# GATE ME 2014 SET-2

 Question 1
One of the eigenvectors of the matrix $\begin{bmatrix} -5 & 2\\ -9 & 6 \end{bmatrix}$ is
 A $\begin{Bmatrix} -1\\ 1 \end{Bmatrix}$ B $\begin{Bmatrix} -2\\ 9 \end{Bmatrix}$ C $\begin{Bmatrix} 2\\ -1 \end{Bmatrix}$ D $\begin{Bmatrix} 1\\ 1 \end{Bmatrix}$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
The characteristic equation $| A-\lambda D=0$
$\text{i.e.} \left|\begin{array}{cc}-5-\lambda & 2 \\ -9 & 6-\lambda\end{array}\right|=0$
$\text{or } (\lambda-6)(\lambda+5)+18=0$
$\text{or } \quad \lambda^{2}-6 \lambda+5 \lambda-30+18=0$
$\text{or } \quad \lambda^{2}-\lambda-12=0$
$\text{or } \lambda=\frac{1 \pm \sqrt{1+48}}{2}=\frac{1 \pm 7}{2}=4,-3$
Corresponding to $\lambda=4,$ we have
$\; [A-\lambda I] x=\left[\begin{array}{cc}-5-\lambda & 2 \\-9 & 6-\lambda\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]$
$=0$
$\text{Or, } \left[\begin{array}{ll}-9 & 2 \\ -9 & 2\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=0$which gives only one independent equation,
$-9 x+2 y=0$
$\therefore \frac{x}{2}=\frac{y}{9}$ gives eigen vector (2,9)
Corresponding to $\lambda=-3,$
$=\left[\begin{array}{ll}-2 & 2 \\-9 & 9\end{array}\right]\left[\begin{array}{l}x \\y \end{array}\right]=0$
which gives -x+y=0 (only one independent equation)
$\therefore \quad \frac{x}{1}=\frac{y}{1} \quad$ which gives (1,1)
So, the eigen vectors are $\left\{\begin{array}{l}2 \\ 9\end{array}\right\} and \left\{\begin{array}{l}1 \\ 1\end{array}\right\}$
 Question 2
$\lim_{x\rightarrow 0}\left ( \frac{e^{2x}-1}{\sin(4x)} \right )$ is equal to
 A 0 B 0.5 C 1 D 2
Engineering Mathematics   Calculus
Question 2 Explanation:
$\lim_{x \rightarrow 0} \frac{\left(e^{2 x}-1\right)}{\sin 4 x},$ it is of$\left(\frac{0}{0}\right)$from
Applying L' Hospital's rule
$\lim_{x \rightarrow 0} \frac{2 e^{2 x}}{4 \cos 4 x}=\frac{2 \times 1}{4 \times 1}=\frac{1}{2}$

 Question 3
Curl of vector $\vec{F}=x^{2}z^{2}\hat{i}-2xy^{2}z\hat{j}+2y^{2}z^{3}\hat{k}$ is
 A $(4yz^{3}+2xy^{2})\hat{i}+2x^{2}z\hat{j}-2y^{2}z\hat{k}$ B $(4yz^{3}+2xy^{2})\hat{i}-2x^{2}z\hat{j}-2y^{2}z\hat{k}$ C $2xz^{2}\hat{i}-4xyz\hat{j}+6y^{2}z^{2}\hat{k}$ D $2xz^{2}\hat{i}+4xyz\hat{j}+6y^{2}z^{2}\hat{k}$
Engineering Mathematics   Calculus
Question 3 Explanation:
$\vec{F}=x^{2} z^{2} \hat{i}-2 x y^{2} z \hat{j}+2 y^{2} z^{3} \hat{k}$
$\nabla \times \vec{F}=\left|\begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x^{2} z^{2}&-2 x y^{2} z^{2} & 2 y^{2} z^{3} \end{array}\right|$
$=\hat{i}\left[\frac{\partial}{\partial y}\left(2 y^{2} z^{3}\right)+\frac{\partial}{\partial z}\left(2 x y^{2} z\right)\right]$
$-\hat{j}\left[\frac{\partial}{\partial x}\left(2 y^{2} z^{3}\right)-\frac{\partial}{\partial z}\left(x^{2} z^{2}\right)\right]$
$+\hat{k}\left[\frac{\partial}{\partial x}\left(-2 x y^{2} z\right)-\frac{\partial}{\partial y}\left(x^{2} z^{2}\right)\right]$
$\nabla \times \vec{F}=\hat{i}\left[4 y z^{3}+2 x y^{2}\right]-\hat{j}\left[-2 z x^{2}\right]$
$=\left(4 y z^{3}+2 x y^{2}\right) \hat{i}+\left(2 x^{2} z\right) \hat{j}-\left(2 y^{2} z\right) \hat{k}$
 Question 4
A box contains 25 parts of which 10 are defective. Two parts are being drawn simultaneously in a random manner from the box. The probability of both the parts being good is
 A $7/20$ B $42/125$ C $25/29$ D $5/9$
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
required prob $=\frac{^{15} C_{2}}{^{25} C_{2}}=\frac{14 \times 15}{25 \times 24}=\frac{7}{20}$
 Question 5
The best approximation of the minimum value attained by $e^{-x}sin(100x)$ for $x\geq 0$ is _______
 A -0.9844 B -1.2652 C -9.2658 D -8.2654
Engineering Mathematics   Calculus
Question 5 Explanation:
$f(x)=e^{-x} \sin (100 x)$
$P(x)=-e^{-x} \sin (100 x)+e^{-x} \cos (100 x) \times 100$
for minima $f^{\prime}(x)=0$
from here $\tan (100 x)=100$
\begin{aligned} x &=\frac{1}{100} \tan ^{-1}(100)=0.0156 \\ f(x) &=e^{-0.0156} \sin (100 \times 0.0156)=-0.9844 \end{aligned}

There are 5 questions to complete.