List of top Mathematics Questions

Let the mean and variance of 8 numbers -10, -7, -1, x, y, 9, 2, 16 be \( 2 \) and \( \frac{293}{4} \), respectively. Then the mean of 4 numbers x, y, x+y+1, |x-y| is:

The letters of the word ``UDAYPUR'' are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word ``UDAYPUR'' is

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 \( f : [1,\infty) \to \mathbb{R} \) be a differentiable function. If
\[ 6\int_{1}^{x} f(t)\,dt = 3x f(x) + x^3 - 4 \] for all \( x \ge 1 \), then the value of \( f(2) - f(3) \) is

Find the domain of \(p(x)=\sin^{-1}(1-2x^2)\). Hence, find the value of \(x\) for which \(p(x)=\frac{\pi}{6}\). Also, write the range of \(2p(x)+\frac{\pi}{2}\).

The number of elements in the relation \[ R=\{(x,y): 4x^2+y^2<52,\; x,y\in\mathbb{Z}\} \] is

Let a circle of radius $4$ pass through the origin $O$, the points $A(-\sqrt{3}a,0)$ and $B(0,-\sqrt{2}b)$, where $a$ and $b$ are real parameters and $ab\neq0$. Then the locus of the centroid of $\triangle OAB$ is a circle of radius

Let \[ f(x)=\int \frac{dx}{2\left(\frac{3}{2}\right)^x+2x\left(\frac12\right)^x} \] such that \(f(0)=-26+24\log_e(2)\). If \(f(1)=a+b\log_e(3)\), where \(a,b\in\mathbb{Z}\), then \(a+b\) is equal to:

Let
\[ A=\begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} \] and \( B \) be two matrices such that
\[ A^{100}-100B+I=0. \]
Then the sum of all the elements of \( B^{100} \) is _______.

\(6 \int_{0}^{\pi} (\sin 3x + \sin 2x + \sin x)dx\) is equal to:

Let \[ f(x)=x^{2025}-x^{2000},\quad x\in[0,1] \] and the minimum value of the function $f(x)$ in the interval $[0,1]$ be \[ (80)^{80}(n)^{-81}. \] Then $n$ is equal to

Let the arithmetic mean of \(\frac{1}{a}\) and \(\frac{1}{b}\) be \(\frac{5}{16}\), where \(a>2\). If \(a,4,b\) are in A.P., then the equation \[ ax^2-ax+2(a-2b)=0 \] has:

The sum of all the roots of the equation \((x-1)^2 - 5|x-1| + 6 = 0\), is:

The value of \[ \sum_{k=1}^{\infty} (-1)^{k+1}\left(\frac{k(k+1)}{k!}\right) \] is:

The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are \( 2, 3, 5, 10, 11, 13, 15, 21 \), then the mean deviation about the median of all the 10 observations is:

Let \(a_1, a_2, a_3, \dots\) be a G.P. of increasing positive terms such that \(a_2 \cdot a_3 \cdot a_4=64\) and \(a_1 + a_3 + a_5 = \frac{813}{7}\). Then \(a_3 + a_5 + a_7\) is equal to:

An SBI health insurance agent found the following data for distribution of ages of 100 policy holders. Find the modal age and median age of the policy holders.

Let the length of the latus rectum of an ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ $(a>b)$ be $30$. If its eccentricity is the maximum value of the function $f(t)=-\dfrac{3}{4}+2t-t^2$, then $(a^2+b^2)$ is equal to

An equilateral triangle OAB is inscribed in the parabola $y^2 = 4x$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is

The natural number 1 is :

If $P(E) = 0.6$, $P(F) = 0.3$ and $P(E \cap F) = 0.2$, find $P(E|F)$ and $P(F|E)$.

A random variable \( X \) takes values \( 0, 1, 2, 3 \) with probabilities \( \frac{2a+1}{30}, \frac{8a-1}{30}, \frac{4a+1}{30}, b \) respectively, where \( a, b \in \mathbb{R} \). Let \( \mu \) and \( \sigma \) respectively be the mean and standard deviation of \( X \) such that \( \sigma^2 + \mu^2 = 2 \). Then \( \frac{a}{b} \) is equal to :

Let \[ I(x) = \int \frac{3\,dx}{(4x+6)\sqrt{4x^2 + 8x + 3}} \] and \[ I(0) = \frac{\sqrt{3}}{4} + 20. \] If \[ I\left(\frac{1}{2}\right) = \frac{a\sqrt{2}}{b} + c, \] where \(a, b, c \in \mathbb{N}\) and \(\gcd(a,b)=1\), then find the value of \[ a + b + c. \]

In the given figure, PA and PB are tangents to a circle centred at O. If \( \angle OAB = 15^\circ \), then \( \angle APB \) equals :