# GATE ME 2009

 Question 1
For a matrix $[M]=\begin{bmatrix} \frac{3}{5} & \frac{4}{5}\\ x & \frac{3}{5} \end{bmatrix}$ the transpose of the matrix is equal to the inverse of the matrix $[M]^{T}= [M]^{-1}$. The value of x is given by
 A $-\frac{4}{5}$ B $-\frac{3}{5}$ C $\frac{3}{5}$ D $\frac{4}{5}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
$\text{If }\quad A^{T}=A^{-1}$
then A is orthogonal matrix.
Therefore $A . A^{-1}=A^{-1} A=I$
$\text{and }A^{T} A=A A^{T}=I$
since $\mathrm{M}$ is orthogonal matrix
$M^{T} M=I$
$=\left[\begin{array}{cc}\frac{3}{5} & x \\ \frac{4}{5} & \frac{3}{5}\end{array}\right]\left[\begin{array}{cc}\frac{3}{5} & \frac{4}{5} \\ x & \frac{3}{5}\end{array}\right]=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$
\left[\begin{aligned} \left(\frac{3}{5}\right)^{2}+x^{2} \quad\left(\frac{3}{5} \cdot \frac{4}{5}\right)+\frac{3}{5} x \\ \left(\frac{4}{5} \cdot \frac{3}{5}\right)+\frac{3}{5} \cdot x\left(\frac{4}{5}\right)^{2}+\left(\frac{3}{5}\right)^{2} \end{aligned} \right]
$=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right] \Rightarrow \text{Compare both sides} a_{12}$
$a_{12}=\left(\frac{3}{5}\right)\left(\frac{4}{5}\right)+\left(\frac{3}{5}\right) x=0$
$\Rightarrow \quad \frac{3}{5} x=-\frac{3}{5} \cdot \frac{4}{5}$
$\Rightarrow x=-\frac{4}{5}$
 Question 2
The divergence of the vector field $3xz\hat{i}+2xy\hat{j}-yz^{2}\hat{k}$ at a point (1,1,1 ) is equal to
 A 7 B 4 C 3 D 0
Engineering Mathematics   Calculus
Question 2 Explanation:
$\text{Vector field,}$
$\vec{f}=3 x z \hat{i}+2 x y \hat{j}-y z^{2} \hat{k}$
$\vec{f}=\left[f_{1} \hat{i}+f_{2} \hat{j}+f_{3} \hat{k}\right]$
\begin{aligned}&\text{Divergence of vector field} \vec{f}=\nabla \cdot \vec{f} \\ &\text{Div}(f)=\left(i \frac{\partial}{\partial x}+\hat{j} \frac{\partial}{\partial y}+\hat{k} \frac{\partial}{\partial z}\right) \cdot\left(t_{1} \hat{i}+f_{2} \hat{j}+f_{3} \hat{k}\right) \\ &\text{Div}(f)=\frac{\partial f_{1}}{\partial x}+\frac{\partial f_{2}}{\partial y}+\frac{\partial f_{3}}{\partial z} \\ &\text{Div}(f)=\frac{\partial}{\partial x}[3 x z]+\frac{\partial}{\partial y}[2 x y]+\frac{\partial}{\partial z}\left[-2 y z^{2}\right] \\ &\text{Div}(f)=3 z+2 x-2 z y \\ &\text{Div}(f)|_{(1,1,1)}=3(1)+2(1)-2(1)(1)=3 \end{aligned}

 Question 3
The inverse Laplace transform of $\frac{1}{(s^{2}+s)}$ is
 A $1+e^{t}$ B $1-e^{t}$ C $1-e^{-t}$ D $1+e^{-t}$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
\begin{aligned} L^{-1}\left(\frac{1}{s^{2}+s}\right) &=? \\ \frac{1}{s^{2}+s} &=\frac{1}{s(s+1)}=\frac{1}{s}-\frac{1}{s+1} \\ L^{-1}\left(\frac{1}{s^{2}+s}\right) &=L^{-1}\left(\frac{1}{s}\right)--L^{-1}\left(\frac{1}{s+1}\right) \\ &=1-e^{-t} \end{aligned}
[Using standard formulae]
Standard formula:
\begin{aligned} L^{-1}\left(\frac{1}{s}\right) &=1 \\ L^{-1}\left(\frac{1}{s+a}\right) &=e^{-a t} \\ L^{-1}\left(\frac{1}{s-a}\right) &=e^{a t} \end{aligned}
 Question 4
If three coins are tossed simultaneously, the probability of getting at least one head
 A $1/8$ B $3/8$ C $1/2$ D $7/8$
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
Three coins are tossed simultaneously, the total
number of ways $=2^{3}=8$
Number of ways to get all tails = 1
$Probability to get all tails$
$=\frac{\text { Number of ways for tails }}{\text { Total number of ways }}=\frac{1}{8}$
Therefore probability to get at least one head is
$=1-\frac{1}{8}=\frac{7}{8}$
or, Total number of ways with at least one head is
$\left.\begin{matrix} & &\left.\begin{array}{lll} H & T & T \\ T & H & T \\ T & T & H \end{array} \right] \\ &&\left.\begin{array}{lll} H & T & T \\ T & H & T \\ T & T & H \end{array} \right] \\ &&\left.\begin{array}{lll} H & H & H \end{array} \right] \end{matrix}\right\}$
Therefore probability to get at least one head $=\frac{7}{8}$
 Question 5
If a closed system is undergoing an irreversible process, the entropy of the system
 A Must increase B Always remains constant C Must decrease D Can increase, decrease or remain constant
Thermodynamics   Thermodynamic System and Processes
Question 5 Explanation:
If a closed system is undergoing an irreversible process, the entropy of the system can increase, decrease or remain constant

There are 5 questions to complete.