PYQs Based Test - 17 : Engineering Mathematics

A real (4 × 4) matrix A satisfies the equation A² = I, where I is a (4 × 4) identity matrix. The positive eigenvalue of A is ________.

Let α, β be two non-zero real numbers and v₁, v₂ be two non-zero real vectors of size 3 × 1. Suppose that v₁ and v₂ satisfy $v_1^T{v}_2=0,v_1^T{v}_1=1$, and $v_2^T{v}_2=1$. Let A be the 3 × 3 matrix given by:

$\mathrm{A}=\alpha {\mathrm{v}}_1{\mathrm{v}}_1^{\mathrm{T}}+\beta {\mathrm{v}}_2{\mathrm{v}}_2^{\mathrm{T}}$

The eigenvalues of A are _______

The integral $\underset{0}{\overset{1}{\int }}\frac{\mathrm{dx}}{\sqrt{1-\mathrm{x}}}$ is equal to _______

Consider the following series:

${\Sigma }_{n=1}^{\infty }\frac{n^d}{c^n}$

For which of the following combinations of c, d values does this series converge?

The value of the contour integral $\frac{1}{2\pi j}\oint {\left(z+\frac{1}{z}\right)}^{2}dz$ evaluated over the unit circle |z| = 1 is _______.