# GATE EC 2015 SET-1

 Question 1
Consider a system of linear equations:
x - 2 y + 3z = -1,
x - 3y + 4z = 1, and
-2x + 4 y - 6z = k.
The value of k for which the system has infinitely many solutions is ______.
 A 1 B 2 C 3 D 4
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
\begin{aligned} x-2 y+3 z&=-1 \\ x-3 y+1 z&=1, \text { and } \\ -2 x+1 y-6 z&= k \\ \left[A:B \right]&=\left[\begin{array}{rrrrr} 1 & -2 & 3 :& -1 \\ 1 & -3 & 4 :& 1 \\ -2 & 4 & 6 :& k \end{array}\right]\\ R_{2} \rightarrow R_{2}-R_{1}, R_{3} &\rightarrow R_{1}+2 R_{1} \\ \left[\begin{array}{cccc} 1 & -2 & 3 :& -1 \\ 0 & -1 & 1 :& 2 \\ 0 & 0 & 0 :& k-2 \end{array}\right] \end{aligned}
For Infinlto may solulion
\begin{aligned} \rho(A: B)&=p(A)\\ &= r \lt \text{ number of variables}\\ \rho(A: B)&=2 \\ k-2&=0 \\ k&=2 \end{aligned}
 Question 2
A function $f(x) =1- x^{2}+x^{3}$ is defined in the closed interval [-1,1]. The value of x , in the open interval (-1,1) for which the mean value theorem is satisfied, is
 A $-1/2$ B $-1/3$ C $1/3$ D $1/2$
Engineering Mathematics   Calculus
Question 2 Explanation:
Since $f(1)\neq f(-1)$ Rolle's mean value theorem does not apply
By Lagrange moon voluo theorem
\begin{aligned} f^{\prime}(x) &=\frac{((1)-f(-1)}{1-(-1)}=\frac{2}{2}=1 \\ -2 x+3 x^{2} &=1 \\ x &=1-\frac{1}{3}\\ x \text{ lies in }(-1,1) \\ \Rightarrow \quad x&=-\frac{1}{3} \end{aligned}

 Question 3
Suppose A and B are two independent events with probabilities $P(A) \neq 0$ and $P(B) \neq 0$. Let $\bar{A}$ and $\bar{B}$ be their complements. Which one of the following statements is FALSE?
 A $P(A\cap B)=P(A)P(B)$ B $P(A|B)=P(A)$ C $P(A\cup B)=P(A) + P(B)$ D $P(\bar{A}\cup \bar{B})=P(\bar{A}) + P(\bar{B})$
Engineering Mathematics   Probability and Statistics
Question 3 Explanation:
$P(A \cup B)=P(A)+P(B)-P(A \cap B)$
since $P(A \cap B)=P(A) \rho(B)$
(not necessarily equal to zero).
So, $P(A \cup B)=P(A)+P(B)$ is false.
 Question 4
Let z = x + iy be a complex variable. Consider that contour integration is performed along the unit circle in anticlockwise direction. Which one of the following statements is NOT TRUE?
 A The residue of $\frac{z}{z^{2}-1}$ at z=1 is 1/2 B $\oint_{c}z^{2}dz=0$ C $\frac{1}{2\pi i} \oint_{c} \frac{1}{z} dz=1$ D $\bar{z}$ (complex conjugate of z ) is an analytical function
Engineering Mathematics   Complex Analysis
Question 4 Explanation:
$f(z)= \bar{z} =x-i y$
$\begin{array}{cc} u=x & v=-y \\ \Rightarrow u_{x}=1 & v_{x}=0 \\ u_{y}=0 & v_{y}=-1 \end{array}$
$u_{x} \neq v_{y}$ i.e. C-R not satisfied
$\Rightarrow \bar{z}$ is not analytic function.
 Question 5
The value of p such that the vector $\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$ is an eigenvector of the matrix $\begin{bmatrix} 4 & 1& 2\\ p& 2& 1\\ 14&-4 &10 \end{bmatrix}$ is _______.
 A 12 B 17 C 20 D 22
Engineering Mathematics   Linear Algebra
Question 5 Explanation:
$\begin{array}{r} A x=\lambda N \\ {\left[\begin{array}{lll} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 \end{array}\right]\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]=\lambda\left[\begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right]} \\ {\left[\begin{array}{c} 12 \\ p+7 \\ 36 \end{array}\right]=\lambda\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]} \\ \frac{p+7}{12}=2 \Rightarrow p=17 \end{array}$

There are 5 questions to complete.