List of top Mathematics Questions

Let $A$ and $B$ be two invertible matrices of order $3 \times 3$. If $\text{det} (ABA^T) = 8$ and $\text{det} (AB^{-1}) = 8$, then $\text{det} (BA^{-1} BT) $ is equal to :

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 :

Let $f \left(x\right)= \int\limits_0^{x} g (t) dt$ , where g is a non-zero even function. If $f \left(x+5\right)=g\left(x\right)$, then $ \int\limits_0^{x}f (t) dt $ equals-

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 $S_n$ denote the sum of the first $n$ terms of an $A.P$. If $S_4 = 16$ and $S_6 = -48$, then $S_{10}$ is equal to :

Let $(x + 10)^{50} + (x - 10)^{50} = a_0 + a_1 x + a_2x^2 + ..... + a_{50} \; x^{50} , $ for all $x \in R , $ the $\frac{a_2}{a_0} $ is equal to :-

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 $z$ be a complex number such that $|z| + z = 3 + i$ (where i = $\sqrt{-1}$ ). Then $|z|$ is equal to :

Let $Z$ be the set of integers. If $A \, = \, \{ x \in Z : 2^{ (x+2) (x^2 - 5x + 6)} \}=1$ and $B \, = \, \{ \, x \in Z: -3 <2x -1 <9 \}$, then the number of subsets of the set $A \times B$, is :

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

$\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 :

$\displaystyle\lim_{x\to0} \frac{\sin^{2}x}{\sqrt{2} - \sqrt{1+\cos x}} $ equals :

The integral $\int sec^{2/3} x cosec^{4/3}x\, dx$ is equal to (Hence $C$ is a constant of integration)

The mean of five observations is $5$ and their variance is $9.20.$ If three of the given five observations are $1, 3$ and $8$, then a ratio of other two observations is :

The number of natural numbers less than $7,000$ which can be formed by using the digits $0,1,3,7,9$ (repitition of digits allowed) is equal to :

The value of $\cot\left(\sum\limits^{19}_{n=1} \cot^{-1} \left(1+ \sum\limits^{n}_{p=1} 2p\right)\right) $ is :

The value of $\int\limits^{\pi/2}_{-\pi/2} \frac{dx}{\left[x\right]+\left[\sin x\right]+4} $ where [t] denotes the greatest integer less than or equal to t, is :

Two vertices of a triangle are $(0,2)$ and $(4,3)$ . If its orthocentre is at the origin, then its third vertex lies in which quadrant ?

In a \(\Delta ABC\). with usual notation, match the items in List - I with the items in List - II and choose the correct option.

List I		List II
(A)	\(r_1r_2\sqrt{\bigg(\frac{4R-r_1-r_2}{r_1+r_2}\bigg)}\)	1	\(b\)
(B)	\(\frac{r_2(r_3+r_1)}{\sqrt{r_1r_2+r_2r_3+r_3r_1}}\)	2	\(a^2,b^2,c^2 are \;in \;AP\)
(C)	\(\frac{a}{c}=\frac{sin(A-B)}{sin(B-C)}\)	3	\(\triangle\)
(D)	\(bc\;cos^2\frac{A}{2}\)	4	\(R\; r_1r_2r_3\)
		5	\(s(s-a)\)

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If $99$ more identical balls are addded to the total number of balls used in forming the equilaterial triangle, then all these balls can be arranged in a square whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is :

Suppose that $20$ pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number beams is :

The area (in s units) of the smaller of the two circles that touch the parabola, $y^2 = 4x$ at the point $(1, 2)$ and the x-axis is :

The contrapositive of the statement "If you are born in India, then you are a citizen of India", is :

The integral $ \int \frac{2x^{3} -1}{x^{4} +x} dx $ is equal to : (Here C is a constant of integration)

The integral $\int \cos( \log \; x)dx$ is equal to : (where C is a constant of integration)