List of top Mathematics Questions

If \(C\) is the midpoint of the line segment \(AB\) and \(P\) is any point outside the line \(AB\), then:

If \(3\) vectors \(\overline{a},\overline{b},\overline{c}\) are such that \(\overline{a}\neq \overline{0}\) and \(\overline{a}\times \overline{b}=2(\overline{a}\times \overline{c})\), \(|\overline{a}|=1\), \(|\overline{c}|=1\), \(|\overline{b}|=4\), and angle between \(\overline{b}\) and \(\overline{c}\) is \(\cos^{-1}\left(\dfrac{1}{4}\right)\) and \(\overline{b}-2\overline{c}=\lambda \overline{a}\), then \(\lambda=\)

In a triangle \(ABC\), if \(a=2\), \(b=3\), and \(\sin A=\dfrac{2}{3}\), then \(\angle B=\)

In a \(\triangle ABC\), \(a:b:c=4:5:6\). The ratio of the radius of the circumcircle to that of the incircle is:

The sides of a triangle \(ABC\) are given by \(a=3\), \(b=5\), and \(c=3\). Then \(\cos A=\)

The value of \[ \sin(x+y)\sec x\sec y \] is

In \(\triangle ABC\), if \[ 3\sin A+4\cos B=6 \] and \[ 4\sin B+3\cos A=1, \] then the angle \(C\) is

If \(\cosh(x-\log 3)=\sinh x\), then \(x=\)

If \[ \sin\left(x+\frac{\pi}{3}\right)+\sin\left(x-\frac{\pi}{3}\right)=1, \] then the value of \(x\) in the interval \([0,\pi]\) is

The smallest integer \(n\) such that \[ \frac{1}{\sin45^\circ\sin46^\circ} + \frac{1}{\sin47^\circ\sin48^\circ} +\cdots+ \frac{1}{\sin133^\circ\sin134^\circ} = \frac{1}{\sin(n^\circ)} \] is

Let \(P_1,P_2,\ldots,P_{15}\) be \(15\) points on a circle. The number of distinct triangles formed by points \(P_i,P_j,P_k\) such that \(i+j+k\neq 15\), is

The value of \[ 1+\cot^2 30^\circ-\sec^2 45^\circ \] is

What is the sum of the first \(n\)-terms of the series, whose \(k\)-term is \(k!\times k\)?

If \[ \frac{4x^3+16x+7}{(x^2+4)^2} = \frac{Ax+B}{x^2+4} + \frac{Cx+D}{(x^2+4)^2}, \] then the number of non-zero values in \(A,B,C,D\) is

The remainder when the polynomial \[ 2x^5-3x^4+5x^3-3x^2+7x-9 \] is divided by \[ x^2-x-3 \] is

Which of the following quadratic equations whose real roots \(x_1,x_2\) satisfy the conditions \[ x_1^2+x_2^2=5, \quad 3(x_1^5+x_2^5)=11(x_1^3+x_2^3)? \]

If \[ {}^nC_{r-1}=36,\qquad {}^nC_r=84,\qquad {}^nC_{r+1}=126, \] then the value of \(nr^2\) is

The range of the function \[ f(x)=\frac{x^2+x+1}{x^2-x+1} \] is

The total number of positive integral solutions \((x,y,z)\) of \[ xyz=24 \] is

If \(1,\omega,\omega^2\) are the cube roots of unity, then \[ (2-\omega)^2(2-\omega^2)^2(2-\omega^{10})^2(2-\omega^{11})^2= \]

Let \(z=x+iy\) be a complex number with \(x,y\in \mathbb{Z}\). Then the area (in square units) of the rectangle whose vertices are the roots of the equation \[ z\overline{z}^{\,3}+\overline{z}z^{3}=350 \] is:

Let \[ f(x)=x^2+\frac{1}{x^2} \] and \[ g(x)=x-\frac{1}{x} \] for \(x\in \mathbb{R}-\{-1,0,+1\}\), then the local minimum of \[ \frac{f(x)}{g(x)} \] is

If the complex numbers \(z_1\), \(z_2\), \(0\) are vertices of an equilateral triangle, then \[ z_1^2+z_2^2= \] is equal to:

Let \[ A=\begin{bmatrix} n & 0 & 0\\ 0 & n & 0\\ 0 & 0 & n \end{bmatrix} \] and \[ B=\begin{bmatrix} 0 & 0 & n\\ 0 & n & 0\\ n & 0 & 0 \end{bmatrix}. \] Then \(A^2+B^2+AB=\)

Let \(z\) and \(w\) be two complex numbers such that \[ \overline{z}+i\overline{w}=0 \] and \[ \operatorname{Arg}(zw)=\pi. \] Then \(\operatorname{Arg}(z)=\)