List of top Mathematics Questions

Let \(\alpha,\beta\) be two real numbers such that \[ \pi\lt (\alpha-\beta)\lt 3\pi. \] If \[ \sin\alpha+\sin\beta=-\frac{21}{65} \] and \[ \cos\alpha+\cos\beta=-\frac{2}{65}, \] then \[ \cos\left(\frac{\beta-\alpha}{2}\right)= \]

The number of solutions of the equations \[ x+y+z=1;\quad x^2+y^2+z^2=1;\quad x^3+y^3+z^3=1 \] is:

Given \(5\) different green toys, \(4\) different blue toys, and \(3\) different red toys, how many combinations of toys can be chosen taking at least one green and one blue toy?

If \[ {}^{2n}C_3 : {}^{n}C_3 = 12:1, \] then \(n=\)

There are \(4\) oranges, \(5\) apples, \(7\) mangoes in a fruit basket. The number of ways of selecting at least one fruit from among the fruits in the basket is

If \[ {}^{9}C_3+{}^{9}C_5={}^{10}C_r \] for some \(r\in \mathbb{N}\), then \(r=\)

If the equivalent partial fraction of \[ \frac{x^3}{(2x-1)(x+2)(x-3)} \] is of the form \[ A+\frac{B}{2x-1}+\frac{C}{x+2}+\frac{D}{x-3}, \] then the value of \(A+B+C\) is:

All the pairs \((x,y)\) that satisfy the inequality \[ 2^{\sqrt{\sin^2 x-2\sin x+5}}-\frac{1}{4^{\sin y}}\leq 1 \] also satisfy the equation:

\(\alpha,\beta\) are the roots of \[ x^2-10x-8=0 \] with \(\alpha\gt \beta\). If \[ a_n=\alpha^n-\beta^n \] for \(n\in \mathbb{N}\), then the value of \[ \frac{a_{10}-8a_8}{5a_9} \] is:

The number of real values of \(m\) so that the equation \[ x^2+(2m+1)x+m=0 \] has equal roots is:

If \[ x=-5+2\sqrt{-4}, \] then the value of \[ x^4+9x^3+35x^2-x+4 \] is:

A true statement among the following identities is:

If \[ A=\begin{bmatrix} x & 0\\ 1 & 1 \end{bmatrix} \quad \text{and} \quad B=\begin{bmatrix} 8 & 0\\ 7 & 1 \end{bmatrix} \] and \(A^3=B\), then \(x=\)

If the solution of the system of simultaneous equations \[ \frac{1}{x}+\frac{2}{y}-\frac{3}{z}-1=0, \] \[ \frac{2}{x}-\frac{4}{y}+\frac{3}{z}-1=0 \] and \[ \frac{3}{x}+\frac{6}{y}-\frac{6}{z}-4=0 \] is \(x=\alpha,\ y=\beta,\ z=\gamma\), then \(\alpha^2+\gamma^2=\)

If \[ (x-iy)^{1/3}=a-ib, \] then the value of \[ \frac{x}{2a}+\frac{y}{2b} \] is:

If \(b\) and \(c\) are non-zero real numbers, \[ A= \begin{bmatrix} 1 & b & c\\ b & 2 & 3\\ c & 3 & 4 \end{bmatrix} \quad \text{and} \quad B= \begin{bmatrix} 0 & b & c\\ -b & 0 & 2\\ -c & -2 & 0 \end{bmatrix}, \] then \(\det(A+B)=\)

Evaluate \[ \sum_{k=1}^{6}\sin\left(\frac{2\pi k}{7}\right) -i\cos\left(\frac{2\pi k}{7}\right) \]

If the rank of the matrix \[ A= \begin{bmatrix} 1 & 2 & 1 & -1 \\ -1 & 2 & 3 & 5 \\ 0 & 1 & k & k \end{bmatrix} \] is \(2\) and \(k\) is a real number, then \(k\) is a root of the following quadratic equation:

The range of the real valued function \(f(x)=\dfrac{x^2+x+1}{x}\) is

If a function \(f:\mathbb{R}-\{l\}\to \mathbb{R}-\{m\}\) defined by \[ f(x)=\frac{x+3}{x-2} \] is a bijection, then \(3l+2m=\)

If \(a_1, a_2, a_3\) ….. and \(b_1, b_2, b_3\) ….. are A.P., and \(a_1 = 2, a_{10} = 3, a_1b_1 = 1 = a_{10}b_{10}\), then \(a_4b_4\) is equal to

Assertion (A): Order of the differential equations of a family of circles with constant radius is two.
Reason (R): An algebraic equation having two arbitrary constants is general solution of a \(2^{\text{nd}}\) order differential equation.

The general solution of the differential equation \[ \frac{dy}{dx}=\cos^2(3x+y) \] is \[ \tan^{-1}\left(\frac{\sqrt{3}}{2}\tan(3x+y)\right)=f(x). \] Then \(f(x)=\)

If the general solution of the differential equation \[ \cos^2x\frac{dy}{dx}+y=\tan x \] is \[ y=\tan x-1+Ce^{-\tan x} \] and it satisfies \[ y\left(\frac{\pi}{4}\right)=1, \] then \(C=\)

If \[ \int_1^3 x\sqrt[n]{x^2-1}\,dx=6, \] then \(n=\)