List of top Mathematics Questions

The physical quantity having the dimensions of the square root of the ratio of the kinetic energy and surface tension is
If \( Ax^3 + Bxy = 4 \) (A and B are arbitrary constants) is the general solution of the differential equation \[ F(x)\frac{d^2y}{dx^2} + G(x)\frac{dy}{dx} - 2y = 0, \] then \( F(1) + G(1) = \)
Evaluate the limit: \[ \lim_{n \to \infty} \left[ \frac{1}{n^2} \sec^2 \frac{1}{n^2} + \frac{2}{n^2} \sec^2 \frac{4}{n^2} + \frac{3}{n^2} \sec^2 \frac{9}{n^2} + .s + \frac{1}{n^2} \sec^2 1 \right] = \]
The general solution of the differential equation \( \left(x \sin \frac{y}{x} \right) dy = \left( y \sin \frac{y}{x} - x \right) dx \) is
The general solution of the differential equation \( \cos(x + y) \, dy = dx \) is
Evaluate the integral: \[ \int_0^1 x^{5/2} (1 - x)^{3/2} \, dx = \]
Evaluate the definite integral: \[ \int_0^1 \frac{2x + 5}{x^2 + 3x + 2} \, dx = \]
If \( \int \frac{x}{x \tan x + 1} \, dx = \log f(x) + k \), then \( f\left(\frac{\pi}{4}\right) = \)
\( \int x \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) \, dx \, (x>0) = \)
The area (in sq. units) of the region given by \( R = \left\{ (x, y) : \dfrac{y^2}{2} \leq x \leq y + 4 \right\} \) is
\( \int \sin^{-1}\left(\frac{x}{\sqrt{a+x}}\right) \, dx = \)
\( \int \frac{dx}{(1+\sqrt{x}) \sqrt{x-x^2}} = \)
The angle between the curves \( y^2 = x \) and \( x^2 = y \) at the point \( (1,1) \) is:
If \( f(x) = x e^{x(1-x)}, \, x \in \mathbb{R} \), then \( f(x) \) is:
If \( (a + \sqrt{2}b \cos x)(a - \sqrt{2}b \cos y) = a^2 - b^2 \), where \( a>b>0 \), then at \( \left( \dfrac{\pi}{4}, \dfrac{\pi}{4} \right) \), \( \dfrac{dy}{dx} = \)
If \( x = \sqrt{2 \cosec^{-1} t} \) and \( y = \sqrt{2 \sec^{-1} t} \), \( |t| \geq 1 \), then \( \dfrac{dy}{dx} = \)
If \( \int \frac{5 \tan x}{\tan x - 2} \, dx = a x + b \log |\sin x - 2 \cos x| + c \), then \( a + b = \)
The difference between the absolute maximum and absolute minimum values of the function \( f(x) = 2x^3 - 15x^2 + 36x - 30 \) on \( [-1, 4] \) is:
Consider the quadratic equation \( ax^2 + bx + c = 0 \), where \( 2a + 3b + 6c = 0 \) and let \[ g(x) = \frac{a x^3}{3} + \frac{b x^2}{2} + c x \] Statement-I: The given quadratic equation \( ax^2 + bx + c = 0 \) has at least one root in \( (0, 1) \). Statement-II: Rolle's theorem is applicable to \( g(x) \) on \( [0, 1] \). Then:
Equation of the plane passing through the origin and perpendicular to the planes \( x + 2y - z = 1 \) and \( 3x - 4y + z = 5 \) is
The circumradius of the triangle formed by the points \( (2, -1, 1) \), \( (1, -3, -5) \), and \( (3, -4, -4) \) is
The domain of the derivative of the function \( f(x) = \dfrac{x}{1 + |x|} \) is
Evaluate the limit: \[ \lim_{x \to \frac{\pi}{4}} \frac{2\sqrt{2} - \left(\cos x + \sin x\right)^3}{1 - \sin 2x} \]
Let \( A(2, 3, 5), B(-1, 3, 2), C(\lambda, 5, \mu) \) be the vertices of \( \triangle ABC \). If the median through the vertex \( A \) is equally inclined to the coordinate axes, then
Let \([x]\) denote the greatest integer less than or equal to \(x\). Then \[ \lim_{x \to 2^+} \left( \frac{[x]^3}{3} - \left[ \frac{x^3}{3} \right] \right) \]