Question 1 |
Which one of the following statements is true for all real symmetric matrices?
All the eigenvalues are real. | |
All the eigenvalues are positive. | |
All the eigenvalues are distinct. | |
Sum of all the eigenvalues is zero. |
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____.
0.1 | |
0.33 | |
0.14 | |
0.66 |
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
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
-\infty | |
0 | |
1 | |
\infty |
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.
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
purely imaginary | |
real and non-negative | |
on the unit circle | |
equal in real and imaginary parts |
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.
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 ?
e^{x} | |
x^{2} | |
1/x | |
ln x |
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.
One independent solution is \frac{1}{x}
Another independent solution is x.
There are 5 questions to complete.