# GATE EC 2014 SET-2

 Question 1
The determinant of matrix A is 5 and the determinant of matrix B is 40. The determinant of matrix AB is ______.
 A 200 B 100 C 50 D 45
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Determinant of A=5
Determinant of B=40
Determinant of AB=|A||B|
$\begin{array}{l} \quad=5 \times 40 \\ \quad=200 \end{array}$
 Question 2
Let X be a random variable which is uniformly chosen from the set of positive odd numbers less than 100. The expectation, E [X], is _____.
 A 100 B 50 C 25 D 10
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
$E[X]=\frac{1+2+3+\cdots 99}{50}=\frac{2500}{50}=50$

 Question 3
For $0\leq t \leq \infty$ , the maximum value of the function $f(t) =e^{-t}-2e^{-2t}$ occurs at
 A $t=log_{e}4$ B $t=log_{e}2$ C t=0 D $t=log_{e}8$
Engineering Mathematics   Calculus
Question 3 Explanation:
\begin{aligned} f(t)&=e^{-t}-2 e^{-2 t} \\ f^{\prime}(t)&=-e^{-t}+4 e^{-2 t} \\ \text{For maximum value} &P(t)=0 \\ f^{\prime}(t)&=0=-e^{-t}+4 e^{-2 t} \\ \Rightarrow \quad 4 e^{-2 t}&=e^{t} \\ 4 e^{t}&=1 \\ \therefore \quad t&=\log _{e} 4 \end{aligned}
 Question 4
The value of
$\lim_{x\rightarrow \infty }(1+\frac{1}{x})^{x}$
is
 A ln 2 B 1 C e D $\infty$
Engineering Mathematics   Calculus
Question 4 Explanation:
$\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e^{\lim _{x \rightarrow \infty} \frac{1}{x} \cdot x}=e^{1}=e$
 Question 5
If the characteristic equation of the differential equation
$\frac{d^{2}y}{dx^{2}}+2\alpha \frac{dy}{dx}+y=0$
has two equal roots, then the values of a are
 A $\pm 1$ B 0,0 C $\pm j$ D $\pm 1/2$
Engineering Mathematics   Differential Equations
Question 5 Explanation:
$\frac{d^{2} y}{d x^{2}}+2 \alpha\frac{d y}{d x}+y=0$
The characteristic equation is given as
\begin{aligned} \left(m^{2}+2(x)+1\right) &=0 \\ m_{1}, m_{2} &=\frac{-2 x_{1} \pm \sqrt{4 x^{2}-4}}{2} \end{aligned}\\ \text{since both roots are equal i.e.} \\ \begin{aligned} m_{1}=& m_{2} \\ \frac{-2 \alpha+\sqrt{4 \alpha^{2}-4}}{2} &=\frac{-2\alpha \cdot-\sqrt{4 a^{2}-4}}{2} \\ \sqrt{4\left(1^{2}-4\right.} &=-\sqrt{4 c^{2}-4} \\ 2 \sqrt{4 c^{2}-4} &=0 \\ 4 \alpha^{2}-4 &=0 \\ \alpha^{2} &=1 \\ \alpha &=\pm 1 \end{aligned}

There are 5 questions to complete.