# GATE EE 2014 SET 3

 Question 1
Two matrices A and B are given below:
$A=\begin{bmatrix} p &q \\ r& s \end{bmatrix};$ $B=\begin{bmatrix} p^2+q^2 & pr +qs \\ pr+qs & r^2+s^2 \end{bmatrix}$
If the rank of matrix A is N, then the rank of matrix B is
 A N/2 B N-1 C N D 2N
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
\begin{aligned} A &=\begin{bmatrix} p & q\\ r & s \end{bmatrix} \\ A \times A=A^2&=\begin{bmatrix} p^2+q^2 & pr+qs\\ pr+qs & r^2+s^2 \end{bmatrix} =B \\ A^2 &=B \end{aligned}
Rank of amtrix does not change when we squaring the matrix, hence rank of B = rank of A=N.
 Question 2
A particle, starting from origin at t=0s, is traveling along x-axis with velocity
$v=\frac{\pi}{2}\cos (\frac{\pi}{2}t)m/s$
At t=3s, the difference between the distance covered by the particle and the magnitude of displacement from the origin is_____
 A 1 B 2 C 3 D 4
Engineering Mathematics   Calculus

 Question 3
Let $\triangledown \cdot (f v)=x^2y+y^2z+z^2x$, where f and v are scalar and vector fields respectively. If $v=yi+zj+xk$, then $v\cdot \triangledown f$ is
 A $x^2y+y^2z+z^2x$ B $2xy+2yz+2zx$ C $x+y+z$ D 0
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} \vec{V}&=y\hat{i}+z\hat{j}+x\hat{k}\\ \hat{i}\frac{\partial (fV)}{\partial x}+\hat{j}\frac{\partial (fV)}{\partial y}+\hat{k}\frac{\partial (fV)}{\partial z}&=x^2y+y^2z+z^2x\\ y\frac{\partial f}{\partial x}+z\frac{\partial f}{\partial y}+x\frac{\partial f}{\partial z}&=x^2y+y^2z+z^2x\;\;...(i)\\ \vec{V}\cdot \Delta f&=y\frac{\partial f}{\partial x}+z\frac{\partial f}{\partial y}+x\frac{\partial f}{\partial z}\;\;...(ii)\\ \text{From equations (i) and (ii)}\\ \vec{V}\cdot \Delta f&=x^2y+y^2z+z^2x \end{aligned}
 Question 4
Lifetime of an electric bulb is a random variable with density $f(x)=kx^2$, where x is measured in years. If the minimum and maximum lifetimes of bulb are 1 and 2 years respectively, then the value of k is _____
 A 0.85 B 0.42 C 0.25 D 0.75
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
Life time of an electric bulb with density
$f(x)=Kx^2$
If minimum and maximum lifetimes of bulb are 1 and 2 years respectively then
\begin{aligned} \int_{1}^{2}Kx^2dx &=1\\ \left.\begin{matrix} K\frac{x^3}{3} \end{matrix}\right|_1^2&=1\\ K\left ( \frac{8}{3}-\frac{1}{3} \right )&=1\\ \frac{7K}{3}&=1\\ K&=\frac{3}{7}=0.42 \end{aligned}
 Question 5
A function f(t) is shown in the figure. The Fourier transform F($\omega$) of f(t) is
 A real and even function of w B real and odd function of w C imaginary and odd function of w D imaginary and even function of w
Signals and Systems   Fourier Transform
Question 5 Explanation:
Fiven signal $f(t)$ is an odd signal. Hence, $F(\omega )$ is imaginary and odd function of $\omega$.

There are 5 questions to complete.