GATE ME 2015 SET-2

 Question 1
At least one eigenvalue of a singular matrix is
 A positive B zero C negative D imaginary
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
For singular matrix
$|A|=0$
According to properties of eigen value
Product of eigen values $=|A|=0$
$\Rightarrow$ Atleast one of the eigen value is zero.
 Question 2
At x = 0, the function $f(x)=\left | x \right |$ has
 A a minimum B a maximum C a point of inflexion D neither a maximum nor minimum
Engineering Mathematics   Calculus
Question 2 Explanation:
The graph of |x| is

from the graph we can say that
|x| has minimum at x=0

 Question 3
Curl of vector $V(x,y,z)= 2x^{2}i+3z^{2}j+y^{3}k$ at $x=y=z=1$ is
 A $-3i$ B $3i$ C $3i-4j$ D $3i-6k$
Engineering Mathematics   Calculus
Question 3 Explanation:
Curl of vector$=\left|\begin{array}{ccc} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 2 x^{2} & 3 z^{2} & y^{3} \end{array}\right|$
$=i\left[\frac{\partial}{\partial x}\left(y^{3}\right) \frac{\partial}{\partial z}\left(3 z^{2}\right)\right]+j\left[\frac{\partial}{\partial x}\left(y^{3}\right) \frac{\partial}{\partial z}\left(2 x^{2}\right)\right]$
$+k\left[\frac{\partial}{\partial x}\left(3z^{2}\right) \frac{\partial}{\partial z}\left(2 x^{2}\right)\right]$
$=i\left[3 y^{2}-6 z\right]-[10]+k[0+0]$
$\text { At } x=1, y=1 \text { and } z=1$
$\text { Curl }=i\left(3 \times 1^{2}-6 \times 1\right)=-3 i$
 Question 4
The Laplace transform of $e^{i5t}$ where $i=\sqrt{-1}$ is
 A $(s-5i)/(s^{2}-25)$ B $(s+5i)/(s^{2}+25)$ C $(s+5i)/(s^{2}-25)$ D $(s-5i)/(s^{2}+25)$
Engineering Mathematics   Differential Equations
Question 4 Explanation:
\begin{aligned} e^{j 5 t} &=\cos 5 t+i \sin 5 t \\ L\left\{e^{(i 5 f)}\right\}&=\frac{s}{s^{2}+25}+\frac{5 i}{s^{2}+25} \\ &=\frac{s+5 i}{s^{2}+25} 2 e^{-x^{2}} \end{aligned}
 Question 5
Three vendors were asked to supply a very high precision component. The respective probabilities of their meeting the strict design specifications are 0.8, 0.7 and 0.5. Each vendor supplies one component. The probability that out of total three components supplied by the vendors, at least one will meet the design specification is ___________
 A 0.12 B 0.97 C 0.65 D 1
Engineering Mathematics   Probability and Statistics
Question 5 Explanation:
Probability of atleast one meet the specification
\begin{aligned} &=1-(\bar{A} \cap \bar{B} \cap \bar{C}) \\ &=1-(0.2 \times 0.3 \times 0.5) \\ &=0.97 \end{aligned}

There are 5 questions to complete.