List of top Mathematics Questions

Let $f(x)=\begin{cases}ax+3 & x<1\\ \frac{4x}{a} & x\ge 1\end{cases}$. If $\lim_{x\rightarrow 1}f(x)$ exists, then the possible values of $a$ are

The range of the function $f(x)=\left(\frac{1}{3}\right)^{3+\sin x}$ is

A box contains 5 white balls, one red ball and 4 black balls. If three balls are drawn at random, then the probability of getting no red ball is

If $P(A/B)=\frac{1}{2}, P(B/A)=\frac{1}{3}$ and $P(A\cap B)=\frac{1}{6}$ then $P(A\cup B)$ is

If the mean and standard deviation of 10 observations are 24 and 4 respectively, then the sum of the squares of all observations is

The vector form of the straight line $\frac{x-2}{1}=\frac{y-1}{-1}=\frac{z-1}{-2}$ is

The combined mean age of a group of boys and girls in a school is 12. If the mean of the boys in that group is 14 and the mean of the girls is 9, then the percentage of boys in that group is

Let $\vec{OP}=2\hat{i}-2\hat{j}-\hat{k}$ and $\vec{OQ}=2\hat{i}+\hat{j}+2\hat{k}$. If the point $R$ lies on $\vec{PQ}$ and $\vec{OR}$ bisects the angle $\angle POQ$, then $2\vec{OR}$ is} \textit{Note: The initial vector has been mathematically corrected from the exam's typo ($2\hat{i}-2\hat{j}-2\hat{k}$) to standard format ($2\hat{i}-2\hat{j}-\hat{k}$) to permit a valid solution.

Let the three vectors $\vec{a}, \vec{b}$, and $\vec{c}$ be pairwise non-collinear vectors. If $\vec{a}+2\vec{b}$ is collinear with $\vec{c}$ and if $\vec{b}+2\vec{c}$ is collinear with $\vec{a}$, then $\vec{a}+2\vec{b}+5\vec{c}$ is equal to

If $\vec{a}\times(\hat{i}-\hat{j}+\hat{k})=(\hat{i}-\hat{j}+\hat{k})\times\vec{b}$ and $|\vec{a}+\vec{b}|=3\sqrt{3}$, then the possible values of $(\vec{a}+\vec{b})\cdot(3\hat{i}+2\hat{j}+\hat{k})$ are

The shortest distance between the lines $\vec{r}=\hat{i}+\hat{j}+3\hat{k}+\lambda(2\hat{i}+2\hat{j}+\hat{k})$ and $\vec{r}=(2\mu+1)\hat{i}+(2\mu-1)\hat{j}+(\mu+1)\hat{k}$ where $\lambda$ and $\mu$ are parameters, is

The equation of line which is parallel to $\frac{2-x}{-3}=\frac{y-2}{2}=\frac{z-4}{1}$ and passing through the point $(1,1,1)$, is

The angle between the lines $\vec{r}=(3+\alpha)\hat{i}+2(1+\alpha)\hat{j}+2(-2+\alpha)\hat{k}$ and $\vec{r}=(5+3\beta)\hat{i}+2(1+\beta)\hat{j}+6\beta\hat{k}$ , where $\alpha$ and $\beta$ are parameters, is

If the foci and vertices of an ellipse are respectively $(\pm 2,0)$ and $(\pm 3,0)$ then its eccentricity is

If the angle between the vectors $\vec{a}=x\hat{i}+3\hat{j}+\hat{k}$ and $\vec{b}=x\hat{i}-x\hat{j}+2\hat{k}$ is acute, then $x$ lies is

The equation of a hyperbola is $9x^{2}-16y^{2}=144$. If $A$ and $S$ are, respectively, the focus and the vertex of one section of the hyperbola, then the length of $AS$ is

If the focus and vertex of a parabola are at a distance of 3 units and 6 units, respectively, from the origin on the positive x-axis, then the equation of the parabola is

If the circle $x^{2}+y^{2}-6x-12y-55=0$ intercepts the x-axis at two points $A$ and $B$, then $|AB|$ is equal to

If the foot of the perpendicular drawn from the origin to the line $y=mx+c$ is $(1,1)$ then the value of $m$ and $c$ are, respectively,

The perpendicular distance between the lines $3x+4y-6=0$ and $6x+8y+18=0$ is

The value of $\tan^{-1}\left(\frac{\cos x-\sqrt{3}\sin x}{\sqrt{3}\cos x+\sin x}\right)$ , where $0<x<\frac{\pi}{2}$ is

The x-intercept and y-intercept of a line are three times and four times of the x-intercept and y-intercept of the line $3x+2y=6$, respectively. Then the equation of the line is

The value of $\tan^{2}(\sec^{-1}(3))$ is

Let $f(x)=(8\sin x+15\cos x+3)^{2}-15$, $x\in\mathbb{R}.$ Then the maximum value of $f$ is

The value of $\cos^{-1}\left(\cos\frac{2\pi}{3}\right)+\sin^{-1}\left(\sin\frac{2\pi}{3}\right)$ is equal to