List of top Mathematics Questions

Let $z = (1+i)(1+2i)(1+3i)\cdots(1+ni)$ and $|z|^2 = 44200$. Find the value of $n$.

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

If domain of $f(x)=\sin^{-1}\!\left(\dfrac{1}{x^2-2x-2}\right)$ is $(-\infty,\alpha]\cup[\beta,\gamma]\cup[\delta,\infty)$, then $(\alpha+\beta+\gamma+\delta)$ is

If the equation \[ x^4-ax^2+9=0 \] has four real and distinct roots, then the least possible integral value of $a$ is

If dataset $A=\{1,2,3,\ldots,19\}$ and dataset $B=\{ax+b;\,x\in A\}$. If mean of $B$ is $30$ and variance of $B$ is $750$, then sum of possible values of $b$ is

If $f(a)$ is the area bounded in the first quadrant by $x=0$, $x=1$, $y=x^2$ and $y=|ax-5|-|1-ax|+ax^2$, then find $f(0)+f(1)$.

Find the number of solutions of the equation \[ \tan(x+100^\circ)=\tan(x+50^\circ)\tan(x-50^\circ) \] where $x\in(0,\pi)$.

Let $P$ and $Q$ be any two $3\times3$ matrices where $P=[p_{ij}]_{3\times3}$, $Q=[q_{ij}]_{3\times3}$ such that $q_{ij}=2^{\,i+j-1}p_{ij}$. Find $|\operatorname{adj}(\operatorname{adj}P)|$.

Let the ellipse \[ E=\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \] have eccentricity equal to the greatest value of the function \[ f(t)=-\frac34+2t-t^2 \] and the length of its latus rectum is $30$. Find the value of $a^2+b^2$.

If two lines drawn from a point $P(2,3)$ intersect the line $x+y=6$ at a distance $\sqrt{\dfrac{2}{3}}$, then the angle between the lines is

If shortest distance between the lines \[ \frac{x+1}{\alpha}=\frac{y-2}{-2}=\frac{z-4}{-2\alpha} \quad \text{and} \quad \frac{x}{\alpha}=\frac{y-1}{1}=\frac{z-1}{\alpha} \] is $\sqrt{2}$, then find the sum of all possible values of $\alpha$.

Let a function $f(x)$ satisfy \[ 3f(x)+2f\!\left(\frac{m}{19x}\right)=5x \] where $m=\sum_{i=1}^{9} i^2$. Find $f(5)+f(2)$.

Let $f(x)=|\log_e x|-|x-1|+5$. Statement 1: $f(x)$ is differentiable for all $x\in(0,\infty)$
Statement 2: $f(x)$ is increasing in $(1,\infty)$
Statement 3: $f(x)$ is decreasing in $(0,1)$
Which of the following is correct?

The value of \[ \int_{\frac{\pi}{2}}^{\pi} \frac{1}{\left\lfloor x \right\rfloor + 4} \, dx \] where \( \left\lfloor x \right\rfloor \) denotes the greatest integer function, is:

If the area of the region \( \{ (x, y) : x^2 + 1 \leq y \leq 3 - x \} \) is divided by the line \( x = -1 \) in the ratio \( m : n \) (where \( m \) and \( n \) are coprime natural numbers), then the value of \( m + n \) is:

If \[ A = \begin{pmatrix} 2 & 3 \\ 3 & 5 \end{pmatrix} \] then the value of \[ |A^{2025} - 3A^{2024} + A^{2023}| \] is:

If the image of the point \( P(3, 2, a) \) reflected about the line \[ \frac{x-3}{2} = \frac{y-5}{5} = \frac{z-2}{-2} \] is \( (5, b, c) \), then the value of \( \sigma^2 + b^2 + c^2 \) is:

If \[ \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \sqrt{5\beta}}{2} \] then the value of \( (\alpha + \beta) \) is:

The number of values of \( x \) satisfying \( \tan^{-1}(4x) + \tan^{-1}(6x) = \frac{\pi}{6} \) and \( x<\left[ \frac{-1}{2\sqrt{6}} , \frac{1}{2\sqrt{6}} \right] \) is:

If \[ \int \left( \cos x \right)^{-\frac{5}{2}} \left( \sin x \right)^{-\frac{1}{2}} \, dx \] \[ = \frac{P_1}{q_1} \left( \cot x \right)^{\frac{3}{2}} + \frac{P_2}{q_2} \left( \cot x \right)^{\frac{3}{2}} + \frac{P_3}{q_3} \left( \cot x \right)^{\frac{1}{2}} + \frac{P_4}{q_4} \left( \cot x \right)^{-\frac{3}{2}} + c \] (where \( c \) is the constant of integration), then value of \[ \frac{15P_1P_2P_3P_4}{q_1q_2q_3q_4} \] is:

If \( A \) is a square matrix of order 3 and \( |A| = 6 \), it is given that \[ \left| \text{adj} \left( 3 \, \text{adj} \left( A^2 \, \text{adj} (2A) \right) \right) \right| = 2^m \cdot 3^n \] where \( m \) and \( n \) are natural numbers. Then find \( m + n \). Here, \( \text{adj}(X) \) denotes the adjoint of matrix \( X \), and \( |X| \) denotes the determinant of matrix \( X \).