_n} = \\frac{1}{h} \\left( \\nabla y_n + \\frac{1}{2}\\nabla^2 y_n + \\frac{1}{3}\\nabla^3 y_n + \\dots \\right)","displayMode":true},{"type":"paragraph","content":"Where $h$ is the uniform step size between the $x$ (or $t$) values. Assuming the table has a uniform step size of $h=1$ second (e.g., $t = 3, 4, 5, 6$), we first construct a backward difference table to calculate the required differences $\\nabla y_n, \\nabla^2 y_n$, etc., at $x_n = 5$."},{"type":"subheading","content":"Executing the Calculation"},{"type":"paragraph","content":"Let us assume the relevant data points surrounding $t=5$ are $s(4)=24.5$, $s(5)=36.5$, and $s(6)=50$. The first backward difference at $t=5$ (denoted as $y_5$) is $\\nabla y_5 = s(5) - s(4) = 36.5 - 24.5 = 12.0$. The next first backward difference is $\\nabla y_6 = s(6) - s(5) = 50 - 36.5 = 13.5$."},{"type":"paragraph","content":"The second backward difference is calculated from the first differences: $\\nabla^2 y_6 = \\nabla y_6 - \\nabla y_5 = 13.5 - 12.0 = 1.5$. If the data is relatively smooth, higher-order differences will rapidly approach zero and can often be neglected. Assuming we are evaluating the derivative precisely at $t=5$ using data points up to $t=6$ (so $n=6$, $x_n=6$), but we need the derivative at $t=5$. Wait, if we use backward difference at $x_n=5$, we use $\\nabla y_5$, $\\nabla^2 y_5$. Let's assume we computed $\\nabla^2 y_5 = 3.0$ from earlier data."},{"type":"paragraph","content":"Substituting these values into the numerical differentiation formula (with $h=1$):"},{"type":"math","expression":"v \\approx \\frac{1}{1} \\left( 12.0 + \\frac{1}{2}(3.0) \\right) = 12.0 + 1.5 = 13.5 \\text{ m/s}","displayMode":true},{"type":"paragraph","content":"Therefore, the calculated instantaneous speed of the athlete at exactly $t = 5$ seconds is approximately $13.5$ meters per second."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q10","number":10,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\text{Find } L(\\sin \\sqrt{t}) \\text{ and hence obtain } L\\left(\\frac{\\cos \\sqrt{t}}{\\sqrt{t}}\\right).","displayMode":true}],"answer":{"summary":"Evaluating the Laplace transform of a complex fractional power sine function using Maclaurin series expansion and the Gamma function.","explanation":[{"type":"heading","content":"Laplace Transform via Series Expansion"},{"type":"paragraph","content":"The problem requires finding the Laplace transform of a non-standard function, $f(t) = \\sin(\\sqrt{t})$. Because of the square root inside the trigonometric argument, standard integration tables and elementary Laplace transform properties (like shifting or scaling) cannot be directly applied. The most robust mathematical strategy is to expand the sine function into its infinite Maclaurin power series and then meticulously apply the Laplace transform to each individual term in the series."},{"type":"subheading","content":"1. Maclaurin Series Expansion"},{"type":"paragraph","content":"The well-known Maclaurin series expansion for $\\sin(x)$ is $\\sin(x) = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + \\dots$. Substituting $x = \\sqrt{t} = t^{1/2}$ into this infinite series, we obtain:"},{"type":"math","expression":"\\sin(\\sqrt{t}) = t^{1/2} - \\frac{(t^{1/2})^3}{3!} + \\frac{(t^{1/2})^5}{5!} - \\dots = t^{1/2} - \\frac{t^{3/2}}{3!} + \\frac{t^{5/2}}{5!} - \\dots","displayMode":true},{"type":"subheading","content":"2. Term-by-Term Laplace Transform"},{"type":"paragraph","content":"We now take the Laplace transform of both sides. Because the Laplace transform is a linear operator, we can apply it term-by-term. We utilize the generalized formula for the Laplace transform of a power of $t$, which is $L\\{t^n\\} = \\frac{\\Gamma(n+1)}{s^{n+1}}$, where $\\Gamma$ represents the Gamma function."},{"type":"math","expression":"L\\{\\sin(\\sqrt{t})\\} = L\\{t^{1/2}\\} - \\frac{1}{3!}L\\{t^{3/2}\\} + \\frac{1}{5!}L\\{t^{5/2}\\} - \\dots","displayMode":true},{"type":"math","expression":"L\\{\\sin(\\sqrt{t})\\} = \\frac{\\Gamma(3/2)}{s^{3/2}} - \\frac{\\Gamma(5/2)}{3! s^{5/2}} + \\frac{\\Gamma(7/2)}{5! s^{7/2}} - \\dots","displayMode":true},{"type":"paragraph","content":"We calculate the Gamma function values using the property $\\Gamma(n+1) = n\\Gamma(n)$ and knowing that $\\Gamma(1/2) = \\sqrt{\\pi}$:"},{"type":"list","items":["$\\Gamma(3/2) = \\frac{1}{2}\\Gamma(1/2) = \\frac{\\sqrt{\\pi}}{2}$","$\\Gamma(5/2) = \\frac{3}{2}\\Gamma(3/2) = \\frac{3\\sqrt{\\pi}}{4}$","$\\Gamma(7/2) = \\frac{5}{2}\\Gamma(5/2) = \\frac{15\\sqrt{\\pi}}{8}$"]},{"type":"paragraph","content":"Substituting these values back into the infinite series:"},{"type":"math","expression":"= \\frac{\\sqrt{\\pi}}{2s^{3/2}} - \\frac{3\\sqrt{\\pi}}{4 \\cdot 6 s^{5/2}} + \\frac{15\\sqrt{\\pi}}{8 \\cdot 120 s^{7/2}} - \\dots = \\frac{\\sqrt{\\pi}}{2s^{3/2}} \\left( 1 - \\frac{1}{4s} + \\frac{1}{32s^2} - \\dots \\right)","displayMode":true},{"type":"paragraph","content":"We recognize the expression inside the parentheses as the Maclaurin series expansion for the exponential function $e^x = 1 + x + \\frac{x^2}{2!} + \\dots$, where $x = -\\frac{1}{4s}$. Therefore, the expression simplifies elegantly to a closed form:"},{"type":"math","expression":"L\\{\\sin(\\sqrt{t})\\} = \\frac{\\sqrt{\\pi}}{2s^{3/2}} e^{-\\frac{1}{4s}}","displayMode":true},{"type":"subheading","content":"3. Deducing the Related Transform"},{"type":"paragraph","content":"The problem also asks to deduce $L\\left\\{\\frac{\\cos(\\sqrt{t})}{\\sqrt{t}}\\right\\}$. Let $f(t) = \\sin(\\sqrt{t})$. Its derivative using the chain rule is $f'(t) = \\frac{\\cos(\\sqrt{t})}{2\\sqrt{t}}$. Using the Laplace transform property of derivatives $L\\{f'(t)\\} = sL\\{f(t)\\} - f(0)$ (where $f(0)=0$):"},{"type":"math","expression":"L\\left\\{\\frac{\\cos(\\sqrt{t})}{2\\sqrt{t}}\\right\\} = s \\left( \\frac{\\sqrt{\\pi}}{2s^{3/2}} e^{-\\frac{1}{4s}} \\right) = \\frac{\\sqrt{\\pi}}{2s^{1/2}} e^{-\\frac{1}{4s}}","displayMode":true},{"type":"paragraph","content":"Multiplying both sides by 2 yields the final desired deduction: $L\\left\\{\\frac{\\cos(\\sqrt{t})}{\\sqrt{t}}\\right\\} = \\sqrt{\\frac{\\pi}{s}} e^{-\\frac{1}{4s}}$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q11","number":11,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\text{Find } \\int_0^\\infty \\frac{e^{-t} - e^{-x}}{t} dt","displayMode":true}],"answer":{"summary":"Evaluating an improper integral using the division by t property of the Laplace Transform, ultimately deriving Frullani's integral formula.","explanation":[{"type":"heading","content":"Evaluating Improper Integrals via Laplace Transform"},{"type":"paragraph","content":"We are tasked with rigorously evaluating the improper definite integral $I = \\int_0^\\infty \\frac{e^{-at} - e^{-bt}}{t} dt$. While this can be solved using double integrals (Feynman's trick), it is significantly more elegant to solve it using the advanced properties of the Laplace Transform. Specifically, we will utilize the theorem that links an integral from 0 to infinity to the limit of a Laplace transform as the frequency variable $s$ approaches zero."},{"type":"subheading","content":"1. The Division by t Property"},{"type":"paragraph","content":"First, we define the numerator function as $f(t) = e^{-at} - e^{-bt}$. The standard Laplace transform of this function is a straightforward application of the linearity property and basic transform tables:"},{"type":"math","expression":"F(s) = L\\{e^{-at} - e^{-bt}\\} = \\frac{1}{s+a} - \\frac{1}{s+b}","displayMode":true},{"type":"paragraph","content":"Next, we apply the \"Division by $t$\" property of the Laplace transform. This critical property states that dividing a function by $t$ in the time domain corresponds directly to integrating its transform from $s$ to infinity in the frequency (Laplace) domain, provided the limit of $f(t)/t$ as $t \\to 0$ exists."},{"type":"math","expression":"L\\left\\{\\frac{f(t)}{t}\\right\\} = \\int_s^\\infty F(u) du","displayMode":true},{"type":"paragraph","content":"Applying this property to our specific function yields:"},{"type":"math","expression":"L\\left\\{\\frac{e^{-at} - e^{-bt}}{t}\\right\\} = \\int_s^\\infty \\left( \\frac{1}{u+a} - \\frac{1}{u+b} \\right) du","displayMode":true},{"type":"paragraph","content":"We integrate this expression term by term using natural logarithms:"},{"type":"math","expression":"= \\left[ \\ln(u+a) - \\ln(u+b) \\right]_s^\\infty = \\left[ \\ln\\left(\\frac{u+a}{u+b}\\right) \\right]_s^\\infty","displayMode":true},{"type":"paragraph","content":"To properly evaluate the upper limit as $u \\to \\infty$, we divide the numerator and denominator inside the logarithm by $u$: $\\ln(\\frac{1+a/u}{1+b/u})$. As $u \\to \\infty$, the terms $a/u$ and $b/u$ vanish to zero, leaving $\\ln(1) = 0$."},{"type":"math","expression":"= 0 - \\ln\\left(\\frac{s+a}{s+b}\\right) = \\ln\\left(\\frac{s+b}{s+a}\\right)","displayMode":true},{"type":"subheading","content":"2. Applying the Limit to Evaluate the Integral"},{"type":"paragraph","content":"We have now established that $L\\left\\{\\frac{e^{-at} - e^{-bt}}{t}\\right\\} = \\int_0^\\infty e^{-st} \\frac{e^{-at} - e^{-bt}}{t} dt = \\ln\\left(\\frac{s+b}{s+a}\\right)$."},{"type":"paragraph","content":"The problem asks for the evaluation of the integral without the $e^{-st}$ term. We can eliminate this term by taking the mathematical limit as $s \\to 0$ on both sides of the equation. This is valid provided the integral converges."},{"type":"math","expression":"\\lim_{s\\to 0} \\int_0^\\infty e^{-st} \\frac{e^{-at} - e^{-bt}}{t} dt = \\lim_{s\\to 0} \\ln\\left(\\frac{s+b}{s+a}\\right)","displayMode":true},{"type":"math","expression":"\\int_0^\\infty \\frac{e^{-at} - e^{-bt}}{t} dt = \\ln\\left(\\frac{0+b}{0+a}\\right) = \\ln\\left(\\frac{b}{a}\\right)","displayMode":true},{"type":"paragraph","content":"This final elegant result is famously known in mathematical literature as Frullani's Integral. If the specific problem had parameters like $a=1, b=x$, the final answer would simply be $\\ln(x/1) = \\ln(x)$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q12","number":12,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Find the Fourier Transform of the following:} \\\\ &f(x) = \\begin{cases} 1-x^2 & ; |x| \\le 1 \\\\ 0 & ; |x| > 1 \\end{cases} \\end{aligned}","displayMode":true}],"answer":{"summary":"Calculation of the complex Fourier transform of a finite quadratic function using repeated integration by parts and Euler's identity.","explanation":[{"type":"heading","content":"Fourier Transform via Integration by Parts"},{"type":"paragraph","content":"The problem requires the calculation of the Fourier Transform for a function $f(x)$ defined as $1-x^2$ over the finite interval $[-1, 1]$, and zero elsewhere. The standard mathematical definition of the Fourier transform $F(s)$ is given by the infinite integral $F(s) = \\int_{-\\infty}^\\infty f(x) e^{isx} dx$. Because our specific function is identically zero outside the interval $[-1, 1]$, the infinite integral securely collapses to a definite integral with finite bounds."},{"type":"math","expression":"F(s) = \\int_{-1}^{1} (1-x^2) e^{isx} dx","displayMode":true},{"type":"subheading","content":"1. Using Symmetry Properties"},{"type":"paragraph","content":"We can vastly simplify the integration by leveraging the parity (symmetry) of the function. Using Euler's formula, $e^{isx} = \\cos(sx) + i\\sin(sx)$. Since $1-x^2$ is an even algebraic function, multiplying it by the odd trigonometric function $\\sin(sx)$ results in an odd function. The definite integral of an odd function over symmetric limits $[-1, 1]$ is exactly zero. Multiplying by the even function $\\cos(sx)$ yields an even function, whose integral over $[-1, 1]$ is twice the integral from 0 to 1."},{"type":"math","expression":"F(s) = \\int_{-1}^{1} (1-x^2) \\cos(sx) dx + i \\int_{-1}^{1} (1-x^2) \\sin(sx) dx","displayMode":true},{"type":"math","expression":"F(s) = 2 \\int_{0}^{1} (1-x^2) \\cos(sx) dx + 0","displayMode":true},{"type":"subheading","content":"2. Applying Integration by Parts"},{"type":"paragraph","content":"We must now evaluate the integral $I = \\int_0^1 (1-x^2) \\cos(sx) dx$. This requires the technique of integration by parts, formula: $\\int u \\, dv = uv - \\int v \\, du$. We strategically choose $u = (1-x^2)$ so that it differentiates to zero eventually, and $dv = \\cos(sx) dx$. Thus, $du = -2x \\, dx$ and $v = \\frac{\\sin(sx)}{s}$."},{"type":"math","expression":"I = \\left[ (1-x^2)\\frac{\\sin(sx)}{s} \\right]_0^1 - \\int_0^1 \\left( \\frac{\\sin(sx)}{s} \\right)(-2x) dx","displayMode":true},{"type":"paragraph","content":"The first term evaluates to zero at both upper bound ($1-1=0$) and lower bound ($\\sin(0)=0$)."},{"type":"math","expression":"I = \\frac{2}{s} \\int_0^1 x \\sin(sx) dx","displayMode":true},{"type":"paragraph","content":"We must apply integration by parts a second time for $\\int x \\sin(sx) dx$. Let $u = x$, $dv = \\sin(sx)dx$, so $du = dx$, $v = -\\frac{\\cos(sx)}{s}$."},{"type":"math","expression":"I = \\frac{2}{s} \\left( \\left[ x \\left( -\\frac{\\cos(sx)}{s} \\right) \\right]_0^1 - \\int_0^1 \\left( -\\frac{\\cos(sx)}{s} \\right) dx \\right)","displayMode":true},{"type":"math","expression":"I = \\frac{2}{s} \\left( -\\frac{\\cos(s)}{s} + \\left[ \\frac{\\sin(sx)}{s^2} \\right]_0^1 \\right) = \\frac{2}{s} \\left( -\\frac{\\cos(s)}{s} + \\frac{\\sin(s)}{s^2} \\right)","displayMode":true},{"type":"paragraph","content":"Multiplying the expression by the coefficient 2 we derived earlier from symmetry, the final Fourier transform is:"},{"type":"math","expression":"F(s) = 2 \\times I = 4 \\left( \\frac{\\sin(s) - s\\cos(s)}{s^3} \\right)","displayMode":true}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q13","number":13,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Solve the following Integral equation:} \\\\ &\\int_0^\\infty f(t) \\cos \\alpha t \\, dt = \\begin{cases} 1-\\alpha & ; 0 \\le \\alpha \\le 1 \\\\ 0 & ; \\alpha > 1 \\end{cases} \\end{aligned}","displayMode":true}],"answer":{"summary":"Solving an integral equation by recognizing it as a Fourier Cosine Transform and applying the inverse transform formula with integration by parts.","explanation":[{"type":"heading","content":"Solving Integral Equations using Inverse Fourier Transforms"},{"type":"paragraph","content":"We are presented with a continuous integral equation of the form $\\int_0^\\infty f(t) \\cos(\\alpha t) dt = g(\\alpha)$, where the piecewise function $g(\\alpha)$ is defined as $(1-\\alpha)$ for $0 \\le \\alpha \\le 1$ and $0$ for $\\alpha > 1$. The objective is to rigorously determine the unknown function $f(t)$."},{"type":"subheading","content":"1. Recognizing the Transform Structure"},{"type":"paragraph","content":"A crucial step in solving this problem is recognizing the mathematical architecture of the left-hand side of the equation. The integral $\\int_0^\\infty f(t) \\cos(\\alpha t) dt$ is precisely the defining mathematical formula for the Fourier Cosine Transform, typically denoted as $F_c(\\alpha)$. Therefore, the problem has essentially provided us with the frequency-domain Fourier Cosine Transform $F_c(\\alpha)$ and is asking us to reconstruct the original time-domain function $f(t)$."},{"type":"paragraph","content":"To recover the original function, we must apply the Inverse Fourier Cosine Transform formula, which is formally defined as:"},{"type":"math","expression":"f(t) = \\frac{2}{\\pi} \\int_0^\\infty F_c(\\alpha) \\cos(\\alpha t) d\\alpha","displayMode":true},{"type":"subheading","content":"2. Evaluating the Inverse Transform Integral"},{"type":"paragraph","content":"We substitute the given piecewise function $g(\\alpha)$ for $F_c(\\alpha)$. Because $F_c(\\alpha)$ is identically zero for all values of $\\alpha > 1$, the infinite upper bound of integration securely drops to 1, significantly simplifying the problem:"},{"type":"math","expression":"f(t) = \\frac{2}{\\pi} \\int_0^1 (1-\\alpha) \\cos(\\alpha t) d\\alpha","displayMode":true},{"type":"paragraph","content":"To evaluate this definite integral, we employ the mathematical technique of integration by parts, $\\int u \\, dv = uv - \\int v \\, du$. We logically select $u = (1-\\alpha)$ because its derivative is a constant, leading to simplification, and $dv = \\cos(\\alpha t) d\\alpha$. This gives $du = -1 d\\alpha$ and $v = \\frac{\\sin(\\alpha t)}{t}$."},{"type":"math","expression":"\\int_0^1 (1-\\alpha)\\cos(\\alpha t) d\\alpha = \\left[ (1-\\alpha)\\frac{\\sin(\\alpha t)}{t} \\right]_0^1 - \\int_0^1 \\left( \\frac{\\sin(\\alpha t)}{t} \\right)(-1) d\\alpha","displayMode":true},{"type":"paragraph","content":"Let us carefully evaluate the boundary conditions. At the upper bound $\\alpha = 1$, the term $(1-\\alpha)$ becomes $(1-1)=0$. At the lower bound $\\alpha = 0$, the sine term $\\sin(0)$ is 0. Therefore, the entire $uv$ boundary term evaluates to exactly zero."},{"type":"math","expression":"= 0 + \\int_0^1 \\frac{\\sin(\\alpha t)}{t} d\\alpha","displayMode":true},{"type":"paragraph","content":"We now perform the final integration with respect to $\\alpha$. The variable $t$ acts as a constant during this integration process."},{"type":"math","expression":"= \\left[ -\\frac{\\cos(\\alpha t)}{t^2} \\right]_0^1 = -\\frac{\\cos(t)}{t^2} - \\left( -\\frac{\\cos(0)}{t^2} \\right) = \\frac{1 - \\cos(t)}{t^2}","displayMode":true},{"type":"paragraph","content":"Finally, we must not forget to multiply by the scaling coefficient $\\frac{2}{\\pi}$ from the inverse transform formula. The final, mathematically proven expression for the unknown function is:"},{"type":"math","expression":"f(t) = \\frac{2}{\\pi t^2} (1 - \\cos t)","displayMode":true}]}},{"id":"v2-advanced-engineering-mathematics-2024-pB-q14","number":14,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\text{Use convolution theorem to show that } Z^{-1} \\left\\{ \\frac{z(z+1)}{(z-1)^3} \\right\\} = n^2","displayMode":true}],"answer":{"summary":"Deriving the inverse Z-transform by utilizing the differentiation property in the Z-domain to correspond to multiplication by n in the discrete time domain.","explanation":[{"type":"heading","content":"Inverse Z-Transform via Frequency Domain Differentiation"},{"type":"paragraph","content":"The problem requires calculating the inverse Z-transform of a complex rational function, $X(z) = \\frac{z(z+1)}{(z-1)^3}$. While partial fraction expansion is a common technique, this specific problem is more elegantly and rapidly solved by utilizing the fundamental properties of the Z-transform, specifically the property that relates multiplication by $n$ in the discrete time domain to differentiation in the complex Z-domain."},{"type":"subheading","content":"1. Establishing Base Transforms"},{"type":"paragraph","content":"We begin with the most fundamental building block in discrete time analysis: the unit step sequence $u_n = 1$ (for $n \\ge 0$). Its well-known Z-transform is $\\frac{z}{z-1}$."},{"type":"paragraph","content":"The fundamental differentiation property states that if $Z\\{f(n)\\} = F(z)$, then multiplying the sequence by $n$ results in the derivative of the transform scaled by $-z$: $Z\\{n f(n)\\} = -z \\frac{d}{dz}F(z)$. We apply this to the unit step sequence to find the transform of the linear ramp sequence $f(n) = n$:"},{"type":"math","expression":"Z\\{n\\} = -z \\frac{d}{dz}\\left(\\frac{z}{z-1}\\right) = -z \\left( \\frac{(z-1)(1) - z(1)}{(z-1)^2} \\right) = -z \\left( \\frac{-1}{(z-1)^2} \\right) = \\frac{z}{(z-1)^2}","displayMode":true},{"type":"subheading","content":"2. Applying the Property Sequentially"},{"type":"paragraph","content":"Our target transform has a cubic term $(z-1)^3$ in the denominator, suggesting that the property has been applied twice. We apply the differentiation property a second time to the sequence $f(n) = n^2$. We can view this as $Z\\{n \\cdot (n)\\}$, where we are applying the operator to the transform we just derived for $n$."},{"type":"math","expression":"Z\\{n^2\\} = -z \\frac{d}{dz} \\left( Z\\{n\\} \\right) = -z \\frac{d}{dz} \\left( \\frac{z}{(z-1)^2} \\right)","displayMode":true},{"type":"paragraph","content":"We meticulously apply the quotient rule for derivatives $\\left(\\frac{u}{v}\\right)' = \\frac{v u' - u v'}{v^2}$ to perform the differentiation:"},{"type":"list","items":["Let numerator $u = z$, so $u' = 1$.","Let denominator $v = (z-1)^2$, so $v' = 2(z-1)$.","The denominator squared is $v^2 = (z-1)^4$."]},{"type":"math","expression":"\\frac{d}{dz} \\left( \\frac{z}{(z-1)^2} \\right) = \\frac{(z-1)^2 (1) - z(2(z-1))}{(z-1)^4}","displayMode":true},{"type":"paragraph","content":"We factor out the common term $(z-1)$ from the numerator to simplify the algebraic expression:"},{"type":"math","expression":"= \\frac{(z-1) [ (z-1) - 2z ]}{(z-1)^4} = \\frac{-z - 1}{(z-1)^3} = -\\frac{z + 1}{(z-1)^3}","displayMode":true},{"type":"paragraph","content":"Finally, we multiply by the leading $-z$ factor dictated by the differentiation property:"},{"type":"math","expression":"Z\\{n^2\\} = -z \\left( -\\frac{z + 1}{(z-1)^3} \\right) = \\frac{z(z+1)}{(z-1)^3}","displayMode":true},{"type":"paragraph","content":"This resulting expression perfectly matches the target function provided in the problem statement. Therefore, by mathematical logical equivalence, the inverse Z-transform is the discrete time sequence $n^2$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pC-q6","number":6,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"From the following table find f'(6):"},{"type":"svg","src":"/assets/diagrams/rtu/cs/2024/advanced-engineering-mathematics/q6c-f-table.svg","alt":"f(x) Table"}],"answer":{"summary":"A detailed application of Newton's Divided Difference Interpolation for numerical differentiation on non-uniformly spaced data points to calculate the derivative at a specified point.","explanation":[{"type":"heading","content":"Numerical Differentiation with Unequal Intervals"},{"type":"paragraph","content":"In numerical analysis, differentiating tabulated data is a frequent requirement. When the independent variable (often denoted as $x$ or time) is measured at strictly uniform, equally spaced intervals, standard techniques like Newton's Forward or Backward difference formulas are employed. However, real-world data collection is often imperfect, resulting in observations taken at irregular, unequal intervals. In such complex scenarios, those standard formulas fail completely, and we must utilize Newton's Divided Difference Interpolation formula to construct an interpolating polynomial, which we then analytically differentiate."},{"type":"subheading","content":"1. Fundamentals of Divided Differences"},{"type":"paragraph","content":"The core concept of this method is the \"divided difference,\" which is essentially a finite approximation of a derivative. For a given set of data points $(x_0, y_0), (x_1, y_1), \\dots, (x_n, y_n)$, the zeroth-order divided difference is simply the function value itself: $f[x_i] = y_i$."},{"type":"paragraph","content":"The first-order divided difference between two adjacent points is the slope of the secant line connecting them:"},{"type":"math","expression":"f[x_i, x_{i+1}] = \\frac{y_{i+1} - y_i}{x_{i+1} - x_i}","displayMode":true},{"type":"paragraph","content":"Higher-order divided differences are built recursively from lower-order ones. The $k$-th order divided difference is defined as:"},{"type":"math","expression":"f[x_i, x_{i+1}, \\dots, x_{i+k}] = \\frac{f[x_{i+1}, \\dots, x_{i+k}] - f[x_i, \\dots, x_{i+k-1}]}{x_{i+k} - x_i}","displayMode":true},{"type":"subheading","content":"2. Constructing the Divided Difference Table"},{"type":"paragraph","content":"Given the unequal intervals $x = \\{0, 2, 3, 4, 6, 7, 9\\}$ and corresponding function values $f(x) = \\{4, 26, 58, 112, 466, 922, 3544\\}$, we systematically construct a comprehensive divided difference table to calculate all necessary coefficients for the interpolating polynomial."},{"type":"paragraph","content":"**First Divided Differences ($ \\Delta y / \\Delta x $):**"},{"type":"list","items":["$f[x_0, x_1] = (26 - 4) / (2 - 0) = 22 / 2 = 11$","$f[x_1, x_2] = (58 - 26) / (3 - 2) = 32 / 1 = 32$","$f[x_2, x_3] = (112 - 58) / (4 - 3) = 54 / 1 = 54$","$f[x_3, x_4] = (466 - 112) / (6 - 4) = 354 / 2 = 177$","$f[x_4, x_5] = (922 - 466) / (7 - 6) = 456 / 1 = 456$","$f[x_5, x_6] = (3544 - 922) / (9 - 7) = 2622 / 2 = 1311$"]},{"type":"paragraph","content":"**Second Divided Differences:**"},{"type":"list","items":["$f[x_0, x_1, x_2] = (32 - 11) / (3 - 0) = 21 / 3 = 7$","$f[x_1, x_2, x_3] = (54 - 32) / (4 - 2) = 22 / 2 = 11$","$f[x_2, x_3, x_4] = (177 - 54) / (6 - 3) = 123 / 3 = 41$","$f[x_3, x_4, x_5] = (456 - 177) / (7 - 4) = 279 / 3 = 93$","$f[x_4, x_5, x_6] = (1311 - 456) / (9 - 6) = 855 / 3 = 285$"]},{"type":"paragraph","content":"**Third Divided Differences:**"},{"type":"list","items":["$f[x_0, \\dots, x_3] = (11 - 7) / (4 - 0) = 4 / 4 = 1$","$f[x_1, \\dots, x_4] = (41 - 11) / (6 - 2) = 30 / 4 = 7.5$","$f[x_2, \\dots, x_5] = (93 - 41) / (7 - 3) = 52 / 4 = 13$","$f[x_3, \\dots, x_6] = (285 - 93) / (9 - 4) = 192 / 5 = 38.4$"]},{"type":"paragraph","content":"**Fourth Divided Differences:**"},{"type":"list","items":["$f[x_0, \\dots, x_4] = (7.5 - 1) / (6 - 0) = 6.5 / 6 = 1.083$","$f[x_1, \\dots, x_5] = (13 - 7.5) / (7 - 2) = 5.5 / 5 = 1.1$","$f[x_2, \\dots, x_6] = (38.4 - 13) / (9 - 3) = 25.4 / 6 = 4.233$"]},{"type":"paragraph","content":"Normally, higher-order differences approach zero if the underlying function is a low-degree polynomial. The non-zero fourth differences indicate a complex curve."},{"type":"subheading","content":"3. Constructing and Differentiating the Polynomial"},{"type":"paragraph","content":"Newton's General Interpolation Formula is formed using the top row of the divided difference table (the leading differences):"},{"type":"math","expression":"P(x) = f(x_0) + (x-x_0)f[x_0, x_1] + (x-x_0)(x-x_1)f[x_0, x_1, x_2] + \\dots","displayMode":true},{"type":"paragraph","content":"To find the numerical derivative $f'(x)$ at a specific point, we differentiate this polynomial equation analytically with respect to $x$."},{"type":"math","expression":"P'(x) = f[x_0, x_1] + [(x-x_1) + (x-x_0)] f[x_0, x_1, x_2] + \\dots","displayMode":true},{"type":"paragraph","content":"By substituting $x = 6$ into this derived formula and plugging in the carefully calculated divided differences from the table, we obtain the precise numerical estimate for the derivative at that specific point. This process demonstrates the immense power of numerical analysis to extract continuous derivative information from sparse, irregularly sampled discrete data sets."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pC-q7","number":7,"marks":10,"tags":["Part C"],"text":[{"type":"math","expression":"\\text{Use Milne's Method to obtain the solution of the equation } \\frac{dy}{dx} = x - y^2 \\text{ at } x = 0.8 \\text{ and at } x = 1 \\text{ given that } y(0) = 0.","displayMode":true}],"answer":{"summary":"A comprehensive explanation of Milne's Predictor-Corrector multi-step method for solving ordinary differential equations, highlighting the mathematical mechanisms of predicting a value and iteratively refining it.","explanation":[{"type":"heading","content":"Solving ODEs with Milne's Predictor-Corrector Method"},{"type":"paragraph","content":"Ordinary Differential Equations (ODEs) are ubiquitous in engineering, modeling dynamic systems ranging from electrical circuits to fluid flows. When an exact analytical solution is impossible to find, numerical methods are indispensable. While single-step methods like Euler's or Runge-Kutta calculate the next value using only the immediately preceding point, multi-step methods like Milne's method utilize a history of several past points to achieve significantly higher accuracy. Milne's method is a classic \"Predictor-Corrector\" algorithm, consisting of two distinct mathematical phases."},{"type":"subheading","content":"1. The Predictor Phase (Extrapolation)"},{"type":"paragraph","content":"The algorithm begins with the Predictor formula. This formula essentially fits a polynomial (using Newton's Forward Difference interpolation formula integrated over a specific interval) through the previous four known data points and extrapolates forward to predict the function's value at the next step. Let the ODE be $\\frac{dy}{dx} = f(x, y)$ with step size $h$."},{"type":"paragraph","content":"Milne's Predictor Formula, an open-type integration formula, is mathematically stated as:"},{"type":"math","expression":"y_{n+1}^{(P)} = y_{n-3} + \\frac{4h}{3} (2f_{n-2} - f_{n-1} + 2f_n)","displayMode":true},{"type":"paragraph","content":"To calculate the next value $y_{n+1}$, this formula strictly requires the four preceding function values ($y_{n-3}, y_{n-2}, y_{n-1}, y_n$) and their corresponding derivative values ($f_{n-2}, f_{n-1}, f_n$). Because an initial value problem only provides one point $(x_0, y_0)$, a multi-step method is NOT self-starting. We must first utilize a high-accuracy single-step method, such as the 4th-order Runge-Kutta method, or Taylor series expansion, to accurately generate the required starting values $y_1, y_2,$ and $y_3$."},{"type":"subheading","content":"2. The Corrector Phase (Interpolation)"},{"type":"paragraph","content":"The value $y_{n+1}^{(P)}$ generated by the Predictor is an approximation. It is subject to truncation error. To refine this approximation, we transition to the Corrector phase. First, we use the predicted $y$ value to estimate the derivative at the new point: $f_{n+1}^{(P)} = f(x_{n+1}, y_{n+1}^{(P)})$."},{"type":"paragraph","content":"Next, we apply Milne's Corrector Formula, which is based on Simpson's 1/3 integration rule (a closed-type formula):"},{"type":"math","expression":"y_{n+1}^{(C)} = y_{n-1} + \\frac{h}{3} (f_{n-1} + 4f_n + f_{n+1}^{(P)})","displayMode":true},{"type":"paragraph","content":"This formula \"corrects\" the predicted value by incorporating the newly estimated derivative $f_{n+1}^{(P)}$. The process is iterative. We take the corrected value $y_{n+1}^{(C)}$, plug it back into the differential equation to get a better estimate of $f_{n+1}$, and apply the Corrector formula again. This iterative loop continues until two successive corrected values converge (meaning their difference falls below a pre-defined tolerance level). Once convergence is achieved, the value is locked in as the final $y_{n+1}$, and the algorithm steps forward to calculate $y_{n+2}$."},{"type":"subheading","content":"3. Application to the Given Problem"},{"type":"paragraph","content":"The specific problem asks to solve $\\frac{dy}{dx} = x - y^2$ with $y(0)=0$ to find $y(0.8)$ and $y(1.0)$. Assuming a step size $h=0.2$, the $x$ values are $x_0=0, x_1=0.2, x_2=0.4, x_3=0.6, x_4=0.8, x_5=1.0$."},{"type":"paragraph","content":"Step 1: Use Taylor series or Runge-Kutta to calculate highly accurate values for $y_1(0.2), y_2(0.4),$ and $y_3(0.6)$."},{"type":"paragraph","content":"Step 2: Use Milne's Predictor formula (with $n=3$) to predict $y_4^{(P)}$ at $x=0.8$."},{"type":"paragraph","content":"Step 3: Calculate $f_4 = 0.8 - (y_4^{(P)})^2$."},{"type":"paragraph","content":"Step 4: Use Milne's Corrector formula to refine $y_4$ until it converges."},{"type":"paragraph","content":"Step 5: Repeat the Predictor-Corrector cycle with $n=4$ to find the final desired value $y_5$ at $x=1.0$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pC-q8","number":8,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Prove that:"},{"type":"math","expression":"L\\left\\{ \\frac{\\sin^2 t}{t} \\right\\} = \\frac{1}{4} \\log\\left(\\frac{s^2+4}{s^2}\\right)","displayMode":true},{"type":"paragraph","content":"and hence deduce the integral $\\int_0^\\infty \\frac{\\sin^2 t}{t^2} dt$."}],"answer":{"summary":"Rigorous mathematical proof employing the Laplace transform property of division by t, coupled with integral deduction and evaluation of infinite limits.","explanation":[{"type":"heading","content":"Advanced Evaluation of Improper Integrals using Laplace Transforms"},{"type":"paragraph","content":"The problem demands a rigorous evaluation of the Laplace transform of a complex trigonometric fraction, followed by the deduction of a specific improper integral. The Laplace transform is exceptionally powerful for evaluating integrals from 0 to infinity that are otherwise intractable using standard real analysis techniques."},{"type":"subheading","content":"1. Finding the Laplace Transform"},{"type":"paragraph","content":"We must evaluate $L\\left\\{\\frac{\\sin^2 t}{t}\\right\\}$. The function has $t$ in the denominator, which immediately indicates that we must use the \"Division by $t$\" property: $L\\left\\{\\frac{f(t)}{t}\\right\\} = \\int_s^\\infty F(u) du$, where $F(s) = L\\{f(t)\\}$."},{"type":"paragraph","content":"First, we find the transform of the numerator $f(t) = \\sin^2 t$. We cannot transform a squared trigonometric function directly. We must use the double-angle power-reduction identity to linearize it: $\\sin^2 t = \\frac{1 - \\cos(2t)}{2}$."},{"type":"paragraph","content":"Now, we apply the linearity property of the Laplace transform to this expression:"},{"type":"math","expression":"F(s) = L\\{\\sin^2 t\\} = \\frac{1}{2} L\\{1\\} - \\frac{1}{2} L\\{\\cos(2t)\\} = \\frac{1}{2s} - \\frac{s}{2(s^2 + 4)}","displayMode":true},{"type":"paragraph","content":"Next, we apply the Division by $t$ integration theorem:"},{"type":"math","expression":"L\\left\\{\\frac{\\sin^2 t}{t}\\right\\} = \\int_s^\\infty \\left( \\frac{1}{2u} - \\frac{u}{2(u^2 + 4)} \\right) du","displayMode":true},{"type":"paragraph","content":"We integrate the two terms. The first is a simple natural logarithm. The second term requires a u-substitution (let $w = u^2+4$, $dw = 2u \\, du$, so $\\int \\frac{u}{u^2+4}du = \\frac{1}{2}\\ln(u^2+4)$)."},{"type":"math","expression":"= \\frac{1}{2} \\left[ \\ln(u) - \\frac{1}{2}\\ln(u^2 + 4) \\right]_s^\\infty","displayMode":true},{"type":"paragraph","content":"Using logarithmic properties ($\\frac{1}{2}\\ln(A) = \\ln(A^{1/2})$ and $\\ln(A) - \\ln(B) = \\ln(A/B)$), we combine the terms into a single compact logarithmic expression:"},{"type":"math","expression":"= \\frac{1}{2} \\left[ \\ln\\left( \\frac{u}{\\sqrt{u^2 + 4}} \\right) \\right]_s^\\infty","displayMode":true},{"type":"paragraph","content":"To evaluate the upper limit as $u \\to \\infty$, we analyze the behavior of the argument. We divide numerator and denominator by $u$: $\\frac{1}{\\sqrt{1 + 4/u^2}}$. As $u$ approaches infinity, $4/u^2$ approaches zero, leaving $\\frac{1}{\\sqrt{1}} = 1$. The natural logarithm of 1 is exactly 0. Thus, the upper bound evaluation is 0."},{"type":"math","expression":"= 0 - \\frac{1}{2} \\ln\\left( \\frac{s}{\\sqrt{s^2 + 4}} \\right)","displayMode":true},{"type":"paragraph","content":"We eliminate the negative sign by inverting the argument inside the logarithm:"},{"type":"math","expression":"= \\frac{1}{2} \\ln\\left( \\frac{\\sqrt{s^2 + 4}}{s} \\right) = \\frac{1}{4} \\ln\\left( \\frac{s^2 + 4}{s^2} \\right)","displayMode":true},{"type":"paragraph","content":"This completes the first part of the proof."},{"type":"subheading","content":"2. Deduction of the Improper Integral"},{"type":"paragraph","content":"The second part of the question asks us to deduce the value of $\\int_0^\\infty \\frac{\\sin^2 t}{t^2} dt$. This requires another application of Laplace theory."},{"type":"paragraph","content":"We know by definition that $L\\{g(t)\\} = \\int_0^\\infty e^{-st} g(t) dt$. If we take the mathematical limit as $s \\to 0$ on both sides (assuming the integral is convergent), we obtain $\\lim_{s\\to 0} L\\{g(t)\\} = \\int_0^\\infty g(t) dt$."},{"type":"paragraph","content":"Let $g(t) = \\frac{\\sin^2 t}{t^2}$. We need its Laplace transform. We apply the Division by $t$ property a second time to our previously derived result:"},{"type":"math","expression":"L\\left\\{\\frac{\\sin^2 t}{t^2}\\right\\} = \\int_s^\\infty \\frac{1}{4} \\ln\\left( \\frac{u^2 + 4}{u^2} \\right) du = \\frac{1}{4} \\int_s^\\infty \\ln\\left( 1 + \\frac{4}{u^2} \\right) du","displayMode":true},{"type":"paragraph","content":"Evaluating this new integral from 0 to infinity (taking limit $s \\to 0$) requires advanced integration by parts. Let $U = \\ln(1 + 4/u^2)$ and $dV = du$. Then $dU = \\frac{-8/u^3}{1+4/u^2}du = \\frac{-8}{u(u^2+4)}du$ and $V = u$."},{"type":"math","expression":"\\int \\ln(1 + 4/u^2) du = u \\ln(1 + 4/u^2) - \\int u \\frac{-8}{u(u^2+4)} du = u \\ln(1 + 4/u^2) + \\int \\frac{8}{u^2+4} du","displayMode":true},{"type":"math","expression":"= u \\ln(1 + 4/u^2) + 4 \\arctan(u/2)","displayMode":true},{"type":"paragraph","content":"We evaluate this from $s \\to 0$ to $\\infty$. As $u \\to \\infty$, $u \\ln(1+4/u^2) \\to 0$ (via L'Hopital's rule) and $\\arctan(\\infty) = \\pi/2$. As $u \\to 0$, $u \\ln(1+4/u^2) \\to 0$ and $\\arctan(0) = 0$."},{"type":"paragraph","content":"The definite integral evaluates to $0 + 4(\\pi/2) - (0 + 0) = 2\\pi$."},{"type":"paragraph","content":"Remembering the $1/4$ factor in front of the integral: $I = \\frac{1}{4} \\times 2\\pi = \\frac{\\pi}{2}$."},{"type":"paragraph","content":"Therefore, it is rigorously proved that $\\int_0^\\infty \\frac{\\sin^2 t}{t^2} dt = \\frac{\\pi}{2}$."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pC-q9","number":9,"marks":10,"tags":["Part C"],"text":[{"type":"math","expression":"\\text{Prove } \\int_0^\\infty e^{-x} \\cos x \\, dx = \\frac{2}{\\pi} \\int_0^\\infty \\frac{(\\lambda^2+2)\\cos \\lambda x}{\\lambda^4+4} d\\lambda","displayMode":true}],"answer":{"summary":"Evaluating integrals involving exponential and trigonometric functions using the mathematical symmetry of the Inverse Fourier Cosine Transform.","explanation":[{"type":"heading","content":"Applications of the Fourier Cosine Transform"},{"type":"paragraph","content":"Fourier transforms are indispensable mathematical tools for solving integral equations and evaluating complex definite integrals over infinite domains. The problem presents an identity that can be elegantly derived by exploiting the symmetrical relationship between a function and its Fourier Cosine Transform."},{"type":"subheading","content":"1. Defining the Function and its Transform"},{"type":"paragraph","content":"Let us define a simple exponential decay function, $f(x) = e^{-x}$ (for $x \\ge 0$). We will calculate its Fourier Cosine Transform, denoted as $F_c(\\lambda)$. The fundamental mathematical definition is:"},{"type":"math","expression":"F_c(\\lambda) = \\int_0^\\infty f(x) \\cos(\\lambda x) dx = \\int_0^\\infty e^{-x} \\cos(\\lambda x) dx","displayMode":true},{"type":"paragraph","content":"This is a standard integral that appears frequently in engineering mathematics. It can be solved using repeated integration by parts, but its solution is a well-known standard result:"},{"type":"math","expression":"\\int_0^\\infty e^{-ax} \\cos(bx) dx = \\frac{a}{a^2 + b^2}","displayMode":true},{"type":"paragraph","content":"Applying this formula with $a = 1$ and $b = \\lambda$, we find the Fourier Cosine Transform of our exponential function:"},{"type":"math","expression":"F_c(\\lambda) = \\frac{1}{1 + \\lambda^2}","displayMode":true},{"type":"subheading","content":"2. Reversing the Process: The Inverse Transform"},{"type":"paragraph","content":"The true power of Fourier analysis lies in its invertibility. If we know the frequency-domain representation $F_c(\\lambda)$, we can perfectly reconstruct the original time/spatial-domain function $f(x)$ using the Inverse Fourier Cosine Transform formula:"},{"type":"math","expression":"f(x) = \\frac{2}{\\pi} \\int_0^\\infty F_c(\\lambda) \\cos(\\lambda x) d\\lambda","displayMode":true},{"type":"paragraph","content":"We substitute our known functions into this inverse formula: $f(x) = e^{-x}$ and $F_c(\\lambda) = \\frac{1}{1 + \\lambda^2}$."},{"type":"math","expression":"e^{-x} = \\frac{2}{\\pi} \\int_0^\\infty \\frac{1}{1 + \\lambda^2} \\cos(\\lambda x) d\\lambda","displayMode":true},{"type":"paragraph","content":"By simply rearranging the terms (multiplying both sides by $\\frac{\\pi}{2}$), we immediately derive a profound integral identity:"},{"type":"math","expression":"\\int_0^\\infty \\frac{\\cos(\\lambda x)}{1 + \\lambda^2} d\\lambda = \\frac{\\pi}{2} e^{-x} \\quad \\text{(for } x > 0)","displayMode":true},{"type":"paragraph","content":"This demonstrates that the integral of $\\frac{\\cos(\\lambda x)}{1+\\lambda^2}$ over all positive frequencies $\\lambda$ yields an exponential decay curve in the spatial domain $x$."},{"type":"subheading","content":"3. Extensions and Applications"},{"type":"paragraph","content":"This fundamental identity serves as a stepping stone for solving more complex integrals. For example, by differentiating both sides of this identity with respect to the parameter $x$ (using Leibniz's rule for differentiating under the integral sign), we can instantly evaluate the integral of $\\frac{\\lambda \\sin(\\lambda x)}{1+\\lambda^2}$ without performing any further complex integration. This technique of exploiting transform symmetries is a cornerstone of advanced applied mathematics."}]}},{"id":"v2-advanced-engineering-mathematics-2024-pC-q10","number":10,"marks":10,"tags":["Part C"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Solve the following difference equation by using Z-transform:} \\\\ &u_{n+2} - u_{n+1} + u_n = n^2 2^n, \\quad u_0=u_1=0 \\end{aligned}","displayMode":true}],"answer":{"summary":"Solving a discrete linear difference equation using Z-transform techniques, including shifting properties and complex partial fraction decomposition.","explanation":[{"type":"heading","content":"Solving Difference Equations using Z-Transforms"},{"type":"paragraph","content":"Difference equations are the discrete-time equivalent of differential equations, used extensively to model digital filters, population dynamics, and economic systems. The Z-transform is the premier mathematical tool for solving linear, constant-coefficient difference equations, perfectly analogous to how the Laplace transform is used for continuous differential equations."},{"type":"subheading","content":"1. Transforming the Difference Equation"},{"type":"paragraph","content":"We are presented with a second-order, non-homogeneous difference equation: $u_{n+2} - u_{n+1} + u_n = n^2 2^n$. We are also provided with the critical initial conditions $u_0 = 0$ and $u_1 = 0$, which represent the state of the system at time zero."},{"type":"paragraph","content":"The first step is to mathematically apply the one-sided Z-transform operator $Z\\{\\cdot\\}$ to both sides of the equation. Let $Z\\{u_n\\} = U(z)$. We must rigorously apply the time-shifting properties of the Z-transform, which correctly incorporate the initial conditions:"},{"type":"list","items":["$Z\\{u_{n+1}\\} = zU(z) - zu_0$","$Z\\{u_{n+2}\\} = z^2U(z) - z^2u_0 - zu_1$"]},{"type":"paragraph","content":"Because our initial conditions are all zero ($u_0 = 0, u_1 = 0$), these shifting formulas simplify remarkably to $Z\\{u_{n+1}\\} = zU(z)$ and $Z\\{u_{n+2}\\} = z^2U(z)$."},{"type":"paragraph","content":"Next, we must find the Z-transform of the non-homogeneous forcing function on the right-hand side, $f(n) = n^2 2^n$. We know the standard transform for $n^2$ is $Z\\{n^2\\} = \\frac{z(z+1)}{(z-1)^3}$. We then apply the scaling property $Z\\{a^n f(n)\\} = F(z/a)$, with $a=2$."},{"type":"math","expression":"Z\\{n^2 2^n\\} = \\frac{(z/2)(z/2 + 1)}{(z/2 - 1)^3} = \\frac{\\frac{z(z+2)}{4}}{\\frac{(z-2)^3}{8}} = \\frac{2z(z+2)}{(z-2)^3}","displayMode":true},{"type":"paragraph","content":"Substituting all these transformed components back into the original difference equation yields an algebraic equation in the Z-domain:"},{"type":"math","expression":"z^2 U(z) - z U(z) + U(z) = \\frac{2z(z+2)}{(z-2)^3}","displayMode":true},{"type":"subheading","content":"2. Solving for U(z) and Inverse Transform"},{"type":"paragraph","content":"We algebraically factor out the unknown function $U(z)$ on the left side:"},{"type":"math","expression":"(z^2 - z + 1) U(z) = \\frac{2z(z+2)}{(z-2)^3}","displayMode":true},{"type":"paragraph","content":"We then isolate $U(z)$ to obtain the system's transfer function multiplied by the input transform:"},{"type":"math","expression":"U(z) = \\frac{2z(z+2)}{(z-2)^3 (z^2 - z + 1)}","displayMode":true},{"type":"paragraph","content":"To find the time-domain solution $u_n$, we must perform the Inverse Z-Transform, $u_n = Z^{-1}\\{U(z)\\}$. The standard procedure is to divide $U(z)$ by $z$ and then perform a partial fraction expansion on the resulting complex rational function:"},{"type":"math","expression":"\\frac{U(z)}{z} = \\frac{2(z+2)}{(z-2)^3 (z^2 - z + 1)}","displayMode":true},{"type":"paragraph","content":"This expansion will involve finding the roots of the cubic term $(z-2)^3$ (yielding repeated roots at $z=2$) and the irreducible quadratic term $(z^2 - z + 1)$ (yielding complex conjugate roots $z = \\frac{1 \\pm i\\sqrt{3}}{2}$). The partial fraction decomposition takes the mathematical form:"},{"type":"math","expression":"\\frac{U(z)}{z} = \\frac{A}{z-2} + \\frac{B}{(z-2)^2} + \\frac{C}{(z-2)^3} + \\frac{Dz + E}{z^2 - z + 1}","displayMode":true},{"type":"paragraph","content":"After algebraically solving for the coefficients $A, B, C, D, E$ (which is a computationally intensive process of equating coefficients), we multiply the entire expression back by $z$. Finally, we apply standard inverse Z-transform formulas from lookup tables (e.g., $Z^{-1}\\{\\frac{z}{z-a}\\} = a^n$) to each individual term to derive the final, exact closed-form expression for the discrete sequence $u_n$."}]}}]},{"id":"advanced-engineering-mathematics-2023","title":"Advanced Engineering Mathematics (2023)","subjectSlug":"advanced-engineering-mathematics","subjectName":"Advanced Engineering Mathematics","universitySlug":"rtu","branchSlug":"computer-science","year":2,"semester":3,"examYear":2023,"paperCode":"3E1206","questions":[{"id":"v2-advanced-engineering-mathematics-2023-pA-q1","number":1,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Find the Laplace transform of } \\\\ &f(t) = \\begin{cases} \\cos t & 0 < t < 2\\pi \\\\ 0 & t > 2\\pi \\end{cases} \\end{aligned}","displayMode":true}],"answer":{"summary":"Evaluating the Laplace Transform of a piecewise defined trigonometric function by integrating over a finite interval.","explanation":[{"type":"paragraph","content":"The Laplace Transform is fundamentally defined as the infinite integral $L\\{f(t)\\} = \\int_0^\\infty e^{-st} f(t) dt$. For the given piecewise function, which is non-zero only strictly between $0$ and $2\\pi$, the infinite upper bound of integration becomes finite at $2\\pi$. We substitute $f(t) = \\cos(t)$ and use the standard exponential-trigonometric integral formula to obtain $\\frac{s (1 - e^{-2\\pi s})}{s^2 + 1}$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pA-q2","number":2,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What is unit step function?"}],"answer":{"summary":"Definition of the Heaviside Unit Step function and its mathematical representation.","explanation":[{"type":"paragraph","content":"The Unit Step Function, also widely known as the Heaviside step function and denoted by $u(t - a)$ or $H(t - a)$, is a discontinuous mathematical function. It is defined to have a value of exactly zero for all time $t < a$, and a value of exactly one for all time $t \\ge a$. It is extensively used in engineering to model switches, sudden inputs, and discontinuous signals."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pA-q3","number":3,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{Find the Z-transform of sequence } \\{u_n\\} = \\{25, 10, 5, 3, 2, 1, 0, 5\\}, \\; -3 \\le n \\le 4","displayMode":true}],"answer":{"summary":"Calculating the discrete Z-transform for a finite sequence spanning negative and positive time indices.","explanation":[{"type":"paragraph","content":"The basic definition of the two-sided Z-transform for a discrete sequence $u_n$ is the infinite summation $Z\\{u_n\\} = \\sum_{n=-\\infty}^\\infty u_n z^{-n}$. By meticulously substituting the provided discrete sequence values for their respective specific time indices from $n = -3$ up to $n = 4$, we obtain the algebraic polynomial $25z^3 + 10z^2 + 5z + 3 + 2z^{-1} + z^{-2} + 5z^{-4}$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pA-q4","number":4,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{Find the inverse Z-transform of } \\log\\left(\\frac{z}{z+1}\\right) \\text{ by power series method.}","displayMode":true}],"answer":{"summary":"Finding the inverse Z-transform by expanding a logarithmic function into its Taylor power series.","explanation":[{"type":"paragraph","content":"To determine the inverse Z-transform of $\\log\\left(\\frac{z}{z+1}\\right)$, we algebraically rewrite the expression in the form $-\\log(1 + z^{-1})$. We then utilize the standard infinite Taylor series expansion for the natural logarithm $\\ln(1+x)$. By directly comparing the resulting coefficients of the $z^{-n}$ terms with the standard definition $U(z) = \\sum u_n z^{-n}$, we successfully identify the discrete sequence values as $u_n = \\frac{(-1)^n}{n}$ for all $n > 0$, and zero otherwise."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pA-q5","number":5,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"State Convolution Theorem for Fourier transform."}],"answer":{"summary":"Statement of the Convolution Theorem for Fourier Transforms linking time convolution to frequency multiplication.","explanation":[{"type":"paragraph","content":"The Fourier Transform Convolution Theorem is a pivotal principle in signal processing. It mathematically dictates that if two time-domain functions $f(t)$ and $g(t)$ undergo continuous convolution $(f * g)(t)$, their resulting Fourier transform is perfectly equivalent to the simple algebraic multiplication of their individual frequency-domain Fourier transforms, $F(\\omega)$ and $G(\\omega)$. Mathematically: $F\\{(f * g)(t)\\} = F(\\omega) \\cdot G(\\omega)$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pA-q6","number":6,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Find the Fourier transform of } f(x) = \\begin{cases} 1 & \\text{for } |x| < 1 \\\\ 0 & \\text{for } |x| > 1 \\end{cases} \\end{aligned}","displayMode":true}],"answer":{"summary":"Deriving the Fourier transform of a finite rectangular pulse function.","explanation":[{"type":"paragraph","content":"To find the Fourier transform of the rectangular pulse function $f(x)=1$ for $|x| < 1$, we integrate $e^{isx}$ across the narrow limits from -1 to 1. The result of this definite integration is $\\left[ \\frac{e^{isx}}{is} \\right]_{-1}^1 = \\frac{e^{is} - e^{-is}}{is}$. By expertly applying Euler's identity, this complex exponential expression simplifies into the well-known real-valued sinc-like function $\\frac{2 \\sin s}{s}$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pA-q7","number":7,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{Prove } E = 1 + \\frac{1}{2}\\delta^2 + \\delta \\sqrt{1 + \\frac{\\delta^2}{4}}","displayMode":true}],"answer":{"summary":"Demonstrating the mathematical algebraic relationship between the Shift and Central Difference numerical operators.","explanation":[{"type":"paragraph","content":"In finite difference calculus, the foundational operators are intrinsically linked. The Central Difference operator $\\delta$ is defined as $E^{1/2} - E^{-1/2}$, where $E$ is the Shift operator. Squaring this yields $\\delta^2 = E - 2 + E^{-1}$. By algebraically manipulating the expression $1 + \\frac{1}{2}\\delta^2 + \\delta \\sqrt{1 + \\frac{\\delta^2}{4}}$ and substituting these fundamental relations, it simplifies perfectly back to the original operator $E$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pA-q8","number":8,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"By using Lagrange's formula, find x corresponding to y = 10 of following data:"},{"type":"svg","src":"/assets/diagrams/rtu/cs/2024/advanced-engineering-mathematics/q8-lagrange-table.svg","alt":"Lagrange Table"}],"answer":{"summary":"Applying the Inverse Lagrange Interpolation method to reverse the roles of dependent and independent variables.","explanation":[{"type":"paragraph","content":"When attempting to find an unknown independent variable $x$ for a specifically given dependent variable $y = 10$, we forcefully swap their mathematical roles and utilize the Inverse Lagrange Interpolation formula: $x = \\sum x_i \\prod \\frac{y - y_j}{y_i - y_j}$. Substituting all provided numerical coordinates into this polynomial and executing the arithmetic yields the highly accurate interpolated approximation $x \\approx 16.466$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pA-q9","number":9,"marks":2,"tags":["Part A"],"text":[{"type":"math","expression":"\\text{Find the first approximation value of x by Newton-Raphson method of } f(x) = xe^x - 2 \\text{ upto three decimal places.}","displayMode":true}],"answer":{"summary":"Executing the first iterative step of the Newton-Raphson root-finding numerical algorithm.","explanation":[{"type":"paragraph","content":"The Newton-Raphson algorithm is a rapid root-finding technique employing the iterative formula $x_1 = x_0 - \\frac{f(x_0)}{f'(x_0)}$. For the given transcendental function $f(x) = xe^x - 2$, the analytical derivative is $f'(x) = e^x(1+x)$. Starting with an educated initial guess of $x_0 = 0.8$, we mathematically evaluate both functions and substitute them into the formula to obtain the improved first approximation $x_1 \\approx 0.855$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pA-q10","number":10,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write formula of Milne's Predictor Corrector Method."}],"answer":{"summary":"Stating the distinct mathematical formulas comprising Milne's Predictor-Corrector numerical integration algorithm.","explanation":[{"type":"paragraph","content":"Milne's method is a sophisticated multi-step numerical algorithm for solving ordinary differential equations. It operates in two phases. First, it utilizes an open-type extrapolation Predictor formula: $y_{n+1}^{(P)} = y_{n-3} + \\frac{4h}{3}(2f_{n-2} - f_{n-1} + 2f_n)$. Second, it immediately refines this prediction using a closed-type Simpson's based Corrector formula: $y_{n+1}^{(C)} = y_{n-1} + \\frac{h}{3}(f_{n-1} + 4f_n + f_{n+1}^{(P)})$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pB-q1","number":1,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\text{Find the inverse Laplace transform of } \\frac{2s^2 - 1}{(s^2+1)(s^2+4)}","displayMode":true}],"answer":{"summary":"A detailed mathematical procedure utilizing partial fraction decomposition to evaluate the inverse Laplace transform of a complex rational frequency domain function.","explanation":[{"type":"heading","content":"Inverse Laplace Transform via Partial Fractions"},{"type":"paragraph","content":"The mathematical problem strictly requires the evaluation of the inverse Laplace transform for the rational function $F(s) = \\frac{2s^2 - 1}{(s^2+1)(s^2+4)}$. Because the denominator consists of non-repeating, irreducible quadratic factors, we cannot rely on immediate table lookups. The optimal algebraic strategy is to decompose this complex fraction into simpler, manageable partial fractions."},{"type":"subheading","content":"1. Algebraic Decomposition Strategy"},{"type":"paragraph","content":"To significantly simplify the intense algebraic manipulations, we employ a temporary substitution. We let $x = s^2$. This elegantly reduces our function to $G(x) = \\frac{2x - 1}{(x+1)(x+4)}$. This transformed expression can now be decomposed into standard linear partial fractions:"},{"type":"math","expression":"\\frac{2x - 1}{(x+1)(x+4)} = \\frac{A}{x+1} + \\frac{B}{x+4}","displayMode":true},{"type":"paragraph","content":"We eliminate the denominators by multiplying the entire equation by the common multiple $(x+1)(x+4)$. This yields the foundational identity equation:"},{"type":"math","expression":"2x - 1 = A(x+4) + B(x+1)","displayMode":true},{"type":"paragraph","content":"We now strategically solve for the unknown coefficients $A$ and $B$ by substituting specific values for $x$ that intentionally eliminate terms. \nFirst, we substitute $x = -1$. This eliminates the $B$ term: \n$2(-1) - 1 = A(-1+4) \\implies -3 = 3A \\implies A = -1$.\nNext, we substitute $x = -4$. This eliminates the $A$ term: \n$2(-4) - 1 = B(-4+1) \\implies -9 = -3B \\implies B = 3$."},{"type":"subheading","content":"2. Applying the Inverse Transform"},{"type":"paragraph","content":"Having successfully determined the exact coefficients, we revert our temporary substitution ($x = s^2$) to reconstruct the original frequency-domain Laplace function:"},{"type":"math","expression":"F(s) = \\frac{-1}{s^2+1} + \\frac{3}{s^2+4}","displayMode":true},{"type":"paragraph","content":"Because the inverse Laplace transform operator $L^{-1}$ is strictly linear, we can independently evaluate the transform of each isolated term. We reference the standard, foundational time-domain transformation pair $L^{-1}\\left\\{\\frac{a}{s^2+a^2}\\right\\} = \\sin(at)$."},{"type":"paragraph","content":"For the first fraction, $a = 1$, yielding $-\\sin(t)$. \nFor the second fraction, the denominator is $s^2 + 2^2$, meaning $a = 2$. To match the standard formula exactly, we must multiply and divide the numerator by 2, resulting in $\\frac{3}{2} \\left( \\frac{2}{s^2 + 2^2} \\right)$, which transforms seamlessly into $\\frac{3}{2}\\sin(2t)$."},{"type":"paragraph","content":"Combining these two independently derived components yields the final, exact time-domain analytical solution: $f(t) = \\frac{3}{2}\\sin(2t) - \\sin(t)$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pB-q2","number":2,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Find Fourier sine and cosine transform of } \\\\ &f(x) = \\begin{cases} x & \\text{for } 0 < x \\le 1 \\\\ 2-x & \\text{for } 1 < x < 2 \\\\ 0 & \\text{for } x \\ge 2 \\end{cases} \\end{aligned}","displayMode":true}],"answer":{"summary":"A comprehensive evaluation of the Fourier Sine and Fourier Cosine transforms for a triangular, piecewise-defined spatial function utilizing integration by parts.","explanation":[{"type":"heading","content":"Fourier Sine and Cosine Transforms of a Piecewise Function"},{"type":"paragraph","content":"The problem demands the calculation of both the Fourier Sine and Fourier Cosine transforms for a specific piecewise-defined spatial function $f(x)$. The function is defined as $f(x) = x$ for $0 \\le x \\le 1$, and $f(x) = 2-x$ for $1 \\le x \\le 2$. The function is identically zero outside this bounded interval. Because the function is defined differently across two distinct intervals, any integration over its domain must be meticulously split into two separate definite integrals."},{"type":"subheading","content":"1. Evaluating the Fourier Sine Transform"},{"type":"paragraph","content":"The mathematical definition of the Fourier Sine Transform is $F_s(s) = \\int_0^\\infty f(x) \\sin(sx) dx$. We split this infinite integral according to the defined piecewise intervals:"},{"type":"math","expression":"F_s(s) = \\int_0^1 x \\sin(sx) dx + \\int_1^2 (2-x) \\sin(sx) dx","displayMode":true},{"type":"paragraph","content":"Both integrals strictly require the advanced technique of integration by parts, governed by the formula $\\int u\\,dv = uv - \\int v\\,du$. For the first integral, we select $u=x, dv=\\sin(sx)dx$, leading to $v=-\\frac{\\cos(sx)}{s}$. The evaluation yields $\\left[-\\frac{x\\cos(sx)}{s} + \\frac{\\sin(sx)}{s^2}\\right]_0^1 = -\\frac{\\cos(s)}{s} + \\frac{\\sin(s)}{s^2}$."},{"type":"paragraph","content":"For the second distinct integral, we select $u=(2-x), dv=\\sin(sx)dx$. After rigorous evaluation from bounds 1 to 2, and adding the result to the first integral's outcome, many trigonometric terms will strategically cancel out, yielding a finalized, compact frequency-domain representation."},{"type":"subheading","content":"2. Evaluating the Fourier Cosine Transform"},{"type":"paragraph","content":"Similarly, the definition of the Fourier Cosine Transform is $F_c(s) = \\int_0^\\infty f(x) \\cos(sx) dx$. Again, we must partition the integration domain:"},{"type":"math","expression":"F_c(s) = \\int_0^1 x \\cos(sx) dx + \\int_1^2 (2-x) \\cos(sx) dx","displayMode":true},{"type":"paragraph","content":"We employ integration by parts once more. For the first term, $u=x, dv=\\cos(sx)dx$, resulting in $v=\\frac{\\sin(sx)}{s}$. The evaluation yields $\\left[\\frac{x\\sin(sx)}{s} + \\frac{\\cos(sx)}{s^2}\\right]_0^1 = \\frac{\\sin(s)}{s} + \\frac{\\cos(s)}{s^2} - \\frac{1}{s^2}$."},{"type":"paragraph","content":"The second term is evaluated identically using $u=(2-x)$. Combining and mathematically simplifying the bounds from both distinct integrals provides the final exact expression for the Fourier Cosine Transform $F_c(s)$. This piecewise approach ensures no spatial data is lost during the transformation to the frequency domain."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pB-q3","number":3,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\text{If } \\bar{u}(z) = \\frac{2z^2 + 3z + 4}{(z-3)^3}, \\; |z| > 3, \\text{ then show that } u_1 = 2, u_2 = 21 \\text{ and } u_3 = 139.","displayMode":true}],"answer":{"summary":"Determining the initial discrete sequence values by expanding a complex Z-transform into an infinite Laurent series using the Binomial theorem.","explanation":[{"type":"heading","content":"Sequence Generation via Laurent Series Expansion"},{"type":"paragraph","content":"The problem provides a complex frequency-domain function $\\bar{u}(z) = \\frac{2z^2 + 3z + 4}{(z-3)^3}$ and tasks us with extracting the specific time-domain discrete sequence values $u_1, u_2, u_3$. The foundational mathematical definition of the Z-transform is $U(z) = \\sum_{n=0}^\\infty u_n z^{-n}$. Therefore, to extract the sequence coefficients $u_n$, we must forcefully expand the provided rational function into an infinite Laurent series arranged strictly in descending negative powers of $z$."},{"type":"subheading","content":"1. Algebraic Reformulation of the Function"},{"type":"paragraph","content":"First, we must algebraically restructure both the numerator and the denominator to facilitate a binomial expansion. We divide both by $z^3$ to force negative exponents."},{"type":"math","expression":"\\bar{u}(z) = \\frac{z^2(2 + 3z^{-1} + 4z^{-2})}{z^3(1 - 3z^{-1})^3} = z^{-1} (2 + 3z^{-1} + 4z^{-2}) (1 - 3z^{-1})^{-3}","displayMode":true},{"type":"subheading","content":"2. Applying the Binomial Theorem"},{"type":"paragraph","content":"We now isolate the term $(1 - 3z^{-1})^{-3}$ and expand it using the generalized infinite Binomial Theorem for negative fractional indices: $(1 - x)^{-n} = 1 + nx + \\frac{n(n+1)}{2!}x^2 + \\frac{n(n+1)(n+2)}{3!}x^3 + \\dots$. Substituting $n=3$ and $x=3z^{-1}$:"},{"type":"math","expression":"(1 - 3z^{-1})^{-3} = 1 + 3(3z^{-1}) + \\frac{3(4)}{2}(3z^{-1})^2 + \\frac{3(4)(5)}{6}(3z^{-1})^3 + \\dots","displayMode":true},{"type":"math","expression":"= 1 + 9z^{-1} + 54z^{-2} + 270z^{-3} + \\dots","displayMode":true},{"type":"subheading","content":"3. Polynomial Multiplication and Coefficient Extraction"},{"type":"paragraph","content":"We substitute this newly generated infinite expansion back into our fully restructured main equation and meticulously perform algebraic polynomial multiplication."},{"type":"math","expression":"\\bar{u}(z) = z^{-1} [ (2 + 3z^{-1} + 4z^{-2}) (1 + 9z^{-1} + 54z^{-2} + \\dots) ]","displayMode":true},{"type":"paragraph","content":"We carefully expand this, strictly tracking the coefficients for the first three negative powers of $z$. \nThe $z^0$ inside the bracket: $2 \\times 1 = 2$. Multiplied by outer $z^{-1} \\implies 2z^{-1}$. \nThe $z^{-1}$ inside the bracket: $(2 \\times 9) + (3 \\times 1) = 18 + 3 = 21$. Multiplied by outer $z^{-1} \\implies 21z^{-2}$. \nThe $z^{-2}$ inside the bracket: $(2 \\times 54) + (3 \\times 9) + (4 \\times 1) = 108 + 27 + 4 = 139$. Multiplied by outer $z^{-1} \\implies 139z^{-3}$."},{"type":"paragraph","content":"The final truncated series is mathematically $2z^{-1} + 21z^{-2} + 139z^{-3} + \\dots$. By directly comparing these generated terms with the definitive Z-transform definition $\\sum u_n z^{-n}$, we decisively extract the required sequence values: **$u_1 = 2$, $u_2 = 21$, and $u_3 = 139$.**"}]}},{"id":"v2-advanced-engineering-mathematics-2023-pB-q4","number":4,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Evaluate } \\int_0^1 \\frac{dx}{1+x^2} \\text{ by using:} \\\\ &\\text{(i) Trapezoidal rule} \\\\ &\\text{(ii) Simpson 1/3 rule} \\end{aligned}","displayMode":true}],"answer":{"summary":"A detailed application of Trapezoidal and Simpson's 1/3 numerical integration rules to approximate the definite integral of a rational function.","explanation":[{"type":"heading","content":"Numerical Integration Techniques"},{"type":"paragraph","content":"The objective is to numerically evaluate the definite integral $\\int_0^1 \\frac{1}{1+x^2} dx$. While this integral has a simple exact analytical solution ($\\arctan(x)$), numerical methods are essential when dealing with non-integrable functions or empirical data. We are tasked with employing both the Trapezoidal Rule and Simpson's 1/3 Rule."},{"type":"subheading","content":"1. Discretizing the Domain"},{"type":"paragraph","content":"The fundamental first step in any numerical integration scheme is to partition the continuous integration domain $[0, 1]$ into a finite number of equally spaced discrete sub-intervals. We will choose a step size (interval width) of $h = 0.25$. This creates exactly $n = 4$ sub-intervals and $5$ distinct data points. We calculate the corresponding $y$-values ($y = \\frac{1}{1+x^2}$) for each $x$-coordinate:"},{"type":"list","items":["$x_0 = 0.00 \\implies y_0 = 1.0000$","$x_1 = 0.25 \\implies y_1 = 0.9412$","$x_2 = 0.50 \\implies y_2 = 0.8000$","$x_3 = 0.75 \\implies y_3 = 0.6400$","$x_4 = 1.00 \\implies y_4 = 0.5000$"]},{"type":"subheading","content":"2. Applying the Trapezoidal Rule"},{"type":"paragraph","content":"The Trapezoidal Rule geometrically approximates the area under the curve by connecting adjacent data points with straight lines, forming a series of trapezoids. Its mathematical formulation is $I \\approx \\frac{h}{2} [ (y_0 + y_n) + 2(y_1 + y_2 + \\dots + y_{n-1}) ]$."},{"type":"math","expression":"I_{Trap} \\approx \\frac{0.25}{2} [ (1.0000 + 0.5000) + 2(0.9412 + 0.8000 + 0.6400) ]","displayMode":true},{"type":"math","expression":"I_{Trap} \\approx 0.125 [ 1.5000 + 2(2.3812) ] = 0.125 [ 1.5000 + 4.7624 ] = 0.125 [ 6.2624 ] = 0.7828","displayMode":true},{"type":"subheading","content":"3. Applying Simpson's 1/3 Rule"},{"type":"paragraph","content":"Simpson's 1/3 Rule provides a superior approximation by fitting quadratic parabolic curves through sets of three adjacent points. This rule strictly requires an even number of intervals ($n$ must be even), which is satisfied here ($n=4$). Its formulation is $I \\approx \\frac{h}{3} [ (y_0 + y_n) + 4(y_{odd}) + 2(y_{even}) ]$."},{"type":"math","expression":"I_{Simp} \\approx \\frac{0.25}{3} [ (1.0000 + 0.5000) + 4(0.9412 + 0.6400) + 2(0.8000) ]","displayMode":true},{"type":"math","expression":"I_{Simp} \\approx 0.08333 [ 1.5000 + 4(1.5812) + 1.6000 ] = 0.08333 [ 1.5000 + 6.3248 + 1.6000 ] = 0.7854","displayMode":true},{"type":"paragraph","content":"The exact analytical value is $\\arctan(1) - \\arctan(0) = \\frac{\\pi}{4} \\approx 0.785398$. As theoretically expected due to its higher-order parabolic interpolation, Simpson's 1/3 Rule ($0.7854$) yields a significantly closer approximation to the true exact value than the linear Trapezoidal Rule ($0.7828$)."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pB-q5","number":5,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\text{Given } \\frac{dy}{dx} = x^2 + y, \\; y(0) = 1. \\text{ Determine } y(0.02) \\text{ and } y(0.04) \\text{ by using modified Euler's method.}","displayMode":true}],"answer":{"summary":"A step-by-step execution of the Modified Euler numerical method to iteratively solve an ordinary differential equation with a small step size.","explanation":[{"type":"heading","content":"Solving ODEs via Modified Euler's Method"},{"type":"paragraph","content":"The task requires approximating the solution to the non-linear first-order Ordinary Differential Equation (ODE) $\\frac{dy}{dx} = x^2 + y$, given the initial boundary condition $y(0)=1$, utilizing a precise step size of $h=0.02$. The Modified Euler's Method (also known as Heun's Method or a 2nd-order Runge-Kutta method) is a predictor-corrector algorithm that provides significantly better accuracy than the standard Forward Euler method by averaging the slopes at the beginning and end of each interval."},{"type":"subheading","content":"1. Iteration Step 1: Approximating y at x = 0.02"},{"type":"paragraph","content":"We commence at $x_0 = 0, y_0 = 1$. The function defines the slope: $f(x,y) = x^2 + y$."},{"type":"paragraph","content":"**Predictor Phase:** We use standard Euler to make an initial rough estimate ($y_1^{(0)}$) of the next point."},{"type":"math","expression":"y_1^{(0)} = y_0 + h \\cdot f(x_0, y_0) = 1 + 0.02(0^2 + 1) = 1.02","displayMode":true},{"type":"paragraph","content":"**Corrector Phase (First Pass):** We refine this estimate by averaging the slope at the initial point $(0,1)$ with the newly estimated slope at the predicted point $(0.02, 1.02)$."},{"type":"math","expression":"y_1^{(1)} = y_0 + \\frac{h}{2} [f(x_0, y_0) + f(x_1, y_1^{(0)})]","displayMode":true},{"type":"math","expression":"y_1^{(1)} = 1 + 0.01 [ (0^2 + 1) + (0.02^2 + 1.02) ] = 1 + 0.01 [ 1 + 1.0204 ] = 1.020204","displayMode":true},{"type":"paragraph","content":"**Corrector Phase (Second Pass):** To ensure maximum numerical stability, we iteratively apply the corrector formula until the digits converge."},{"type":"math","expression":"y_1^{(2)} = 1 + 0.01 [ (0^2 + 1) + (0.02^2 + 1.020204) ] = 1 + 0.01 [ 2.020604 ] = 1.020206","displayMode":true},{"type":"paragraph","content":"The value has essentially converged. We establish $y_1 \\approx 1.0202$ at $x_1 = 0.02$."},{"type":"subheading","content":"2. Iteration Step 2: Approximating y at x = 0.04"},{"type":"paragraph","content":"We now step forward to $x_2 = 0.04$, utilizing our newly established values $x_1 = 0.02$ and $y_1 = 1.0202$."},{"type":"paragraph","content":"**Predictor Phase:**"},{"type":"math","expression":"y_2^{(0)} = y_1 + h \\cdot f(x_1, y_1) = 1.0202 + 0.02(0.02^2 + 1.0202) = 1.0202 + 0.02(1.0206) = 1.0406","displayMode":true},{"type":"paragraph","content":"**Corrector Phase:**"},{"type":"math","expression":"y_2^{(1)} = y_1 + \\frac{h}{2} [f(x_1, y_1) + f(x_2, y_2^{(0)})]","displayMode":true},{"type":"math","expression":"y_2^{(1)} = 1.0202 + 0.01 [ (0.02^2 + 1.0202) + (0.04^2 + 1.0406) ] = 1.0202 + 0.01 [ 1.0206 + 1.0422 ]","displayMode":true},{"type":"math","expression":"y_2^{(1)} = 1.0202 + 0.01 [ 2.0628 ] = 1.0202 + 0.020628 = 1.040828","displayMode":true},{"type":"paragraph","content":"A secondary corrector pass would confirm convergence. Thus, the rigorously calculated approximation for $y(0.04)$ is approximately $1.0408$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pB-q6","number":6,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"By using Stirling formula, find $u_{32}$ from the following data:"},{"type":"math","expression":"\\begin{aligned} &u_{20} = 14.035, \\quad u_{25} = 13.674, \\quad u_{30} = 13.257 \\\\ &u_{35} = 12.734, \\quad u_{40} = 12.089, \\quad u_{45} = 11.309 \\end{aligned}","displayMode":true}],"answer":{"summary":"Employing Stirling's central difference interpolation formula to highly accurately estimate a function value situated near the center of a tabulated dataset.","explanation":[{"type":"heading","content":"Central Interpolation with Stirling's Formula"},{"type":"paragraph","content":"When tasked with interpolating a value that lies firmly in the middle region of a tabulated dataset (rather than at the beginning or the extreme end), Newton's Forward and Backward formulas become less accurate due to truncation of higher-order differences. Central difference formulas, such as Stirling's formula (which is the mathematical arithmetic mean of Gauss's Forward and Backward formulas), provide significantly superior accuracy by symmetrically utilizing data points radiating outward from the center."},{"type":"subheading","content":"1. Structuring the Central Difference Table"},{"type":"paragraph","content":"We are provided with data at uniform intervals $x = \\{20, 25, 30, 35, 40, 45\\}$ (step size $h=5$). We seek the value at $x = 32$. We logically designate the closest tabulated $x$-value as the central origin, $x_0 = 30$. The corresponding index parameter is $p = \\frac{x - x_0}{h} = \\frac{32 - 30}{5} = 0.4$."},{"type":"paragraph","content":"We systematically construct the central difference table, meticulously labeling the entries relative to the chosen $x_0$ origin."},{"type":"list","items":["$y_{-2}(20) = 14.035, y_{-1}(25) = 13.674, y_0(30) = 13.257, y_1(35) = 12.734, y_2(40) = 12.089$","First Differences: $\\Delta y_{-2} = -0.361, \\Delta y_{-1} = -0.417, \\Delta y_0 = -0.523, \\Delta y_1 = -0.645$","Second Differences: $\\Delta^2 y_{-2} = -0.056, \\Delta^2 y_{-1} = -0.106, \\Delta^2 y_0 = -0.122$","Third Differences: $\\Delta^3 y_{-2} = -0.050, \\Delta^3 y_{-1} = -0.016$","Fourth Difference: $\\Delta^4 y_{-2} = 0.034$"]},{"type":"subheading","content":"2. Applying Stirling's Formula"},{"type":"paragraph","content":"Stirling's mathematical interpolation formula utilizes the central differences located symmetrically horizontally across the $y_0$ row of the table:"},{"type":"math","expression":"y_p = y_0 + p \\left( \\frac{\\Delta y_0 + \\Delta y_{-1}}{2} \\right) + \\frac{p^2}{2!} \\Delta^2 y_{-1} + \\frac{p(p^2-1)}{3!} \\left( \\frac{\\Delta^3 y_{-1} + \\Delta^3 y_{-2}}{2} \\right) + \\dots","displayMode":true},{"type":"paragraph","content":"We carefully substitute the precisely extracted values from our table and the parameter $p = 0.4$ into the formula:"},{"type":"math","expression":"y_{0.4} = 13.257 + 0.4 \\left( \\frac{-0.523 - 0.417}{2} \\right) + \\frac{(0.4)^2}{2} (-0.106) + \\frac{0.4(0.16 - 1)}{6} \\left( \\frac{-0.016 - 0.050}{2} \\right)","displayMode":true},{"type":"math","expression":"y_{0.4} = 13.257 + 0.4(-0.470) + 0.08(-0.106) + (-0.056)(-0.033)","displayMode":true},{"type":"math","expression":"y_{0.4} = 13.257 - 0.188 - 0.00848 + 0.001848 \\approx 13.062","displayMode":true},{"type":"paragraph","content":"The rigorous mathematical calculation confirms that the highly accurate interpolated value at $x = 32$ is approximately $13.062$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pB-q7","number":7,"marks":4,"tags":["Part B"],"text":[{"type":"math","expression":"\\text{Solve linear difference equation } u_{n+2} + 6u_{n+1} + 9u_n = 2^n \\text{ given } u_0 = 1, \\; u_1 = 1.","displayMode":true}],"answer":{"summary":"Solving a non-homogeneous linear difference equation by distinctly calculating the Complementary Function and the Particular Integral.","explanation":[{"type":"heading","content":"Solving Second-Order Non-Homogeneous Difference Equations"},{"type":"paragraph","content":"The problem requires solving the second-order, linear, constant-coefficient, non-homogeneous difference equation $u_{n+2} + 6u_{n+1} + 9u_n = 2^n$. The total comprehensive solution is mathematically constructed by independently finding two distinct components: the Complementary Function (CF) which represents the natural unforced response of the system, and the Particular Integral (PI) which represents the forced response to the specific input $2^n$. The final solution is $u_n = CF + PI$."},{"type":"subheading","content":"1. Determining the Complementary Function"},{"type":"paragraph","content":"To find the CF, we set the right-hand side forcing function exactly to zero, creating a homogeneous equation. We then utilize the mathematical shift operator $E$ (where $Eu_n = u_{n+1}$) to rewrite the equation algebraically: $(E^2 + 6E + 9)u_n = 0$."},{"type":"paragraph","content":"This yields the quadratic characteristic auxiliary equation $E^2 + 6E + 9 = 0$. Factoring this perfect square gives $(E+3)^2 = 0$. This indicates real, repeated roots at $E_1 = E_2 = -3$. According to the fundamental theorems of difference equations, the structural form of the CF for repeated roots is:"},{"type":"math","expression":"u_n^{(c)} = (C_1 + C_2 n)(-3)^n","displayMode":true},{"type":"subheading","content":"2. Determining the Particular Integral"},{"type":"paragraph","content":"The PI addresses the specific non-homogeneous forcing function $f(n) = 2^n$. We apply inverse operator methods to find it. The PI is mathematically defined as $\\frac{1}{F(E)} f(n)$."},{"type":"math","expression":"PI = \\frac{1}{E^2 + 6E + 9} (2^n) = \\frac{1}{(E+3)^2} (2^n)","displayMode":true},{"type":"paragraph","content":"When the forcing function is an exponential of the form $a^n$, the inverse operator rule allows us to directly substitute $E$ with the constant base $a$, provided the resulting denominator does not equal zero. Here, $a=2$."},{"type":"math","expression":"PI = \\frac{1}{(2+3)^2} (2^n) = \\frac{1}{5^2} (2^n) = \\frac{1}{25} 2^n","displayMode":true},{"type":"subheading","content":"3. Constructing the General Solution"},{"type":"paragraph","content":"Combining both distinct components, the general overall solution to the difference equation is:"},{"type":"math","expression":"u_n = (C_1 + C_2 n)(-3)^n + \\frac{1}{25} 2^n","displayMode":true},{"type":"paragraph","content":"The problem provides specific initial boundary conditions $u_0 = 1$ and $u_1 = 1$. We rigorously substitute these into our general solution to uniquely determine the unknown constants $C_1$ and $C_2$."},{"type":"paragraph","content":"At $n=0$: $u_0 = (C_1 + 0)(-3)^0 + \\frac{1}{25}(1) = C_1 + \\frac{1}{25} = 1 \\implies C_1 = \\frac{24}{25}$."},{"type":"paragraph","content":"At $n=1$: $u_1 = (C_1 + C_2)(-3)^1 + \\frac{1}{25}(2) = -3(\\frac{24}{25} + C_2) + \\frac{2}{25} = 1$."},{"type":"math","expression":"-\\frac{72}{25} - 3C_2 + \\frac{2}{25} = 1 \\implies -3C_2 - \\frac{70}{25} = 1 \\implies -3C_2 = 1 + \\frac{70}{25} = \\frac{95}{25}","displayMode":true},{"type":"math","expression":"C_2 = -\\frac{95}{75} = -\\frac{19}{15}","displayMode":true},{"type":"paragraph","content":"Substituting the exact calculated constants back yields the final, definitively proven specific solution sequence:"},{"type":"math","expression":"u_n = \\left(\\frac{24}{25} - \\frac{19}{15} n\\right)(-3)^n + \\frac{1}{25} 2^n","displayMode":true}]}},{"id":"v2-advanced-engineering-mathematics-2023-pC-q1","number":1,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"From the following table of values of x and y find dy/dx and d²y/dx² at x = 1.2 :"},{"type":"svg","src":"/assets/diagrams/rtu/cs/2024/advanced-engineering-mathematics/q1c-diff-table.svg","alt":"Differentiation Table"}],"answer":{"summary":"A rigorous derivation and application of numerical differentiation using Newton's Forward Difference interpolation formula for uniformly spaced data.","explanation":[{"type":"heading","content":"Numerical Differentiation with Newton's Forward Differences"},{"type":"paragraph","content":"In engineering applications, we frequently encounter experimental data presented as discrete sets of points rather than continuous analytical functions. When the objective is to analyze the rate of change (velocity) or acceleration of the underlying physical process, we must perform numerical differentiation. The choice of mathematical algorithm is strictly governed by the location of the point of interest relative to the entire dataset and the spacing of the independent variables. When the data is equally spaced and the point of interest lies near the beginning (upper region) of the tabulated data, Newton's Forward Difference Interpolation formula provides the most robust and theoretically sound basis for differentiation."},{"type":"subheading","content":"1. Theoretical Foundation of the Forward Formula"},{"type":"paragraph","content":"Consider a rigorous dataset of $n+1$ distinct points: $(x_0, y_0), (x_1, y_1), \\dots, (x_n, y_n)$, where the independent variable $x$ increments by a strictly constant step size $h$. Therefore, $x_i = x_0 + i \\cdot h$ for any integer index $i$. Newton's Forward Difference formula constructs an interpolating polynomial $P(x)$ that perfectly passes through all these discrete points."},{"type":"paragraph","content":"To formulate this polynomial, we introduce a dimensionless parameter $p$, defined as the normalized distance from the starting node: $p = \\frac{x - x_0}{h}$. The continuous continuous polynomial is mathematically expressed in terms of forward differences ($\\Delta$):"},{"type":"math","expression":"y(x) \\approx P(x) = y_0 + p \\Delta y_0 + \\frac{p(p-1)}{2!} \\Delta^2 y_0 + \\frac{p(p-1)(p-2)}{3!} \\Delta^3 y_0 + \\dots","displayMode":true},{"type":"subheading","content":"2. Deriving the First Derivative (Velocity)"},{"type":"paragraph","content":"To extract the derivative $\\frac{dy}{dx}$, we must apply the chain rule of differential calculus. Since our polynomial is expressed entirely in terms of the parameter $p$, the chain rule dictates that $\\frac{dy}{dx} = \\frac{dy}{dp} \\cdot \\frac{dp}{dx}$."},{"type":"paragraph","content":"From our precise definition $p = \\frac{x - x_0}{h}$, differentiating with respect to $x$ yields the constant scaling factor: $\\frac{dp}{dx} = \\frac{1}{h}$."},{"type":"paragraph","content":"Next, we meticulously differentiate the interpolating polynomial $y(p)$ with respect to $p$ term by term:"},{"type":"list","items":["The derivative of the constant $y_0$ is exactly 0.","The derivative of $p \\Delta y_0$ is simply $\\Delta y_0$.","The term $\\frac{p(p-1)}{2!} = \\frac{p^2 - p}{2}$. Its derivative is $\\frac{2p - 1}{2}$.","The term $\\frac{p(p-1)(p-2)}{3!} = \\frac{p^3 - 3p^2 + 2p}{6}$. Its derivative is $\\frac{3p^2 - 6p + 2}{6}$."]},{"type":"paragraph","content":"Combining these derived components via the chain rule yields the general, continuous numerical first derivative formula valid for any point $x$ within the interpolation interval:"},{"type":"math","expression":"\\frac{dy}{dx} = \\frac{1}{h} \\left[ \\Delta y_0 + \\frac{2p - 1}{2} \\Delta^2 y_0 + \\frac{3p^2 - 6p + 2}{6} \\Delta^3 y_0 + \\dots \\right]","displayMode":true},{"type":"paragraph","content":"Crucially, if we need to evaluate the derivative exactly at the starting data point $x = x_0$, the parameter $p$ becomes exactly zero. Substituting $p=0$ into the general equation simplifies it remarkably into the standard pointwise formula:"},{"type":"math","expression":"\\left. \\frac{dy}{dx} \\right|_{x=x_0} = \\frac{1}{h} \\left[ \\Delta y_0 - \\frac{1}{2} \\Delta^2 y_0 + \\frac{1}{3} \\Delta^3 y_0 - \\frac{1}{4} \\Delta^4 y_0 + \\dots \\right]","displayMode":true},{"type":"subheading","content":"3. Deriving the Second Derivative (Acceleration)"},{"type":"paragraph","content":"The second derivative $\\frac{d^2y}{dx^2}$ is obtained by strictly applying the chain rule a second time to the first derivative equation: $\\frac{d^2y}{dx^2} = \\frac{d}{dp}\\left(\\frac{dy}{dx}\\right) \\cdot \\frac{dp}{dx}$."},{"type":"paragraph","content":"We differentiate the bracketed expression of the first derivative with respect to $p$:"},{"type":"list","items":["The term $\\Delta y_0$ is constant with respect to $p$, so its derivative is 0.","The term $\\frac{2p-1}{2} \\Delta^2 y_0$ differentiates to exactly $\\Delta^2 y_0$.","The term $\\frac{3p^2-6p+2}{6} \\Delta^3 y_0$ differentiates to $\\frac{6p-6}{6} \\Delta^3 y_0 = (p-1)\\Delta^3 y_0$."]},{"type":"paragraph","content":"Multiplying by the second $\\frac{1}{h}$ factor from the chain rule yields the general second derivative:"},{"type":"math","expression":"\\frac{d^2y}{dx^2} = \\frac{1}{h^2} \\left[ \\Delta^2 y_0 + (p - 1) \\Delta^3 y_0 + \\frac{6p^2 - 18p + 11}{12} \\Delta^4 y_0 + \\dots \\right]","displayMode":true},{"type":"paragraph","content":"Evaluating this explicitly at the base point $x = x_0$ (where $p = 0$) results in the final formula:"},{"type":"math","expression":"\\left. \\frac{d^2y}{dx^2} \\right|_{x=x_0} = \\frac{1}{h^2} \\left[ \\Delta^2 y_0 - \\Delta^3 y_0 + \\frac{11}{12} \\Delta^4 y_0 - \\frac{5}{6} \\Delta^5 y_0 + \\dots \\right]","displayMode":true},{"type":"subheading","content":"4. Methodological Execution"},{"type":"paragraph","content":"To solve the specific problem requiring the derivatives at $x=1.2$, the procedure is as follows: First, mathematically confirm the uniform step size $h=0.2$. Second, construct a comprehensive forward difference table calculating all successive differences $\\Delta^n y$. Third, select $x_0 = 1.2$ as the base node, identifying the specific diagonal row of forward differences ($\\Delta y_0, \\Delta^2 y_0, \\dots$) corresponding to this exact starting point. Finally, substitute these precise numerical values into the derived theoretical formulas to compute both the gradient and concavity at $x=1.2$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pC-q2","number":2,"marks":10,"tags":["Part C"],"text":[{"type":"math","expression":"\\text{If } \\frac{dy}{dx} = x + y^2 \\text{ use Runge-Kutta method to find an approximate value of y for } x = 0.2, \\text{ given that } y = 1 \\text{ when } x = 0. \\text{ Use Laplace transform to solve.}","displayMode":true}],"answer":{"summary":"A detailed exposition of the fourth-order Runge-Kutta numerical method, outlining its algorithmic execution for solving complex non-linear initial value problems.","explanation":[{"type":"heading","content":"Algorithmic Mastery of the Runge-Kutta 4th Order Method"},{"type":"paragraph","content":"The Runge-Kutta family of algorithms represents the gold standard for numerically solving Ordinary Differential Equations (ODEs) within engineering applications. While single-step Euler methods are conceptually simple, their high truncation error mandates impractically minuscule step sizes. Multi-step methods like Milne's are highly accurate but suffer from the severe disadvantage of not being self-starting. The 4th Order Runge-Kutta method (RK4) beautifully balances these constraints: it is a robust, self-starting single-step method that achieves exceptional fourth-order accuracy ($O(h^4)$) by strategically probing the derivative function at multiple intermediate points within each interval."},{"type":"subheading","content":"1. Theoretical Architecture of RK4"},{"type":"paragraph","content":"Consider the standard Initial Value Problem (IVP): $\\frac{dy}{dx} = f(x, y)$ subject to the strict initial condition $y(x_0) = y_0$. We intend to step forward by a finite increment $h$ to find the corresponding value $y_1$ at $x_1 = x_0 + h$."},{"type":"paragraph","content":"The RK4 algorithm constructs a weighted average of four distinct, carefully sampled slopes ($k_1, k_2, k_3, k_4$) across the interval to predict the final trajectory."},{"type":"list","items":["**$k_1$ (Initial Slope):** This is the exact slope evaluated exactly at the starting node $(x_0, y_0)$. It is essentially a standard Euler prediction. $k_1 = h \\cdot f(x_0, y_0)$.","**$k_2$ (First Midpoint Slope):** The algorithm uses $k_1$ to project halfway across the interval (to $x_0 + h/2$) and calculates the slope at this tentatively estimated midpoint. $k_2 = h \\cdot f(x_0 + \\frac{h}{2}, y_0 + \\frac{k_1}{2})$.","**$k_3$ (Second Midpoint Slope):** The algorithm refines its estimate of the midpoint by utilizing the newly calculated $k_2$ slope to project from the start. It recalculates the slope there. $k_3 = h \\cdot f(x_0 + \\frac{h}{2}, y_0 + \\frac{k_2}{2})$.","**$k_4$ (Terminal Slope):** Finally, the algorithm utilizes $k_3$ to project all the way to the end of the interval (to $x_0 + h$) and evaluates the slope at this final predicted boundary. $k_4 = h \\cdot f(x_0 + h, y_0 + k_3)$."]},{"type":"paragraph","content":"The final formula ingeniously combines these four slopes using a weighted average derived precisely from Simpson's 1/3 integration rule, heavily favoring the mathematically superior midpoint estimations (weighting them twice as heavily as the boundary slopes):"},{"type":"math","expression":"y_1 = y_0 + \\frac{1}{6} (k_1 + 2k_2 + 2k_3 + k_4)","displayMode":true},{"type":"subheading","content":"2. Step-by-Step Execution for the Given Problem"},{"type":"paragraph","content":"The specific mathematical problem presented is $\\frac{dy}{dx} = x + y^2$, with initial boundaries $x_0 = 0$, $y_0 = 1$, and a specified step size $h = 0.2$. The objective function is $f(x,y) = x + y^2$."},{"type":"paragraph","content":"**Calculating $k_1$:**"},{"type":"math","expression":"k_1 = (0.2) \\cdot [ (0) + (1)^2 ] = (0.2) \\cdot [ 1 ] = 0.2000","displayMode":true},{"type":"paragraph","content":"**Calculating $k_2$:** The coordinates shift to $x = 0 + 0.1 = 0.1$ and $y = 1 + 0.2/2 = 1.1$."},{"type":"math","expression":"k_2 = (0.2) \\cdot [ (0.1) + (1.1)^2 ] = (0.2) \\cdot [ 0.1 + 1.21 ] = (0.2) \\cdot [ 1.31 ] = 0.2620","displayMode":true},{"type":"paragraph","content":"**Calculating $k_3$:** The coordinates shift to $x = 0 + 0.1 = 0.1$ and $y = 1 + 0.262/2 = 1.131$."},{"type":"math","expression":"k_3 = (0.2) \\cdot [ (0.1) + (1.131)^2 ] = (0.2) \\cdot [ 0.1 + 1.279161 ] = (0.2) \\cdot [ 1.379161 ] = 0.2758322","displayMode":true},{"type":"paragraph","content":"**Calculating $k_4$:** The coordinates shift to the terminal boundary $x = 0 + 0.2 = 0.2$ and $y = 1 + 0.2758322 = 1.2758322$."},{"type":"math","expression":"k_4 = (0.2) \\cdot [ (0.2) + (1.2758322)^2 ] = (0.2) \\cdot [ 0.2 + 1.627747 ] = (0.2) \\cdot [ 1.827747 ] = 0.3655494","displayMode":true},{"type":"subheading","content":"3. Final Synthesis"},{"type":"paragraph","content":"With all four constituent slope approximations meticulously calculated to high precision, we blend them into the final Simpson's weighting equation to determine the ultimate function value at the new coordinate $x = 0.2$."},{"type":"math","expression":"y(0.2) = y_0 + \\frac{1}{6} (k_1 + 2k_2 + 2k_3 + k_4)","displayMode":true},{"type":"math","expression":"y(0.2) = 1.0 + \\frac{1}{6} (0.2000 + 2(0.2620) + 2(0.2758322) + 0.3655494)","displayMode":true},{"type":"math","expression":"y(0.2) = 1.0 + \\frac{1}{6} (0.2000 + 0.5240 + 0.5516644 + 0.3655494) = 1.0 + \\frac{1}{6} (1.6412138)","displayMode":true},{"type":"math","expression":"y(0.2) = 1.0 + 0.2735356 \\approx 1.2735","displayMode":true},{"type":"paragraph","content":"Thus, the highly accurate approximated solution for the non-linear differential equation evaluated precisely at $x=0.2$ is $1.2735$."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pC-q3","number":3,"marks":10,"tags":["Part C"],"text":[{"type":"math","expression":"(D^2 + 9)y = \\cos 2t, \\; y(0) = 1, \\; y(\\pi/2) = -1.","displayMode":true}],"answer":{"summary":"A rigorous solution to a second-order linear boundary value differential equation utilizing the algebraic powers of the Laplace Transform and complex partial fraction decomposition.","explanation":[{"type":"heading","content":"Solving Boundary Value ODEs via Laplace Transforms"},{"type":"paragraph","content":"The Laplace transform provides a profoundly elegant algebraic method for solving linear ordinary differential equations (ODEs). While traditional methods (like undetermined coefficients or variation of parameters) require calculating a general solution and subsequently applying boundary conditions, the Laplace transform seamlessly incorporates initial conditions directly into the algebraic formulation from the very first step. The problem presented is a second-order, linear, constant-coefficient non-homogeneous equation: $y'' + 9y = \\cos(2t)$, subject to the split boundary conditions $y(0) = 1$ and $y(\\pi/2) = -1$."},{"type":"subheading","content":"1. Initialization and Transformation"},{"type":"paragraph","content":"Crucially, this is a Boundary Value Problem (BVP), not an Initial Value Problem (IVP). The Laplace transform explicitly requires both initial state values at $t=0$: specifically $y(0)$ and the derivative $y'(0)$. Because the initial derivative $y'(0)$ is completely unknown, we must assign it a temporary constant, let $y'(0) = c$. This constant will be resolved at the very end of the mathematical process."},{"type":"paragraph","content":"We systematically apply the one-sided Laplace operator $L\\{\\cdot\\}$ across the entire differential equation. Let $L\\{y(t)\\} = Y(s)$. We utilize the fundamental derivative transform property $L\\{y''(t)\\} = s^2 Y(s) - s y(0) - y'(0)$."},{"type":"math","expression":"L\\{y'' + 9y\\} = L\\{\\cos(2t)\\}","displayMode":true},{"type":"math","expression":"[s^2 Y(s) - s(1) - c] + 9 Y(s) = \\frac{s}{s^2 + 4}","displayMode":true},{"type":"paragraph","content":"We now mathematically isolate the transformed algebraic function $Y(s)$ by grouping terms:"},{"type":"math","expression":"(s^2 + 9) Y(s) - s - c = \\frac{s}{s^2 + 4}","displayMode":true},{"type":"math","expression":"Y(s) = \\frac{s + c}{s^2 + 9} + \\frac{s}{(s^2 + 9)(s^2 + 4)}","displayMode":true},{"type":"subheading","content":"2. Complex Partial Fraction Decomposition"},{"type":"paragraph","content":"The first term, $\\frac{s+c}{s^2+9}$, cleanly separates into standard inverse forms $\\frac{s}{s^2+9} + \\frac{c}{s^2+9}$. However, the second term $\\frac{s}{(s^2+9)(s^2+4)}$ is a complex rational fraction requiring rigorous partial fraction decomposition into irreducible quadratic components."},{"type":"math","expression":"\\frac{s}{(s^2+9)(s^2+4)} = \\frac{As + B}{s^2+9} + \\frac{Cs + D}{s^2+4}","displayMode":true},{"type":"paragraph","content":"By cross-multiplying to eliminate the denominators, we obtain the identity $s = (As+B)(s^2+4) + (Cs+D)(s^2+9)$. By methodically equating coefficients of corresponding powers of $s$, we solve the linear system to find exactly $A = -1/5, B = 0, C = 1/5, D = 0$."},{"type":"paragraph","content":"Therefore, the completely decomposed, unified expression for $Y(s)$ is:"},{"type":"math","expression":"Y(s) = \\frac{s}{s^2+9} + \\frac{c}{s^2+9} - \\frac{1}{5}\\left(\\frac{s}{s^2+9}\\right) + \\frac{1}{5}\\left(\\frac{s}{s^2+4}\\right)","displayMode":true},{"type":"math","expression":"Y(s) = \\frac{4}{5}\\left(\\frac{s}{s^2+9}\\right) + \\frac{c}{3}\\left(\\frac{3}{s^2+9}\\right) + \\frac{1}{5}\\left(\\frac{s}{s^2+4}\\right)","displayMode":true},{"type":"subheading","content":"3. Inverse Transformation and Resolving Constants"},{"type":"paragraph","content":"We now apply the inverse Laplace transform $L^{-1}$ to shift back from the $s$-domain into the $t$-domain, utilizing standard trigonometric transformation pairs."},{"type":"math","expression":"y(t) = \\frac{4}{5}\\cos(3t) + \\frac{c}{3}\\sin(3t) + \\frac{1}{5}\\cos(2t)","displayMode":true},{"type":"paragraph","content":"This represents the generalized solution containing the unknown constant $c$. To ascertain the specific unique solution, we apply the final boundary condition provided in the problem statement: $y(\\pi/2) = -1$."},{"type":"math","expression":"-1 = \\frac{4}{5}\\cos\\left(\\frac{3\\pi}{2}\\right) + \\frac{c}{3}\\sin\\left(\\frac{3\\pi}{2}\\right) + \\frac{1}{5}\\cos\\left(2 \\cdot \\frac{\\pi}{2}\\right)","displayMode":true},{"type":"paragraph","content":"We meticulously evaluate the trigonometric functions at these specific boundary angles: $\\cos(3\\pi/2) = 0$, $\\sin(3\\pi/2) = -1$, and $\\cos(\\pi) = -1$. Substituting these values:"},{"type":"math","expression":"-1 = \\frac{4}{5}(0) + \\frac{c}{3}(-1) + \\frac{1}{5}(-1) \\implies -1 = -\\frac{c}{3} - \\frac{1}{5}","displayMode":true},{"type":"math","expression":"\\frac{c}{3} = 1 - \\frac{1}{5} = \\frac{4}{5} \\implies c = \\frac{12}{5}","displayMode":true},{"type":"paragraph","content":"By inserting the exactly determined value $c = 12/5$ back into our generalized expression, we finalize the mathematical solution:"},{"type":"math","expression":"y(t) = \\frac{4}{5}\\cos(3t) + \\frac{4}{5}\\sin(3t) + \\frac{1}{5}\\cos(2t)","displayMode":true}]}},{"id":"v2-advanced-engineering-mathematics-2023-pC-q4","number":4,"marks":10,"tags":["Part C"],"text":[{"type":"math","expression":"\\begin{aligned} &\\text{Obtain the Fourier transform of } f(x) = \\begin{cases} x^2 & |x| \\le a \\\\ 0 & |x| > a \\end{cases} \\\\ &\\text{Hence evaluate } \\int_0^\\infty \\cos\\left(\\frac{as}{2}\\right) \\frac{(a^2 s^2 - 2)\\sin as + 2as \\cos as}{s^3} ds \\end{aligned}","displayMode":true}],"answer":{"summary":"Deriving the complex Fourier Transform of a bounded quadratic function using iterative integration by parts and exploiting functional symmetry.","explanation":[{"type":"heading","content":"Fourier Transform of Bounded Polynomials"},{"type":"paragraph","content":"The problem mandates calculating the Fourier Transform of the specific spatial function $f(x) = x^2$ constrained strictly within the symmetric interval $-a < x < a$, and identically zero everywhere else. The fundamental mathematical definition for the continuous Fourier Transform is the infinite improper integral $F(s) = \\int_{-\\infty}^\\infty f(x) e^{isx} dx$. Because our specified function abruptly vanishes outside the bounded region, the infinite limits mathematically collapse precisely to $[-a, a]$."},{"type":"subheading","content":"1. Exploiting Functional Parity and Symmetry"},{"type":"paragraph","content":"We start with the bounded complex integral: $F(s) = \\int_{-a}^a x^2 e^{isx} dx$. To avoid convoluted complex algebra, we immediately decompose the exponential term into its real and imaginary trigonometric constituents using Euler's identity: $e^{isx} = \\cos(sx) + i\\sin(sx)$."},{"type":"math","expression":"F(s) = \\int_{-a}^a x^2 [\\cos(sx) + i\\sin(sx)] dx = \\int_{-a}^a x^2 \\cos(sx) dx + i \\int_{-a}^a x^2 \\sin(sx) dx","displayMode":true},{"type":"paragraph","content":"We now analyze the mathematical parity (even/odd nature) of the integrands. The function $x^2$ is an even function. The function $\\cos(sx)$ is an even function. Their product, $x^2\\cos(sx)$, is therefore purely even. Conversely, $\\sin(sx)$ is an odd function. The product $x^2\\sin(sx)$ is purely odd. A fundamental theorem of calculus states that the definite integral of any odd function over perfectly symmetric bounds $[-a, a]$ evaluates exactly to zero. Furthermore, the integral of an even function over symmetric bounds is exactly twice the integral from 0 to the upper bound $a$. Therefore, the entire complex imaginary part identically vanishes, leaving a strictly real integral:"},{"type":"math","expression":"F(s) = 2 \\int_0^a x^2 \\cos(sx) dx","displayMode":true},{"type":"subheading","content":"2. Iterative Integration by Parts"},{"type":"paragraph","content":"To evaluate this polynomial-trigonometric product integral, we must employ the mathematical technique of integration by parts ($\\int u \\, dv = uv - \\int v \\, du$) iteratively twice, as the polynomial is of degree 2."},{"type":"paragraph","content":"Let $u = x^2$ and $dv = \\cos(sx) dx$. Differentiating yields $du = 2x \\, dx$ and integrating yields $v = \\frac{\\sin(sx)}{s}$."},{"type":"math","expression":"\\int_0^a x^2 \\cos(sx) dx = \\left[ x^2 \\frac{\\sin(sx)}{s} \\right]_0^a - \\int_0^a \\frac{\\sin(sx)}{s} (2x) dx = \\frac{a^2 \\sin(sa)}{s} - \\frac{2}{s} \\int_0^a x \\sin(sx) dx","displayMode":true},{"type":"paragraph","content":"We perform a second iteration of integration by parts on the remaining integral $\\int x \\sin(sx) dx$. We let $u = x$ and $dv = \\sin(sx) dx$. Thus, $du = dx$ and $v = -\\frac{\\cos(sx)}{s}$."},{"type":"math","expression":"\\int_0^a x \\sin(sx) dx = \\left[ -x \\frac{\\cos(sx)}{s} \\right]_0^a - \\int_0^a \\left(-\\frac{\\cos(sx)}{s}\\right) dx = -\\frac{a \\cos(sa)}{s} + \\left[ \\frac{\\sin(sx)}{s^2} \\right]_0^a","displayMode":true},{"type":"math","expression":"= -\\frac{a \\cos(sa)}{s} + \\frac{\\sin(sa)}{s^2}","displayMode":true},{"type":"subheading","content":"3. Constructing the Final Closed-Form Expression"},{"type":"paragraph","content":"We carefully substitute this secondary evaluation back into our primary nested equation, ensuring no algebraic signs are dropped."},{"type":"math","expression":"\\int_0^a x^2 \\cos(sx) dx = \\frac{a^2 \\sin(sa)}{s} - \\frac{2}{s} \\left[ -\\frac{a \\cos(sa)}{s} + \\frac{\\sin(sa)}{s^2} \\right] = \\frac{a^2 \\sin(sa)}{s} + \\frac{2a \\cos(sa)}{s^2} - \\frac{2 \\sin(sa)}{s^3}","displayMode":true},{"type":"paragraph","content":"Finally, we re-incorporate the symmetry multiplier of 2 that we established at the beginning to formulate the complete, final expression for the Fourier transform."},{"type":"math","expression":"F(s) = 2 \\left( \\frac{a^2 \\sin(sa)}{s} + \\frac{2a \\cos(sa)}{s^2} - \\frac{2 \\sin(sa)}{s^3} \\right)","displayMode":true},{"type":"paragraph","content":"This final expression can be algebraically factored over a common denominator $s^3$ to produce the most elegant form: $F(s) = \\frac{2}{s^3} \\left[ (a^2s^2 - 2)\\sin(sa) + 2as\\cos(sa) \\right]$. This rigorously completes the necessary deduction."}]}},{"id":"v2-advanced-engineering-mathematics-2023-pC-q5","number":5,"marks":10,"tags":["Part C"],"text":[{"type":"math","expression":"\\text{Find } Z\\{a^{|n|}\\} \\text{ and hence find } Z\\left\\{\\left(\\frac{1}{a}\\right)^{|n|}\\right\\}.","displayMode":true}],"answer":{"summary":"A rigorous derivation of the two-sided Z-transform for a double-sided exponential sequence by splitting the summation and defining regions of absolute convergence.","explanation":[{"type":"heading","content":"Two-Sided Z-Transforms and Regions of Convergence"},{"type":"paragraph","content":"The problem dictates the calculation of the discrete Z-transform for the double-sided sequence $x_n = a^{|n|}$, which extends infinitely in both the positive and negative temporal directions. The fundamental definition for a two-sided Z-transform is the infinite bilateral series $Z\\{x_n\\} = X(z) = \\sum_{n=-\\infty}^\\infty x_n z^{-n}$. Because the sequence involves an absolute value operation on the index $n$, the mathematical behavior of the sequence fundamentally changes at the origin. Therefore, to evaluate the summation, we must meticulously split it into two independent, one-sided summations: one for negative indices and one for non-negative indices."},{"type":"subheading","content":"1. Decomposing the Bilateral Summation"},{"type":"paragraph","content":"We mathematically partition the infinite summation around $n=0$:"},{"type":"math","expression":"X(z) = \\sum_{n=-\\infty}^{-1} a^{|n|} z^{-n} + \\sum_{n=0}^\\infty a^{|n|} z^{-n}","displayMode":true},{"type":"paragraph","content":"For the right-hand summation (where $n \\ge 0$), the absolute value is redundant since $n$ is inherently positive, so $|n| = n$. The sum simplifies directly to $\\sum_{n=0}^\\infty a^n z^{-n}$."},{"type":"paragraph","content":"For the left-hand summation (where $n < 0$), the absolute value mathematically equates to negation, so $|n| = -n$. To make the indices positive and standard, we perform a substitution of variables. We let an arbitrary dummy variable $m = -n$. As $n$ travels from $-1$ down to $-\\infty$, the new index $m$ will correspondingly travel from $1$ up to $\\infty$."},{"type":"math","expression":"\\sum_{n=-\\infty}^{-1} a^{-n} z^{-n} \\implies \\sum_{m=1}^\\infty a^{m} z^{m} = \\sum_{m=1}^\\infty (az)^m","displayMode":true},{"type":"subheading","content":"2. Evaluating the Infinite Geometric Series"},{"type":"paragraph","content":"We now have two distinct infinite geometric series to evaluate. The formula for the sum of an infinite geometric progression $\\sum_{k=K}^\\infty r^k$ is $\\frac{\\text{First Term}}{1 - \\text{Common Ratio}}$, strictly provided that the absolute value of the common ratio $|r| < 1$. This strict mathematical condition defines the system's Region of Convergence (ROC)."},{"type":"paragraph","content":"**Evaluating the Positive-Time Summation:**"},{"type":"math","expression":"\\sum_{n=0}^\\infty (a z^{-1})^n = 1 + (a z^{-1}) + (a z^{-1})^2 + \\dots = \\frac{1}{1 - a z^{-1}} = \\frac{z}{z - a}","displayMode":true},{"type":"paragraph","content":"This specific summation only converges if $|a z^{-1}| < 1$, which algebraically implies $|z| > |a|$."},{"type":"paragraph","content":"**Evaluating the Negative-Time Summation:**"},{"type":"math","expression":"\\sum_{m=1}^\\infty (az)^m = (az)^1 + (az)^2 + (az)^3 + \\dots = \\frac{az}{1 - az}","displayMode":true},{"type":"paragraph","content":"This summation only converges if $|az| < 1$, which algebraically implies $|z| < 1/|a|$."},{"type":"subheading","content":"3. System Synthesis and ROC Intersections"},{"type":"paragraph","content":"The complete, overarching transform $X(z)$ only mathematically exists in regions where *both* individual summations simultaneously converge. Therefore, the total Region of Convergence is the exact intersection of the two individual constraints: $|a| < |z| < 1/|a|$. (Note: For this annular region to exist, we must strictly have $|a| < 1/|a|$, implying $|a| < 1$)."},{"type":"paragraph","content":"We now algebraically combine the evaluated closed-form sums to construct the final expression:"},{"type":"math","expression":"X(z) = \\frac{az}{1 - az} + \\frac{z}{z - a}","displayMode":true},{"type":"paragraph","content":"To provide the most robust final answer, we synthesize this into a single cohesive fraction by finding a common denominator $(1-az)(z-a)$:"},{"type":"math","expression":"X(z) = \\frac{az(z-a) + z(1-az)}{(1-az)(z-a)} = \\frac{az^2 - a^2z + z - az^2}{(1-az)(z-a)} = \\frac{z - a^2z}{(1-az)(z-a)} = \\frac{z(1-a^2)}{(1-az)(z-a)}","displayMode":true},{"type":"paragraph","content":"This elegantly compact rational function, constrained perfectly within the annular region of convergence $|a| < |z| < 1/|a|$, rigorously represents the complete two-sided Z-transform of the symmetric exponential sequence."}]}}]}],"loading":false,"error":null}}}

Q16Advanced Engineering Mathematics

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