# GATE EE 2015 SET 2

 Question 1
Given f(z)=g(z)+h(z), where f, g, h are complex valued functions of a complex variable z. Which one of the following statements is TRUE?
 A If f(z) is differentiable at $z_0$, then g(z) and g(z) are also differentiable at $z_0$. B If g(z) and h(z) are differentiable at $z_0$, then f(z) is also differentiable at $z_0$. C If f(z) is continuous at $z_0$, then it is differentiable at $z_0$. D If f(z) is differentiable at $z_0$, then so are its real and imaginary parts.
Engineering Mathematics   Complex Variables
 Question 2
We have a set of 3 linear equations in 3 unknowns. '$X \equiv Y$' means X and Y are equivalent statements and '$X\not\equiv Y$' means X and Y are not equivalent statements.

P: There is a unique solution.
Q: The equations are linearly independent.
R: All eigenvalues of the coefficient matrix are nonzero.
S: The determinant of the coefficient matrix is nonzero.

Which one of the following is TRUE?
 A $P\equiv Q\equiv R\equiv S$ B $P\equiv R\not\equiv Q\equiv S$ C $P\equiv Q\not\equiv R\equiv S$ D $P\not\equiv Q\not\equiv R\not\equiv S$
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
\begin{aligned} a_{11}x_1+a_{12}x_2+a_{13}x_3 &=b_1 \\ a_{21}x_1+a_{22}x_2+a_{23}x_3 &=b_2 \\ a_{31}x_1+a_{32}x_2+a_{33}x_3 &=b_3 \\ \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22}& a_{21}\\ a_{31}& a_{32} &a_{33} \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}&= \begin{bmatrix} b_1\\ b_2\\ b_3 \end{bmatrix} \end{aligned}
if $|A|\neq 0$ then AX=B can be written as $X=A^{-1}B$
If $|A|\neq 0$ then $\lambda _1 \cdot \lambda _2\cdot \lambda _3\neq 0$ each $\lambda _i$ is non-zero.
If $|A|\neq 0$ then all the row (column) vectors of A are linearly independent.

 Question 3
Match the following. A P-2 Q-1 R-4 S-3 B P-4 Q-1 R-3 S-2 C P-4 Q-3 R-1 S-2 D P-3 Q-4 R-2 S-1
Electromagnetic Fields   Coordinate Systems and Vector Calculus
Question 3 Explanation:
Stokes theorem $\oint \vec{A}\cdot dl=\int \int (\triangledown \times A)\cdot \hat{n}ds$
Gauss theorem $\int \int D\cdot ds=Q$
Divergence theorem $\oint \oint A\cdot \hat{n}ds=\int \int \int \triangledown \cdot \bar{A}dV$
Cauchy integral theorem $\oint _cf(z)dz=0$
 Question 4
The Laplace transform of $f(t)=2\sqrt{t/\pi }$ is $s^{-3/2}$. The Laplace transform of $g(t)=\sqrt{1/\pi t}$ is
 A $3s^{-5/2}/2$ B $s^{-1/2}$ C $s^{1/2}$ D $s^{3/2}$
Signals and Systems   Laplace Transform
Question 4 Explanation:
Given that,
$f(t)=2\sqrt{\frac{t}{\pi}}\rightleftharpoons F(s)=s^{-3/2}$
By using property of differentiation In time,
\begin{aligned} \frac{df(t)}{dt}&\rightleftharpoons sF(s) \\ \Rightarrow \; \frac{2}{\sqrt{\pi}}\cdot \frac{1}{2}t^{-1/2}&\rightleftharpoons sF(s) \\ \Rightarrow \; \frac{1}{\sqrt{\pi t}}&\rightleftharpoons s\cdot s^{-3/2} \\ \Rightarrow \; \frac{1}{\sqrt{\pi t}}&\rightleftharpoons s^{-1/2} \end{aligned}
 Question 5
Match the following. A P-1 Q-2 R-1 S-3 B P-1 Q-2 R-1 S-3 C P-1 Q-2 R-3 S-3 D P-3 Q-1 R-2 S-1
Electrical and Electronic Measurements   Galvanometers, Voltmeters and Ammeters

There are 5 questions to complete.