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[3]{x-1})^2
As f(x) is square of \sqrt[3]{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.