Question 1 
The parabolic arc y=\sqrt{x}, 1\leq x\leq 2
is revolved around the xaxis. The volume of the solid of revolution is
\frac{\pi }{4}  
\frac{\pi }{2}  
\frac{3\pi }{4}  
\frac{3\pi }{2} 
Question 1 Explanation:
The volume of a solid generated by revolution about the xaxis, of the area bounded by curve y=f(x),the xaxis and the ordinates
\text{x=a, y=b is}
\text { Volume }=\int_{a}^{b} \pi y^{2} d x
Here, a=1, b=2 and y=\sqrt{x} \Rightarrow y^{2}=x
\begin{aligned} \therefore Volume &=\int_{1}^{2} \pi \cdot x \cdot d x \\ &=\pi \cdot\left[\frac{x^{2}}{2}\right]_{1}^{2}=\frac{\pi}{2}\left[x^{2}\right]_{1}^{2} \\ &=\frac{\pi}{2}\left[2^{2}1^{2}\right]=\frac{3}{2} \pi \end{aligned}
\text{x=a, y=b is}
\text { Volume }=\int_{a}^{b} \pi y^{2} d x
Here, a=1, b=2 and y=\sqrt{x} \Rightarrow y^{2}=x
\begin{aligned} \therefore Volume &=\int_{1}^{2} \pi \cdot x \cdot d x \\ &=\pi \cdot\left[\frac{x^{2}}{2}\right]_{1}^{2}=\frac{\pi}{2}\left[x^{2}\right]_{1}^{2} \\ &=\frac{\pi}{2}\left[2^{2}1^{2}\right]=\frac{3}{2} \pi \end{aligned}
Question 2 
The Blasius equation, \frac{\mathrm{d}^{3}f }{\mathrm{d} \eta ^{3}}+\frac{f}{2}\frac{\mathrm{d} ^{2}f }{\mathrm{d} \eta ^{2}}=0 is a
Second order nonlinear ordinary differential equation  
Third order nonlinear ordinary differential equation  
Third order linear ordinary differential equation  
Mixed order nonlinear ordinary differential equation 
Question 2 Explanation:
\frac{d^{3} f}{d n^{3}}+\frac{f}{2} \frac{d^{2} f}{d n^{2}}=0 is third order \left(\frac{d^{3} f}{d n^{3}}\right) and it is non linear, since the product f \times \frac{d^{2} f}{d n^{2}} is not allowed in linear differential equation.
Question 3 
The value of the integral \int_{\infty }^{\infty }\frac{\mathrm{d} x}{1+x^{2}}
is
\pi  
\frac{\pi }{2}  
\frac{\pi }{2}  
\pi 
Question 3 Explanation:
\begin{aligned} \int_{\infty}^{\infty} \frac{d x}{1+x^{2}} &=\left[\tan ^{1} x\right]_{\infty}^{\infty} \\ &=\tan ^{1}(\infty)\tan ^{1}(\infty) \\ &=\frac{\pi}{2}\left[\frac{\pi}{2}\right]=\pi \end{aligned}
Question 4 
The modulus of the complex number \left ( \frac{3+4i}{12i} \right )
is
5  
\sqrt{5}  
\frac{1}{\sqrt{5}}  
\frac{1}{5} 
Question 4 Explanation:
\begin{aligned} Z &=\frac{3+4 i}{12 i}=\frac{(3+4 i)(1+2 i)}{(12 i)(1+2 i)} \\ &=\frac{5+10 i}{5}=1+2 i \\ Z &=\sqrt{(1)^{2}+(2)^{2}}=\sqrt{5} \end{aligned}
Question 5 
The function y=\left  23x \right 
is continuous \forall x\in R
and differentiable \forall x\in R
 
is continuous \forall x\in R
and differentiable \forall x\in R
except at x=\frac{3}{2}
 
is continuous \forall x\in R
and differentiable \forall x\in R
except at x=\frac{2}{3}
 
is continuous \forall x\in R except at x=3
and differentiable \forall x\in R

Question 5 Explanation:
\begin{aligned} y=23 x &=23 x \quad 23 x \geq 0 \\ &=3 x2 \quad 23 x \lt 0 \end{aligned}
Therefore
y=23 x \quad x \leq \frac{2}{3}
=3 x2 \quad x gt \frac{2}{3}
since 23 x and 3 x2 are polynomials, these are continuous at all points. The only concern is at
x=\frac{2}{3}
Left limit at x=\frac{2}{3} is 23 \times \frac{2}{3}=0
Right limit at x=\frac{2}{3} is 3 \times \frac{2}{3}2=0
f\left(\frac{2}{3}\right)=23 \times \frac{2}{3}=0
since, Left limit = Right limit =f\left(\frac{2}{3}\right)
Function is continuous at \frac{2}{3}
y is therefore continuous \forall x \in R
Now since 23 x and 3 x2 are polynomials,they are differentiable.
only concern is at x =\frac{2}{3}
Now, at x=\frac{2}{3}, L D= Left derivative =3
R D= Right derivative =+3
LD \neq RD
\therefore The function y is not differentiable at
x=\frac{2}{3}
So, we can say that y is differential \forall \bar{x} \in R , except at x=\frac{2}{3}
Therefore
y=23 x \quad x \leq \frac{2}{3}
=3 x2 \quad x gt \frac{2}{3}
since 23 x and 3 x2 are polynomials, these are continuous at all points. The only concern is at
x=\frac{2}{3}
Left limit at x=\frac{2}{3} is 23 \times \frac{2}{3}=0
Right limit at x=\frac{2}{3} is 3 \times \frac{2}{3}2=0
f\left(\frac{2}{3}\right)=23 \times \frac{2}{3}=0
since, Left limit = Right limit =f\left(\frac{2}{3}\right)
Function is continuous at \frac{2}{3}
y is therefore continuous \forall x \in R
Now since 23 x and 3 x2 are polynomials,they are differentiable.
only concern is at x =\frac{2}{3}
Now, at x=\frac{2}{3}, L D= Left derivative =3
R D= Right derivative =+3
LD \neq RD
\therefore The function y is not differentiable at
x=\frac{2}{3}
So, we can say that y is differential \forall \bar{x} \in R , except at x=\frac{2}{3}
Question 6 
Mobility of a statically indeterminate structure is
\leq 1  
0  
1  
\geq 2 
Question 6 Explanation:
Mobility or degree of freedom for a statically indeterminate structure is always less than zero.
i.e.,F \lt 0
\therefore only option (A) is negative value
\therefore F \leq1
i.e.,F \lt 0
\therefore only option (A) is negative value
\therefore F \leq1
Question 7 
There are two points P and Q on a planar rigid body. The relative velocity between the two points
should always be along PQ  
Can be oriented along any direction  
should always be perpendicular to PQ  
should be along QP when the body undergoes pure translation 
Question 7 Explanation:
V_{P Q}= Relative velocity between P and Q.
V_{P Q}=V_{P}V_{Q} Always perpendicular to PQ.
Question 8 
The state of planestress at a point is given by \sigma _{x}= 200MPa, \sigma _{y}=100MPa
and \tau _{xy}=100MPa. The maximum shear stress in MPa is
111.8  
150.1  
180.3  
223.6 
Question 8 Explanation:
\begin{aligned} \ln \operatorname{plane}, \tau_{\max } &=\frac{1}{2} \sqrt{\left(\sigma_{x}\sigma_{y}\right)^{2}+4 \tau_{x y}^{2}} \\ &=\frac{1}{2} \sqrt{100^{2}+4 \times 100^{2}} \\ &=111.8033 \mathrm{MPa} \end{aligned}
Question 9 
Which of the following statements is INCORRECT?
Grashof's rule states that for a planar crankrocker four bar mechanism, the sum of the shortest and longest link lengths cannot be less than the sum of the remaining two link lengths.  
Inversions of a mechanism are created by fixing different links one at a time.  
Geneva mechanism is an intermittent motion device  
Gruebler's criterion assumes mobility of a planar mechanism to be one. 
Question 9 Explanation:
Grashof's rule states that for a planar crank rocker four bar mechanism, the sum of shortest and longest link length is less than the sum of remaining two link length i.e.
s+l \leq p+q
s+l \leq p+q
Question 10 
The natural frequency of a springmass system on earth is \omega _{n}. The natural frequency of this system on the moon (g_{moon}= g_{earth}/6)
is
\omega _{n}  
0.408\omega _{n}  
0.204\omega _{n}  
0.167\omega _{n} 
Question 10 Explanation:
\omega_{n}=\sqrt{\frac{K}{m}}
There are 10 questions to complete.