List of top Mathematics Questions

If $ \sum_{k=0}^{n+1} C_k^n = 512 $, find $ \sum_{k=0}^{n} C_k^n $.

Find the domain of the function: $ f(x) = \sqrt{7 - 11x} $

If $ a, a r, a r^2 $ are in a geometric progression (G.P.), then find the value of: $ \left| a \quad a r \right| \quad \left| a r^2 \quad a r^3 \right| = \left| a r^3 \quad a r^6 \right| $

Find the limit: $ \lim_{x \to 0^+} 2 \left\lfloor x \right\rfloor - \frac{x}{|x|} $

Find the limit: $ \lim_{x \to 0^+} 2 \left\lfloor x \right\rfloor - \frac{x}{|x|} $

Solve the following differential equation and integrate: $ \frac{dy}{dx} + \frac{2x}{1 + x^2} \cdot y = x $

If $ f(x) = \frac{\sqrt{x^4}}{\sqrt{x^2}} $, find $ f'(27) $.

Equation of parabola having focii (-3, 1) and (3, 1)

Find the value of $ \sin 60^\circ - \sin 80^\circ + \sin 100^\circ - \sin 120^\circ $

If $ f(x) = \frac{1}{x^2} $, $ u = f(x) $, and $ f'(x) $, then find $ \frac{du}{dx} $.

Given the system of equations: \[ Z + \bar{Z} = 4 \quad \text{and} \quad Z - \bar{Z} = 6 \] Find \( |Z| \).

If \( z_1, z_2, z_3 \in \mathbb{C} \) are the vertices of an equilateral triangle, whose centroid is \( z_0 \), then \( \sum_{k=1}^{3} (z_k - z_0)^2 \) is equal to

Let $ y = y(x) $ be the solution of the differential equation $$ \frac{dy}{dx} + 2y \sec^2 x = 2 \sec^2 x + 3 \tan x \sec^2 x $$ such that $ y(0) = \frac{5}{4} $. Then $ 12 \left( y \left( \frac{\pi}{4} \right) - e^2 \right) $ is equal to:

Let the area of the triangle formed by a straight line $ L: x + by + c = 0 $ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line $ L $ makes an angle of $ 45^\circ $ with the positive x-axis, then the value of $ b^2 + c^2 $ is:

Let the point $ P $ of the focal chord $ PQ $ of the parabola $ y^2 = 16x $ be $ (1, -4) $. If the focus of the parabola divides the chord $ PQ $ in the ratio $ m : n $, gcd($m, n$) = 1, then $ m^2 + n^2 $ is equal to:

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is:

$$ \int \frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}} \, dx - 3 \log \left( \sqrt{3} \right) $$ is equal to:

Given three identical bags each containing 10 balls, whose colours are as follows:
 

Bag I	3 Red	2 Blue	5 Green
Bag II	4 Red	3 Blue	3 Green
Bag III	5 Red	1 Blue	4 Green

A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from Bag I is $ p $ and if the ball is Green, the probability that it is from Bag III is $ q $, then the value of $ \frac{1}{p} + \frac{1}{q} $ is:

If $ \theta \in \left[ \frac{7\pi}{6}, \frac{4\pi}{3} \right] $, then the number of solutions of $$ \sqrt{3} \csc^2\theta - 2(\sqrt{3} - 1) \csc\theta - 4 = 0, $$ is equal to:

If $ y = \cos \left( \frac{\pi}{3} + \cos^{-1} \frac{x}{2} \right) $, then $ (x - y)^2 + 3y^2 $ is equal to ________.