# GATE ME 2016 SET-2

 Question 1
The condition for which the eigenvalues of the matrix
$A=\begin{bmatrix} 2&1 \\ 1& k \end{bmatrix}$
are positive, is
 A k$\gt$1/2 B k$\gt$-2 C k$\gt$0 D k$\lt$-1/2
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
All Eigen values of $A=\left[\begin{array}{ll}2 & 1 \\ 1 & k\end{array}\right]$ are positive
$2>0$
$\therefore 2 \times 2$leading minor must be greater than zero
$\left|\begin{array}{ll}2 & 1 \\ 1 & k\end{array}\right|>0$
$2 k-1>0$
$2 k>1$
$k>\frac{1}{2}$
 Question 2
The values of x for which the function
$f( x)=\frac{x^{2}-3x-4}{x^{2}+3x-4}$
is NOT continuous are
 A 4 and -1 B 4 and 1 C -4 and 1 D -4 and -1
Engineering Mathematics   Calculus
Question 2 Explanation:
$f(x)=\frac{x^{2}-3 x-4}{x^{2}+3 x-4}$ is not continous
when
\begin{aligned} x^{2}+3 x-4 &=0 \\ (x+4)(x-1) &=0 \\ x &=-4,1 \end{aligned}

 Question 3
Laplace transform of $cos(\omega t)$ is
 A $\frac{s}{s^{2}+\omega ^{2}}$ B $\frac{\omega }{s^{2}+\omega ^{2}}$ C $\frac{s}{s^{2}\omega ^{2}}$ D $\frac{\omega }{s^{2}-\omega ^{2}}$
Engineering Mathematics   Calculus
Question 3 Explanation:
$L(\cos \omega t)=\frac{s}{s^{2}+\omega^{2}}$
 Question 4
A function of the Complex Variables z=x+iy, is given as f(x,y)=u(x,y)+i v(x,y) , where u(x,y)=2kxy and v(x,y)=$x^{2}-y^{2}$. The value of k, for which the function is analytic, is _____
 A -5 B -1 C -8 D 1
Engineering Mathematics   Complex Variables
Question 4 Explanation:
Given that $( f(z)=u+i v)$ is analytic
\begin{aligned} u(x, y) &=2 k x y & & v=x^{2}-y^{2} \\ u_{x} &=2 k y & & v_{y}=-2 y \\ u_{x} &=v_{y} & & \\ k &=-1 & & v_{x}=2 x \\ u_{y} &=2 k x & & & \\ u_{y} &=-v_{x} & & & \\ 2 k x &=-2 x & & & \\ k &=-1 & & \end{aligned}
 Question 5
Numerical integration using trapezoidal rule gives the best result for a single variable function
 A linear B parabolic C logarithmic D hyperbolic
Engineering Mathematics   Numerical Methods
Question 5 Explanation:
Trapezoidal rule gives the best result in single variable function when the function is linear (degree 1)

There are 5 questions to complete.