PYQs Based Test - 10 : Engineering Mathematics

A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain a unique solution by multiplying both left and right sides of the equation by A${}^T$ (the super script T denotes the transpose) and inverting the matrix A${}^T$ A? Answer is

Let $f\left(z\right)=\frac{1}{z+a},a>0$. The value of the integral ∮ f(z) dz over a circle C with center (-a, 0) and radius R > 0 evaluated in the anti-clock wise direction is

Consider a sequence of tossing of a fair coin where the outcomes of tosses are independent. The probability of getting the head for the third time in the fifth toss is

The double integral $\underset{0}{\overset{a}{\int }}\underset{0}{\overset{y}{\int }}f\left(x,y\right)$ dx dy is equivalent to

The function p(x) is given by p(x) = A/x${}^{\mu }$ where A and μ are constants with μ > 1 and 1 < x < ∞ and p (x) = 0 for -∞ < x < 1. For p(x) to be a probability density function, the value of A should be equal to