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\geq0, 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.

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