PYQs Based Test - 18 : Engineering Mathematics

The maximum area (in square units) of a rectangle whose vertices lie on the ellipse x² + 4y² = 1 is ____________________.

Consider the following system of linear equations,

x₁ + 2x₂ = b₁

2x1 + 4x2 = b2

3x1 + 7x2 = b3

3x1 + 9x2 = b4

Which one of the following conditions ensures that a solution exists for the above system?

Consider a channel over which either symbol x${}_A$ or symbol x${}_B$ is transmitted. Let the output of channel Y be the input to a maximum likelihood (ML) detector at the receiver. The conditional probability density functions for 𝑌 given x${}_A$ and x${}_B$ are:

${\mathrm{f}}_{Y|{\mathrm{x}}_{\mathrm{A}}}(\mathrm{y})={\mathrm{e}}^{-(\mathrm{y}+1)}\mathrm{u}(\mathrm{y}+1)$

${\mathrm{f}}_{Y|{\mathrm{x}}_{\mathrm{B}}}(\mathrm{y})={\mathrm{e}}^{(\mathrm{y}-1)}(1-\mathrm{u}(\mathrm{y}-1))$

where u(⋅) is the standard unit step function. The probability of symbol error for this system is _________ (rounded off to two decimal places).

A simple closed path 𝐶 in the complex plane is shown in the figure. If

$\oint \frac{2^z}{z^2-1}dz=-i\pi A$

where i = √-1 then the value of A is _______ (rounded off to two decimal places)

Consider the following partial differential equation (PDE)

$\mathrm{a}\frac{{\partial }^2\mathrm{f}(\mathrm{x},\mathrm{y})}{\partial {\mathrm{x}}^2}+\mathrm{b}\frac{{\partial }^2\mathrm{f}(\mathrm{x},\mathrm{y})}{\partial {\mathrm{y}}^2}=\mathrm{f}(\mathrm{x},\mathrm{y})$

where a and b are distinct positive real numbers. Select the combination(s) of values of the real parameters 𝜉 and 𝜂 such that f(x, y) = e${}^{(\text{ξ}x+ny)}$ is a solution of the given PDF.