# GATE EC 2008

 Question 1
All the four entries of the 2x2 matrix $P=\begin{bmatrix} P_{11} &P_{12} \\ P_{21} & P_{22} \end{bmatrix}$ are nonzero, and one of its eigenvalue is zero. Which of the following statements is true?
 A $P_{11}P_{22}-P_{12}P_{21}=1$ B $P_{11}P_{22}-P_{12}P_{21}=-1$ C $P_{11}P_{22}-P_{12}P_{21}=0$ D $P_{11}P_{22}+P_{12}P_{21}=0$
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Eigenvalues are the roots of the determinant formed by matrix [sI-P]
$\begin{array}{l} [sI-P]=\left[\begin{array}{cc}s-P_{11} & -P_{12} \\-P_{21} & s-P_{22}\end{array}\right] \\\; [s I - P]=0 \\ \Rightarrow \quad \left(s-P_{11}\right)\left(s-P_{22}\right)-P_{12} P_{21}=0\\ \Rightarrow \quad s^{2}-\left(P_{11}+P_{22}\right) s+P_{11} P_{22}-P_{12} P_{21}=0 \end{array}$
Since, one of the its eigenvalues is zero, therefore,
putting s=0
$P_{11} P_{22}-P_{12} P_{21}=0$
which is the desired condition.
 Question 2
The system of linear equations
4x + 2y = 7
2x + y = 6
has
 A a unique solution B no solution C an infinite number of solutions D exactly two distinct solutions
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
The system can be written in matrix from as
$\left[\begin{array}{ll}4 & 2 \\ 2 & 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}7 \\6 \end{array}\right]$
The Augmentod matrix [AIB] is given by
$\left[\begin{array}{ll|l}4 & 2 & 7 \\ 2 & 1 & 6\end{array}\right]$
Performing Gauss elimination on this [A \mid B] as follows:
$\left[\begin{array}{ll|l}4 & 2 & 7 \\2 & 1 & 6 \end{array}\right] \frac{R_{2}-\frac{2}{4} R_{1}}{=R_{2}-\frac{1}{2} R_{1}}\left[\begin{array}{ll|l}4 & 2 & 7 \\0 & 0 & 5 / 2 \end{array}\right]$
Now Rank $[A \mid B]=2$
(The number of non-zero rows in [ A|B]
Rank [A]=1
(The number of non-zero rows in [A])
since, Rank $[A \mid B] \neq \text{Rank}[A].$
The system has no solution.
 Question 3
The equation sin(z) = 10 has
 A no real or complex solution B exactly two distinct complex solutions C a unique solution D an infinite number of complex solutions
Engineering Mathematics   Complex Analysis
Question 3 Explanation:
sinz can have value between -1 to +1. Thus no solution.
 Question 4
For real values of x, the minimum value of the function f(x)=exp(x)+exp(-x) is
 A 2 B 1 C 0.5 D 0
Engineering Mathematics   Calculus
Question 4 Explanation:
$f(x)=e^{x}+e^{-x}=e^{x}+\frac{1}{e^{x}}$
Arithmetic mean of $e^{x}$ and $\frac{1}{e^{x}}$ is
$\mathrm{A.M},=\frac{e^{x}+\frac{1}{e^{x}}}{2}$
Geometric mean of $e^{x}$ and $\frac{1}{e^{x}}$ is
G.M. $=e^{x} \cdot \frac{1}{e^{x}}=1$
It is known that A.M. $\geq$ G.M
$\frac{\left(e^{x}+\frac{1}{e^{x}}\right)}{2} \geq 1$
$e^{x}+\frac{1}{e^{x}} \geq 2$
Therefore, $\left(e^{x}+e^{-x}\right)_{\min }=2$
 Question 5
Which of the following functions would have only odd powers of x in its Taylor series expansion about the point x = 0 ?
 A $sin(x^{3})$ B $sin(x^{2})$ C $cos(x^{3})$ D $cos(x^{2})$
Engineering Mathematics   Calculus
Question 5 Explanation:
$\begin{array}{l} \sin x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\ldots . \\ \cos x=1-\frac{x^{2}}{2}+\frac{x^{4}}{4}-\frac{x^{6}}{6}+\ldots . \end{array}$
 Question 6
Which of the following is a solution to the differential equation $\frac{dx(t)}{dt}+3x(t)=0$ ?
 A $x(t)=3e^{-t}$ B $x(t)=2e^{-3t}$ C $x(t)=-\frac{3}{2}t^{2}$ D $x(t)=3t^{2}$
Engineering Mathematics   Differential Equations
Question 6 Explanation:
$\begin{array}{rr} & (D+3) x(t)=0 \\ \Rightarrow \quad & D=-3 \\ \text { So, } & x(t)=C e^{-3 t} \end{array}$
 Question 7
In the following graph, the number of trees (P) and the number of cut-set (Q) are A P = 2,Q = 2 B P = 2,Q = 6 C P = 4,Q = 6 D P = 4,Q = 10
Network Theory   Graph Theory and State Equations
Question 7 Explanation:
Different trees (P) are shown below Different cut-sets (Q) are shown below: Question 8
In the following circuit, the switch S is closed at t = 0. The rate of change of current $\frac{di}{dt}(0^{+})$ is given by A 0 B $\frac{R_{s}I_{s}}{L}$ C $\frac{(R+R_{s})I_{s}}{L}$ D $\infty$
Network Theory   Transient Analysis
Question 8 Explanation:
At t = o, the inductor behaves as an open circuit
\begin{aligned} \text { So, } \quad V_{L}&=I_{S} R_{S} \\ \qquad V_{L}&=L \frac{d i}{d t}\left(0^{+}\right) \\ \Rightarrow \quad \frac{d i}{d t}\left(0^{+}\right)&=\frac{V_{L}}{L}=\frac{I_{s} R_{S}}{L} \end{aligned}
 Question 9
The input and output of a continuous time system are respectively denoted by x(t) and y(t). Which of the following descriptions corresponds to a causal system ?
 A y(t) = x(t - 2) + x(t + 4) B y(t) = (t - 4)x(t + 1) C y(t) = (t + 4)x(t - 1) D y(t) = (t + 5)x(t + 5)
Signals and Systems   Basics of Signals and Systems
Question 9 Explanation:
A system is casual if the output at any time depends only on values of the input at the present time and in the past.
 Question 10
The impulse response h(t) of a linear time invariant continuous time system is described by h(t) = exp($\alpha$t)u(t) + exp($\beta$t)u(-t), where u(-t) denotes the unit step function, and $\alpha$ and $\beta$ are real constants. This system is stable if
 A $\alpha$ is positive and $\beta$ is positive B $\alpha$ is negative and $\beta$ is negative C $\alpha$ is positive and $\beta$ is negative D $\alpha$ is negatine and $\beta$ is positive
Signals and Systems   LTI Systems Continuous and Discrete
Question 10 Explanation:
$h(t)=e^{\alpha t} u(t)+e^{\beta t} u(-t)$
For the system to be stable, $\int_{-\infty}^{\infty} h(t) d t \lt \infty$
For the above condition, h(t) should be as shown below. There are 10 questions to complete.