List of top Mathematics Questions

There are 7 men and 5 women in a park. The number of ways of arranging them around a circular path such that 4 particular persons which include 2 particular men and 2 particular women never stand together is:

If \(\alpha\) and \(\beta\) are acute angles and \[ \cos\alpha(1+\tan\alpha\tan\beta)=1, \] then \[ \sin\left(\frac{\alpha-2\beta}{3}\right)= \]

If \[ \frac{x^{2}+1}{(x^{4}+5x^{2}+6)(x^{6}+x^{4})} = \frac{A}{x^{4}}+\frac{B}{x^{2}}+\frac{C}{x^{2}+2}+\frac{D}{x^{2}+3}, \] then \(A-B=\):

If \(\theta=\frac{11\pi}{7}\), then \[ \frac{1+\cos 8\theta}{\cot^{2}4\theta} + \frac{1-\cos 8\theta}{\tan^{2}4\theta} = \]

The number of real roots of the equation \[ x^7+3x^5-13x^3-15x=0 \] is:

If \(\alpha,\beta\) are the rational roots and \(l,m\) are the irrational roots of \[ (x^2-9x+11)^2-(x-4)(x-5)=3, \] then \(\alpha+\beta+lm=\)

The number of integral solutions of \[ x+y+z=13 \] such that \[ 1\le x\le9,\qquad 0\le y\le9,\qquad 0\le z\le9 \] is

If \(\dfrac{{}^{\,n-1}C_{r-1}}{{}^{\,n}C_r}=\dfrac{3}{5}\) and \(\dfrac{{}^{\,n+1}C_{r+1}}{{}^{\,n}C_r}=\dfrac{11}{7}\), then \({}^{\,n}C_{r+3}\div{}^{\,r}C_{n/2}\) is equal to:

Let \(\alpha,\beta,\gamma,\delta\) be the roots of \[ 4x^4+8x^3-17x^2-12x+9=0. \] If \[ 4(\alpha+4)(\beta+4)(\gamma+4)(\delta+4)=k, \] then \(k=\)

If \(a\in \mathbb{Z}\) and the equation \[ (x-a)(x-10)+1=0 \] has integral roots, then the values of \(a\) are:

$m,n$ and $k$ are integers and $9.5\le n\le12$. If \[ \frac{(\cos\theta+i\sin\theta)^m} {(\sin\theta+i\cos\theta)^n} = k(\sin17\theta-i\cos17\theta), \] then $n-m-k=$

If \(\omega\) is a complex cube root of unity, then \[ (1+\omega)(1+\omega^2)(1+\omega^4)(1+\omega^5)(1+\omega^7)(1+\omega^8)\cdots \] (2n factors) is equal to

Let \(A\) be a \(3\times3\) matrix. If \[ A \begin{bmatrix} 001 \end{bmatrix} = \begin{bmatrix} 123 \end{bmatrix}, \quad A \begin{bmatrix} 101 \end{bmatrix} = \begin{bmatrix} 10-1 \end{bmatrix}, \quad A \begin{bmatrix} 110 \end{bmatrix} = \begin{bmatrix} 110 \end{bmatrix}, \] then the rank of \((A-I)\) is

Consider the system of linear equations (L): \[ 2x-y-z=-3,\qquad x+2y+z=4,\qquad 3x+y+kz=3 \] Let \(k\in N\) and \(1\le k\le2026\). If A={k:{ no solution}},
B={k:{ unique solution}},
C={k:{ infinite solutions} then \(n(A)+n(B)+n(C)=\)}

If \[ A=\{z=x+iy:|z-4||z-3|\}, \] \[ B=\{z=x+iy:-3\le y\le3,\;x\in N,\;y\in N\}, \] and \(C=A\cap B\), then \(n(C)=\)

If \(z_1\) and \(z_2\) are two complex numbers such that \[ |z_1-a|=|z_2-a| \] for \(a\in\mathbb R\), and \[ Arg(z_1-a)+Arg(z_2-a)=\frac{\pi}{2}, \] then \[ \frac{z_1-a}{z_2-a}= \]

Let \(A\) be a \(3\times3\) matrix such that \[ \det(A)=-1. \] If \[ B^{-1}=Adj\!\left(A\,Adj(A^2)\right), \] then find \[ \det((\det A)B). \]

If \[ A= \begin{bmatrix} \cos\alpha& 0 &\sin\alpha\\ 0& 1& 0 -\sin\alpha& 0 &\cos\alpha\\ \end{bmatrix} \] and \(A^2=A^T\) for one value of \(\alpha\in(0,\pi)\), then \(A^3=\):

If \(S(n):2n<n!\), \(n\in\mathbb N\), then \(S(n+1)\) is true for all \(n\ge\):

If \(f:\mathbb R\to\mathbb R\) is defined by \[ f(x)=|x| \] and \(A=(0,1)\), then \(f^{-1}(A)\) is:

The domain of the real valued function \[ f(x)=\cos^{-1}\!\left(\log_{5}\frac{x}{5}\right)+\log_{5}\!\left(\cos^{-1}\frac{x}{5}\right) \] is:

Evaluate: \[ \int_{-2\pi}^{2\pi}(1+\cos x)^3(1-\cos x)^4\,dx \]

Evaluate: \[ \int_{\pi/2}^{4051\pi/2}\frac{\cos^22x}{1+\sin2x}\,dx \]

Find the area of the region bounded by the curve \[ y=x^2-4, \] the \(x\)-axis and the lines \(x=-2\) and \(x=3\).

The general solution of the differential equation \[ (x+y-1)\,dy=(x-y+1)\,dx \] is