# GATE ME 2006

 Question 1
Match the items in columns I and II.
 A P-1 Q-4 R-3 S-2 B P-1 Q-4 R-2 S-3 C P-1 Q-3 R-2 S-4 D P-4 Q-1 R-2 S-3

 Question 2
The solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{d} x}+ 2xy = e^{-x^{2}}$ with $y(0)=1$ is:
 A $(1+x) e^{+x^{2}}$ B $(1+x) e^{-x^{2}}$ C $(1-x) e^{+x^{2}}$ D $(1-x) e^{-x^{2}}$
Engineering Mathematics   Differential Equations
Question 2 Explanation:
Given equation
$\frac{d y}{d x}+2 x y=\sigma^{x^{2}}$
Integrating factor
\begin{aligned} \text { I.F. } &=\mathrm{e}^{\int 2 x \;dx}=e^{x^{2}} \\ \text { Solution is } y x^{e^{x^{2}}}&=\int e^{-x^{2}} \cdot e^{x^{2}} d x \\ \therefore \quad y x^{ e^{x^{2}}}&=x+c\\ d t x &=0 \\ y &=1\\ \therefore \; \; c&=1 \\ y&=(1+x) e^{-x^{2}} \end{aligned}
 Question 3
Let x denote a real number. Find out the INCORRECT statement.
 A $S = \left \{ x : x \gt 3 \right \}$ represents the set of all real numbers greater than 3 B $S=\left \{ x : x^{2} \lt 0 \right \}$ represents the empty set C $S=\left \{ x : x \in A \; and \; x \in B \right \}$ represents the union of set A and set B D $S = \left \{ x : a \lt x \lt b \right \}$ represents the set of all real numbers between a and b, where a and b are real numbers.
Engineering Mathematics   Linear Algebra
Question 3 Explanation:
The incorrect statement is, $S=\left \{ x : x \in A \; and \; x \in B \right \}$ represents the union of set A and set B
As and represents the intersection not union.
 Question 4
A box contains 20 defective items and 80 non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective.?
 A $\frac{1}{5}$ B $\frac{1}{25}$ C $\frac{20}{99}$ D $\frac{19}{495}$
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
Probability of first item being defective is
$P_{1}=\frac{20}{100}$
Probability that second item being defective is
$P_{2}=\frac{19}{99}$
Probability that both are defective
$P=P_{1} P_{2}=\frac{20}{100} \times \frac{19}{99}=\frac{19}{495}$
 Question 5
For a circular shaft of diameter d subjected to torque T, the maximum value of the shear stress is:
 A $\frac{64T}{\pi d^{3}}$ B $\frac{32T}{\pi d^{3}}$ C $\frac{16T}{\pi d^{3}}$ D $\frac{8T}{\pi d^{3}}$
Strength of Materials   Torsion of Shafts
Question 5 Explanation:
\begin{aligned} \frac{T}{J} &=\frac{\tau}{d / 2} \\ \Rightarrow \qquad \frac{T}{\frac{\pi}{32} d^{4}} &=\frac{\tau}{d / 2} \\ \therefore \qquad & \tau=\frac{16 T}{\pi d^{3}} \end{aligned}
 Question 6
For a four-bar linkage in toggle position, the value of mechanical advantage is:
 A 0 B 0.5 C 1 D $\infty$
Theory of Machine   Dynamic Analysis of Slider-crank
Question 6 Explanation:

$\omega_{4}$ of the output link DC becomes zero at the extreme positions. The extreme positions of the linkage are known as "Toggle position".
$\therefore$ Mechanical advantage $=\frac{\omega_{\text {input }}}{\omega_{\text {output }}}$
$\because \quad \omega_{\text {output }}=0$
$\therefore$Mechanical advantage $=\infty$
 Question 7
The differential equation governing the vibrating system is:
 A $m\ddot{x} + c\dot{x} + k(x-y) = 0$ B $m(\ddot{x}-\ddot{y}) + c(\dot{x}-\dot{y}) + kx = 0$ C $m \ddot{x} + c(\dot{x}-\dot{y}) + kx = 0$ D $m(\ddot{x}-\ddot{y}) + c(\dot{x}-\dot{y}) + k(x-y) = 0$
Theory of Machine   Vibration
Question 7 Explanation:

Differential equation governing the above vibration
system is given by
$\begin{array}{l} \Rightarrow \frac{m d^{2} x}{d t^{2}}+C\left(\frac{d x}{d t}-\frac{d y}{d t}\right)+k x=0 \\ \Rightarrow m \ddot{x}+c(\dot{x}-\dot{y})+k x=0 \end{array}$
 Question 8
A pin-ended column of length L, modulus of elasticity E and second moment of the cross-sectional area I is loaded centrically by a compressive load P. the critical buckling load $(P_{cr})$ is given by
 A $P_{cr}= \frac{EI}{\pi ^{2}L^{2}}$ B $P_{cr}= \frac{\pi ^{2} EI}{3L^{2}}$ C $P_{cr}= \frac{\pi EI}{L^{2}}$ D $P_{cr}= \frac{\pi^{2} EI}{L^{2}}$
Strength of Materials   Euler's Theory of Column
Question 8 Explanation:
$P_{E}=\frac{n \pi^{2} E I}{L^{2}}$
For pin-ended column
n=1
Here n= End fixity coefficient
E = Modulus of elasticity
I = Second moment of area
L = Actual length of column
 Question 9
The number of inversions for a slider crank mechanism is:
 A 6 B 5 C 4 D 3
Theory of Machine   Planar Mechanisms
Question 9 Explanation:
No. of links of a slider crank mechanism =4
So there are four inversion of slider crank mechanism.
 Question 10
For a Newtonian fluid
 A shear stress is proportional to shear strain B rate of shear stress is proportional to shear strain C shear stress is proportional to rate of shear strain D rate of shear stress is proportional to rate of shear strain
Fluid Mechanics   Fluid Properties
Question 10 Explanation:
For a Newtonian fluid
$\tau \propto \frac{d u}{d y}$
$\tau=\mu \frac{d u}{d y}$
where $\quad \tau=$ shear stress
$\frac{d u}{d y}=$ rate of shear strain
There are 10 questions to complete.