# GATE EE 2014 SET 1

 Question 1
Given a system of equations
x + 2y + 2z = $b_1$
5x + y + 3z = $b_2$
What of the following is true regarding its solutions
 A The system has a unique solution for any given $b_1 \; and \; b_2$ B The system will have infinitely many solutions for any given $b_1 \; and \; b_2$ C Whether or not a solution exists depends on the given $b_1 \; and \; b_2$ D The systems would have no solution for any values of $b_1 \; and \; b_2$
Engineering Mathematics   Linear Algebra
 Question 2
Let $f(x)=xe^{-x}$. The maximum value of the function in the interval $(0,\infty )$ is
 A $e^{-1}$ B e C $1-e^{-1}$ D $1+e^{-1}$
Engineering Mathematics   Calculus
Question 2 Explanation:
\begin{aligned} f(x)&=xe^{-x}\\ f'(x)&=e^{-x}-xe^{-x}=0\\ e^{-x}(1-x)&=0\\ x&=1\\ f''(x)&=-e^{-x}-e^{-x}+xe^{-x}\\ &=e^{-x}(x-2)\\ f''(1)&=e^{-1}(-1)=-e^{-1} \lt 0 \end{aligned}
Hence f(x) has maximum value at x=1
$f(1)=1\cdot e^{-1}=e^{-1}$

 Question 3
The solution for the differential equation
$\frac{d^{2}x}{dt^{2}}=-9x$
with initial conditions x(0)=1 and $\frac{dx}{dt}|_{t=0}=1$, is
 A $t^{2}+t+1$ B $sin 3t+ \frac{1}{3} cos3t+\frac{2}{3}$ C $\frac{1}{3} sin 3t+cos3t$ D $cos3t+t$
Engineering Mathematics   Differential Equations
Question 3 Explanation:
\begin{aligned} \frac{d^2x}{dt^2}&=-9x\\ \frac{d^2x}{dt^2}+9x&=0\\ (D^2+9)x &=0 \end{aligned}
Auxiliary equation is $m^2+9=0$
\begin{aligned} m&= \pm 3i\\ x&= C_1 \cos 3t+C_2 \sin 3t\;\;...(i)\\ x(0) &=1\;\;i.e.\;x\rightarrow 1\; when t\rightarrow 0 \\ 1&=C_1 \\ \frac{dx}{dt}&=-3C_1 \sin 3t+3C_2 \cos 3t\;\;...(ii) \\ x'(0)&=1\;\; i.e.\;x'\rightarrow 1\; when \; t\rightarrow 0 \\ 1&=3C_2 \\ C_2&=\frac{1}{3}\\ \therefore \;x&=\cos 3t+\frac{1}{3}\sin 3t \end{aligned}
 Question 4
Let $X(s)=\frac{3s+5}{s^{2}+10s+21}$ be the Laplace Transform of a signal x(t). Then, x($0^+$) is
 A 0 B 3 C 5 D 21
Signals and Systems   Laplace Transform
Question 4 Explanation:
Given, $X(s)=\left [ \frac{3s+5}{s^2+10s+21} \right ]$
Using initial value theorem,
\begin{aligned} x(0^+) &= \lim_{s \to \infty }[sX(s)]\\ x(0^+) &= \lim_{s \to \infty } \left [ \frac{s(3s+5)}{s^2+10+21} \right ] \\ &= \lim_{s \to \infty }\left [ \frac{3+\frac{5}{s}}{1+\frac{10}{s}+\frac{21}{s^2}} \right ]=3 \end{aligned}
 Question 5
Let S be the set of points in the complex plane corresponding to the unit circle. (That is, S={z:|z|=1}). Consider the function f(z)=zz* where z* denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane
 A unit circle B horizontal axis line segment from origin to (1, 0) C the point (1, 0) D the entire horizontal axis
Engineering Mathematics   Complex Variables
Question 5 Explanation:

\begin{aligned} z&=x+iy\\ z^*&=x-iy\\ zz^*&=(x+iy)(x-iy)\\ &=x^2+y^2 \end{aligned}
which is equal to (1) always as given
\begin{aligned} |z|&=1\\ zz^*&=x^2+y^2 \end{aligned}

There are 5 questions to complete.