List of top Mathematics Questions

A value of $\theta \ \ \in \ (0, \pi /3)$ for which $\begin {vmatrix} 1 + cos^2 \theta & sin^2\theta & 4 cos 6\theta \\ cos^2 \theta & 1+sin^2 \theta & 4 cos6 \theta \\ cos^2 \theta & sin^2 \theta & 1+4 cos6 \theta \end {vmatrix} $ =0 , is :

If [$x$] denotes the greatest integer $\le x$, then the system of linear equations [$sin\,\theta$] x + [$-cos\,\theta$]y=0 [$cot\,\theta$] $x + y = 0$

If three distinct numbers a,b,c are in G.P. and the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then which one of the following statements is correct?

$\displaystyle\lim_{n\to\infty} \left(\frac{\left(n+1\right)^{\frac{1}{3}} }{n^{\frac{4}{3}}} + \frac{\left(n+2\right)^{\frac{1}{3}}}{n^{\frac{4}{3}}} + ..... + \frac{\left(2n\right)^{\frac{1}{3}}}{n^{\frac{4}{3}}}\right) $ equal to :

If the sum of the deviations of $50$ observations from $30$ is $50$, then the mean of these observation is :

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1| = 4 |z_2|$. If $z = \frac{3z_{1}}{2z_{2}} + \frac{2z_{2}}{3z_{1}}$ then :

Let $N$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g : N \to N$ such that : $f(n) = \begin{cases} \frac{n+1}{2} & \quad \text{if } n \text{ is odd}\\ \frac{n}{2} & \quad \text{if } n \text{ is even} \end{cases}$ and $g(n) = n-(-1)^n$. The $fog$ is :

Let $a_1, a_2, ....., a_{10}$ be a G.P. If $\frac{a_3}{a_1} = 25 $, then $\frac{a_9}{a_5}$ equals :

A helicopter is flying along the curve given by $y - x^{3/2} = 7, (x \ge 0)$. A soldier positioned at the point $\left(\frac{1}{2}, 7\right)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is :

The term independent of x in the expansion of $\bigg(\frac{1}{60} - \frac{x^8}{81}\bigg). \bigg(2x^2 - \frac{3}{x^2}\bigg)^6$ is equal to:

If $\begin {bmatrix} 1 & 1 \\ 0 & 1 \end {bmatrix}$ . $\begin {bmatrix} 1 & 2 \\ 0 & 1 \end {bmatrix} $ . $\begin {bmatrix} 1 & 3 \\ 0 & 1 \end {bmatrix}$ ..... $\begin {bmatrix} 1 & n-1 \\ 1 & 1 \end {bmatrix}$ = $\begin {bmatrix} 1 & 78 \\ 0 & 1 \end {bmatrix}$, then the inverse of $\begin {bmatrix} 1 & n \\ 0 & 1 \end {bmatrix}$ is

Consider a triangular plot $ABC$ with sides $AB = 7m, BC = 5m$ and $CA = 6m$. A vertical lamp-post at the mid point $D$ of $AC$ subtends an angle $30^{\circ}$ at $B$. The height (in $m$) of the lamp-post is:

All possible numbers are formed using the digits \(1,1,2,2,2,2,3,4,4\) taken all at a time. The number of such numbers in which the odd digits occupy even places is :

If $f\left(x\right) = \frac{2- x\cos x}{2+x \cos x}$ and $ g\left(x\right) =\log_{e}x ., \left(x>0\right) $ then the value of integral $\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}} g\left(f\left(x\right)\right)dx $ is :

If $A = \begin{bmatrix}e^{t}&e^{t} \cos t&e^{-t}\sin t\\ e^{t}&-e^{t} \cos t -e^{-t}\sin t&-e^{-t} \sin t+ e^{-t} \cos t\\ e^{t}&2e^{-t} \sin t&-2e^{-t} \cos t\end{bmatrix} $ Then $A$ is -

Let $f : (-1,1) \to R$ be a function defined by $f(x) = max \{- | x |,- \sqrt{ 1- x^2} \}$. If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly :

The maximum volume (in $cu. m$) of the right circular cone having slant height $3\,m$ is :

In a class of $140$ students numbered $1$ to $140$, all even numbered students opted mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is :

Let S and S' be the foci of the ellipse and B be any one of the extremities of its minor axis. If 'S'BS is a right angled triangle with right angle at B and area ($\Delta$S'BS) = 8 s units, then the length of a latus rectum of the ellipse is :

The value of $\cos \frac{\pi}{2^{2}} .\cos \frac{\pi}{2^{3} } . ...... .\cos \frac{\pi}{2^{10}} .\sin \frac{\pi}{2^{10}} $ is :

The solution of the differential equation , $\frac{dy}{dx} = \left(x-y\right)^{2} , $ when $y(1) = 1$, is :

For $x > 1$, if $(2x)^{2y} = 4e^{2x - 2y}$, then $\left( 1 +\log_{e} 2x\right)^{2} \frac{dy}{dx} $ is equal to :

Let $z \in C$ with Im(z) = 10 and it satisfies $\frac{2z-n}{2z+n} = 2i -1$ for some natural number $n$.Then :

In a game, a man wins Rs. $100$ if he gets $5$ of $6$ on a throw of a fair die and loses Rs. $50$ for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/ loss (in rupees) is :

Let $a_1, a_2 , ....... , a_{30}$ be an $A. P$., $S =\displaystyle \sum^{30}_{i=1} a_i $ and $T = \displaystyle\sum^{15}_{i=1} a_{(2i -1)}$. If $a_5 = 27$ and $S - 2T = 75, $ then $a_{10}$ is equal to :