GATE ME 2010

Question 1
The parabolic arc y=\sqrt{x}, 1\leq x\leq 2 is revolved around the x-axis. The volume of the solid of revolution is
A
\frac{\pi }{4}
B
\frac{\pi }{2}
C
\frac{3\pi }{4}
D
\frac{3\pi }{2}
Engineering Mathematics   Calculus
Question 1 Explanation: 
The volume of a solid generated by revolution about the x-axis, of the area bounded by curve y=f(x),the x-axis 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}
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
A
Second order nonlinear ordinary differential equation
B
Third order nonlinear ordinary differential equation
C
Third order linear ordinary differential equation
D
Mixed order nonlinear ordinary differential equation
Engineering Mathematics   Differential Equations
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
A
-\pi
B
\frac{-\pi }{2}
C
\frac{\pi }{2}
D
\pi
Engineering Mathematics   Calculus
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}{1-2i} \right ) is
A
5
B
\sqrt{5}
C
\frac{1}{\sqrt{5}}
D
\frac{1}{5}
Engineering Mathematics   Complex Variables
Question 4 Explanation: 
\begin{aligned} Z &=\frac{3+4 i}{1-2 i}=\frac{(3+4 i)(1+2 i)}{(1-2 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 | 2-3x \right |
A
is continuous \forall x\in R and differentiable \forall x\in R
B
is continuous \forall x\in R and differentiable \forall x\in R except at x=\frac{3}{2}
C
is continuous \forall x\in R and differentiable \forall x\in R except at x=\frac{2}{3}
D
is continuous \forall x\in R except at x=3 and differentiable \forall x\in R
Engineering Mathematics   Calculus
Question 5 Explanation: 
\begin{aligned} y=|2-3 x| &=2-3 x \quad 2-3 x \geq 0 \\ &=3 x-2 \quad 2-3 x \lt 0 \end{aligned}
Therefore
y=2-3 x \quad x \leq \frac{2}{3}
=3 x-2 \quad x gt \frac{2}{3}
since 2-3 x and 3 x-2 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 2-3 \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)=2-3 \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 2-3 x and 3 x-2 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
A
\leq -1
B
0
C
1
D
\geq 2
Theory of Machine   Planar Mechanisms
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 \leq-1
Question 7
There are two points P and Q on a planar rigid body. The relative velocity between the two points
A
should always be along PQ
B
Can be oriented along any direction
C
should always be perpendicular to PQ
D
should be along QP when the body undergoes pure translation
Theory of Machine   Planar Mechanisms
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 plane-stress at a point is given by \sigma _{x}= -200MPa, \sigma _{y}=100MPa and \tau _{xy}=100MPa. The maximum shear stress in MPa is
A
111.8
B
150.1
C
180.3
D
223.6
Strength of Materials   Stress-strain Relationship and Elastic Constants
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?
A
Grashof's rule states that for a planar crank-rocker four bar mechanism, the sum of the shortest and longest link lengths cannot be less than the sum of the remaining two link lengths.
B
Inversions of a mechanism are created by fixing different links one at a time.
C
Geneva mechanism is an intermittent motion device
D
Gruebler's criterion assumes mobility of a planar mechanism to be one.
Theory of Machine   Planar Mechanisms
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
Question 10
The natural frequency of a spring-mass system on earth is \omega _{n}. The natural frequency of this system on the moon (g_{moon}= g_{earth}/6) is
A
\omega _{n}
B
0.408\omega _{n}
C
0.204\omega _{n}
D
0.167\omega _{n}
Theory of Machine   Vibration
Question 10 Explanation: 
\omega_{n}=\sqrt{\frac{K}{m}}
There are 10 questions to complete.

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