VITEEE 2025

The period of the function $f(x)=3 \sin \frac{\pi x}{3}+4 \cos \frac{\pi x}{4}$ is

$\int \frac{e^{x^2}\left(2 x+x^3\right)}{\left(3+x^2\right)^2} d x$ is equal to

$S=\cot ^{-1}\left(\frac{1+2 \times 6}{4}\right)+\cot ^{-1}\left(\frac{1+3 \times 8}{10}\right)+\cot ^{-1} \left(\frac{1+4 \times 10}{18}\right) \ldots \ldots \ldots$ upto $\infty$ terms is equal to

If $f(x)=\left(\frac{x^2+5 x+3}{x^2+x+2}\right)^x$, then $\lim _{x \rightarrow \infty} f(x)$ is

The set of real value of $x$ for which $\log _{0.2} \frac{x+2}{x} \leq 1$ is

The value of $(016)^{\log _{25}\left(\frac{1}{3}+\frac{1}{3^2}+\ldots+\infty\right)}$ is equal to
$..........$

Let $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right], x \in R$ and $A^4=\left[a_{i j}\right]$.
It $a_{11}=109$, then $a_{22}$ is equal to

The point $(-2 m, m+1)$ is an interior point of the smaller region bounded by circle $x^2+y^2=4$ and the parabola $y^2=4 x$, then

The focal chord of $y^2=16 x$ is a tangent to $(x-6)^2+y^2=2$, then the possible values of the slope of this chord are

For the parabola $y^2=16 x$, length of a focal chord, whose on end point is $(16,16)$ is $L^2$, then absolute value of $L$ is

Total number of 3 letters word that can be formed from the letters of the word 'SAHARANPUR' is equal to

The number of numbers greater than a million that can be formed with the digits 2 , $3,0,3,4,2$ and 3 is

A five digit number (having all different digits) is formed using the digits $1,2,3,4,5$, 6,78 and 9 . The probability that the formed number either begins or ends with an odd digit is equal to

In a $\triangle A E X, T$ is the mid-point of $X E$ and $P$ is the mid-point of $E T$. If the $\triangle A P E$ is equilateral of side length equal to unity, then which of the following alternative is not correct?

If $[x]$ denotes the integral part of $x$ and $k=\sin ^{-1}\left(\frac{1+t^2}{2 t}\right)>0$, then number of values of $\alpha$ for which the equation $(x-[k])(x+\alpha)-1$ has integral roots

If $\alpha$ and $\beta$ are the roots of equation $x^2+p x+2=0$ and $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ are roots of equation $2 x^2+2 q x+1=0$, then $\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\frac{1}{\beta}\right)\left(\alpha+\...

Sum to 10 terms of the series $1+2(1 \cdot 1)+3(1 \cdot 1)^2+4(1 \cdot 1)^3+$ $\_\_\_\_$ is

$S_n=1 \cdot 3+2 \cdot 2^2+3 \cdot 3^3+4 \cdot 2^4+\ldots \ldots \ldots$ upto $n$ terms. If $S_{20}=a \cdot 3^{21}+b \cdot 2^{22}+\frac{391}{288}$, then value of $32 a-9 b$ is

The equation of the line, where length of the perpendicular segment from the origin to the line is 4 and the inclination of the perpendicular segment with the positive direction of $X$-axis is $30^{\circ}$.

The equation of a straight line passing through $(1,2)$ and having intercept of length 3 between the straight lines $3 x+4 y=24$ and $3 x+4 y=12$ is

The distance between the point with position vector $-\hat{i}-5 \hat{j}-10 \hat{k}$ and the point of intersection of the line $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ with the plane $x-y+z=5$, is $\_\_\_\_$ (units)

The equation of the straight line through the origin and parallel to the line
$$ \begin{aligned} & (b+c) x+(c+a) y+(a+b) z=k= \\ & (b-c) x+(c-a) y+(a-b) z \text { is } \end{aligned} $$

The least positive non-integral solution of the equation $\sin \pi\left(x^2+x\right)=\sin \pi x^2$ is

For any four vectors $\mathbf{a , b , c , d}$ the expression $(\mathrm{b} \times \mathrm{c}) \cdot(\mathrm{a} \times \mathrm{d})+(\mathrm{c} \times \mathrm{a}) \cdot(\mathrm{b} \times \mathrm{d})+(\mathrm{a} \times \mathrm{b}) \cdot(\mathrm...

If $\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}=0$, where $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ are unit vectors and the unit vector $\hat{\complement}$ is inclined at an angle $\theta$ to both $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$. If ...

Which of the following statement is correct regarding the AC circuit shown in the adjacent figure?

In the fusion reaction,
\({ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \longrightarrow{ }_2^3 \mathrm{He}+{ }_0^1 \mathrm{n}\)
the masses of deuteron, helium and neutron expressed in amu are 2.015, 3.017 and 1.009, respectively. If 1 kg of deute...

The radius of the orbit of an electron in a Hydrogen-like atom is $45 a_0$, where $a_0$ is the Bohr radius. Its orbital angular momentum is $\frac{3 h}{2 \pi}$. It is given that $h$ is Planck constant and $R$ is Rydberg constant. The possib...

\(\text { A } 2 \mu \mathrm{~F} \text { capacitor is charged as shown, }\)

When switch $S$ is shifted to position $2, \%$ of energy lost is

A network of four capacitors of capacity equal to $C_1=C, C_2=2 C, C_3=3 C$ and $C_4=4 C$ are connected to a battery, as shown in the figure. The ratio of the charges on $C_2$ and $C_4$ is $\_\_\_\_$

Capacity of a parallel plate capacitor in the absence of dielectric medium is $C_0$. A sheet of dielectric constant $k$ and thickness of one-fourth of the plate separation is inserted between the plates. If the new capacity is $C$, then $\f...

A ball suspended by a thread swings in a vertical plane, so that its acceleration in the extreme position and lowest position are equal. The angle $\theta$ of thread deflection in the extreme position will be

Three electric bulbs of 200 W and 400 W are shown in figure. The resultant power of the combination, if rated voltage is applied across the combination is

A parallel beam of fast moving electrons is incident normally on a narrow slit. If the speed of the electrons is decreased, the angular width of central maxima

When a beam of 10.6 eV photons of intensity $2.0 \mathrm{~W} / \mathrm{m}^2$ falls on a platinum surface of area $1.0 \times 10^{-4} \mathrm{~m}^2$ and work function $5.6 \mathrm{eV} .0 .53 \%$ of the incident photons eject photoelectrons. ...

The average depth of Indian ocean is about 3000 m . The fractional compression, $\frac{\Delta V}{V}$ of water at the bottom of the ocean (given, that the Bulk modulus of the water $=2.2 \times 10^9 \mathrm{Nm} { }^{-2}$ and $g=10 \mathrm{~m...

The following four wires of length $L$ and radius $r$ are made of the same material. Which of these will have the largest extension, when the same tension is applied?

A conductor of length 0.6 m is moving with a speed of $5 \mathrm{~m} / \mathrm{s}$ perpendicular to a magnetic field of intensity $0.8 \mathrm{~Wb} / \mathrm{m}^2$. The induced emf across the conductor is

If the magnetic field in plane electromagnetic wave is
\(\mathbf{B}=3 \times 10^{-8} \sin \left(1.6 \times 10^3 x+48 \times 10^{10} t\right) \hat{\mathbf{j}} \mathrm{T},\)
then find the expression of electric field?

A laser beam of cross-sectional area $5 \mathrm{~mm}^2$ is 30 mW . Find the magnitude of maximum electric field in this electromagnetic wave.

In given situation, force on charge $Q_3$ is

A liquid film is formed over a frame $A B C D$ as shown in figure. Wire $C D$ can slide without friction. Maximum value of mass that can be hanged from $C D$ without breaking the liquid film is

A car of mass 500 kg is lifted by a hydraulic jack that consist of two pistons. If the diameter of large and small pistons are 2 m and 20 cm , respectively. Then force required (in N ) to lift the car by smaller piston is (take, $g=10 \math...

An object is projected from surface of Earth with a kinetic energy twice that of escape energy ' $K$ ', from surface of Earth. It's kinetic energy when it reaches far away from Earth is

If the Earth is assumed to be a sphere of uniform mass density and its weight on the surface is 200 N , then the weight of the body at a depth halfway to the Earth's center will be

When the ideal monoatomic gas is heated at constant pressure fraction of heat energy supplied which increases the internal energy of gas is

Two identical long solid cylinders are used to conduct heat from temperature $T_1$ to temperature $T_2$. Originally, the cylinders are connected in series and the rate of heat transfer is $H$. If the cylinders are connected in parallel, the...

A flask contains argon and chlorine in the ratio of $2: 1$ by mass, the temperature of mixture is $27^{\circ} \mathrm{C}$, the ratio of root mean square ( $U_{\mathrm{rms}}$ ) of the molecule of two gases (Given, Atomic mass of argon $=39.9...

When a ball of mass 5 kg hits a bat with a velocity $3 \mathrm{~m} / \mathrm{s}$, in positive direction and it moves back with a velocity $4 \mathrm{~m} / \mathrm{s}$, find the impulse in SI units.

A cylindrical magnet has a length of 15 cm and diameter 3 cm . If it has uniform magnetisation of $4.0 \times 10^3 \mathrm{Am}^{-1}$, then magnetic dipole moment is