List of top Mathematics Questions

Let \( f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f^{(3)}(3) \) for all \( x \in \mathbb{R} \). Then the value of \( f'(5) \) is:

The largest value of \( n \in \mathbb{N} \) such that \( 7^n \) divides \( 101! \) is ______.

The minimum value of \( (\sin^{-1} x)^2 + (\cos^{-1} x)^2 \) in \( x \in \left[ \frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}} \right] \) is \( \frac{a \pi^2}{b} \), then \( a + b \) is:

If \( a_1 = 1 \) and for \( n \geq 1 \), \[ a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2} \] then \[ \left| \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \right| \] is equal to

The value of \[ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\pi + 4x^{11}}{1 - \sin\left( \left| x \right| + \frac{\pi}{6} \right)} dx \] is

The value of cosec10° - √3 sec10°

If the mean and variance of observations \( x, y, 12, 14, 4, 10, 2, 8 \) and 16 respectively where \( x>y \), then the value of \( 3x - y \) is

If \( A = \left[ \begin{array}{cc} \alpha & 2 \\ 1 & 2 \end{array} \right] \), \( B = \left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right] \) and \( A^2 - 4A + 2I = 0, B^2 - 2B + I = 0 \), then \( \text{adj}(A^3 - B^3) \) is equal to

Area enclosed by \[ x^2 + 4y^2 \leq 4, \quad |x| \leq 1, \quad y \geq 1 - |x| \text{ is} \]

The value of \[ \int_0^{\frac{\pi}{2}} \left( \sin x + \sin 2x + \sin 3x \right) dx \] is

If \( x^2 + x + 1 = 0 \), then \[ \left( x + \frac{1}{x} \right)^4 + \left( x^2 + \frac{1}{x^2} \right)^4 + \left( x^3 + \frac{1}{x^3} \right)^4 + \dots + \left( x^{25} + \frac{1}{x^{25}} \right)^4 \] is

If \( a_1, a_2, a_3, \dots \) are the terms of an increasing geometric progression such that \[ a_1 + a_3 + a_5 = 21, \quad a_1a_3a_5 = 64 \] then \[ a_1 + a_2 + a_3 \] is

Let $5000<N<9000$ and $N$ has digits from the set $\{0,1,2,5,9\}$. If digits can be repeated, then find the number of such numbers $N$ which are divisible by $3$.

If $A=\{1,2,3,4\}$. A relation from set $A$ to $A$ is defined as $(a,b)\,R\,(c,d)$ such that $2a+3b=3c+4d$. Find the number of elements in the relation.

If matrices \( A \) and \( B \) are such that \[ A = \begin{bmatrix} 0 & -2 & 3 \\ -2 & 0 & 1 \\ -1 & 1 & 0 \end{bmatrix} \] and \( B(I - A) = (I + A) \), then find \( B \).

Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{x}\left(e^{\tan x - x} - 1\right) + \ell n(\sec x + \tan x) - x}{\tan x - x} \]

Let \[ f(x)=\int \frac{1-\sin(\ell n t)}{1-\cos(\ell n t)} \, dt \] and \[ f\left(e^{\pi/2}\right)=-e^{\pi/2} \] then find $f\left(e^{\pi/4}\right)$.

Let the mean and variance of 10 numbers be $10$ and $2$ respectively. If one number $\alpha$ is replaced by another number $\beta$, then the new mean and variance are $10.1$ and $1.99$ respectively. Find $(\alpha+\beta)$.

The value of \[ \frac{\sqrt{3}\,\cosec 20^\circ-\sec 20^\circ}{\cos 20^\circ \cos 40^\circ \cos 60^\circ \cos 80^\circ} \] is:

If $\cot x=\dfrac{5}{12}$ for some $x\in\left(\pi,\dfrac{3\pi}{2}\right)$, then \[ \sin 7x\left(\cos\frac{13x}{2}+\sin\frac{13x}{2}\right) +\cos 7x\left(\cos\frac{13x}{2}-\sin\frac{13x}{2}\right) \] is equal to:

Find the number of real solutions of \[ x|x-3|+|x-1|+3=0 \]

Maximum value of $n$ for which $40^n$ divides $60!$ is

Let $f(x)$ be a differentiable function satisfying \[ f(x)=e^x+\int_0^1 (y+xe^x)f(y)\,dy \] Find $f(0)+e$, where $e$ is Napier's constant.

Let 4 integers $a_1, a_2, a_3, a_4$ are in A.P. with integral common difference $d$ such that $a_1+a_2+a_3+a_4=48$ and $a_1a_2a_3a_4+d^4=361$. Then the greatest term in this A.P. is

Evaluate \[ \left(\frac{4}{7}+\frac{1}{3}\right)+\left(\frac{4}{7}+\frac{4}{3}\right)\left(\frac{1}{3}\right) +\left(\frac{4}{7}+\frac{4}{3}\right)^2\left(\frac{1}{3}\right)^2+\cdots \]