# 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
Engineering Mathematics   Numerical Methods
Question 1 Explanation:
(P) Gauss- Seidal method $\rightarrow$ Linear algebraic equation
(Q) Forward Newton $\rightarrow$ Gauss method Interpolation
(R) Runge-Kutta method $\rightarrow$ Non-linear differential equations
(S) Trapezoidal Aule $\rightarrow$ Numerical integration
 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}

There are 5 questions to complete.