List of top Mathematics Questions

If the co-domain of the function \[ f(x)= \begin{cases} 3\sin x-4\cos x, & x\in \left(\tan^{-1}\frac{4}{3}-\frac{\pi}{2},\,\tan^{-1}\frac{4}{3}+\frac{\pi}{2}\right)\\[2mm] \log(\sin x), & x\in \left(\frac{5\pi}{6},\pi\right) \end{cases} \] is \((-\infty,5)\), then \(f\) is

Assertion (A): If \[ 2+6+16+40+\cdots \text{ to } k \text{ terms}=4608, \] then \(k=9\). Reason (R): \[ 2+3\cdot2+4\cdot2^2+\cdots+n\cdot2^{\,n-1}=n\cdot2^n, \qquad \forall n\in\mathbb N \] Which one of the following options is correct?

If \[ A= \begin{bmatrix} 2 & -3 & 1\\ -3 & 1 & -2\\ 1 & -2 & 3 \end{bmatrix}, \] then \(\operatorname{Adj}(A)=\)

If \[ A= \begin{bmatrix} c & -a & b\\ a & b & -c\\ -b & c & a \end{bmatrix}, \qquad B= \begin{bmatrix} 1 & 0 & 2\\ 0 & 1 & 2\\ 1 & 2 & 0 \end{bmatrix} \] and \[ AB= \begin{bmatrix} 4 & 4 & -2\\ 1 & 1 & 10\\ -1 & 5 & -4 \end{bmatrix}, \] then \(a^2+b^2+c^2=\)

Let \[ f(x)=\frac{|x|-1}{|x-1|} \] be a real valued function from \( \mathbb{R}-\{1\} \) to \(B\). If \(f\) is a surjection, then \(B=\)

Let $\vec{a}=2\hat{i}-\hat{j}-\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$. Let $\vec{v}$ be the vector in the plane of $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\dfrac{1}{\sqrt{14}}$. Then $|\vec{v}|$ is equal to

Let \(A\) and \(B\) be \(n\times n\) real matrices. Which of the following statements is correct?

If a polygon has 54 diagonals, find the number of sides of the polygon.

The number of real solution of equation x$|$x+4$|$+3$|$x+2$|$+10=0 is/are :

If \(A\) is a \(3 \times 3\) matrix and \( |A| = 5 \), find the value of \( |adj(A)| \).

The number of critical points of the function} \[ f(x)= \begin{cases} \dfrac{|\sin x|}{x}, & x\neq0\\ 1, & x=0 \end{cases} \] in the interval \((-2\pi,2\pi)\) is equal to:

If the zeroes of a polynomial p(x) are $-3$ and 8, then p(x) equals

Let \(f(x)\) be defined by: \[ f(x)= \begin{cases} \displaystyle \int_x^6 (|t-2|+3)\,dt, & x>4 2x+8, & x\le4 \end{cases} \] Then at \(x=4\), \(f(x)\) is:

Find sum up to 8 terms of the series
\[ \frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\cdots \]

If the sum of the first four terms of an A.P. is \(6\) and the sum of its first six terms is \(4\), then the sum of its first twelve terms is

The coefficient of \(x^{48}\) in \[ (1+x) + 2(1+x)^2 + 3(1+x)^3 + \cdots + 100(1+x)^{100} \] is equal to

Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

A chord of a circle, of radius 14 cm, subtends an angle of $60^\circ$ at the centre. Find the area of the smaller sector and perimeter of the smaller segment.

PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

In the given figure, point D divides the side BC of $\Delta ABC$ in the ratio $1 : 2$. Find length AD.