Question 1 |

For matrices of same dimension M, N and scalar c, which one of these properties DOES NOT ALWAYS hold ?

(M^{T})^{T}=M | |

(cM)^{T}=c(M)^{T} | |

(M+N)^{T}=M^{T}+N^{T} | |

MN=NM |

Question 1 Explanation:

Matrix multiplication is not commutative.

Question 2 |

In a housing society, half of the families have a single child per family, while the
remaining half have two children per family. The probability that a child picked
at random, has a sibling is _____

1.5 | |

0.67 | |

2 | |

3 |

Question 2 Explanation:

Required probability

=\frac{\frac{1}{2} \times \frac{2}{3}}{\frac{1}{2} \times \frac{1}{3}+\frac{1}{2} \times \frac{2}{3}}=\frac{2}{3}=0.666

=\frac{\frac{1}{2} \times \frac{2}{3}}{\frac{1}{2} \times \frac{1}{3}+\frac{1}{2} \times \frac{2}{3}}=\frac{2}{3}=0.666

Question 3 |

C is a closed path in the z -plane by |z| = 3. The value of the integral \oint_{c}(\frac{z^{2}-z+4j}{z+2j})dz is

-4\pi (1+ j2 ) | |

4\pi (3- j2 ) | |

-4\pi (3+ j2 ) | |

4\pi (1- j2) |

Question 3 Explanation:

\oint_{c} \frac{z^{2}-z+4 j}{z+2 j}

Pole =z=-2 j

which is inside of |z|=3

From Cauchy integral formula

\begin{aligned} \oint \frac{z^{2}-z+4 j}{z+2 j} &=2 \pi i\left[\lim _{z \rightarrow-2 j} z^{2}-z+4 j\right] \\ &=2 \pi j[-4+2 j+4 j] \\ &=2 \pi j[-4+6 j] \\ &=-4 \pi[2 j+3] \end{aligned}

Question 4 |

A real (4x4) matrix A satisfies the equation A_{2} = I, where I is the (4x4)
identity matrix. The positive eigen value of A is _____.

1 | |

2 | |

3 | |

4 |

Question 4 Explanation:

\begin{aligned} \text{since, }A^{2}=l, \text{eig}\left(A^{2}\right)&=\text{eig}(I)=1 \\ \Rightarrow \quad \text{eig}(A)^{2}&=1\\ \Rightarrow \quad \text{eig}(A)&=\pm 1 \end{aligned}

Therefore, the positive eigen value of A is +1.

Therefore, the positive eigen value of A is +1.

Question 5 |

Let X_{1} , X_{2}, \; and \; X_{3} be independent and identically distributed random variables with the uniform distribution on [0,1]. The probability P{X_{1} is the largest} is

0.5 | |

0.33 | |

0.25 | |

0.75 |

Question 5 Explanation:

If multiple independent random variables are uniformly distributed in the same interval then each random variable will have equal chances to be largest and to be lowest.

P\left(X_{1} \text { is the largest) }=\frac{1}{3}\right.

P\left(X_{1} \text { is the largest) }=\frac{1}{3}\right.

There are 5 questions to complete.