List of practice Questions

Let a relation $ R $ on $ \mathbb{N} \times \mathbb{N} $ be defined as: $$(x_1, y_1) \, R \, (x_2, y_2) \text{ if and only if } x_1 \leq x_2 \text{ or } y_1 \leq y_2.$$
Consider the two statements:
[(I)] $ R $ is reflexive but not symmetric.
[(II)] $ R $ is transitive.
Then which one of the following is true:

The sum of all rational terms in the expansion of $$ \left( \frac{1}{2^5} + \frac{1}{5^3} \right)^{15} $$ is equal to:

If $ A $ and $ B $ are two non-zero square matrices of the same order such that: $$ (A + B)^2 = A^2 + B^2, $$ then:

Evaluate: $$ \lim_{n \to \infty} \frac{1}{n^{k+1}} \left[ 2^k + 4^k + 6^k + \dots + (2n)^k \right]. $$

Find the general solution of the differential equation: $$ (xy + y^2)dx - (x^2 - 2xy)dy = 0. $$ is

Which of the following causal analog transfer functions is used to design causal IIR digital filter with transfer function? $$ H(z) = \frac{0.05z}{z-e^{-0.42}} + \frac{0.05z}{z-e^{-0.2}} $$ Assume impulse invariance transformation with $ T = 0.1 \, \text{s} $.

In this question, there are three statements followed by conclusions numbered I and II. You have to take the given statements to be true even if they seem to be at variance from commonly known facts and then decide which of the given conclusions logically follow from the three statements.
Statements:

All books are ledgers.
All pens are keys.
Some pens are books.

Conclusions:

I. Some ledgers are keys.
II. Some keys are books.

If the lines $ \frac{x-1}{2} = \frac{y-2}{\alpha} = \frac{z-3}{2} $ and $ \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-3}{-2} $ are perpendicular, then the value of $ \alpha $ is:

Let $ f : \mathbb{R} \rightarrow \mathbb{R} $ be a function defined by
$$ f(x) = \frac{x}{(1 + x^4)^{1/4}} $$
and $ g(x) = f(f(f(x))) $. Then
$$ 18 \int_{\sqrt[3]{\frac{8}{3}}}^{\sqrt[3]{\frac{4}{3}}} x^3 g(x) \, dx $$
equals:

The inverse Laplace transform of $$ \frac{1}{2s^2 + 3s + 1} $$

If a real-valued function $$ f(x) = \begin{cases} \frac{2x^2 + k(2x) + 9}{3x^2 - 7x - 6}, & \text{for } x \neq 3, \\ l, & \text{for } x = 3 \end{cases} $$ is continuous at $ x = 3 $ and $ l $ is a finite value, then $ l - k = $:

The general solution of the differential equation $$ (y^2 + x + 1) \, dy = (y + 1) \, dx $$ is:

If the differential equation $$ x \, dy + (y + y^2 x) \, dx = 0 $$ with condition $ y = 1 $ at $ x = 1 $, then the solution is:

If \[\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} \, dx = \frac{1}{12} \tan^{-1}(3 \tan x) + \text{constant},\]then the maximum value of $a \sin x + b \cos x$ is:

If $$ y = \tan^{-1} \left[ \frac{\log_e\left(\frac{e}{x}\right)}{\log_e(e x^2)} \right] + \tan^{-1} \left[ \frac{3 + 2\log_e x}{1 - 6\cdot\log_e x} \right], $$ then $ \frac{d^2y}{dx^2} = ? $

Given below are two statements:
Statement I: When the speed of liquid is zero everywhere, the pressure difference at any two points depends on the equation $$ P_1 - P_2 = \rho g (h_2 - h_1). $$ Statement II: In the ventury tube shown, $$ 2gh = v_1^2 - v_2^2. $$

In the light of the above statements, choose the most appropriate answer from the options given below.

The shortest distance between the lines
$$ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} $$ and
$$ \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5} $$ is:

If the circles $ S = x^2 + y^2 - 14x + 6y + 33 = 0 $ and $ S' = x^2 + y^2 - a^2 = 0 $ (where $ a \in \mathbb{N} $) have 4 common tangents, then the possible number of values of $ a $ is:

If $$ y = 1 + x + x^2 + x^3 + \dots \quad \text{and} \quad |x| < 1, \text{ then } y'' = $$

Rank the following functions by order of growth (Highest running time to lowest running time):
1. $n \times 3^n$
2. $2^{\log(n)}$
3. $n^{2/3}$
4. $4^n$
5. $4^{\log(n)}$
Choose the correct answer from the options given below:

The following system of equations: $$ x_1 + x_2 + x_3 = 1, \quad x_1 + 2x_2 + 3x_3 = 2, \quad x_1 + 4x_2 + \alpha x_3 = 4 $$ has a unique solution. Possible value(s) for $\alpha$ is/are:

Let $ \alpha, \beta $ be the distinct roots of the equation $$ x^2 - (t^2 - 5t + 6)x + 1 = 0, \, t \in \mathbb{R} \, \text{and} \, a_n = \alpha^n + \beta^n. $$ Then the minimum value of $ \frac{a_{2023} + a_{2025}}{a_{2024}} $ is:

Evaluate the limit: $$ \lim_{x \to 4} \left( \frac{1}{x - 4} - \frac{5}{x^2 - 3x - 4} \right) $$ is equal to

If the equation of the circle whose radius is 3 units and which touches internally the circle $$ x^2 + y^2 - 4x - 6y - 12 = 0 $$ at the point $(-1, -1)$ is $$ x^2 + y^2 + px + qy + r = 0, $$ then $p + q - r$ is:

The integral $$ \int \frac{x^8 - x^2}{(x^{12} + 3x^6 + 1) \tan^{-1}\left( \frac{x^3 + 1}{x^3} \right)} \, dx $$ is equal to: