Advanced Engineering Mathematics I

Evaluate, ${\Delta }^6(ax-1)(b{x}^2-1)(c{x}^3-1)$.

Prove that, $\left(\frac{{\Delta }^2}{E}\right)x^3=6x$ (if $h=1$).

Using Newton-Raphson's method, find the root of $x^4-12x+7=0$ which is near to $x=2$.

Find the z-transform of unit impulse function which is given by ${\delta }_n=\{\begin{array}{ll}1& \text{if }n=0\\ 0& \text{if }n\ne 0\end{array}$.

Find inverse Z Transform of $\frac{5z}{(2-z)(3z-1)}$.

Find the Laplace transform of $f(t)=\{\begin{array}{ll}\sin t& 0<t<\pi \\ 0& t>\pi \end{array}$.

Find inverse Laplace transform of $\frac{s+2}{(s-2{)}^3}$.

Write the Formulae of Fourier complex transform Fourier cosine transform and their inverse also.

Write the formulae of Simpson 1/3 rule and Simpson 3/8 rule.

By using Picard's method, solve the equation $\frac{dy}{dx}=y-x$ with $x=0$, $y=2$ upto third order of approximation.

From the following table find the number of students who obtained: a) Less than 45 marks. b) More than 45 marks. | Marks obtained | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | |---|---|---|---|---|---| | No's of students | 31 | 42 | 51 | 35 | 31 |

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.

Find the Fourier Sine and Cosine transform of $f(x)=\{\begin{array}{ll}x& \text{for }0<x\le 1\\ 2-x& \text{for }1<x<2\\ 0& \text{for }x\ge 2\end{array}$.

If $\bar{u}(z)=\frac{2z^2+5z+14}{(z-1{)}^4}$ for the sequence $\{u_n\}$, $n\ge 0$. Evaluate $u_2$ and $u_3$.

Find Inverse Laplace transform of $\frac{S}{S^4+4a^4}$.

Solve $(D^2+9)y=\cos 2t$, given that $y(0)=1$, $y(\pi /2)=-1$.

Obtain Fourier transform of $f(x)=\{\begin{array}{ll}{x}^2& \text{for }|x|\le a\\ 0& \text{for }|x|>a\end{array}$. Hence evaluate ${\int }_0^{\infty }\cos \left(\frac{as}{2}\right)\left[\frac{(a^2{s}^2-2)\sin as+2as\cos as}{s^3}\right]ds$.

Solve by z transform of $u_{n+2}-6u_{n+1}+8u_n=2^n+6n$.

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: | x | 0 | 0.1 | 0.2 | 0.3 | |---|---|-----|-----|-----| | y | 2 | 2.01 | 2.04 | 2.09 |

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): | $t\Rightarrow$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | |---|---|---|---|---|---|---|---| | $x\Rightarrow$ | 30.28 | 31.43 | 32.98 | 33.54 | 33.97 | 33.48 | 32.13 | Evaluate: i) Velocity for $t=0.1,0.5$ and $0.3$ ii) Acceleration for $t=.02,.33$ and $.58$