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.