# GATE EE 2014 SET 2

 Question 1
Which one of the following statements is true for all real symmetric matrices?
 A All the eigenvalues are real. B All the eigenvalues are positive. C All the eigenvalues are distinct. D Sum of all the eigenvalues is zero.
Engineering Mathematics   Linear Algebra
 Question 2
Consider a dice with the property that the probability of a face with n dots showing up proportional to n. The probability of the face with three dots showing up is____.
 A 0.1 B 0.33 C 0.14 D 0.66
Engineering Mathematics   Probability and Statistics
Question 2 Explanation:
Let probability of occurence of one dot is P.
So, writing total probability
P+2P+3P+4P+5P+6P=1
$P=\frac{1}{21}$
Hence, problem of occurrence of 3 dot is $=3P=\frac{3}{21}=\frac{1}{7}=0.142$

 Question 3
Minimum of the real valued function $f(x)=(x-1)^{2/3}$ occurs at x equal to
 A $-\infty$ B 0 C 1 D $\infty$
Engineering Mathematics   Calculus
Question 3 Explanation:
$f(x)=(x-1)^{2/3}=(\sqrt{x-1})^2$
As f(x) is square of $\sqrt{x-1}$, hence its minimum value be 0 where at x=1.
 Question 4
All the values of the multi-valued complex function $1^i$, where $i=\sqrt{-1}$, are
 A purely imaginary B real and non-negative C on the unit circle D equal in real and imaginary parts
Engineering Mathematics   Complex Variables
Question 4 Explanation:
Let $z=1^i=1^{e^{i(4n+1)\pi/2}}\;\;\;n \in I$
z=1 which is purly real and non negative.
 Question 5
Consider the differential equation $x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-y=0$. Which of the following is a solution to this differential equation for $x \gt 0$ ?
 A $e^{x}$ B $x^{2}$ C 1/x D ln x
Engineering Mathematics   Differential Equations
Question 5 Explanation:
\begin{aligned} x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y&=0\\ \text{Let, }x=e^z\leftrightarrow z&=\log x\\ s\frac{d}{dx}=xD=\theta &=\frac{d}{dz}\\ x^2D^2&=\theta (\theta -1)\\ (x^2D^2+xD-1)y&=0\\ [\theta (\theta -1)+\theta -1]y&=0\\ (\theta ^2-\theta +\theta -1)&=0\\ (\theta ^2-1)y&=0\\ \text{Auxiliary equation is }m^2-1&=0\\ m&=\pm 1\\ \text{CF is }C_1e^{-z}+C_2e^z&\\ \text{Solution is }y&=C_1e^{-z}+C_2e^z\\ y&=C_1x^{-1}+C_2x\\ y&=C_1\frac{1}{x}+C_2x \end{aligned}
One independent solution is $\frac{1}{x}$
Another independent solution is x.

There are 5 questions to complete.