List of top Mathematics Questions

Let $\vec a,\vec b$ and $\vec c$ be three non-zero vectors such that no two of them are collinear. If the vector $\vec a+\vec b$ is collinear with $\vec c$ and $\vec b+\vec c$ is collinear with $\vec a$, then \[ \vec a+\vec b+\vec c= \]

The sides of a triangle are in the ratio \[ 1:\sqrt3:2. \] Then the angles opposite to these sides are in the ratio

Let \[ \vec a=4\hat i-\hat j+\alpha\hat k \] and \[ \vec b=\hat i+\alpha\hat j-4\hat k \] be two vectors. If $\alpha_1,\alpha_2$ ($\alpha_1<\alpha_2$) are two different values of $\alpha$ such that \[ (\vec a,\vec b)=\cos^{-1}\left(-\frac{2}{7}\right), \] then \[ \alpha_1+2\alpha_2= \]

The shortest distance between the two lines \[ \vec r=(\hat i-\hat j)+s(\hat j+2\hat k) \] and \[ \vec r=(2\hat i+\hat k)+t(\hat i-\hat j+\hat k) \] is

If $\Delta$ denotes the area of triangle $ABC$ and $s$ is its semi-perimeter, then

In a triangle $ABC$, $C=90^\circ$. If $r$ is the inradius and $R$ is the circumradius of the triangle, then \[ 2(r+R)= \]

The number of real solutions of the equation \[ \tan^{-1}\sqrt{x(x+1)} + \sin^{-1}\sqrt{x^2+x+1} = \frac{\pi}{2} \] is

The value of \[ \tan81^\circ-\tan63^\circ-\tan27^\circ+\tan9^\circ \] is

If \[ u=\log\tan\left(\frac{\pi}{4}+\frac{\theta}{2}\right), \] then \[ \tanh\frac{u}{2} = \]

The number of distinct real roots of \[ \begin{vmatrix} \sin x & \cos x & \cos x\\ \cos x & \sin x & \cos x\\ \cos x & \cos x & \sin x \end{vmatrix}=0 \] in the interval \[ \left(-\frac{\pi}{4},\frac{\pi}{4}\right) \] is

Evaluate \[ \cos\frac{6\pi}{17}\cos\frac{10\pi}{17}\cos\frac{12\pi}{17}\cos\frac{14\pi}{17}. \]

For $n\in\mathbb Z$, a set of values of $\theta$ satisfying \[ \sec\theta-1=(\sqrt2-1)\tan\theta \] is

The value of $n$ for which the sum \[ \frac{{}^nC_0}{2^n} + 2\frac{{}^nC_1}{2^n} + 3\frac{{}^nC_2}{2^n} +\cdots+ (n+1)\frac{{}^nC_n}{2^n} =16 \] is

If all the letters of the word RANKS are permutated in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order, then the rank of the word RANKS is

If \[ \frac{x}{(x^2+1)^2(x-1)} = \frac{Ax+B}{x^2+1} + \frac{Cx+D}{(x^2+1)^2} + \frac{E}{x-1}, \] then \[ A+B-C+2D= \]

In the binomial expansion of $(x+a)^{15}$, if the eleventh term is the geometric mean of the eighth and twelfth terms, then the numerically greatest term in its expansion is

The number of different nine-digit numbers that can be formed by rearranging all the digits of the number $223355888$ so that odd digits always occupy even positions is

If $\alpha \neq \beta$ and $\alpha^2=5\alpha-3,\ \beta^2=5\beta-3$, then the equation whose roots are $\dfrac{\alpha}{\beta}$ and $\dfrac{\beta}{\alpha}$ is

A student is allowed to select at most $n$ books from a collection of $2n+1$ books. If the total number of ways in which he can select at least one book is $255$, then the value of $n$ is

If the roots of the equation \[ x^3-px^2+qx-s=0 \] are in geometric progression, then

If $\alpha,\beta,\gamma$ are the roots of \[ x^3-10x^2+7x+8=0, \] match List-I with List-II and choose the correct option.

If $z_1,z_2$ are two roots of the equation \[ z^2+az+b=0 \] and on the Argand plane the points represented by $z_1,z_2$ and the origin form an equilateral triangle, then $a^2=$

If \[ \frac{3}{2+\cos\theta+i\sin\theta}=x+iy, \] then \[ (x-1)(x-3)= \]

If \[ y= \begin{vmatrix} \sin x & \cos x & \sin x\\ \cos x & -\sin x & \cos x\\ x & 1 & 1 \end{vmatrix}, \] then \[ \frac{dy}{dx}= \]

If $\alpha,\beta$ are the roots of the equation \[ x^2-p(x+1)-c=0, \] then \[ \frac{\alpha^2+2\alpha+1}{\alpha^2+2\alpha+c} + \frac{\beta^2+2\beta+1}{\beta^2+2\beta+c} = \]