VITEEE 2024

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...

The primary of a transformer has 400 turns while the secondary has 2000 turns. If the power output from the secondary at 1000 V is 12 kW . The resistance of the primary is $0.2 \Omega$ and that of the secondary is $2 \Omega$. If the efficie...

A transformer has 200 windings in the primary and 400 windings in the secondary. The primary is connected to an AC supply of 110 V and a current of 10 A flows in it. The voltage across the secondary and the current in it, respectively, are

In an electric circuit, a capacitor of reactance $55 \Omega$ is connected across the source of 220 V . The value of displacement current is

Which of the following graphs represents the correct variation of inductive reactance $X_L$ with angular frequency $\omega$ ?

The ratio of minimum wavelengths of Balmer and Paschen series of hydrogen atom will be

36 cells each of internal resistance $0.5 \Omega$ and emf 1.5 V each are used to send current through an external circuit of $2 \Omega$ resistor. For getting maximum current, $n$ cells are combined in series in each row. If total number of ...

The maximum power rating of a $20 \Omega$ resistor is 2 kW . Then, it cannot be used with 300 V DC source because the

The magnitude of resistance $X$ in the circuit shown below in the figure, when no current flows through the $5 \Omega$ resistor, is

A metal surface is illuminated with photons of energies $E_1=4 \mathrm{eV}$ and $E_2=2.5 \mathrm{eV}$ respectively. The ratio of maximum speeds of the photoelectrons in two cases is 2 .
Work function of metal surface will be

The magnetic field in a electromagnetic wave is given by
\(B_y=2 \times 10^{-7} \sin \left(0.5+10^3 x+1.5 \times 10^{11} t\right) \mathrm{T}\)
The frequency of the wave is (in GHz)

An electromagnetic wave with poynting vector $6 \mathrm{Wm}^{-2}$ is absorbed by a surface area $12 \mathrm{~m}^2$. The force on the surface is

Eight dipoles of charges of magnitude $e$ each are placed inside a cube. The total electric flux coming out of the cube will be

Two point charges of $+3 \mu \mathrm{C}$ and $-3 \mu \mathrm{C}$ are placed at a certain distance apart from each other. Electric potential at a distance 60 cm from the mid-point of dipole on axial line is 150 V , then the value of dipole l...

The work done in moving a charge particle of charge $2 \times 10^{-8} \mathrm{C}$ between two points having potential difference of 36 V is

Three charges, each of $+4 \mu \mathrm{C}$, are placed at the corners $A, B, C$ of a square $A B C D$ of side 1 m . What is the magnitude (in $\mathrm{NC}^{-1}$ ) of electric field at point $O$ towards point $D$ ?

In a car lift, compressed air exerts a force $F_1$ on a small piston having a radius of 5.0 cm . This pressure is transmitted to a second piston of radius 15.0 cm . If the mass of the car to be lifted is 1350 kg . The force on a small pisto...

If gravitational attraction between two points masses be given by $F=G \frac{m_1 m_2}{r^n}$, then the period of a satellite in a circular orbit will be proportional to

The distance of the centres of Moon and the Earth is $D$. The mass of the Earth is 81 times the mass of the Moon. At what distance from the centre of the Earth, the gravitational force on a particle will be zero?

Variation of internal energy with density of one mole of monoatomic gas is depicted in the below figure, corresponding variation of pressure with volume can be depicted as (assuming the curve is rectangular hyperbola)

Two different ideal diatomic gases $A$ and $B$ are initially in the same state. $A$ and $B$ are then expanded to same final volume through adiabatic and isothermal process, respectively. If $p_A, p_B$ and $T_A, T_B$ represent the final pres...

A cyclic process for 1 mole of an ideal is shown in the $V-T$ diagram. The work done in $A B, B C$ and $C A$ respectively is

A disc of mass 10 g is kept floating horizontally by throwing 10 marbles per second against it from below. If the mass of each marble is 5 g . What will be velocity with which the marble are striking the disc? Assume that, the marble strike...

A short bar magnet of magnetic moment of $0.15 \mathrm{~J} / \mathrm{T}$ is placed in a uniform magnetic field of 0.4 T . The potential energy of the magnet in unstable equilibrium is

The horizontal range of a projectile is $4 \sqrt{3}$ times of its maximum height. The angle of projection will be

A current carrying conductor experiences maximum force in magnetic field when the angle between current element and direction of magnetic field is