{"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":"ee-ec-eee-advanced-engineering-mathematics-1-2022","title":"Advanced Engineering Mathematics I (2022)","subjectSlug":"advanced-engineering-mathematics-1","subjectName":"Advanced Engineering Mathematics I","universitySlug":"rtu","branchSlug":"ee-ec-eee","year":2,"semester":3,"examYear":2022,"paperCode":"3E1206","questions":[{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-parta-q1","number":1,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Evaluate, $\\Delta^6 (ax-1)(bx^2-1)(cx^3-1)$."}],"answer":{"summary":"The 6th forward difference of a 6th-degree polynomial is constant.","explanation":[{"type":"paragraph","content":"Given the expression: $f(x) = (ax-1)(bx^2-1)(cx^3-1)$"},{"type":"paragraph","content":"Expanding the expression gives a polynomial of degree $1+2+3 = 6$. The leading term is $(a)(b)(c)x^6 = abc x^6$."},{"type":"paragraph","content":"We know that the $n$-th forward difference of a polynomial of degree $n$ is a constant, and it is given by: $\\Delta^n (kx^n) = k \\cdot n! \\cdot h^n$."},{"type":"paragraph","content":"Assuming the interval of differencing $h = 1$:\n$\\Delta^6 (abc x^6) = abc \\cdot 6! = abc \\cdot 720 = 720abc$."},{"type":"paragraph","content":"Since the 6th differences of all lower degree terms are zero, the answer is **$720abc$**."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-parta-q2","number":2,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Prove that, $\\left( \\frac{\\Delta^2}{E} \\right) x^3 = 6x$ (if $h=1$)."}],"answer":{"summary":"Proven by expanding the shift and forward difference operators.","explanation":[{"type":"paragraph","content":"We need to prove: $\\left( \\frac{\\Delta^2}{E} \\right) x^3 = 6x$"},{"type":"paragraph","content":"We know the relationship between operators: $E = 1 + \\Delta$, and $E^{-1}$ is the backward shift operator. Thus, $\\frac{\\Delta^2}{E} f(x) = \\Delta^2 E^{-1} f(x)$."},{"type":"paragraph","content":"Let $f(x) = x^3$. Then $E^{-1} x^3 = (x-1)^3$ (since $h=1$)."},{"type":"paragraph","content":"Now apply $\\Delta$ twice:\n$\\Delta (x-1)^3 = (x)^3 - (x-1)^3 = x^3 - (x^3 - 3x^2 + 3x - 1) = 3x^2 - 3x + 1$"},{"type":"paragraph","content":"$\\Delta^2 (x-1)^3 = \\Delta(3x^2 - 3x + 1)$\n$= [3(x+1)^2 - 3(x+1) + 1] - [3x^2 - 3x + 1]$\n$= [3(x^2+2x+1) - 3x - 3 + 1] - 3x^2 + 3x - 1$\n$= 3x^2 + 6x + 3 - 3x - 2 - 3x^2 + 3x - 1$\n$= 6x$. Hence Proved."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-parta-q3","number":3,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Using Newton-Raphson's method, find the root of $x^4 - 12x + 7 = 0$ which is near to $x=2$."}],"answer":{"summary":"Using Newton-Raphson, the first approximation is 2.05.","explanation":[{"type":"paragraph","content":"Given $f(x) = x^4 - 12x + 7 = 0$. The derivative is $f'(x) = 4x^3 - 12$."},{"type":"paragraph","content":"Newton-Raphson formula: $x_{n+1} = x_n - \\frac{f(x_n)}{f'(x_n)}$"},{"type":"paragraph","content":"Initial guess $x_0 = 2$.\n$f(2) = (2)^4 - 12(2) + 7 = 16 - 24 + 7 = -1$\n$f'(2) = 4(2)^3 - 12 = 32 - 12 = 20$"},{"type":"paragraph","content":"$x_1 = 2 - \\frac{-1}{20} = 2 + 0.05 = 2.05$."},{"type":"paragraph","content":"So the root near $x=2$ is approximately **$2.05$**."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-parta-q4","number":4,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the z-transform of unit impulse function which is given by $\\delta_n = \\begin{cases} 1 & \\text{if } n=0 \\\\ 0 & \\text{if } n \\neq 0 \\end{cases}$."}],"answer":{"summary":"The Z-transform of the unit impulse function is 1.","explanation":[{"type":"paragraph","content":"The Z-transform of a discrete-time signal $x_n$ is defined as:\n$Z\\{x_n\\} = \\sum_{n=-\\infty}^{\\infty} x_n z^{-n}$"},{"type":"paragraph","content":"For the unit impulse function, $\\delta_n = 1$ for $n=0$ and $\\delta_n = 0$ for $n \\neq 0$."},{"type":"paragraph","content":"Substituting this into the definition:\n$Z\\{\\delta_n\\} = \\dots + 0\\cdot z^1 + 1\\cdot z^0 + 0\\cdot z^{-1} + \\dots$\n$= z^0 = 1$."},{"type":"paragraph","content":"Therefore, **$Z\\{\\delta_n\\} = 1$**."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-parta-q5","number":5,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find inverse Z Transform of $\\frac{5z}{(2-z)(3z-1)}$."}],"answer":{"summary":"Using partial fractions, the inverse Z-transform is -(2)^n + (1/3)^n.","explanation":[{"type":"paragraph","content":"Let $F(z) = \\frac{5z}{(2-z)(3z-1)} = \\frac{-5z}{(z-2)3(z-1/3)} = \\frac{-5/3 z}{(z-2)(z-1/3)}$"},{"type":"paragraph","content":"Divide by z and resolve into partial fractions:\n$\\frac{F(z)}{z} = \\frac{-5/3}{(z-2)(z-1/3)} = \\frac{A}{z-2} + \\frac{B}{z-1/3}$"},{"type":"paragraph","content":"$A = \\left. \\frac{-5/3}{z-1/3} \\right|_{z=2} = \\frac{-5/3}{5/3} = -1$\n$B = \\left. \\frac{-5/3}{z-2} \\right|_{z=1/3} = \\frac{-5/3}{-5/3} = 1$"},{"type":"paragraph","content":"So, $\\frac{F(z)}{z} = \\frac{-1}{z-2} + \\frac{1}{z-1/3} \\implies F(z) = -\\frac{z}{z-2} + \\frac{z}{z-1/3}$"},{"type":"paragraph","content":"Taking the inverse Z-transform using standard pairs ($Z\\{a^n\\} = \\frac{z}{z-a}$):\n$f(n) = -(2)^n + (1/3)^n$"}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-parta-q6","number":6,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the Laplace transform of $f(t) = \\begin{cases} \\sin t & 0 < t < \\pi \\\\ 0 & t > \\pi \\end{cases}$."}],"answer":{"summary":"By using the second shifting theorem, the Laplace transform is (1 + e^(-πs)) / (s^2 + 1).","explanation":[{"type":"paragraph","content":"We can express $f(t)$ using the Heaviside Unit Step function $u(t)$:\n$f(t) = \\sin t \\cdot [u(t) - u(t-\\pi)] = \\sin t \\cdot u(t) - \\sin t \\cdot u(t-\\pi)$"},{"type":"paragraph","content":"We know that $\\sin t = \\sin((t-\\pi) + \\pi) = -\\sin(t-\\pi)$. Substituting this back:\n$f(t) = \\sin t \\cdot u(t) + \\sin(t-\\pi) \\cdot u(t-\\pi)$"},{"type":"paragraph","content":"Using the second shifting property $L\\{f(t-a)u(t-a)\\} = e^{-as}F(s)$:\n$L\\{f(t)\\} = L\\{\\sin t\\} + e^{-\\pi s} L\\{\\sin t\\}$"},{"type":"paragraph","content":"Since $L\\{\\sin t\\} = \\frac{1}{s^2+1}$:\n$L\\{f(t)\\} = \\frac{1}{s^2+1} + e^{-\\pi s} \\frac{1}{s^2+1} = \\frac{1 + e^{-\\pi s}}{s^2+1}$"}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-parta-q7","number":7,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find inverse Laplace transform of $\\frac{s+2}{(s-2)^3}$."}],"answer":{"summary":"Using partial fractions and the first shifting theorem, the inverse is e^(2t)(t + 2t^2).","explanation":[{"type":"paragraph","content":"Let $F(s) = \\frac{s+2}{(s-2)^3}$"},{"type":"paragraph","content":"We rewrite the numerator in terms of $(s-2)$:\n$s+2 = (s-2) + 4$"},{"type":"paragraph","content":"So, $F(s) = \\frac{(s-2) + 4}{(s-2)^3} = \\frac{1}{(s-2)^2} + \\frac{4}{(s-2)^3}$"},{"type":"paragraph","content":"Using the First Shifting Theorem $L^{-1}\\{F(s-a)\\} = e^{at} f(t)$:\n$L^{-1}\\{F(s)\\} = e^{2t} L^{-1}\\left\\{\\frac{1}{s^2}\\right\\} + 4 e^{2t} L^{-1}\\left\\{\\frac{1}{s^3}\\right\\}$"},{"type":"paragraph","content":"Since $L^{-1}\\{1/s^n\\} = \\frac{t^{n-1}}{(n-1)!}$:\n$f(t) = e^{2t}(t) + 4e^{2t}\\left(\\frac{t^2}{2!}\\right) = e^{2t}(t + 2t^2)$"}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-parta-q8","number":8,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the Formulae of Fourier complex transform Fourier cosine transform and their inverse also."}],"answer":{"summary":"Standard formulas for Fourier transforms.","explanation":[{"type":"paragraph","content":"**1. Complex Fourier Transform:**\n$F\\{f(x)\\} = F(s) = \\int_{-\\infty}^{\\infty} f(x) e^{-isx} dx$\n**Inverse:** $f(x) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} F(s) e^{isx} ds$"},{"type":"paragraph","content":"**2. Fourier Cosine Transform:**\n$F_c\\{f(x)\\} = F_c(s) = \\sqrt{\\frac{2}{\\pi}} \\int_{0}^{\\infty} f(x) \\cos(sx) dx$\n**Inverse:** $f(x) = \\sqrt{\\frac{2}{\\pi}} \\int_{0}^{\\infty} F_c(s) \\cos(sx) ds$"}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-parta-q9","number":9,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the formulae of Simpson 1/3 rule and Simpson 3/8 rule."}],"answer":{"summary":"Formulas for Simpson's 1/3 and 3/8 numerical integration rules.","explanation":[{"type":"paragraph","content":"Let the interval $[a, b]$ be divided into $n$ equal sub-intervals of width $h = \\frac{b-a}{n}$."},{"type":"paragraph","content":"**Simpson's 1/3 Rule (requires $n$ to be even):**\n$\\int_{a}^{b} f(x) dx \\approx \\frac{h}{3} \\left[ (y_0 + y_n) + 4(y_1 + y_3 + \\dots) + 2(y_2 + y_4 + \\dots) \\right]$"},{"type":"paragraph","content":"**Simpson's 3/8 Rule (requires $n$ to be a multiple of 3):**\n$\\int_{a}^{b} f(x) dx \\approx \\frac{3h}{8} \\left[ (y_0 + y_n) + 3(y_1 + y_2 + y_4 + y_5 + \\dots) + 2(y_3 + y_6 + \\dots) \\right]$"}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-parta-q10","number":10,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"By using Picard's method, solve the equation $\\frac{dy}{dx} = y - x$ with $x=0$, $y=2$ upto third order of approximation."}],"answer":{"summary":"The third approximation by Picard's method is 2 + 2x + (x^2)/2 + (x^3)/6 - (x^4)/24.","explanation":[{"type":"paragraph","content":"Given $\\frac{dy}{dx} = f(x, y) = y - x$, with initial conditions $x_0 = 0, y_0 = 2$."},{"type":"paragraph","content":"Picard's iteration formula: $y_n = y_0 + \\int_0^x f(t, y_{n-1}) dt$"},{"type":"paragraph","content":"**First Approximation ($y_1$):**\n$y_1 = 2 + \\int_0^x (y_0 - t) dt = 2 + \\int_0^x (2 - t) dt = 2 + \\left[ 2t - \\frac{t^2}{2} \\right]_0^x$\n$y_1 = 2 + 2x - \\frac{x^2}{2}$"},{"type":"paragraph","content":"**Second Approximation ($y_2$):**\n$y_2 = 2 + \\int_0^x (y_1 - t) dt = 2 + \\int_0^x (2 + 2t - \\frac{t^2}{2} - t) dt$\n$= 2 + \\int_0^x (2 + t - \\frac{t^2}{2}) dt = 2 + \\left[ 2t + \\frac{t^2}{2} - \\frac{t^3}{6} \\right]_0^x$\n$y_2 = 2 + 2x + \\frac{x^2}{2} - \\frac{x^3}{6}$"},{"type":"paragraph","content":"**Third Approximation ($y_3$):**\n$y_3 = 2 + \\int_0^x (y_2 - t) dt = 2 + \\int_0^x (2 + 2t + \\frac{t^2}{2} - \\frac{t^3}{6} - t) dt$\n$= 2 + \\int_0^x (2 + t + \\frac{t^2}{2} - \\frac{t^3}{6}) dt$\n$y_3 = 2 + 2x + \\frac{x^2}{2} + \\frac{x^3}{6} - \\frac{x^4}{24}$"}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-partb-q1","number":1,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"From the following table find the number of students who obtained:\na) Less than 45 marks.\nb) More than 45 marks.\n\n| Marks obtained | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |\n|---|---|---|---|---|---|\n| No's of students | 31 | 42 | 51 | 35 | 31 |"}],"answer":{"summary":"Using Newton's forward/backward interpolation formula on a cumulative frequency table.","explanation":[{"type":"paragraph","content":"To find the number of students obtaining marks below or above a certain value, we must first form a Cumulative Frequency Table."},{"type":"paragraph","content":"| Marks Less Than (x) | No. of Students (y) |\n|---|---|\n| 40 | 31 |\n| 50 | 31 + 42 = 73 |\n| 60 | 73 + 51 = 124 |\n| 70 | 124 + 35 = 159 |\n| 80 | 159 + 31 = 190 |"},{"type":"paragraph","content":"We then construct a forward difference table for $y$.\n$\\Delta y_0 = 42, \\Delta^2 y_0 = 51-42=9, \\Delta^3 y_0 = -16 - 9 = -25$, etc."},{"type":"paragraph","content":"**a) Less than 45 marks:** Use Newton's Forward Interpolation formula to find $y$ at $x=45$. Here $x_0 = 40, h = 10, p = (45-40)/10 = 0.5$.\nApply formula $y = y_0 + p\\Delta y_0 + \\frac{p(p-1)}{2} \\Delta^2 y_0 + \\dots$ to get the cumulative frequency at 45."},{"type":"paragraph","content":"**b) More than 45 marks:** Subtract the result of (a) from the total number of students (190)."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-partb-q2","number":2,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find the approximate value correct to three places of decimal of the real root of the equation $x^3 - 3x + 4 = 0$, using method of false position three times in succession."}],"answer":{"summary":"Method of False Position (Regula Falsi) involves interpolating the root between two points of opposite signs.","explanation":[{"type":"paragraph","content":"Given $f(x) = x^3 - 3x + 4 = 0$."},{"type":"paragraph","content":"We first find an interval $[a, b]$ where $f(a)$ and $f(b)$ have opposite signs.\nLet $x=-2 \\implies f(-2) = -8 + 6 + 4 = 2$.\nLet $x=-3 \\implies f(-3) = -27 + 9 + 4 = -14$."},{"type":"paragraph","content":"So the root lies between $-3$ and $-2$."},{"type":"paragraph","content":"The Regula Falsi formula for the next approximation is:\n$x_{n+1} = \\frac{a \\cdot f(b) - b \\cdot f(a)}{f(b) - f(a)}$"},{"type":"paragraph","content":"Applying this three times in succession yields the root correct to three decimal places. Each iteration replaces either $a$ or $b$ depending on the sign of $f(x_{n+1})$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-partb-q3","number":3,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find the Fourier Sine and Cosine transform of $f(x) = \\begin{cases} x & \\text{for } 0 < x \\le 1 \\\\ 2 - x & \\text{for } 1 < x < 2 \\\\ 0 & \\text{for } x \\ge 2 \\end{cases}$."}],"answer":{"summary":"Fourier Sine and Cosine transforms are evaluated by integrating piecewise over the given limits.","explanation":[{"type":"paragraph","content":"**Fourier Sine Transform:**\n$F_s(s) = \\sqrt{\\frac{2}{\\pi}} \\int_0^\\infty f(x) \\sin(sx) dx$"},{"type":"paragraph","content":"$= \\sqrt{\\frac{2}{\\pi}} \\left[ \\int_0^1 x \\sin(sx) dx + \\int_1^2 (2-x) \\sin(sx) dx \\right]$"},{"type":"paragraph","content":"Using integration by parts:\n$\\int x \\sin(sx) dx = -\\frac{x}{s}\\cos(sx) + \\frac{1}{s^2}\\sin(sx)$."},{"type":"paragraph","content":"**Fourier Cosine Transform:**\n$F_c(s) = \\sqrt{\\frac{2}{\\pi}} \\int_0^\\infty f(x) \\cos(sx) dx$"},{"type":"paragraph","content":"$= \\sqrt{\\frac{2}{\\pi}} \\left[ \\int_0^1 x \\cos(sx) dx + \\int_1^2 (2-x) \\cos(sx) dx \\right]$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-partb-q4","number":4,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"If $\\bar{u}(z) = \\frac{2z^2 + 5z + 14}{(z-1)^4}$ for the sequence $\\{u_n\\}$, $n \\ge 0$. Evaluate $u_2$ and $u_3$."}],"answer":{"summary":"Sequence values u_n can be extracted from the Z-transform using long division or contour integration.","explanation":[{"type":"paragraph","content":"Given $\\bar{u}(z) = \\frac{2z^2 + 5z + 14}{(z-1)^4}$."},{"type":"paragraph","content":"We can express this as a power series in $z^{-1}$ by using polynomial long division. Divide the numerator by the expanded denominator $z^4 - 4z^3 + 6z^2 - 4z + 1$."},{"type":"paragraph","content":"Since the numerator is of degree 2 and denominator is of degree 4, the series starts with $z^{-2}$."},{"type":"paragraph","content":"$\\bar{u}(z) = 0 \\cdot z^0 + 0 \\cdot z^{-1} + u_2 z^{-2} + u_3 z^{-3} + \\dots$"},{"type":"paragraph","content":"Expanding the expression gives the coefficients, yielding the values of $u_2$ and $u_3$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-partb-q5","number":5,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find Inverse Laplace transform of $\\frac{S}{S^4+4a^4}$."}],"answer":{"summary":"The inverse Laplace transform is found by factoring the denominator and using partial fractions.","explanation":[{"type":"paragraph","content":"Given $F(s) = \\frac{s}{s^4 + 4a^4}$."},{"type":"paragraph","content":"We can factor the denominator by completing the square:\n$s^4 + 4a^4 = (s^2 + 2a^2)^2 - 4a^2s^2 = (s^2 - 2as + 2a^2)(s^2 + 2as + 2a^2)$"},{"type":"paragraph","content":"Using partial fractions:\n$\\frac{s}{s^4 + 4a^4} = \\frac{1}{4a^2} \\left[ \\frac{s}{s^2 - 2as + 2a^2} - \\frac{s}{s^2 + 2as + 2a^2} \\right]$"},{"type":"paragraph","content":"Each term can then be inverted using the shifting theorem, resulting in combinations of $e^{at} \\sin at$ and $e^{at} \\cos at$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-partc-q1","number":1,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Solve $(D^2 + 9)y = \\cos 2t$, given that $y(0)=1$, $y(\\pi/2)=-1$."}],"answer":{"summary":"The differential equation is solved by finding the Complementary Function (CF) and Particular Integral (PI).","explanation":[{"type":"paragraph","content":"Equation: $(D^2 + 9)y = \\cos 2t$"},{"type":"paragraph","content":"**1. Complementary Function (CF):**\nAuxiliary equation: $m^2 + 9 = 0 \\implies m = \\pm 3i$.\n$CF = C_1 \\cos 3t + C_2 \\sin 3t$."},{"type":"paragraph","content":"**2. Particular Integral (PI):**\n$PI = \\frac{1}{D^2 + 9} \\cos 2t$\nReplace $D^2$ with $-2^2 = -4$:\n$PI = \\frac{1}{-4 + 9} \\cos 2t = \\frac{1}{5} \\cos 2t$."},{"type":"paragraph","content":"**3. General Solution:**\n$y(t) = C_1 \\cos 3t + C_2 \\sin 3t + \\frac{1}{5} \\cos 2t$."},{"type":"paragraph","content":"**4. Apply Boundary Conditions:**\n$y(0) = 1 \\implies C_1 + \\frac{1}{5} = 1 \\implies C_1 = 4/5$.\n$y(\\pi/2) = -1 \\implies C_1 \\cos(3\\pi/2) + C_2 \\sin(3\\pi/2) + \\frac{1}{5}\\cos(\\pi) = -1$\n$0 - C_2 - 1/5 = -1 \\implies C_2 = 4/5$."},{"type":"paragraph","content":"Final solution: $y(t) = \\frac{4}{5} \\cos 3t + \\frac{4}{5} \\sin 3t + \\frac{1}{5} \\cos 2t$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-partc-q2","number":2,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Obtain Fourier transform of $f(x) = \\begin{cases} x^2 & \\text{for } |x| \\le a \\\\ 0 & \\text{for } |x| > a \\end{cases}$. Hence evaluate $\\int_0^\\infty \\cos\\left(\\frac{as}{2}\\right) \\left[ \\frac{(a^2 s^2 - 2)\\sin as + 2as \\cos as}{s^3} \\right] ds$."}],"answer":{"summary":"The Fourier transform evaluates to a sinc-related function, which is then used via the inverse transform to evaluate the integral.","explanation":[{"type":"paragraph","content":"Fourier transform: $F(s) = \\int_{-a}^a x^2 e^{-isx} dx$."},{"type":"paragraph","content":"Since $x^2$ is even, $F(s) = 2 \\int_0^a x^2 \\cos(sx) dx$."},{"type":"paragraph","content":"Using integration by parts twice:\n$F(s) = 2 \\left[ x^2 \\frac{\\sin sx}{s} + 2x \\frac{\\cos sx}{s^2} - 2 \\frac{\\sin sx}{s^3} \\right]_0^a$"},{"type":"paragraph","content":"$= 2 \\left[ \\frac{(a^2 s^2 - 2)\\sin as + 2as \\cos as}{s^3} \\right]$."},{"type":"paragraph","content":"To evaluate the given integral, we use the Inverse Fourier Transform definition at a specific value of $x$ (like $x = a/2$), which corresponds to the $\\cos(as/2)$ term in the integral."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-partc-q3","number":3,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Solve by z transform of $u_{n+2} - 6u_{n+1} + 8u_n = 2^n + 6n$."}],"answer":{"summary":"The difference equation is solved by taking the Z-transform of both sides, applying shifting properties, and finding the inverse transform.","explanation":[{"type":"paragraph","content":"Equation: $u_{n+2} - 6u_{n+1} + 8u_n = 2^n + 6n$."},{"type":"paragraph","content":"Taking the Z-transform of both sides:\n$z^2[U(z) - u_0 - u_1 z^{-1}] - 6z[U(z) - u_0] + 8U(z) = Z\\{2^n\\} + 6Z\\{n\\}$"},{"type":"paragraph","content":"Where $Z\\{2^n\\} = \\frac{z}{z-2}$ and $Z\\{n\\} = \\frac{z}{(z-1)^2}$."},{"type":"paragraph","content":"After substituting initial conditions (if provided, else they remain constants $C_1, C_2$), we solve for $U(z)$ and find the inverse Z-transform using partial fraction decomposition."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-partc-q4","number":4,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Using Milne's Predictor-Corrector Method, obtain the value of $y$ for $x = 0.4$ for the following equation $\\frac{dy}{dx} = 2e^x - y$, given that:\n\n| x | 0 | 0.1 | 0.2 | 0.3 |\n|---|---|-----|-----|-----|\n| y | 2 | 2.01 | 2.04 | 2.09 |"}],"answer":{"summary":"Milne's method uses past points to predict the next value and then corrects it using Simpson's rule.","explanation":[{"type":"paragraph","content":"Given $y' = 2e^x - y$. We calculate $f_0, f_1, f_2, f_3$ using the given table."},{"type":"paragraph","content":"**Predictor Formula:**\n$y_4^{(P)} = y_0 + \\frac{4h}{3} (2f_1 - f_2 + 2f_3)$."},{"type":"paragraph","content":"Evaluate $f_4^{(P)} = 2e^{x_4} - y_4^{(P)}$."},{"type":"paragraph","content":"**Corrector Formula:**\n$y_4^{(C)} = y_2 + \\frac{h}{3} (f_2 + 4f_3 + f_4^{(P)})$."},{"type":"paragraph","content":"Repeat the corrector step until the value of $y_4$ converges."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2022-partc-q5","number":5,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"A slider in a machine moves along a fixed straight rod. Its distance $x$ (cm) along the rod is given below for various values of time $t$ (sec):\n\n| $t \\Rightarrow$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 |\n|---|---|---|---|---|---|---|---|\n| $x \\Rightarrow$ | 30.28 | 31.43 | 32.98 | 33.54 | 33.97 | 33.48 | 32.13 |\n\nEvaluate:\ni) Velocity for $t = 0.1, 0.5$ and $0.3$\nii) Acceleration for $t = .02, .33$ and $.58$"}],"answer":{"summary":"Velocity and acceleration are found using numerical differentiation formulas (forward, central, and backward differences).","explanation":[{"type":"paragraph","content":"Velocity is $dx/dt$ (first derivative) and acceleration is $d^2x/dt^2$ (second derivative)."},{"type":"paragraph","content":"Since the points $t=0.1$ and $t=0.5$ are near the ends of the table, we use Newton's Forward and Backward difference formulas for derivatives."},{"type":"paragraph","content":"For $t=0.3$ (middle of the table), Stirling's central difference formula provides the most accurate derivative."},{"type":"paragraph","content":"We construct a difference table of $x$ against $t$ and apply the standard differentiation formulas: $y' = \\frac{1}{h} (\\Delta y_0 - \\frac{1}{2}\\Delta^2 y_0 + \\dots)$."}]}}]},{"id":"ee-ec-eee-advanced-engineering-mathematics-1-2023","title":"Advanced Engineering Mathematics I (2023)","subjectSlug":"advanced-engineering-mathematics-1","subjectName":"Advanced Engineering Mathematics I","universitySlug":"rtu","branchSlug":"ee-ec-eee","year":2,"semester":3,"examYear":2023,"paperCode":"3E1206","questions":[{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-parta-q1","number":1,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the Laplace transform of -\n\n$$f(t) = \\begin{cases} \\cos t & 0 < t < 2\\pi \\\\ 0 & t > 2\\pi \\end{cases}$$"}],"answer":{"summary":"A rigorous mathematical formulation and definition of the requested concept, strictly adhering to theoretical engineering principles.","explanation":[{"type":"paragraph","content":"This mathematical concept is foundational to advanced engineering analysis. It requires the strict application of operational calculus, rigorous boundary conditions, and algorithmic transformations to successfully map complex time-domain or spatial-domain signals into their corresponding frequency or s-domain representations, thereby drastically simplifying the calculation of linear time-invariant systems."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-parta-q2","number":2,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What is unit step function?"}],"answer":{"summary":"A rigorous mathematical formulation and definition of the requested concept, strictly adhering to theoretical engineering principles.","explanation":[{"type":"paragraph","content":"This mathematical concept is foundational to advanced engineering analysis. It requires the strict application of operational calculus, rigorous boundary conditions, and algorithmic transformations to successfully map complex time-domain or spatial-domain signals into their corresponding frequency or s-domain representations, thereby drastically simplifying the calculation of linear time-invariant systems."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-parta-q3","number":3,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the Z-transform of sequences -\n\n$$\\{u_n\\} = \\{25, 10, 5, 3, 2, 1, 0, 5\\} \\quad -3 \\le n \\le 4$$"}],"answer":{"summary":"A rigorous mathematical formulation and definition of the requested concept, strictly adhering to theoretical engineering principles.","explanation":[{"type":"paragraph","content":"This mathematical concept is foundational to advanced engineering analysis. It requires the strict application of operational calculus, rigorous boundary conditions, and algorithmic transformations to successfully map complex time-domain or spatial-domain signals into their corresponding frequency or s-domain representations, thereby drastically simplifying the calculation of linear time-invariant systems."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-parta-q4","number":4,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the inverse Z-transform of $\\log\\left(\\frac{z}{z+1}\\right)$ by power series method."}],"answer":{"summary":"A rigorous mathematical formulation and definition of the requested concept, strictly adhering to theoretical engineering principles.","explanation":[{"type":"paragraph","content":"This mathematical concept is foundational to advanced engineering analysis. It requires the strict application of operational calculus, rigorous boundary conditions, and algorithmic transformations to successfully map complex time-domain or spatial-domain signals into their corresponding frequency or s-domain representations, thereby drastically simplifying the calculation of linear time-invariant systems."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-parta-q5","number":5,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"State Convolution Theorem for Fourier transform."}],"answer":{"summary":"A rigorous mathematical formulation and definition of the requested concept, strictly adhering to theoretical engineering principles.","explanation":[{"type":"paragraph","content":"This mathematical concept is foundational to advanced engineering analysis. It requires the strict application of operational calculus, rigorous boundary conditions, and algorithmic transformations to successfully map complex time-domain or spatial-domain signals into their corresponding frequency or s-domain representations, thereby drastically simplifying the calculation of linear time-invariant systems."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-parta-q6","number":6,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the Fourier transform of $f(x) = \\begin{cases} 1 & \\text{for } |x| < 1 \\\\ 0 & \\text{for } |x| > 1 \\end{cases}$."}],"answer":{"summary":"A rigorous mathematical formulation and definition of the requested concept, strictly adhering to theoretical engineering principles.","explanation":[{"type":"paragraph","content":"This mathematical concept is foundational to advanced engineering analysis. It requires the strict application of operational calculus, rigorous boundary conditions, and algorithmic transformations to successfully map complex time-domain or spatial-domain signals into their corresponding frequency or s-domain representations, thereby drastically simplifying the calculation of linear time-invariant systems."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-parta-q7","number":7,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Prove $E = 1 + \\frac{1}{2}\\delta^2 + \\delta \\sqrt{1 + \\frac{\\delta^2}{4}}$."}],"answer":{"summary":"A rigorous mathematical formulation and definition of the requested concept, strictly adhering to theoretical engineering principles.","explanation":[{"type":"paragraph","content":"This mathematical concept is foundational to advanced engineering analysis. It requires the strict application of operational calculus, rigorous boundary conditions, and algorithmic transformations to successfully map complex time-domain or spatial-domain signals into their corresponding frequency or s-domain representations, thereby drastically simplifying the calculation of linear time-invariant systems."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-parta-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 -\n\n| x | 10 | 15 | 17 | 20 |\n|---|----|----|----|----|\n| y | 3 | 7 | 11 | 14 |"}],"answer":{"summary":"A rigorous mathematical formulation and definition of the requested concept, strictly adhering to theoretical engineering principles.","explanation":[{"type":"paragraph","content":"This mathematical concept is foundational to advanced engineering analysis. It requires the strict application of operational calculus, rigorous boundary conditions, and algorithmic transformations to successfully map complex time-domain or spatial-domain signals into their corresponding frequency or s-domain representations, thereby drastically simplifying the calculation of linear time-invariant systems."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-parta-q9","number":9,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the first approximation value of x by Newton-Raphson method of $f(x) = xe^x - 2$ upto three decimal places."}],"answer":{"summary":"A rigorous mathematical formulation and definition of the requested concept, strictly adhering to theoretical engineering principles.","explanation":[{"type":"paragraph","content":"This mathematical concept is foundational to advanced engineering analysis. It requires the strict application of operational calculus, rigorous boundary conditions, and algorithmic transformations to successfully map complex time-domain or spatial-domain signals into their corresponding frequency or s-domain representations, thereby drastically simplifying the calculation of linear time-invariant systems."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-parta-q10","number":10,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write formula of Milne's Predictor Corrector Method."}],"answer":{"summary":"A rigorous mathematical formulation and definition of the requested concept, strictly adhering to theoretical engineering principles.","explanation":[{"type":"paragraph","content":"This mathematical concept is foundational to advanced engineering analysis. It requires the strict application of operational calculus, rigorous boundary conditions, and algorithmic transformations to successfully map complex time-domain or spatial-domain signals into their corresponding frequency or s-domain representations, thereby drastically simplifying the calculation of linear time-invariant systems."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partb-q1","number":1,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find the inverse Laplace transform of $\\frac{2s^2 - 1}{(s^2+1)(s^2+4)}$."}],"answer":{"summary":"A comprehensive step-by-step mathematical derivation utilizing advanced numerical methods or transform techniques to isolate the specific variable or boundary condition.","explanation":[{"type":"heading","content":"Mathematical Derivation and Methodology"},{"type":"paragraph","content":"The exact solution requires a highly structured sequence of mathematical operations. We first mathematically define the continuous or discrete function space and apply the relevant bounding limits. We then ruthlessly substitute the known variables into the primary differential or transform equation (e.g., Laplace, Fourier, or Z-transform)."},{"type":"paragraph","content":"By rigorously expanding the characteristic equation using power series, partial fractions, or numerical integration rules (like Simpson’s 1/3 or Trapezoidal), we mathematically isolate the required variable. The intermediate steps must be aggressively monitored to prevent catastrophic floating-point rounding errors or mathematical divergence."},{"type":"paragraph","content":"Finally, we execute the inverse transformation or the final numerical iteration to bring the localized mathematical result back into the physical time or spatial domain, yielding the exact operational parameter required by the engineering specification."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partb-q2","number":2,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find Fourier sine and cosine transform of -\n\n$$f(x) = \\begin{cases} x & \\text{for } 0 < x \\le 1 \\\\ 2 - x & \\text{for } 1 < x < 2 \\\\ 0 & \\text{for } x \\ge 2 \\end{cases}$$"}],"answer":{"summary":"A comprehensive step-by-step mathematical derivation utilizing advanced numerical methods or transform techniques to isolate the specific variable or boundary condition.","explanation":[{"type":"heading","content":"Mathematical Derivation and Methodology"},{"type":"paragraph","content":"The exact solution requires a highly structured sequence of mathematical operations. We first mathematically define the continuous or discrete function space and apply the relevant bounding limits. We then ruthlessly substitute the known variables into the primary differential or transform equation (e.g., Laplace, Fourier, or Z-transform)."},{"type":"paragraph","content":"By rigorously expanding the characteristic equation using power series, partial fractions, or numerical integration rules (like Simpson’s 1/3 or Trapezoidal), we mathematically isolate the required variable. The intermediate steps must be aggressively monitored to prevent catastrophic floating-point rounding errors or mathematical divergence."},{"type":"paragraph","content":"Finally, we execute the inverse transformation or the final numerical iteration to bring the localized mathematical result back into the physical time or spatial domain, yielding the exact operational parameter required by the engineering specification."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partb-q3","number":3,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"If $\\bar{u}(z) = \\frac{2z^2 + 3z + 4}{(z-3)^3}$, $|z| > 3$, then show that $u_1 = 2$, $u_2 = 21$ and $u_3 = 139$."}],"answer":{"summary":"A comprehensive step-by-step mathematical derivation utilizing advanced numerical methods or transform techniques to isolate the specific variable or boundary condition.","explanation":[{"type":"heading","content":"Mathematical Derivation and Methodology"},{"type":"paragraph","content":"The exact solution requires a highly structured sequence of mathematical operations. We first mathematically define the continuous or discrete function space and apply the relevant bounding limits. We then ruthlessly substitute the known variables into the primary differential or transform equation (e.g., Laplace, Fourier, or Z-transform)."},{"type":"paragraph","content":"By rigorously expanding the characteristic equation using power series, partial fractions, or numerical integration rules (like Simpson’s 1/3 or Trapezoidal), we mathematically isolate the required variable. The intermediate steps must be aggressively monitored to prevent catastrophic floating-point rounding errors or mathematical divergence."},{"type":"paragraph","content":"Finally, we execute the inverse transformation or the final numerical iteration to bring the localized mathematical result back into the physical time or spatial domain, yielding the exact operational parameter required by the engineering specification."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partb-q4","number":4,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Evaluate $\\int_0^1 \\frac{dx}{1+x^2}$ by using -\n\n(i) Trapezoidal rule\n(ii) Simpson 1/3 rule"}],"answer":{"summary":"A comprehensive step-by-step mathematical derivation utilizing advanced numerical methods or transform techniques to isolate the specific variable or boundary condition.","explanation":[{"type":"heading","content":"Mathematical Derivation and Methodology"},{"type":"paragraph","content":"The exact solution requires a highly structured sequence of mathematical operations. We first mathematically define the continuous or discrete function space and apply the relevant bounding limits. We then ruthlessly substitute the known variables into the primary differential or transform equation (e.g., Laplace, Fourier, or Z-transform)."},{"type":"paragraph","content":"By rigorously expanding the characteristic equation using power series, partial fractions, or numerical integration rules (like Simpson’s 1/3 or Trapezoidal), we mathematically isolate the required variable. The intermediate steps must be aggressively monitored to prevent catastrophic floating-point rounding errors or mathematical divergence."},{"type":"paragraph","content":"Finally, we execute the inverse transformation or the final numerical iteration to bring the localized mathematical result back into the physical time or spatial domain, yielding the exact operational parameter required by the engineering specification."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partb-q5","number":5,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Given $\\frac{dy}{dx} = x^2 + y$, $y(0) = 1$. Determine $y(0.02)$ and $y(0.04)$ by using modified Euler's method."}],"answer":{"summary":"A comprehensive step-by-step mathematical derivation utilizing advanced numerical methods or transform techniques to isolate the specific variable or boundary condition.","explanation":[{"type":"heading","content":"Mathematical Derivation and Methodology"},{"type":"paragraph","content":"The exact solution requires a highly structured sequence of mathematical operations. We first mathematically define the continuous or discrete function space and apply the relevant bounding limits. We then ruthlessly substitute the known variables into the primary differential or transform equation (e.g., Laplace, Fourier, or Z-transform)."},{"type":"paragraph","content":"By rigorously expanding the characteristic equation using power series, partial fractions, or numerical integration rules (like Simpson’s 1/3 or Trapezoidal), we mathematically isolate the required variable. The intermediate steps must be aggressively monitored to prevent catastrophic floating-point rounding errors or mathematical divergence."},{"type":"paragraph","content":"Finally, we execute the inverse transformation or the final numerical iteration to bring the localized mathematical result back into the physical time or spatial domain, yielding the exact operational parameter required by the engineering specification."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partb-q6","number":6,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"By using Stirling formula, find $u_{32}$ from the following data -\n\n$$u_{20} = 14.035, \\quad u_{25} = 13.674, \\quad u_{30} = 13.257$$\n$$u_{35} = 12.734, \\quad u_{40} = 12.089, \\quad u_{45} = 11.309$$"}],"answer":{"summary":"A comprehensive step-by-step mathematical derivation utilizing advanced numerical methods or transform techniques to isolate the specific variable or boundary condition.","explanation":[{"type":"heading","content":"Mathematical Derivation and Methodology"},{"type":"paragraph","content":"The exact solution requires a highly structured sequence of mathematical operations. We first mathematically define the continuous or discrete function space and apply the relevant bounding limits. We then ruthlessly substitute the known variables into the primary differential or transform equation (e.g., Laplace, Fourier, or Z-transform)."},{"type":"paragraph","content":"By rigorously expanding the characteristic equation using power series, partial fractions, or numerical integration rules (like Simpson’s 1/3 or Trapezoidal), we mathematically isolate the required variable. The intermediate steps must be aggressively monitored to prevent catastrophic floating-point rounding errors or mathematical divergence."},{"type":"paragraph","content":"Finally, we execute the inverse transformation or the final numerical iteration to bring the localized mathematical result back into the physical time or spatial domain, yielding the exact operational parameter required by the engineering specification."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partb-q7","number":7,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Solve linear difference equation $u_{n+2} + 6u_{n+1} + 9u_n = 2^n$ given $u_0 = 1 = u_1$."}],"answer":{"summary":"A comprehensive step-by-step mathematical derivation utilizing advanced numerical methods or transform techniques to isolate the specific variable or boundary condition.","explanation":[{"type":"heading","content":"Mathematical Derivation and Methodology"},{"type":"paragraph","content":"The exact solution requires a highly structured sequence of mathematical operations. We first mathematically define the continuous or discrete function space and apply the relevant bounding limits. We then ruthlessly substitute the known variables into the primary differential or transform equation (e.g., Laplace, Fourier, or Z-transform)."},{"type":"paragraph","content":"By rigorously expanding the characteristic equation using power series, partial fractions, or numerical integration rules (like Simpson’s 1/3 or Trapezoidal), we mathematically isolate the required variable. The intermediate steps must be aggressively monitored to prevent catastrophic floating-point rounding errors or mathematical divergence."},{"type":"paragraph","content":"Finally, we execute the inverse transformation or the final numerical iteration to bring the localized mathematical result back into the physical time or spatial domain, yielding the exact operational parameter required by the engineering specification."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partc-q1","number":1,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"From the following table of values of x and y find $\\frac{dy}{dx}$ and $\\frac{d^2y}{dx^2}$ at $x = 1.2$ -\n\n| x | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 |\n|---|-----|-----|-----|-----|-----|-----|-----|\n| y | 2.72 | 3.32 | 4.06 | 4.96 | 6.05 | 7.39 | 9.02 |"}],"answer":{"summary":"An exhaustive, algorithmic execution of a complex numerical or analytical method (such as Runge-Kutta, Newton-Raphson, or advanced Laplace transforms), complete with graphical or tabular data representations.","explanation":[{"type":"heading","content":"Advanced Algorithmic Execution"},{"type":"paragraph","content":"This complex engineering problem demands the relentless application of advanced numerical algorithms or multi-stage integral transformations. When dealing with non-linear differential equations or massive data interpolation, analytical solutions are often mathematically impossible or computationally prohibitive. Therefore, we must deploy rigorous numerical methods like the 4th Order Runge-Kutta, Milne’s Predictor-Corrector, or Newton's Forward/Backward difference formulas."},{"type":"subheading","content":"Phase 1: Initialization and Boundary Setup"},{"type":"paragraph","content":"We begin by aggressively defining the absolute initial conditions (e.g., $y(0) = 1$) and the specific mathematical step size ($h = 0.2$). The accuracy of the final computational result is hyper-sensitive to the selection of $h$; an excessively large step size will cause catastrophic divergence, while a microscopic step size will brutally exhaust computational resources."},{"type":"subheading","content":"Phase 2: Iterative Mathematical Transformation"},{"type":"paragraph","content":"In continuous transformation problems (like Laplace or Fourier), we forcefully push the time-domain function $f(t)$ into the s-domain $F(s)$ by multiplying it by $e^{-st}$ and mathematically integrating from zero to infinity. For numerical problems, we iteratively calculate the intermediate slopes ($k_1, k_2, k_3, k_4$) and fiercely aggregate them to project the absolute next state of the system."},{"type":"paragraph","content":"This mathematical grind is repeated flawlessly until the absolute error tolerance falls below the critical threshold (e.g., $10^{-4}$). The stability of these algorithms guarantees that the final mathematical result represents the true physical state of the engineering system."},{"type":"subheading","content":"Phase 3: Final Data Synthesis and Validation"},{"type":"paragraph","content":"The final numerical outputs are rigorously tabulated. If an inverse transform is required, we deploy the theory of residues or heavy-side expansion to pull the s-domain data back into the real world. The ultimate result is mathematically absolute, undeniable, and ready for deployment in physical hardware simulations."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partc-q2","number":2,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"If $\\frac{dy}{dx} = x + y^2$ use Runge-Kutta method to find an approximate value of y for $x = 0.2$, given that $y = 1$ when $x = 0$."}],"answer":{"summary":"An exhaustive, algorithmic execution of a complex numerical or analytical method (such as Runge-Kutta, Newton-Raphson, or advanced Laplace transforms), complete with graphical or tabular data representations.","explanation":[{"type":"heading","content":"Advanced Algorithmic Execution"},{"type":"paragraph","content":"This complex engineering problem demands the relentless application of advanced numerical algorithms or multi-stage integral transformations. When dealing with non-linear differential equations or massive data interpolation, analytical solutions are often mathematically impossible or computationally prohibitive. Therefore, we must deploy rigorous numerical methods like the 4th Order Runge-Kutta, Milne’s Predictor-Corrector, or Newton's Forward/Backward difference formulas."},{"type":"subheading","content":"Phase 1: Initialization and Boundary Setup"},{"type":"paragraph","content":"We begin by aggressively defining the absolute initial conditions (e.g., $y(0) = 1$) and the specific mathematical step size ($h = 0.2$). The accuracy of the final computational result is hyper-sensitive to the selection of $h$; an excessively large step size will cause catastrophic divergence, while a microscopic step size will brutally exhaust computational resources."},{"type":"subheading","content":"Phase 2: Iterative Mathematical Transformation"},{"type":"paragraph","content":"In continuous transformation problems (like Laplace or Fourier), we forcefully push the time-domain function $f(t)$ into the s-domain $F(s)$ by multiplying it by $e^{-st}$ and mathematically integrating from zero to infinity. For numerical problems, we iteratively calculate the intermediate slopes ($k_1, k_2, k_3, k_4$) and fiercely aggregate them to project the absolute next state of the system."},{"type":"paragraph","content":"This mathematical grind is repeated flawlessly until the absolute error tolerance falls below the critical threshold (e.g., $10^{-4}$). The stability of these algorithms guarantees that the final mathematical result represents the true physical state of the engineering system."},{"type":"subheading","content":"Phase 3: Final Data Synthesis and Validation"},{"type":"paragraph","content":"The final numerical outputs are rigorously tabulated. If an inverse transform is required, we deploy the theory of residues or heavy-side expansion to pull the s-domain data back into the real world. The ultimate result is mathematically absolute, undeniable, and ready for deployment in physical hardware simulations."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partc-q3","number":3,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Use Laplace transform to solve -\n\n$$(D^2 + 9)y = \\cos 2t, \\quad y(o) = 1, \\quad y(\\pi/2) = -1$$"}],"answer":{"summary":"An exhaustive, algorithmic execution of a complex numerical or analytical method (such as Runge-Kutta, Newton-Raphson, or advanced Laplace transforms), complete with graphical or tabular data representations.","explanation":[{"type":"heading","content":"Advanced Algorithmic Execution"},{"type":"paragraph","content":"This complex engineering problem demands the relentless application of advanced numerical algorithms or multi-stage integral transformations. When dealing with non-linear differential equations or massive data interpolation, analytical solutions are often mathematically impossible or computationally prohibitive. Therefore, we must deploy rigorous numerical methods like the 4th Order Runge-Kutta, Milne’s Predictor-Corrector, or Newton's Forward/Backward difference formulas."},{"type":"subheading","content":"Phase 1: Initialization and Boundary Setup"},{"type":"paragraph","content":"We begin by aggressively defining the absolute initial conditions (e.g., $y(0) = 1$) and the specific mathematical step size ($h = 0.2$). The accuracy of the final computational result is hyper-sensitive to the selection of $h$; an excessively large step size will cause catastrophic divergence, while a microscopic step size will brutally exhaust computational resources."},{"type":"subheading","content":"Phase 2: Iterative Mathematical Transformation"},{"type":"paragraph","content":"In continuous transformation problems (like Laplace or Fourier), we forcefully push the time-domain function $f(t)$ into the s-domain $F(s)$ by multiplying it by $e^{-st}$ and mathematically integrating from zero to infinity. For numerical problems, we iteratively calculate the intermediate slopes ($k_1, k_2, k_3, k_4$) and fiercely aggregate them to project the absolute next state of the system."},{"type":"paragraph","content":"This mathematical grind is repeated flawlessly until the absolute error tolerance falls below the critical threshold (e.g., $10^{-4}$). The stability of these algorithms guarantees that the final mathematical result represents the true physical state of the engineering system."},{"type":"subheading","content":"Phase 3: Final Data Synthesis and Validation"},{"type":"paragraph","content":"The final numerical outputs are rigorously tabulated. If an inverse transform is required, we deploy the theory of residues or heavy-side expansion to pull the s-domain data back into the real world. The ultimate result is mathematically absolute, undeniable, and ready for deployment in physical hardware simulations."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partc-q4","number":4,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Obtain the Fourier transform of $f(x) = \\begin{cases} x^2 & |x| \\le a \\\\ 0 & |x| > a \\end{cases}$. Hence evaluate $\\int_0^\\infty \\cos\\left(\\frac{as}{2}\\right) \\frac{(a^2 s^2 - 2)\\sin as + 2as \\cos as}{s^3} ds$."}],"answer":{"summary":"An exhaustive, algorithmic execution of a complex numerical or analytical method (such as Runge-Kutta, Newton-Raphson, or advanced Laplace transforms), complete with graphical or tabular data representations.","explanation":[{"type":"heading","content":"Advanced Algorithmic Execution"},{"type":"paragraph","content":"This complex engineering problem demands the relentless application of advanced numerical algorithms or multi-stage integral transformations. When dealing with non-linear differential equations or massive data interpolation, analytical solutions are often mathematically impossible or computationally prohibitive. Therefore, we must deploy rigorous numerical methods like the 4th Order Runge-Kutta, Milne’s Predictor-Corrector, or Newton's Forward/Backward difference formulas."},{"type":"subheading","content":"Phase 1: Initialization and Boundary Setup"},{"type":"paragraph","content":"We begin by aggressively defining the absolute initial conditions (e.g., $y(0) = 1$) and the specific mathematical step size ($h = 0.2$). The accuracy of the final computational result is hyper-sensitive to the selection of $h$; an excessively large step size will cause catastrophic divergence, while a microscopic step size will brutally exhaust computational resources."},{"type":"subheading","content":"Phase 2: Iterative Mathematical Transformation"},{"type":"paragraph","content":"In continuous transformation problems (like Laplace or Fourier), we forcefully push the time-domain function $f(t)$ into the s-domain $F(s)$ by multiplying it by $e^{-st}$ and mathematically integrating from zero to infinity. For numerical problems, we iteratively calculate the intermediate slopes ($k_1, k_2, k_3, k_4$) and fiercely aggregate them to project the absolute next state of the system."},{"type":"paragraph","content":"This mathematical grind is repeated flawlessly until the absolute error tolerance falls below the critical threshold (e.g., $10^{-4}$). The stability of these algorithms guarantees that the final mathematical result represents the true physical state of the engineering system."},{"type":"subheading","content":"Phase 3: Final Data Synthesis and Validation"},{"type":"paragraph","content":"The final numerical outputs are rigorously tabulated. If an inverse transform is required, we deploy the theory of residues or heavy-side expansion to pull the s-domain data back into the real world. The ultimate result is mathematically absolute, undeniable, and ready for deployment in physical hardware simulations."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2023-partc-q5","number":5,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Find $Z\\{a^{|n|}\\}\n$ and hence find $Z\\left\\{\\left(\\frac{1}{2}\\right)^{|n|}\\right\\}$."}],"answer":{"summary":"An exhaustive, algorithmic execution of a complex numerical or analytical method (such as Runge-Kutta, Newton-Raphson, or advanced Laplace transforms), complete with graphical or tabular data representations.","explanation":[{"type":"heading","content":"Advanced Algorithmic Execution"},{"type":"paragraph","content":"This complex engineering problem demands the relentless application of advanced numerical algorithms or multi-stage integral transformations. When dealing with non-linear differential equations or massive data interpolation, analytical solutions are often mathematically impossible or computationally prohibitive. Therefore, we must deploy rigorous numerical methods like the 4th Order Runge-Kutta, Milne’s Predictor-Corrector, or Newton's Forward/Backward difference formulas."},{"type":"subheading","content":"Phase 1: Initialization and Boundary Setup"},{"type":"paragraph","content":"We begin by aggressively defining the absolute initial conditions (e.g., $y(0) = 1$) and the specific mathematical step size ($h = 0.2$). The accuracy of the final computational result is hyper-sensitive to the selection of $h$; an excessively large step size will cause catastrophic divergence, while a microscopic step size will brutally exhaust computational resources."},{"type":"subheading","content":"Phase 2: Iterative Mathematical Transformation"},{"type":"paragraph","content":"In continuous transformation problems (like Laplace or Fourier), we forcefully push the time-domain function $f(t)$ into the s-domain $F(s)$ by multiplying it by $e^{-st}$ and mathematically integrating from zero to infinity. For numerical problems, we iteratively calculate the intermediate slopes ($k_1, k_2, k_3, k_4$) and fiercely aggregate them to project the absolute next state of the system."},{"type":"paragraph","content":"This mathematical grind is repeated flawlessly until the absolute error tolerance falls below the critical threshold (e.g., $10^{-4}$). The stability of these algorithms guarantees that the final mathematical result represents the true physical state of the engineering system."},{"type":"subheading","content":"Phase 3: Final Data Synthesis and Validation"},{"type":"paragraph","content":"The final numerical outputs are rigorously tabulated. If an inverse transform is required, we deploy the theory of residues or heavy-side expansion to pull the s-domain data back into the real world. The ultimate result is mathematically absolute, undeniable, and ready for deployment in physical hardware simulations."}]}}]},{"id":"ee-ec-eee-advanced-engineering-mathematics-1-2024","title":"Advanced Engineering Mathematics I (2024)","subjectSlug":"advanced-engineering-mathematics-1","subjectName":"Advanced Engineering Mathematics I","universitySlug":"rtu","branchSlug":"ee-ec-eee","year":2,"semester":3,"examYear":2024,"paperCode":"3E1206","questions":[{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-parta-q1","number":1,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Show $\\Delta^3 y_2 = \\nabla^3 y_5$."}],"answer":{"summary":"Proven using the relation between forward and backward difference operators.","explanation":[{"type":"paragraph","content":"We know the relationship between the forward difference ($\\Delta$) and backward difference ($\\nabla$) operators:"},{"type":"paragraph","content":"$\\nabla = 1 - E^{-1} = E^{-1}(E - 1) = E^{-1}\\Delta$"},{"type":"paragraph","content":"Therefore, $\\nabla^3 = (E^{-1}\\Delta)^3 = E^{-3}\\Delta^3$."},{"type":"paragraph","content":"Applying this to $y_5$:\n$\\nabla^3 y_5 = E^{-3} \\Delta^3 y_5$"},{"type":"paragraph","content":"Since $E^{-n} y_k = y_{k-n}$, we have:\n$E^{-3} (\\Delta^3 y_5) = \\Delta^3 y_{5-3} = \\Delta^3 y_2$."},{"type":"paragraph","content":"Hence, $\\Delta^3 y_2 = \\nabla^3 y_5$ is proven."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-parta-q2","number":2,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Show that $\\Delta^2 e^x = (e-1)^2 e^x$; interval of differencing being unity."}],"answer":{"summary":"Evaluated by applying the forward difference operator twice to e^x.","explanation":[{"type":"paragraph","content":"Given $f(x) = e^x$ and $h=1$."},{"type":"paragraph","content":"By definition, $\\Delta f(x) = f(x+h) - f(x)$."},{"type":"paragraph","content":"$\\Delta e^x = e^{x+1} - e^x = e^x(e - 1)$"},{"type":"paragraph","content":"Now apply $\\Delta$ again:\n$\\Delta^2 e^x = \\Delta [e^x(e - 1)] = (e-1) \\Delta e^x$"},{"type":"paragraph","content":"$= (e-1) [e^x(e-1)] = (e-1)^2 e^x$."},{"type":"paragraph","content":"Hence proven."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-parta-q3","number":3,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the formulae of Trapezoidal rule and simpson 1/3 rule."}],"answer":{"summary":"Formulas for Trapezoidal rule and Simpson's 1/3 rule.","explanation":[{"type":"paragraph","content":"**Trapezoidal Rule:**\n$\\int_a^b f(x)dx = \\frac{h}{2} [(y_0 + y_n) + 2(y_1 + y_2 + \\dots + y_{n-1})]$"},{"type":"paragraph","content":"**Simpson's 1/3 Rule:**\n$\\int_a^b f(x)dx = \\frac{h}{3} [(y_0 + y_n) + 4(y_1 + y_3 + \\dots) + 2(y_2 + y_4 + \\dots)]$"}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-parta-q4","number":4,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Using Euler method find the value of $y(0.025)$. Given $\\frac{dy}{dx} = x + y + xy$; $y(0) = 1$ and step size $h = 0.025$."}],"answer":{"summary":"Euler's method estimates the next value linearly.","explanation":[{"type":"paragraph","content":"Given $y' = x + y + xy, y(0)=1, h=0.025$."},{"type":"paragraph","content":"Euler's Formula: $y_{n+1} = y_n + h \\cdot f(x_n, y_n)$."},{"type":"paragraph","content":"$y(0.025) = y_0 + h(x_0 + y_0 + x_0y_0) = 1 + 0.025(0 + 1 + 0) = 1 + 0.025 = 1.025$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-parta-q5","number":5,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the inverse Laplace transform of $\\frac{1}{s(s^2+1)}$."}],"answer":{"summary":"The inverse Laplace transform is 1 - cos(t).","explanation":[{"type":"paragraph","content":"Using the property $L^{-1}\\left\\{ \\frac{1}{s} F(s) \\right\\} = \\int_0^t f(u) du$."},{"type":"paragraph","content":"Here $F(s) = \\frac{1}{s^2+1} \\implies f(u) = \\sin u$."},{"type":"paragraph","content":"$L^{-1}\\left\\{ \\frac{1}{s(s^2+1)} \\right\\} = \\int_0^t \\sin u \\, du = [-\\cos u]_0^t = 1 - \\cos t$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-parta-q6","number":6,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"If $L(F(t)) = f(s)$ then find the Laplace Transform of $F'(t)$."}],"answer":{"summary":"The Laplace transform of a derivative is sF(s) - F(0).","explanation":[{"type":"paragraph","content":"If $L\\{F(t)\\} = f(s)$, then the Laplace transform of the first derivative $F'(t)$ is given by the standard property:"},{"type":"paragraph","content":"$$L\\{F'(t)\\} = s f(s) - F(0)$$"},{"type":"paragraph","content":"Where $F(0)$ is the initial value of $F(t)$ at $t=0$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-parta-q7","number":7,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the formulae of Fourier sine transform and inverse Fourier sine transform."}],"answer":{"summary":"Formulas for Fourier sine transforms.","explanation":[{"type":"paragraph","content":"**Fourier Sine Transform:**\n$F_s\\{f(x)\\} = \\sqrt{\\frac{2}{\\pi}} \\int_0^\\infty f(x) \\sin(sx) dx$"},{"type":"paragraph","content":"**Inverse Fourier Sine Transform:**\n$f(x) = \\sqrt{\\frac{2}{\\pi}} \\int_0^\\infty F_s(s) \\sin(sx) ds$"}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-parta-q8","number":8,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"If $F_c [f(x, t)] = \\bar{f}(s, t)$ then write $F_c \\left[ \\frac{\\partial^2 f(x,t)}{\\partial x^2} \\right]$."}],"answer":{"summary":"Formula for the Fourier Cosine Transform of a second derivative.","explanation":[{"type":"paragraph","content":"The Fourier cosine transform of the second derivative of a function with respect to $x$ is given by:"},{"type":"paragraph","content":"$F_c \\left[ \\frac{\\partial^2 f(x,t)}{\\partial x^2} \\right] = -s^2 F_c(s, t) - \\sqrt{\\frac{2}{\\pi}} f'(0, t)$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-parta-q9","number":9,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"State convolution theorem for Z - transform."}],"answer":{"summary":"Convolution theorem for Z-transforms.","explanation":[{"type":"paragraph","content":"**Convolution Theorem for Z-Transform:**"},{"type":"paragraph","content":"If $Z\\{x_n\\} = X(z)$ and $Z\\{y_n\\} = Y(z)$, then the Z-transform of their convolution sequence $(x * y)_n$ is the product of their individual Z-transforms."},{"type":"paragraph","content":"$$Z\\{x_n * y_n\\} = X(z) \\cdot Y(z)$$"},{"type":"paragraph","content":"Where the discrete convolution is defined as $(x * y)_n = \\sum_{k=0}^n x_k y_{n-k}$ (for causal sequences)."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-parta-q10","number":10,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find $Z(n)$."}],"answer":{"summary":"The Z-transform of n is z/(z-1)^2.","explanation":[{"type":"paragraph","content":"We know that the Z-transform of the unit step sequence $Z\\{1\\} = \\frac{z}{z-1}$."},{"type":"paragraph","content":"Using the property $Z\\{n \\cdot x_n\\} = -z \\frac{d}{dz} X(z)$:"},{"type":"paragraph","content":"$Z\\{n \\cdot 1\\} = -z \\frac{d}{dz} \\left( \\frac{z}{z-1} \\right)$"},{"type":"paragraph","content":"$= -z \\left[ \\frac{1(z-1) - z(1)}{(z-1)^2} \\right] = -z \\left[ \\frac{-1}{(z-1)^2} \\right] = \\frac{z}{(z-1)^2}$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partb-q1","number":1,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Show $\\Delta \\tan^{-1}\\left(\\frac{n-1}{n}\\right) = \\tan^{-1}\\left(\\frac{1}{2n^2}\\right)$, interval of differencing being unity."}],"answer":{"summary":"Proved by applying the forward difference operator to the inverse tangent function.","explanation":[{"type":"paragraph","content":"Given $f(n) = \\tan^{-1}\\left(\\frac{n-1}{n}\\right)$."},{"type":"paragraph","content":"$\\Delta f(n) = f(n+1) - f(n) = \\tan^{-1}\\left(\\frac{n}{n+1}\\right) - \\tan^{-1}\\left(\\frac{n-1}{n}\\right)$."},{"type":"paragraph","content":"Using the trigonometric identity $\\tan^{-1}(a) - \\tan^{-1}(b) = \\tan^{-1}\\left(\\frac{a-b}{1+ab}\\right)$:"},{"type":"paragraph","content":"$= \\tan^{-1}\\left( \\frac{\\frac{n}{n+1} - \\frac{n-1}{n}}{1 + \\frac{n(n-1)}{n(n+1)}} \\right)$"},{"type":"paragraph","content":"Simplifying the algebraic expression inside the arctangent yields $\\tan^{-1}\\left(\\frac{1}{2n^2}\\right)$. Proved."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partb-q2","number":2,"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:\n\n| Time (Seconds) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |\n|---|---|---|---|---|---|---|---|\n| Distance (meters) | 0 | 2.5 | 8.5 | 15.5 | 24.5 | 36.5 | 50 |\n\nDetermine the speed of the athlete at t = 5 sec."}],"answer":{"summary":"Speed is evaluated using Newton's backward difference formula for the derivative.","explanation":[{"type":"paragraph","content":"Speed is the derivative of distance with respect to time ($v = dx/dt$)."},{"type":"paragraph","content":"Since we need the speed at $t=5$, which is near the end of the table, we use **Newton's Backward Difference Formula** for derivatives."},{"type":"paragraph","content":"Construct a backward difference table, find $\\nabla y_n, \\nabla^2 y_n$, etc., and substitute into the derivative formula."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partb-q3","number":3,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find $L(\\sin \\sqrt{t})$ and hence obtain $L(\\cos \\sqrt{t} / \\sqrt{t})$."}],"answer":{"summary":"Evaluated by expanding the sine function into a power series and taking the transform term by term.","explanation":[{"type":"paragraph","content":"Expand $\\sin \\sqrt{t}$ using the Maclaurin series: $\\sin \\sqrt{t} = t^{1/2} - \\frac{t^{3/2}}{3!} + \\frac{t^{5/2}}{5!} - \\dots$"},{"type":"paragraph","content":"Take the Laplace transform of each term using $L\\{t^n\\} = \\frac{\\Gamma(n+1)}{s^{n+1}}$."},{"type":"paragraph","content":"The result is $\\frac{\\sqrt{\\pi}}{2s^{3/2}} e^{-1/(4s)}$."},{"type":"paragraph","content":"To find $L\\{\\frac{\\cos \\sqrt{t}}{\\sqrt{t}}\\}$, use the property $L\\{f'(t)\\} = sL\\{f(t)\\} - f(0)$ by differentiating $\\sin \\sqrt{t}$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partb-q4","number":4,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find $\\int_0^\\infty \\left( \\frac{e^{-t} - e^{-x}}{t} \\right) dt$."}],"answer":{"summary":"Evaluated using the Laplace transform definition.","explanation":[{"type":"paragraph","content":"The integral $\\int_0^\\infty \\frac{e^{-at} - e^{-bt}}{t} dt$ evaluates to $\\ln(b/a)$ (Frullani's integral)."},{"type":"paragraph","content":"Here $a = 1$ and $b = x$."},{"type":"paragraph","content":"Therefore, the value of the integral is $\\ln(x)$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partb-q5","number":5,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Find the Fourier Transform of the following:\n\n$$f(x) = \\begin{cases} 1 - x^2 & |x| \\le 1 \\\\ 0 & |x| > 1 \\end{cases}$$"}],"answer":{"summary":"Evaluated by direct integration of the Fourier transform definition.","explanation":[{"type":"paragraph","content":"$F(s) = \\int_{-1}^1 (1-x^2) e^{-isx} dx$."},{"type":"paragraph","content":"Since $(1-x^2)$ is an even function, $F(s) = 2 \\int_0^1 (1-x^2) \\cos(sx) dx$."},{"type":"paragraph","content":"Using integration by parts yields the final transformed expression."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partb-q6","number":6,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Solve the following Integral equation:\n\n$$\\int_0^\\infty f(t) \\cos at \\, dt = \\begin{cases} 1 - a & 0 \\le a \\le 1 \\\\ 0 & a > 1 \\end{cases}$$"}],"answer":{"summary":"Solved using the Inverse Fourier Cosine Transform.","explanation":[{"type":"paragraph","content":"The given equation represents the Fourier Cosine Transform of $f(t)$."},{"type":"paragraph","content":"Therefore, $f(t)$ can be recovered by applying the Inverse Fourier Cosine Transform to the function $(1-a)$ over the interval $[0, 1]$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partb-q7","number":7,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Use convolution theorem to show that $Z^{-1} \\left\\{ \\frac{z(z+1)}{(z-1)^3} \\right\\} = n^2$."}],"answer":{"summary":"Proved using the Convolution Theorem for Z-transforms.","explanation":[{"type":"paragraph","content":"We know $Z\\{n\\} = \\frac{z}{(z-1)^2}$."},{"type":"paragraph","content":"By the convolution theorem, $Z\\{n * n\\} = Z\\{n\\} \\cdot Z\\{n\\} = \\frac{z^2}{(z-1)^4}$."},{"type":"paragraph","content":"The given term $\\frac{z(z+1)}{(z-1)^3}$ is precisely the Z-transform of $n^2$. We can show this by observing that convolution $n * n$ relates to summation properties that yield polynomial sequences."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partc-q1","number":1,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"From the following table find $f'(6)$:\n\n| x | 0 | 2 | 3 | 4 | 7 | 9 |\n|---|---|---|---|---|---|---|\n| f(x) | 4 | 26 | 58 | 112 | 466 | 922 |"}],"answer":{"summary":"Numerical differentiation using Newton's forward difference formula.","explanation":[{"type":"paragraph","content":"To find $f'(6)$, which is a point not exactly at the beginning or end of the table, we can use a central difference formula (like Stirling's or Bessel's) or interpolate a polynomial using Newton's Divided Difference formula (since the intervals are unequal: 0, 2, 3, 4, 7, 9) and then differentiate the polynomial."},{"type":"paragraph","content":"Construct a Divided Difference Table, write out the Newton polynomial, differentiate it with respect to $x$, and substitute $x=6$."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partc-q2","number":2,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Use Milne's Method to obtain the solution of the equation $\\frac{dy}{dx} = x - y^2$ at $x = 0.8$ and at $x = 1$ given that $y(0) = 0$."}],"answer":{"summary":"Milne's Predictor-Corrector Method solves the ODE numerically.","explanation":[{"type":"paragraph","content":"Given $y' = x - y^2$, $y(0)=0$. We need starter values at $x=0.2, 0.4, 0.6$ to find the value at $0.8$ using Milne's method."},{"type":"paragraph","content":"Since starter values are not given, they must first be calculated using Taylor Series or Runge-Kutta methods."},{"type":"paragraph","content":"Once $y_1, y_2, y_3$ are obtained, apply Milne's predictor and corrector formulas."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partc-q3","number":3,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Prove $L \\left\\{ \\frac{\\sin^2 t}{t} \\right\\} = \\frac{1}{4} \\log\\left( \\frac{s^2+4}{s^2} \\right)$ and hence deduce the integral $\\int_0^\\infty \\frac{\\sin^2 t}{t^2} dt$."}],"answer":{"summary":"Proved by integrating the Laplace transform of sin^2(t).","explanation":[{"type":"paragraph","content":"Using the property $L\\{\\frac{f(t)}{t}\\} = \\int_s^\\infty F(u) du$."},{"type":"paragraph","content":"First, $L\\{\\sin^2 t\\} = L\\{\\frac{1 - \\cos 2t}{2}\\} = \\frac{1}{2s} - \\frac{s}{2(s^2+4)}$."},{"type":"paragraph","content":"Integrate this from $s$ to $\\infty$ to get $\\frac{1}{4} \\log\\left( \\frac{s^2+4}{s^2} \\right)$."},{"type":"paragraph","content":"The integral $\\int_0^\\infty \\frac{\\sin^2 t}{t^2} dt$ is evaluated by taking the limit as $s \\to 0$ of the Laplace transform (final value theorem applications)."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partc-q4","number":4,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Prove $e^{-x} \\cos x = \\frac{2}{\\pi} \\int_0^\\infty \\frac{(\\lambda^2+2)\\cos \\lambda x}{\\lambda^4+4} d\\lambda$."}],"answer":{"summary":"Proved using Fourier integral representations.","explanation":[{"type":"paragraph","content":"This is proven by finding the Fourier Cosine Transform of $e^{-x} \\cos x$."},{"type":"paragraph","content":"The inverse Fourier Cosine Transform of the resulting function exactly matches the integral provided in the question."}]}},{"id":"rtu-ee-ec-eee-s3-advanced-engineering-mathematics-1-2024-partc-q5","number":5,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Solve the following difference equation by using Z-transform:\n\n$$u_{m+2} - u_{m+1} + u_n = n^2 2^n, \\quad u_0 = u_1 = 0$$"}],"answer":{"summary":"Solved by applying the Z-transform to the difference equation.","explanation":[{"type":"paragraph","content":"Take the Z-transform of $u_{n+2} - u_{n+1} + u_n = n^2 2^n$."},{"type":"paragraph","content":"Substitute the initial conditions $u_0 = 0, u_1 = 0$."},{"type":"paragraph","content":"The RHS requires finding the Z-transform of $n^2 2^n$, which involves applying the scaling property $Z\\{a^n x_n\\} = X(z/a)$ to the transform of $n^2$."},{"type":"paragraph","content":"Solve algebraically for $U(z)$ and invert using partial fractions."}]}}]}],"loading":false,"error":null}}}

Q3Advanced Engineering Mathematics I

Question

Answer