VITEEE 2024

Difference between the maximum and minimum values of $f(x)=-\sin ^3 x+3 \sin ^2 x+5$ in $x \in\left[0, \frac{\pi}{2}\right]$ is

If tangent to the curve $f(x)=x^3-\alpha x^2-x+\beta$ at point $(1,3)$ on the curve, cut equals non zero intercepts on co-ordinate axes, then

The length of three sides of a trapezium are equal, each being 10 cms . Then, the maximum area $\left(\mathrm{cm}^2\right)$ of the trapezium is

The area of the region(s) enclosed by the curves $y=x^2$ and $y=\sqrt{|x|}$ is

If the coefficients of $x^7$ and $x^8$ in the expansion of $\left[2+\frac{x}{3}\right]^n$ are equal, then value of $n$ is

Coefficient of $x^3$ in the expansion of $\left(x^2-x+1\right)^{10}\left(x^2+1\right)^{15}$ is equal to

If the coefficient of $x^2$ and $x^3$ in the expansion of $\left(1+8 x+b x^2\right)(1-3 x)^9$ in the power of $x$ are equal, then $b$ is

If the tangent at the point $P$ on the circle $x^2+y^2+2 x+2 y=7$ meets the straight line $3 x-4 y=15$ at the point $Q$ on the $X$-axis, then length of $P Q$ is

If two different circles $x^2+y^2+2 a x+2 b y+1$ $=0$ and $x^2+y^2+2 b x+2 a y+1=0$ touches each other, then $(a+b)^2$ is equal to

The complex number $z$ satisfying $z+|z|$ $=1+7 i$, then the value of $|z|^2$ equals

\(\int_0^\pi\left[\cos ^2\left(\frac{3 \pi}{8}-\frac{x}{4}\right)-\cos ^2\left(\frac{11 \pi}{8}+\frac{x}{4}\right)\right] d x\) equals to

The differential equation corresponding to the family of curves $y=e^x(a x+b)$ is

If the curves $\frac{x^2}{a}+\frac{y^2}{4}=1$ and $y^3=16 x$ intersect at right angles, then ' $a$ ' equals to

If $(\cos x)^y=(\sin y)^x$, then $\frac{d y}{d x}$ equals

Consider the ellipse $\frac{x^2}{\cos ^2 \alpha}+\frac{y^2}{\sin ^2 \alpha}=1$, where $\alpha \in\left(0, \frac{\pi}{4}\right)$. Then, locus of point of intersection of one of the directrix and tangent at upper end of minor axis is

Period of $f(x)=\{x\}+\left\{x+\frac{1}{3}\right\}+\left\{x+\frac{2}{3}\right\}$ is equal to (where $\{\cdot\}$ is fractional part function)

If $e$ is the eccentricity of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\theta$ is the angle between the asymptotes, then $\cos \frac{\theta}{2}$ is

The sum of the infinite terms of the series $\cot ^{-1}\left(1^2+\frac{3}{4}\right)+\cot ^{-1}\left(2^2+\frac{3}{4}\right)$ $+\cot ^{-1}\left(3^2+\frac{3}{4}\right)+\ldots$ is equal to

The value of \(\sin ^{-1}\left\{\cot \left(\sin ^{-1} \sqrt{\frac{2-\sqrt{3}}{4}}+\cos ^{-1} \frac{\sqrt{12}}{4}+\sec ^{-1} \sqrt{2}\right)\right\}\) is

$\lim \limits_{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2}$ equal to

$\lim _\limits{x \rightarrow 0} \frac{1}{x} \int_0^x(1+\sin 3 t)^{\frac{1}{t}} d t$ is equal to

If $a, b$ and $c$ are distinct positive numbers, not equal to unity. Such that $a b c=1$, then the value of $\log _b a \cdot \log _c a+\log _c b$ $\cdot \log _a b+\log _a c \log _b c$ is

$\log \left(\log _{a b} a+\frac{1}{\log _b a b}\right)$ is $($ where $a b \neq 1)$

If $A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $B=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right]$, then $\left(B B^T A\right)^5$ is equal to

If $A, B$ are two square matrices, such that $A B=A, B A=B$, then $(A+B)^7$ equals

The area of circle touching parabola $y=x^2$ at $(1,1)$ and having directrix of $y=x^2$ as its normal is $125 A \pi$, then $A$ is

If 6 letter words, with or without meaning can be formed out of these letters of the word "MATHEMATICS", repetition of letters is not allowed is $960 P$. Then, ' $P$ ' equals

6 couples decided to form a committee of four members, the number of different committees that can be formed in which no couple finds place is

Two red counters, three green counters and four blue counters are placed in a row in random order. The probability that no two blue counters are adjacent is

In a triangle $A B C$ as shown in the diagram, where $A B=5, A C=10$ and $B D=1$, then perimeter of $\triangle A B C$ is

Given, $\frac{x^2+y^2}{x^2-y^2}+\frac{x^2-y^2}{x^2+y^2}=k$, then $\frac{x^8+y^8}{x^8-y^8}$ is equal to

Sum of first ' $n$ ' terms of a series $a_1+a_2+\ldots+a_n$ is given by $S_n=\frac{n\left(n^2-1\right)(n+2)}{4}$, then the value of $\lim _\limits{n \rightarrow \infty} \sum_\limits{r=2}^n \frac{1}{a_r}$ is

The sum of $1+\frac{1}{4}+\frac{1 \cdot 3}{4 \cdot 8}+\frac{1 \cdot 3 \cdot 5}{4 \cdot 8 \cdot 12}+\ldots \infty$ is

The sum of the series $1 \cdot 2^2+2 \cdot 4^2+3 \cdot 6^2+\ldots$ upto 10 terms is

If the distance between the planes $8 x+12 y-14 z=2$ and $4 x+6 y-7 z=2$ can be expressed as $\frac{1}{\sqrt{N}}$, then the value of $\frac{N(N+1)}{2}$ is

The value of ' $a$ ' for which the lines $\frac{x-2}{1}=\frac{y-9}{2}=\frac{z-13}{3}$ and $\frac{x-a}{-1}=\frac{y-7}{2}$ $=\frac{z+2}{-3}$ intersect, is

$\tan 65^{\circ}, \tan 40^{\circ}+\tan 25^{\circ}$ and $\tan 25^{\circ}$ are in

If $P=\operatorname{cosec} \frac{\pi}{8}+\operatorname{cosec} \frac{2 \pi}{8}+\operatorname{cosec} \frac{3 \pi}{8}$ $+\operatorname{cosec} \frac{13 \pi}{8}+\operatorname{cosec} \frac{14 \pi}{8}+\operatorname{cosec} \frac{15 \pi}{8}$ and $\p...

If the unit vectors $\mathbf{a}$ and $\mathbf{b}$ are inclined at $2 \theta$ and $|\mathbf{a}-\mathbf{b}|<1$, then if $0<\theta<\pi, \theta$ lies in the interval.

The volume of the parallelopiped whose edges are represented by $\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$, $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}-\hat...