# GATE ME 2015 SET-2

 Question 1
At least one eigenvalue of a singular matrix is
 A positive B zero C negative D imaginary
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
For singular matrix
$|A|=0$
According to properties of eigen value
Product of eigen values $=|A|=0$
$\Rightarrow$ Atleast one of the eigen value is zero.
 Question 2
At x = 0, the function $f(x)=\left | x \right |$ has
 A a minimum B a maximum C a point of inflexion D neither a maximum nor minimum
Engineering Mathematics   Calculus
Question 2 Explanation:
The graph of |x| is

from the graph we can say that
|x| has minimum at x=0
 Question 3
Curl of vector $V(x,y,z)= 2x^{2}i+3z^{2}j+y^{3}k$ at $x=y=z=1$ is
 A $-3i$ B $3i$ C $3i-4j$ D $3i-6k$
Engineering Mathematics   Calculus
Question 3 Explanation:
Curl of vector$=\left|\begin{array}{ccc} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 2 x^{2} & 3 z^{2} & y^{3} \end{array}\right|$
$=i\left[\frac{\partial}{\partial x}\left(y^{3}\right) \frac{\partial}{\partial z}\left(3 z^{2}\right)\right]+j\left[\frac{\partial}{\partial x}\left(y^{3}\right) \frac{\partial}{\partial z}\left(2 x^{2}\right)\right]$
$+k\left[\frac{\partial}{\partial x}\left(3z^{2}\right) \frac{\partial}{\partial z}\left(2 x^{2}\right)\right]$
$=i\left[3 y^{2}-6 z\right]-[10]+k[0+0]$
$\text { At } x=1, y=1 \text { and } z=1$
$\text { Curl }=i\left(3 \times 1^{2}-6 \times 1\right)=-3 i$
 Question 4
The Laplace transform of $e^{i5t}$ where $i=\sqrt{-1}$ is
 A $(s-5i)/(s^{2}-25)$ B $(s+5i)/(s^{2}+25)$ C $(s+5i)/(s^{2}-25)$ D $(s-5i)/(s^{2}+25)$
Engineering Mathematics   Differential Equations
Question 4 Explanation:
\begin{aligned} e^{j 5 t} &=\cos 5 t+i \sin 5 t \\ L\left\{e^{(i 5 f)}\right\}&=\frac{s}{s^{2}+25}+\frac{5 i}{s^{2}+25} \\ &=\frac{s+5 i}{s^{2}+25} 2 e^{-x^{2}} \end{aligned}
 Question 5
Three vendors were asked to supply a very high precision component. The respective probabilities of their meeting the strict design specifications are 0.8, 0.7 and 0.5. Each vendor supplies one component. The probability that out of total three components supplied by the vendors, at least one will meet the design specification is ___________
 A 0.12 B 0.97 C 0.65 D 1
Engineering Mathematics   Probability and Statistics
Question 5 Explanation:
Probability of atleast one meet the specification
\begin{aligned} &=1-(\bar{A} \cap \bar{B} \cap \bar{C}) \\ &=1-(0.2 \times 0.3 \times 0.5) \\ &=0.97 \end{aligned}
 Question 6
A small ball of mass 1 kg moving with a velocity of 12 m/s undergoes a direct central impact with a stationary ball of mass 2 kg. The impact is perfectly elastic. The speed (in m/s) of 2 kg mass ball after the impact will be ____________
 A 8m/s B 9m/s C 9m/s D 4m/s
Engineering Mechanics   Impulse and Momentum, Energy Formulations
Question 6 Explanation:

1. Conserving linear momentum
\begin{aligned} 1 \times 12 &=1 \times V_{1}+2 \times V_{2} \\ 12 &=V_{1}+2 V_{2}\qquad \ldots(i) \end{aligned}
2. Velocity of approach = Velocity of seperation
$12-0=v_{2}-v_{1}$
$V_{2}-V_{1}=12 \qquad \ldots(ii)$
From (i) and (ii), we get
$v_{2}=8 \mathrm{m} / \mathrm{s}$
 Question 7
A rod is subjected to a uni-axial load within linear elastic limit. When the change in the stress is 200 MPa, the change in the strain is 0.001. If the Poisson's ratio of the rod is 0.3, the modulus of rigidity (in GPa) is ________________
 A 76.9230GPa B 12.35698GPa C 19.2365GPa D 98.1458GPa
Strength of Materials   Stress-strain Relationship and Elastic Constants
Question 7 Explanation:

With in linear elastic limit
$\sigma=E \in$
$E \rightarrow \text{ slope of } \sigma \text{ vs }\in \text{ curve}$
\begin{aligned} E &=\frac{d \sigma}{d \epsilon}=\frac{200}{0.001}=200 \mathrm{GPa} \\ E &=2 G[1+\mu] \\ G &=\frac{E}{2(1+\mu)}=\frac{200}{2(1+0.3)}=\frac{100}{1.3} \\ &=76.9230 \mathrm{GPa} \end{aligned}
 Question 8
A gas is stored in a cylindrical tank of inner radius 7 m and wall thickness 50 mm. The gage pressure of the gas is 2 MPa. The maximum shear stress (in MPa) in the wall is
 A 35 B 70 C 140 D 280
Strength of Materials   Thin Cylinder
Question 8 Explanation:
Maximum shear stress in the wall
$=\frac{\sigma_{1}}{2}=\frac{p d}{4 t}=\frac{2 \times 14 \times 1000}{4 \times 50}=140 \mathrm{MPa}$
 Question 9
The number of degrees of freedom of the planetary gear train shown in the figure is
 A 0 B 1 C 2 D 3
Theory of Machine   Gear and Gear Train
Question 9 Explanation:
Degree of freedom: $F=3(l-1)-2 j-h$
$=3(4-1)-2 \times 3-1=2$
 Question 10
In a spring-mass system, the mass is m and the spring constant is k. The critical damping coefficient of the system is 0.1 kg/s. In another spring-mass system, the mass is 2m and the spring constant is 8k. The critical damping coefficient (in kg/s) of this system is ____________
 A 0.4kg/s B 1kg/s C 0.9kg/s D 2kg/s
Theory of Machine   Vibration
Question 10 Explanation:
\begin{aligned} \xi &=\frac{C}{2 \sqrt{m k}} \\ C &=2 \xi \sqrt{m k} \\ \frac{C_{2}}{C_{1}} &=\frac{2 \xi_{2} \sqrt{m_{2} k_{2}}}{2 \xi_{1} \sqrt{m_{1} k_{1}}} \\ \frac{C_{2}}{0.1} &=\frac{2 \times 1 \sqrt{2 m \times 8 k}}{2 \times 1 \sqrt{m \times k}} \\ C_{2} &=0.4 \mathrm{kg} / \mathrm{s} \end{aligned}
There are 10 questions to complete.