Question 1 |
Which one of the following equations is a correct identity for arbitrary 3x3 real matrices P, Q and R?
P(Q+R)=PQ+RP | |
(P-Q)^{2}=P^{2}-2PQ+Q^{2} | |
det(P+Q)=det \; P+det \; Q | |
(P+Q)^{2}=P^{2}+PQ+QP+Q^{2} |
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
3 | |
0 | |
-1 | |
-2 |
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
1+e^{-x^{2}} | |
2e^{-x^{2}} | |
1+e^{x^{2}} | |
2e^{x^{2}} |
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}}
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 _______
12% | |
65% | |
98% | |
50% |
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 \%
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
\frac{s-2}{(s-2)^{2}+16} | |
\frac{s+2}{(s-2)^{2}+16} | |
\frac{s-2}{(s+2)^{2}+16} | |
\frac{s+2}{(s+2)^{2}+16} |
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).
\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
j=2m-3 | |
m=2j+1 | |
m=2j-3 | |
m=2j-1 |
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)
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 _______
0.357 | |
0.123 | |
0.2658 | |
1.354 |
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
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?
Bevel gears | |
Double helical gears | |
Worm gears | |
Rack and pinion 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.
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 _______
0.5 | |
1 | |
1.5 | |
2 |
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
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 _______
6.78 | |
9.54 | |
6.31 | |
3.98 |
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.
