# GATE ME 2014 SET-4

 Question 1
Which one of the following equations is a correct identity for arbitrary 3x3 real matrices P, Q and R?
 A $P(Q+R)=PQ+RP$ B $(P-Q)^{2}=P^{2}-2PQ+Q^{2}$ C $det(P+Q)=det \; P+det \; Q$ D $(P+Q)^{2}=P^{2}+PQ+QP+Q^{2}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
\begin{aligned} (P+Q)^{2} &=P^{2}+P Q+Q P+Q^{2} \\ &=P \cdot P+P \cdot Q+Q \cdot P+Q \cdot Q \\ &=P^{2}+P Q+Q P+Q^{2} \end{aligned}
 Question 2
The value of the integral $\int_{0}^{2}\frac{(x-1)^{2}\sin(x-1)}{(x-1)^{2}+\cos(x-1)}dx$ is
 A 3 B 0 C -1 D -2
Engineering Mathematics   Calculus
Question 2 Explanation:
\begin{aligned} I &= \int_{0}^{2}\frac{(x-1)^2 \sin (x-1)}{(x-1)^2 + \cos (x-1)}dx\\ \text{Taking}\; x-1=z&\Rightarrow dx=dz \\ \text{for}\; x=0,&z\rightarrow -1\; \text{and}\; x=2, z \rightarrow 1 \\ \therefore \; I &=\int_{-1}^{1} \frac{z^2 \sin z}{z^2+\cos z}dz\\ \text{let}\;\; f(z)&=\frac{z^2 \sin z}{z^2+ \cos z} dz\\ f(-z) &= \frac{z^2 \sin z}{z^2+ \cos z}\\ f(z)&= -f(z) \; \text{function is ODD}\\ \therefore \;\; I&=0 \end{aligned}

 Question 3
The solution of the initial value problem $\frac{\mathrm{d} y}{\mathrm{d} x}= -2xy;$ y(0)=2 is
 A $1+e^{-x^{2}}$ B $2e^{-x^{2}}$ C $1+e^{x^{2}}$ D $2e^{x^{2}}$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
$\frac{d y}{d x}=2 x y=0 \qquad ...(1)$
$I F.=e^{\int 2 x d x}=e^{x^{2}}$
Multiplying I.F. to both side of equation (1)
$e^{x^{2}}\left[\frac{d y}{d x}+2 x y\right]=0$
$\Rightarrow \frac{d}{d x}\left(e^{x^{2}} y\right)=0$
$e^{x^{2}} y=c$
from the given boundary condition, C=2
$\therefore e^{x^{2}} y=2$
$y=2 e^{-x^{2}}$
 Question 4
A nationalized bank has found that the daily balance available in its savings accounts follows a normal distribution with a mean of Rs. 500 and a standard deviation of Rs. 50. The percentage of savings account holders, who maintain an average daily balance more than Rs. 500 is _______
 A 12% B 65% C 98% D 50%
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
49 to 51
$Z=\frac{x-\mu}{\sigma}$
$\text { Given, } x=500, \mu=500, \sigma=50$
$\therefore \quad z=\frac{500-500}{50}=0$
$\therefore P(0)=50 \%$
 Question 5
Laplace transform of $cos(\omega t)$ is $\frac{s}{s^{2}+\omega ^{2}}$. The Laplace transform of $e^{-2t}cos(4t)$ is
 A $\frac{s-2}{(s-2)^{2}+16}$ B $\frac{s+2}{(s-2)^{2}+16}$ C $\frac{s-2}{(s+2)^{2}+16}$ D $\frac{s+2}{(s+2)^{2}+16}$
Engineering Mathematics   Differential Equations
Question 5 Explanation:
Given $L \cos (\omega t)=\frac{s}{s^{2}+\omega^{2}}$
$\Rightarrow L\left(e^{-2 t} \cos 4 t\right)=?$
By the given formula
$\Rightarrow \quad L(\cos 4 t)=\frac{s}{s^{2}+16}$
$\Rightarrow L\left(e^{2 t} \cos 4 t\right)=\frac{s+2}{(s+2)^{2}+16}$
( $\because$ By using first shifting property of Laplace)
Hence option should be (D).

There are 5 questions to complete.