# 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).
 Question 6
In a statically determinate plane truss, the number of joints (j) and the number of members (m) are related by
 A $j=2m-3$ B $m=2j+1$ C $m=2j-3$ D $m=2j-1$
Theory of Machine   Planar Mechanisms
Question 6 Explanation:
A simple truss is formed by enlarging the basic truss element which contains three members and three joints, by adding two additional members for each additional joint, so the total number of members m in a simple truss is given by
$m=3+2(j-3)=2 j-3$
where m= number of members
j= total number of joints (including those attached to the supports)
 Question 7
If the Poisson's ratio of an elastic material is 0.4, the ratio of modulus of rigidity to Young's modulus is _______
 A 0.357 B 0.123 C 0.2658 D 1.354
Strength of Materials   Stress-strain Relationship and Elastic Constants
Question 7 Explanation:
$(0.35 \text { to } 0.36)$
$E=2 G(1+v)$
$\frac{G}{E}=\frac{1}{2(1+v)}=\frac{1}{2 \times 1.4}=\frac{1}{2.8}=0.357$
 Question 8
Which one of the following is used to convert a rotational motion into a translational motion?
 A Bevel gears B Double helical gears C Worm gears D Rack and pinion gears
Machine Design   Gears
Question 8 Explanation:
Bevel gears: Rotational motion transfer between axes at right angle.
Worm gears: For large reduction ratio in a single stage.
Double helical gears: Rotational motion transfer between parallel axes.
Rack and Pinion gears: Rotational to linear motion conversion.
 Question 9
The number of independent elastic constants required to define the stress-strain relationship for an isotropic elastic solid is _______
 A 0.5 B 1 C 1.5 D 2
Strength of Materials   Stress-strain Relationship and Elastic Constants
Question 9 Explanation:
(1.9 to 2.1)
Either E or G, 2 independent constant
Either G or K, 2 independent constant
Either E or K, 2 independent constant
 Question 10
A point mass is executing simple harmonic motion with an amplitude of 10 mm and frequency of 4 Hz. The maximum acceleration (m/$s^2$) of the mass is _______
 A 6.78 B 9.54 C 6.31 D 3.98
Theory of Machine   Vibration
Question 10 Explanation:
\begin{aligned} f_{n} &=\frac{\omega_{n}}{2 \times \pi} \\ \Rightarrow \omega_{n} &=2 \pi f=2 \times 3.14 \times 4=25.12 \mathrm{rad} / \mathrm{s} \\ a_{\max } &=\omega_{n}^{2} x=(25.12)^{2} \times 10 \times 10^{-3} \\ &=6.31 \mathrm{m} / \mathrm{s}^{2} \end{aligned}
There are 10 questions to complete.