VITEEE 2025

The velocity of a particle moving in a straight line varies with time in such a manner that $v$ versus $t$ graph is velocity is $v_m$ and the total time of motion is $t_0$

(i) Average velocity of the particle is $\frac{\pi}{4} v_m$

(ii) Such motion cannot be realized in practical terms

Choose the correct option on the basis of above two statements.

The height $y$ and the distance $x$ along the horizontal plane of a projectile on a certain planet (with no surrounding atmosphere) are given by $y=8 t-5 t^2 \mathrm{~m}$ and $x=6 t \mathrm{~m}$, where $t$ is in seconds. The velocity with which the projectile is projected is

A ball is projected horizontally with a velocity of $5 \mathrm{~ms}^{-1}$ from the top of a building 19.6 m high. How long will the ball take to hit the ground?

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

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

A body of mass $M$ is moving with a uniform speed of $10 \mathrm{~m} / \mathrm{s}$ on frictionless surface under the influence of two forces $F_1$ and $F_2$. The net power of the system is

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

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.

If the rotational kinetic energy of a body is increased by $300 \%$, then the percentage increase in its angular momentum will be

A steel wire of mass 4 g and length 80 cm is fixed at the two ends. The tension in the wire is 50 N , then the frequency of fourth harmonic 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

The displacement from mean position of a particle in SHM at 3 s is $\frac{\sqrt{3}}{2}$ of the amplitude, then its time period is

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 $g=10 \mathrm{~ms}^{-2}$ ) 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, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

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?

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 $\frac{C}{C_0}$ is

In given situation, force on charge $Q_3$ is

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 $=39.9 v$, Molecular mass of chlorine $=70.9 u$ )

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, then the rate of heat transfer 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

Ratio of powers of a convex and concave lens is $\frac{3}{2}$ and their equivalent focal length (when lenses are kept in constant) is 30 cm . Focal lengths of lenses are

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

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

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

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 $\_\_\_\_$

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?

In hydrogen atom, an electron is revolving in the orbit of radius $0.53 \mathop {\rm{A}}\limits^{\rm{o}} $ with $6.6 \times 10^{15} \mathrm{rps}$. Magnetic field produced at the centre of the orbit is

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 number of photoelectrons emitted per second is

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 deuterium undergoes complete fusion, then find the amount of total energy released. ( $1 \mathrm{amu}=9315 \mathrm{MeV}$ )

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.

To get an output 1 from the circuit shown in the figure, the input must be

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 possible wavelength(s), when the atom de-excites, is (are)

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

Under isothermal condition, a gas at 300 K expands from 0.1 L to 0.25 L against a constant external pressure of 2 bar. The work done by the gas is (Given that, 1 L bar $=100 \mathrm{~J}$ )

A 2 M aqueous solution of HCl has density $1.08 \mathrm{~g} / \mathrm{mL}$. What is the molality of the solution?

The number of radial nodes in $4 s$ and $3 p$ orbitals respectively is

Given, $\Delta G^{\circ}=-502 \mathrm{~kJ} \mathrm{~mol}^{-1}$ for the dissociation of ,

$\mathrm{Mg}(\mathrm{OH})_2(s) \rightleftharpoons \mathrm{Mg}^{2+}(a q)+2 \mathrm{OH}^{-}(a q)$ at 298 K . Find the $K_{\mathrm{sp}}$ value.

$(R=8314 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K})$

A 0.05 M solution of aniline hydrochloride has a pH of 3.85 . Calculate the ionisation constant ( $K_b$ ) of aniline.

Arrange the energy of $2 s$ orbitals in increasing order for the atoms $\mathrm{C}, \mathrm{B}, \mathrm{N}$ and O .

In the following redox reaction, what will be the sum of $x, y$, and $z$ ?

$$ \begin{aligned} x \mathrm{Cr}_2 \mathrm{O}_7^{2-}(a q) & +y \mathrm{SO}_3^{2-}(a q)+z \mathrm{H}^{+}(a q) \longrightarrow \mathrm{Cr}^{3+}(a q)+\mathrm{SO}_4^{2-}(a q)+\mathrm{H}_2 \mathrm{O}(l) \end{aligned} $$

Ozone is quantitatively determined by bubbling it through an excess of acidic potassium iodide solution, which liberates iodine. The liberated iodine is then titrated with sodium thiosulphate solution.

If 25.0 mL of the iodine-containing solution required 20.0 mL of 0.01 M sodium thiosulphate for complete titration, calculate the number of moles of ozone that reacted.

The freezing points of 0.01 M solutions of glucose, $\mathrm{NaNO}_3$, and $\mathrm{AlCl}_3$ would be

How many moles of $\mathrm{KMnO}_4$ are required to oxidise 1 mole of oxalic acid in acidic medium?

6.35 g of copper (Atomic mass $=635 \mathrm{u}$ ) is deposited at the cathode from a $\mathrm{Cu}^{2+}$ solution using a certain quantity of electric charge.

The volume of oxygen gas liberated at STP from water electrolysis using the same quantity of electric charge is

The osmotic pressure of a solution prepared by dissolving 48 mg of $\mathrm{MgCl}_2$ in 500 mL water at $25^{\circ} \mathrm{C}$ is (Assuming complete dissociation).

(Molar mass of $\mathrm{MgCl}_2=95 \mathrm{~g} / \mathrm{mol}, R=0.0821 \mathrm{L} \cdot \mathrm{atm} / \mathrm{mol} \cdot \mathrm{K}$ )

When $\mathrm{H}_2 \mathrm{O}_2$ is added to a solution of acidified sodium dichromate, a deep blue colour is observed due to the formation of a peroxo complex. This change signifies that

In a first-order reaction, the initial concentration of $A$ is $0.5 \mathrm{~mol} \mathrm{~L}^{-1}$. After 20 minutes, the concentration reduces to $0.4 \mathrm{~mol} \mathrm{~L}^{-1}$. The rate constant is (in $\mathrm{min}^{-1}$ ).

Among the following, the species having same bond order are

How long should water be electrolysed using 50 A current to release enough $\mathrm{H}_2$ gas to completely reduce 8.0 g of CuO ?

$\left(\mathrm{CuO}+\mathrm{H}_2 \longrightarrow \mathrm{Cu}+\mathrm{H}_2 \mathrm{O}\right.$; Atomic mass of $\mathrm{Cu}=635 \mathrm{u}$ )

The activation energy for a reaction is $9.0 \mathrm{kcal} / \mathrm{mol}$. The increase in the rate constant when the temperature is increased from 298 K to 308 K is

Which of the following oxides of nitrogen has a lone pair and is paramagnetic?

Arrange the following in the correct increasing order of atomic size.

$\mathrm{F}, \mathrm{Cl}, \mathrm{Br}, \mathrm{I}$

Which of the following complex ions is expected to appear colourless in aqueous solution?

Among the following, the strongest nucleophile in aqueous medium is

$$ \text { Match List-I with List-II. } $$

Choose the correct match from the options given below

Which of the following complexes can show both facial ( $f a c$ ) and meridional (mer) isomerism?

Which of the following molecules has the maximum dipole moment?

Which of the following ethers cannot be prepared efficiently by Williamson synthesis?

Identify the final product $C$ in the following sequence.

$$ \begin{aligned} \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{Br}+(\text { alc. } \mathrm{KOH}) & \xrightarrow{h \nu} A+\mathrm{Br}_2 \longrightarrow B+(\text { alc. } \mathrm{KCN}) \longrightarrow C \end{aligned} $$

Which of the following compounds on reaction with $\mathrm{CH}_3 \mathrm{MgBr}$ followed by acid hydrolysis gives a tertiary alcohol?

Which of the following does not give a white precipitate with sodium bisulphide?

In the diazotisation of aniline with sodium nitrite and hydrochloric acid the excess of hydrochloric acid is used primarily to

Identify the product $Q$ in the following reaction sequence,

Which of the following pairs form the same osazone?

$$ \text { Match List-I with List-II. } $$

Choose the correct match from the options given below

Which of the following is a water-soluble vitamin?

An unknown ester is hydrolysed to give ethanol as one of the products. Which chemical test can be used to confirm the identity of this alcohol?

The product $(X)$ formed in the given reaction.

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

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

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

$..........$

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

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

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 $\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\frac{1}{\beta}\right)\left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)$ is equal to

$S_n=1 \cdot 3+2 \cdot 2^2+3 \cdot 3^3+4 \cdot 2^4+\ldots \ldots \ldots$ 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 value of

$$ 99^{50}-90 \cdot 98^{50}+\frac{99 \cdot 98}{1 \cdot 2}(97)^{50}-\ldots \ldots \ldots \ldots .+99 $$

is

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

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

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

If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x)^{3 / 2}-\left(1+\frac{1}{2} x\right)^3}{(1-x)^{1 / 2}}$ may be approximate as

If $a$ and $b$ are two complex numbers, then the sum of $(n+1)$ terms of the series $a c_0-(a+d) c_1+(a+2 d) c_2-(a+3 d) c_3+$ $\_\_\_\_$ is

If $\alpha$ is a root of $x^4=1$ with negative principal argument, then the principle argument of $\Delta(A)$, where $\Delta(A)=\left|\begin{array}{ccc}1 & 1 & 1 \\ \alpha^n & \alpha^{n+1} & \alpha^{n+3} \\ \frac{1}{\alpha^{n+1}} & \frac{1}{\alpha^n} & 0\end{array}\right|$

If $a$ is the root of equation $z^n+2 z^{n-1}+3 z^{n-2}+12-18 z=0$ which lies inside $|z|=1$, then

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)

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 $\hat{\mathbf{c}}=m \hat{\mathbf{a}}+n \hat{\mathbf{b}}+p(\hat{\mathbf{a}} \times \hat{\mathbf{b}})$, where, $m, n, p \in R$, then

The equation of the straight line through the origin and parallel to the line

$$ \begin{aligned} & (b+c) x+(c+a) y+(a+b) z=k= \\ & (b-c) x+(c-a) y+(a-b) z \text { is } \end{aligned} $$

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

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

Let $f: R \rightarrow R, f(x)=x^3-3 x^2+3 x-2$, then $f^{-1}(x)$ is given by

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

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?

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

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

Let $f(x)=\cos ^{-1}\left(2 x \sqrt{1-x^2}\right)$, then $f^{\prime}(0.6)$ equals to

The minimum value of $(u-v)^2+\left(\sqrt{2-u^2}-\frac{9}{v}\right)^2$, where $0 < u < \sqrt{2}$ and $v > 0$

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

The function, $f(x)=(3 x-7) x^{2 / 3}, x \in R$ is increasing for all $x$ lying in

Let $f$ be a non-negative function defined on the interval $[0,1]$. If $\int_0^x \sqrt{1-(f(t))^2} d t =\int_0^x f(t) d t, 0 \leq x \leq 1$ and $f(0)=0$, then

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 area of the region $\left\{(x, y): x y<8,1 \leq y \leq x^2\right\}$ is

The differential equation for all family of line which are at a unit distance from the origin is

In the given figure, the equation of the large circle is $x^2+y^2+4 y-5=0$ and the distance between centre is 4 . Then, the equation of smaller circle 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 $30^{\circ}$.

Let $c_1: x^2+y^2=1$ and $c_2:(x-10)^2+y^2=9$ be two circles a line touching $c_1$ at $P$ and $c_2$ at $Q$. If $M$ is the mid-point of $P Q$, then $M$ lies on a circle $(x-5)^2+y^2=r^2$ where ' $r$ ' is $(r>0)$

In a code language, 'SERIES' is coded as 87 and 'DISMAY' is coded as 91 . How will 'SPRINT' be coded in the same language?

The total production of computers in 2013, 2015 and 2018 is $x \%$ of the total exports of computers by the company during the 6 yr .

The value of $x$ is

The questions given below consist of a question followed by two statements numbered as I and II. We have to decide whether these statements give enough information required to answer the question or not.

When is Yuvi's wedding anniversary?

Statements

I. On 7th day of a month.

II. The month has 29 days only in a leap year.

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

Based on the English alphabetical order, three of the following four letter-clusters are alike in a certain way and thus form a group. Which letter-cluster does not belong to that group?

Select the option that is related to the fifth letter-cluster in the same way as the second letter-cluster is related to the first letter-cluster and the fourth letter-cluster is related to the third letter-cluster.

ABIDE : GFKDC :: CHARM : OTCJE :: BLOOM : ?

Which of the following numbers will replace the question mark (?) in the given series?

167, 117, ?, 47, 27

Which of the following terms will replace the question mark (?) in the given series?

DIQ, EJR, FKS,?

Two statements are given, followed by three conclusions numbered as I, II and III.

Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the given statements.

Statements

All scanners are printers.Some scanners are copiers.

Conclusions

I. Some printers are copiers.

II. No copier is a printer.

III. Some printers are scanners.

Why have major airline companies pledged to reach net-zero carbon emissions by mid-century?

In a code language, 'TREE' is written as 'WSHF', and 'BALL' is written as 'EBOM'.How will 'WALL' be written in that language?

Select the most appropriate antonym of the underlined word in the following sentence. The woman in the crowd broke down in hysterics.

By what per cent were the total exports of computers, by the company, in 2013, 2014 and 2018 less than the total production of computers in 2015 to 2017? (correct to one decimal place)

Select the most appropriate synonym of the given word.

Prudent

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

Biofuels are not made from $\_\_\_\_$

Select the most appropriate idiom to fill in the blank. The team always $\_\_\_\_$ to come up with unique advertisements for all its clients.

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

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