List of top Mathematics Questions

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 
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: 
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 \( \frac{d^2y}{dx^2} = ? \) 
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 $$ y = 1 + x + x^2 + x^3 + \dots \quad \text{and} \quad |x| < 1, \text{ then } y'' = $$
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: 
Consider the line $L$ passing through the points $(1, 2, 3)$ and $(2, 3, 5)$. The distance of the point $$ \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) $$ from the line $L$ along the line $$ \frac{3x - 11}{2} = \frac{3y - 11}{1} = \frac{3z - 19}{2} $$ is equal to: 
Let \( C: x^2 + y^2 = 4 \) and \( C': x^2 + y^2 - 4\lambda x + 9 = 0 \) be two circles. If the set of all values of \( \lambda \) such that the circles \( C \) and \( C' \) intersect at two distinct points is \( R = [a, b] \), then the point \( (8a + 12, 16b - 20) \) lies on the curve:
Let \( P \) be a point on the hyperbola \( H: \frac{x^2}{9} - \frac{y^2}{4} = 1 \), in the first quadrant such that the area of the triangle formed by \( P \) and the two foci of \( H \) is \( 2 \sqrt{13} \). Then, the square of the distance of \( P \) from the origin is
Let \(f : \mathbb{R} \to \mathbb{R}\) be given by \(f(x) = |x^2 - 1|\), then:
\(f(x) = \cos x - 1 + \frac{x^2}{2!}, \, x \in \mathbb{R}\)
Then \(f(x)\) is:
Let \(f : \mathbb{R} \to \mathbb{R}\) be a differentiable function and \(f(1) = 4\). Then the value of
\[ \lim_{x \to 1} \int_{4}^{f(x)} \frac{2t}{x - 1} \, dt \]
If \(\int \frac{\log(x + \sqrt{1 + x^2})}{1 + x^2} \, dx = f(g(x)) + c\), then:
The equation \(2x^5 + 5x = 3x^3 + 4x^4\) has: