List of top Mathematics Questions

Find the general solution of: $$ (2x - y)^2 dy - 2(2x - y)^2 dx - 2 dx = 0 $$

Evaluate the integral: $$ \int_0^2 x^2 (2 - x)^5 \, dx $$

If $$ y = A t^2 + \frac{B}{t} \quad (A, B \text{ constants}) $$ is a general solution of the differential equation $$ f(t) y'' + g(t) y' + h(t) y = 0, $$ then find the relation between $ g(t), f(t), h(t) $.

Evaluate: $$ \int_{-\pi/2}^{\pi/2} \sin \left(x - [x]\right) dx $$ where $[x]$ is the greatest integer function.

If $ f(x) = \max \{ x^3 - 4, x^4 - 4 \} $ and $ g(x) = \min \{ x^2, x^3 \} $, evaluate: $$ \int_{-1}^1 (f(x) - g(x)) \, dx $$

If $$ \int \frac{x^2 - x + 2}{x^2 + x + 2} dx = x - \log(f(x)) + \frac{2}{\sqrt{7}} \tan^{-1}(g(x)) + c, $$ then find $$ f(-1) + \sqrt{7} g(-1). $$

If $$ \int \frac{dx}{1 - \sin^4 x} = A \tan x + B \tan^{-1}(\sqrt{2} \tan x) + C, $$ then find $ A^2 - B^2 $.

If $$ \int \frac{a \cos x + 3 \sin x}{5 \cos x + 2 \sin x} dx = \frac{26}{29} x - \frac{k}{29} \log |5 \cos x + 2 \sin x| + c, $$ then find $ |a + k| $.

If the function $$ f(x) = \sin x - \cos^2 x $$ is defined on the interval $ [-\pi, \pi] $, then $ f $ is strictly increasing in the interval:

If $$ \int \frac{x^4 + 1}{x^2 + 1} dx = Ax^3 + Bx^2 + Cx + D \tan^{-1} x + E, $$ then find $ A + B + C + D $.

Evaluate the integral: $$ \int \sec \left(x - \frac{\pi}{3}\right) \sec \left(x + \frac{\pi}{6}\right) dx $$

If Lagrange's mean value theorem is applied to the function $$ f(x) = e^x $$ defined on the interval $ [1, 2] $ and the value of $ c \in (1, 2) $ is $ k $, then find $ e^{k-1} $.

If $$ y = (\log_x \sin x)^x, $$ then find $$ \frac{dy}{dx}. $$

If a real valued function $$ f(x) = \begin{cases} \frac{\sin a(x - [x])}{e^{x - [x]}}, & x<1 \\ b + 1, & x = 1 \\ \frac{|x^2 + x - 2|}{x - 1}, & x>1 \end{cases} $$ is continuous at $ x=1 $, then find $ b $. Here, $ [x] $ denotes the greatest integer function.

If the tangent of the curve $$ 4y^3 = 3ax^2 + x^3 $$ drawn at the point $ (a, a) $ forms a triangle of area $\frac{25}{24}$ sq. units with the coordinate axes, then find $ a $.

If $$ \sin x \sqrt{\cos y} - \cos y \sqrt{\sin x} = 0, $$ then find $$ \frac{dy}{dx}. $$

If the area of a square is 575 square units, then the approximate value of its side is:

If $$ f(x) = 2 + |\sin^{-1} x|, $$ and $$ A = \{ x \in \mathbb{R} \mid f'(x) \text{ exists} \}, $$ then find $ A $.

The direction cosines of the line making angles $$ \frac{\pi}{4}, \frac{\pi}{3} $$ and $ \theta $ (where $ 0<\theta<\frac{\pi}{2} $) with X, Y, and Z axes respectively are:

Evaluate: $$ \lim_{x \to \infty} \left[ x - \log(\cosh x) \right] $$

Evaluate: $$ \lim_{x \to \infty} \left( \sqrt[3]{x^3 + 4x^2} - \sqrt{x^2 - 3x} \right) $$

If $ \theta $ is the angle subtended by a latus rectum at the center of the hyperbola having eccentricity $$ \frac{2}{\sqrt{7} - \sqrt{3}}, $$ then find $ \sin \theta $.

If the equation of the plane passing through point $ (3, 2, 5) $ and perpendicular to the planes $$ 2x - 3y + 5z = 7, \quad 5x + 2y - 3z = 11 $$ is $$ x + by + cz + d = 0, $$ then find $ 2b + 3c + d $.

The tangent drawn at an extremity (in the first quadrant) of latus rectum of the hyperbola $$ \frac{x^2}{4} - \frac{y^2}{5} = 1 $$ meets the x-axis and y-axis at $ A $ and $ B $ respectively. If $ O $ is the origin, find $$ (OA)^2 - (OB)^2. $$

The points $ A(-1, 2, 3), B(2, -3, 1), C(3, 1, -2) $ are: