List of top Mathematics Questions

Simplify: $\dfrac{\cos\theta}{1 - \tan\theta} + \dfrac{\sin\theta}{1 - \cot\theta}$
If $4^x - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1}$, then the value of $x$ is:
The total number of permutations of $n$ different things taken not more than $r$ at a time, when each thing may be repeated any number of times is:
If a polygon of $n$ sides has 560 diagonals, then $n$ = ?
How many chords can be drawn through 21 points on a circle?
A person writes letters to 6 friends and addresses envelopes. In how many ways can the letters be placed so that at least two go to wrong envelopes?
If \[ \frac{x^4 + 24x^2 + 28}{(x^2 + 1)^3} = \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{(x^2 + 1)^2} + \frac{Ex + F}{(x^2 + 1)^3} \] then the value of $A + B + C + D + E + F =$ ?
The number of real roots of the equation \[ \frac{\sqrt{x}}{\sqrt{1 - x}} + \frac{\sqrt{1 - x}}{\sqrt{x}} = \frac{13}{6} \] is:
The number of positive real roots of the equation \[ 3^{x+1} + 3^{-x+1} = 10 \] is:
If the identity $\cos^4\theta = a\cos4\theta + b\cos2\theta + c$ holds for some $a, b, c \in \mathbb{Q}$, then $(a, b, c) =$ ?
Solve: $iz^3 + z^2 - z + i = 0 \Rightarrow |z| = $ ?
If $\dfrac{x - 1}{3 + i} + \dfrac{y - 1}{3 - i} = i$, then the true statement among the following is:
The number of integer solutions of the equation $|1 - i|^x = 2^x$ is:
If $f(x) = ax^2 + bx + c$ for some $a, b, c \in \mathbb{R}$ with $a + b + c = 3$ and \[ f(x+y) = f(x) + f(y) + xy,\quad \forall x, y \in \mathbb{R} \] Then $\sum_{n=1}^{10} f(n) = $ ?
Let $A = \begin{bmatrix} 5 & \sin^2\theta & \cos^2\theta \\ -\sin^2\theta & -5 & 1 \\ \cos^2\theta & 1 & 5 \end{bmatrix}$, then the maximum value of $\det(A)$ is:
Given $a_i^2 + b_i^2 + c_i^2 = 1$ and $a_i a_j + b_i b_j + c_i c_j = 0$ for $i \neq j$, and
\[ A = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \] Then $\det(AA^T) = $ ?
If $A = \frac{1}{7} \begin{bmatrix} 3 & -2 & 6 \\-6 & -3 & 2 \\ -2 & 6 & 3 \end{bmatrix}$, then $A^{-1} = $ ?
If $A = \begin{bmatrix} \alpha^2 & 5 \\ 5 & -\alpha \end{bmatrix}$ and $\det(A^{10}) = 1024$, then $\alpha =$ ?
If $f(x) = \sqrt{2 - x^2}$ and $g(x) = \log(1 - x)$ are two real-valued functions, then the domain of the function $(f+g)(x)$ is:
The range of the real valued function \[ f(x) = \sqrt{\frac{x^2 + 2x + 8}{x^2 + 2x + 4}} \] is

Given: \[ \lim_{x \to a} \left\lfloor \left( \frac{x^3}{a} - \left\lfloor \frac{x}{a} \right\rfloor^3 \right) \right\rfloor = k, \quad \lim_{x \to a} \left( \left\lfloor \frac{x^3}{a} \right\rfloor - \left\lfloor \frac{x}{a} \right\rfloor^3 \right) = l \] Then: \[ \boxed{?} \]

The equation √5·y − √8 = 0 is the directrix of a hyperbola:

x² / a² − y² / b² = 1

and the eccentricity is e = √5 / 2. 

Find the value of a.

The mean deviation about the mean for the given data:  

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If \(\cosec\, x = \frac{4}{5}\), then \( \cosh x = \) ?

Let \[ f(x) = \begin{cases} \frac{1 - \sin^3 x}{3 \cos^2 x}, & x<\frac{\pi}{2} \\ \alpha, & x = \frac{\pi}{2} \\ \frac{\beta(1 - \sin x)}{(\pi - 2x)^2}, & x>\frac{\pi}{2} \end{cases} \] If \( f(x) \) is continuous at \( x = \frac{\pi}{2} \), find \( \alpha \beta \).