Question 1 |
At least one eigenvalue of a singular matrix is
positive | |
zero | |
negative | |
imaginary |
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.
|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 minimum | |
a maximum | |
a point of inflexion | |
neither a maximum nor minimum |
Question 2 Explanation:
The graph of |x| is
from the graph we can say that
|x| has minimum at x=0
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
-3i | |
3i | |
3i-4j | |
3i-6k |
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
=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
(s-5i)/(s^{2}-25) | |
(s+5i)/(s^{2}+25) | |
(s+5i)/(s^{2}-25) | |
(s-5i)/(s^{2}+25) |
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 ___________
0.12 | |
0.97 | |
0.65 | |
1 |
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}
\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 ____________
8m/s | |
9m/s | |
9m/s | |
4m/s |
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 ________________
76.9230GPa | |
12.35698GPa | |
19.2365GPa | |
98.1458GPa |
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
35 | |
70 | |
140 | |
280 |
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}
=\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
0 | |
1 | |
2 | |
3 |
Question 9 Explanation:
Degree of freedom: F=3(l-1)-2 j-h
=3(4-1)-2 \times 3-1=2
=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 ____________
0.4kg/s | |
1kg/s | |
0.9kg/s | |
2kg/s |
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.