# GATE ME 2014 SET-2

 Question 1
One of the eigenvectors of the matrix $\begin{bmatrix} -5 & 2\\ -9 & 6 \end{bmatrix}$ is
 A $\begin{Bmatrix} -1\\ 1 \end{Bmatrix}$ B $\begin{Bmatrix} -2\\ 9 \end{Bmatrix}$ C $\begin{Bmatrix} 2\\ -1 \end{Bmatrix}$ D $\begin{Bmatrix} 1\\ 1 \end{Bmatrix}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
The characteristic equation $| A-\lambda D=0$
$\text{i.e.} \left|\begin{array}{cc}-5-\lambda & 2 \\ -9 & 6-\lambda\end{array}\right|=0$
$\text{or } (\lambda-6)(\lambda+5)+18=0$
$\text{or } \quad \lambda^{2}-6 \lambda+5 \lambda-30+18=0$
$\text{or } \quad \lambda^{2}-\lambda-12=0$
$\text{or } \lambda=\frac{1 \pm \sqrt{1+48}}{2}=\frac{1 \pm 7}{2}=4,-3$
Corresponding to $\lambda=4,$ we have
$\; [A-\lambda I] x=\left[\begin{array}{cc}-5-\lambda & 2 \\-9 & 6-\lambda\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]$
$=0$
$\text{Or, } \left[\begin{array}{ll}-9 & 2 \\ -9 & 2\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=0$which gives only one independent equation,
$-9 x+2 y=0$
$\therefore \frac{x}{2}=\frac{y}{9}$ gives eigen vector (2,9)
Corresponding to $\lambda=-3,$
$=\left[\begin{array}{ll}-2 & 2 \\-9 & 9\end{array}\right]\left[\begin{array}{l}x \\y \end{array}\right]=0$
which gives -x+y=0 (only one independent equation)
$\therefore \quad \frac{x}{1}=\frac{y}{1} \quad$ which gives (1,1)
So, the eigen vectors are $\left\{\begin{array}{l}2 \\ 9\end{array}\right\} and \left\{\begin{array}{l}1 \\ 1\end{array}\right\}$
 Question 2
$\lim_{x\rightarrow 0}\left ( \frac{e^{2x}-1}{\sin(4x)} \right )$ is equal to
 A 0 B 0.5 C 1 D 2
Engineering Mathematics   Calculus
Question 2 Explanation:
$\lim_{x \rightarrow 0} \frac{\left(e^{2 x}-1\right)}{\sin 4 x},$ it is of$\left(\frac{0}{0}\right)$from
Applying L' Hospital's rule
$\lim_{x \rightarrow 0} \frac{2 e^{2 x}}{4 \cos 4 x}=\frac{2 \times 1}{4 \times 1}=\frac{1}{2}$
 Question 3
Curl of vector $\vec{F}=x^{2}z^{2}\hat{i}-2xy^{2}z\hat{j}+2y^{2}z^{3}\hat{k}$ is
 A $(4yz^{3}+2xy^{2})\hat{i}+2x^{2}z\hat{j}-2y^{2}z\hat{k}$ B $(4yz^{3}+2xy^{2})\hat{i}-2x^{2}z\hat{j}-2y^{2}z\hat{k}$ C $2xz^{2}\hat{i}-4xyz\hat{j}+6y^{2}z^{2}\hat{k}$ D $2xz^{2}\hat{i}+4xyz\hat{j}+6y^{2}z^{2}\hat{k}$
Engineering Mathematics   Calculus
Question 3 Explanation:
$\vec{F}=x^{2} z^{2} \hat{i}-2 x y^{2} z \hat{j}+2 y^{2} z^{3} \hat{k}$
$\nabla \times \vec{F}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x^{2} z^{2}&-2 x y^{2} z^{2} & 2 y^{2} z^{3} \end{array}\right|$
$=\hat{i}\left[\frac{\partial}{\partial y}\left(2 y^{2} z^{3}\right)+\frac{\partial}{\partial z}\left(2 x y^{2} z\right)\right]$
$-\hat{j}\left[\frac{\partial}{\partial x}\left(2 y^{2} z^{3}\right)-\frac{\partial}{\partial z}\left(x^{2} z^{2}\right)\right]$
$+\hat{k}\left[\frac{\partial}{\partial x}\left(-2 x y^{2} z\right)-\frac{\partial}{\partial y}\left(x^{2} z^{2}\right)\right]$
$\nabla \times \vec{F}=\hat{i}\left[4 y z^{3}+2 x y^{2}\right]-\hat{j}\left[-2 z x^{2}\right]$
$=\left(4 y z^{3}+2 x y^{2}\right) \hat{i}+\left(2 x^{2} z\right) \hat{j}-\left(2 y^{2} z\right) \hat{k}$
 Question 4
A box contains 25 parts of which 10 are defective. Two parts are being drawn simultaneously in a random manner from the box. The probability of both the parts being good is
 A $7/20$ B $42/125$ C $25/29$ D $5/9$
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
required prob $=\frac{^{15} C_{2}}{^{25} C_{2}}=\frac{14 \times 15}{25 \times 24}=\frac{7}{20}$
 Question 5
The best approximation of the minimum value attained by $e^{-x}sin(100x)$ for $x\geq 0$ is _______
 A -0.9844 B -1.2652 C -9.2658 D -8.2654
Engineering Mathematics   Calculus
Question 5 Explanation:
$f(x)=e^{-x} \sin (100 x)$
$P(x)=-e^{-x} \sin (100 x)+e^{-x} \cos (100 x) \times 100$
for minima $f^{\prime}(x)=0$
from here $\tan (100 x)=100$
\begin{aligned} x &=\frac{1}{100} \tan ^{-1}(100)=0.0156 \\ f(x) &=e^{-0.0156} \sin (100 \times 0.0156)=-0.9844 \end{aligned}
 Question 6
A steel cube, with all faces free to deform, has Young's modulus, E, Poisson's ratio, v, and coefficient of thermal expansion, $\alpha$. The pressure (hydrostatic stress) developed within the cube, when it is subjected to a uniform increase in temperature, $\Delta T$, is given by
 A 0 B $\frac{\alpha (\Delta T)E}{1-2v}$ C $-\frac{\alpha (\Delta T)E}{1-2v}$ D $\frac{\alpha (\Delta T)E}{3(1-2v)}$
Strength of Materials   Stress-strain Relationship and Elastic Constants
Question 6 Explanation:
As all forces are free to deform, there is no thermal stress.
 Question 7
A two member truss ABC is shown in the figure. The force (in kN) transmitted in member AB is
 A 15kN B 60kN C 80kN D 20kN
Engineering Mechanics   FBD, Equilirbium, Plane Trusses and Virtual work
Question 7 Explanation:

\begin{aligned} A B &=1 \\ A C &=0.5 \mathrm{m} \\ B C &=\sqrt{1^{2}+0.5^{2}} \\ &=\sqrt{1.25}=1.118 \mathrm{m} \\ A_{x}+C_{x} &=0 \\ A_{y}+C_{y} &=10 \\ \;&\qquad\text{(from force equilibrium)}\\ \Sigma M_{A}&=0 \\ C_{x} \times 0.5&=10 \times 1 \\ \text{or }\quad C_{x}&=20 \mathrm{kN} \\ \text{and }\quad A_{x}&=-20 \mathrm{kN} \\ \end{aligned}

\begin{aligned} \Sigma M_{c}&=0 \\ \Rightarrow \quad F_{A B} \times 0.5&=20 \times 0.5 \\ \therefore \quad F_{A B}&=20 \mathrm{kN} \end{aligned}
 Question 8
A 4-bar mechanism with all revolute pairs has link lengths $l_f$=20mm, $l_{in}$=40mm, $l_{co}$=50mm and $l_{out}$=60mm. The suffixes 'f', 'in', 'co' and 'out' denote the fixed link, the input link, the coupler and output link respectively. Which one of the following statements is true about the input and output links?
 A Both links can execute full circular motion B Both links cannot execute full circular motion C Only the output link cannot execute full circular motion D Only the input link cannot execute full circular motion
Theory of Machine   Planar Mechanisms
Question 8 Explanation:
$S+L \lt P+Q$
In this case, shortest link is fixed and hence, the
resulting mechanism is double crank mechanism.
 Question 9
In vibration isolation, which one of the following statements is NOT correct regarding Transmissibility (T)?
 A T is nearly unity at small excitation frequencies B T can be always reduced by using higher damping at any excitation frequency C T is unity at the frequency ratio of $\sqrt{2}$ D T is infinity at resonance for undamped systems
Theory of Machine   Vibration
 Question 10
In a structure subjected to fatigue loading, the minimum and maximum stresses developed in a cycle are 200 MPa and 400 MPa respectively. The value of stress amplitude (in MPa) is
 A 200 B 100 C 50 D 150
\begin{aligned} \sigma _{min}&=200 \; \text{MPa} \\ \sigma _{max}&=400 \; \text{MPa} \\ \sigma _a&= \frac{\sigma _{max}-\sigma _{min}}{2}\\ &= \frac{400-200}{2}\\ &=100\; \text{MPa} \end{aligned}