# GATE ME 2016 SET-1

 Question 1
The solution to the system of equations
$\begin{bmatrix} 2&5 \\ -4&3 \end{bmatrix} \begin{Bmatrix} x\\y \end{Bmatrix} = \begin{Bmatrix} 2\\-30 \end{Bmatrix}$
 A 6,2 B -6,2 C -6,-2 D 6,-2
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
${\left[\begin{array}{cc}2 & 5 \\-4 & 3\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]=\left[\begin{array}{c}2 \\-30\end{array}\right]}$
${\left[\begin{array}{ccc}2 & 5 & 2 \\-4 & 3 & -30\end{array}\right]}$
$R_{2}+2 R_{1}$
${\left[\begin{array}{ccc}2 & 5 & 2 \\0 & 13 & -26\end{array}\right]}$
\begin{aligned} 13 y&=-26 \\ \text { or } \qquad y&=-2 \\ 2 x+5 y&=2 \\ 2 x+5(-2)&=2 \\ 2 x=2+10&=12 \\ \text{or }\qquad x&=6 \end{aligned}
 Question 2
If $f ( t )$ is a function defined for all t$\geq$0, its Laplace transform F(s) is defined as
 A $\int_{0}^{\infty }e^{st}f( t )dt$ B $\int_{0}^{\infty }e^{-st}f( t )dt$ C $\int_{0}^{\infty }e^{ist}f( t )dt$ D $\int_{0}^{\infty }e^{-ist}f( t )dt$
Engineering Mathematics   Calculus
Question 2 Explanation:
$L(f(t))=\int_{0}^{\infty} e^{-s t} f(t) d t$

 Question 3
$f( z )=u( x ,y )+iv(x,y )$ is an analytic function of Complex Variables $z=x+iy$ where $i=\sqrt{-1}$ If $u(x,y)=2xy$, then $v(x,y)$ may be expressed as
 A $-x^{2}+y^{2}+constant$ B $x^{2}-y^{2}+constant$ C $x^{2}+y^{2}+constant$ D $-(x^{2}+y^{2})+constant$
Engineering Mathematics   Complex Variables
Question 3 Explanation:
$u=2 x y$
$u_{x}=2 y \quad u_{y}=2 x$
In option (A)
$V_{x}=-2 x \quad u_{y}=-V_{x}$
$V_{y}=2 y$
C-R equation are satisfied only in option A
 Question 4
Consider a Poisson distribution for the tossing of a biased coin. The mean for this distribution is $\mu$. The standard deviation for this distribution is given by
 A $\sqrt{\mu }$ B $\mu ^{2}$ C $\mu$ D $1/\mu$
Engineering Mathematics   Probability and Statistics
Question 4 Explanation:
In poisson distribution mean = Variance
Given that mean = Variance =m
Standard deviation $=\sqrt{\text { Variance }}=\sqrt{\mu}$
 Question 5
Solve the equation $x=10\cos( x )$ using the Newton-Raphson method. The initial guess is $x=\pi /4$ The value of the predicted root after the first iteration, up to second decimal, is ________
 A 1.5639 B 2.569 C 9.241 D 7.586
Engineering Mathematics   Numerical Methods
Question 5 Explanation:
\begin{aligned} f(x) &=x-10 \cos x f\left(\frac{\pi}{4}\right) \\ &=\frac{\pi}{4}-\frac{10}{\sqrt{2}}=-6.286 \\ f^{\prime}(x) &=1+10 \sin x f^{\prime}\left(\frac{\pi}{4}\right) \\ &=1+\frac{10}{\sqrt{2}}=8.07 \\ x_{1} &=x_{0}-\frac{f\left(x_{0}\right)}{f^{\prime}\left(x_{0}\right)}=\frac{\pi}{4}-\left(\frac{-6.286}{8.07}\right) \\ &=\frac{\pi}{4}+\frac{6.286}{8.07}=1.5639 \end{aligned}

There are 5 questions to complete.