JEE Main 2024 — 30 January, Shift 1

A line passing through the point $A(9,0)$ makes an angle of $30^{\circ}$ with the positive direction of $x$-axis. If this line is rotated about A through an angle of $15^{\circ}$ in the clockwise direction, then its equation in the new position is

Let $\mathrm{S}_{\mathrm{a}}$ denote the sum of first n terms an arithmetic progression. If $S_{20}=790$ and $S_{10}=145$, then $S_{15}-$ $\mathrm{S}_{5}$ is :

If $z=x+i y, x y \neq 0$, satisfies the equation $z^{2}+i \bar{z}=0$, then $|z|^{2}$ is equal to :

Let $\vec{a}=a_{i} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}$ and $\vec{b}=b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k} \quad$ be two vectors such that $|\vec{a}|=1 ; \vec{a} \cdot \vec{b}=2$ and $|\vec{b}|=4$. If $\vec{c}=2(\vec{a} \times \vec{b})-3 \vec{b}$, then the angle between $\vec{b}$ and $\vec{c}$ is equal to :

The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $\mathrm{y}=-2 \mathrm{x}^{2}+54$ at points $(\mathrm{x}, \mathrm{y})$ and $(-\mathrm{x}, \mathrm{y})$ where $y>0$ is :

The value of $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{n^{3}}{\left(n^{2}+k^{2}\right)\left(n^{2}+3 k^{2}\right)}$ is :

Let $\mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be a non constant twice differentiable such that $g^{\prime}\left(\frac{1}{2}\right)=g^{\prime}\left(\frac{3}{2}\right)$. If a real valued function f is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$, then :

The area (in square units) of the region bounded by the parabola $y^{2}=4(x-2)$ and the line $y=2 x-8$

Let $y=y(x)$ be the solution of the differential equation $\sec \mathrm{x} d \mathrm{~d}+\{2(1-\mathrm{x}) \tan \mathrm{x}+\mathrm{x}(2-\mathrm{x})\}$ $d x=0$ such that $y(0)=2$. Then $y(2)$ is equal to :

Let $(\alpha, \beta, \gamma)$ be the foot of perpendicular from the point $(1,2,3)$ on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$. then $19(\alpha+\beta+\gamma)$ is equal to :

Two integers $x$ and $y$ are chosen with replacement from the set $\{0,1,2,3, \ldots . .10\}$. Then the probability that $|x-y|>5$ is :

If the domain of the function $f(x)=\cos ^{-1}\left(\frac{2-|x|}{4}\right)+\left(\log _{e}(3-x)\right)^{-1}$ is $[-\alpha, \beta)-\{y\}$, then $\alpha+\beta+\gamma$ is equal to :

Consider the system of linear equation $x+y+z=$ $4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^{2} z=\mu^{2}+15$, where $\lambda, \mu \in \mathrm{R}$. Which one of the following statements is NOT correct?

If the circles $(x+1)^{2}+(y+2)^{2}=r^{2}$ and $x^{2}+y^{2}-4 x-4 y+4=0$ intersect at exactly two distinct points, then

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

Let $M$ denote the median of the following frequency distribution.

Then 20 M is equal to :

If $f(x)=\left|\begin{array}{ccc}2 \cos ^{4} x & 2 \sin ^{4} x & 3+\sin ^{2} 2 x\\ 3+2 \cos ^{4} x & 2 \sin ^{4} x & \sin ^{2} 2 x\\ 2 \cos ^{4} x & 3+2 \sin ^{4} x & \sin ^{2} 2 x\end{array}\right|$ then $\frac{1}{5} f^{\prime}(0)$ is equal to $\_\_\_\_$

Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram ABCD if the diagonal $\overrightarrow{B D}=\hat{i}+2 \hat{j}+3 \hat{k}$ then the area of the parallelogram is equal to

If $2 \sin ^{3} x+\sin 2 x \cos x+4 \sin x-4=0$ has exactly 3 solutions in the interval $\left[0, \frac{\mathrm{n} \pi}{2}\right], \mathrm{n} \in \mathrm{N}$, then the roots of the equation $x^{2}+n x+(n-3)=0$ belong to :

Let $\mathrm{f}:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$, If the $\lim _{x \rightarrow 0} \frac{x \int_{0}^{x} f(t) d t}{e^{x^{2}}-1}=\alpha$, then $8 \alpha^{2}$ is equal to :

A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is.

If $d_{1}$ is the shortest distance between the lines $x+1=2 y=-12 z, x=y+2=6 z-6$ and $d_{2}$ is the shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}, \frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}$, then the value of $\frac{32 \sqrt{3} d_{1}}{d_{2}}$ is :

Let the latus rectum of the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{b^{2}}=1$ subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $\mathrm{b}^{2}$ is equal to $\frac{l}{\mathrm{~m}}(1+\sqrt{\mathrm{n}})$, where $l$ and m are co-prime numbers, then $l^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}$ is equal to $\qquad$

Let $\mathrm{A}=\{1,2,3, \ldots .7\}$ and let $\mathrm{P}(1)$ denote the power set of $A$. If the number of functions $\mathrm{f}: A \rightarrow P(A)$ such that $a \in f(a), \forall a \in A$ is $m^{n}, m$ and $\mathrm{n} \in \mathrm{N}$ and m is least, then $\mathrm{m}+\mathrm{n}$ is equal to

The value $9 \int_{0}^{9}\left[\sqrt{\frac{10 x}{x+1}}\right] d x$, where $[t]$ denotes the greatest integer less than or equal to $t$, is

Number of integral terms in the expansion of $\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824}$ is equal to

Let $y = y\left( x \right)$ be the solution of the differential equation $\left( {1 - {x^2}} \right)dy = \left[ {xy + \left( {{x^3} + 2} \right)\sqrt {3\left( {1 - {x^2}} \right)} } \right]dx$, $- 1 < x < 1,y\left( 0 \right) = 0$. If $y\left( {\frac{1}{2}} \right) = \frac{m}{n},m$ and $n$ are coprime numbers, then ${\rm{m}} + {\rm{n}}$ is equal to ${\rm{\;}}$

Let $\alpha, \beta \in N$ be roots of equation $x^{2}-70 x+\lambda=0$, where $\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathrm{~N}$. If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to :

If the function

is differentiable on $\mathbb{R}$, then $48(a+b)$ is equal to

Let $\alpha=1^{2}+4^{2}+8^{2}+13^{2}+19^{2}+26^{2}+\ldots \ldots$ upto 10 terms and $\beta=\sum_{n=1}^{10} n^{4}$. If $4 \alpha-\beta=55 k+40$, then k is equal to

Match List -I with List -II

Young’s modules of material of a wire of length ‘L’ and cross-sectional area A is Y. If the length of the wire is doubled and cross-sectional area is halved the young’s modules will be

The work function of a substance is 3.0 eV . The longest wavelength of light that can cause the emission of photoelectrons from this substance is approximately:

The ratio of the magnitude of the kinetic energy to the potential energy of an electron in the $5^{\text {th }}$ excited state of a hydrogen atom is :

The electric field of an electromagnetic wave in free space is represented as $\vec{E}=E_{0} \cos (\omega t-k z) \hat{i}$. The corresponding magnetic induction vector will be :

The diffraction pattern of a light of wavelength 400 nm diffracting from a slit of width 0.2 mm is focused on the focal plane of a convex lens of focal length 100 cm . The width of the $1^{\text {st }}$ secondary maxima will be :

Primary coil of a transformer is connected to 220 V ac. Primary and secondary turns of the transforms are 100 and 10 respectively. Secondary coil of transformer is connected to two series resistance shown in shown in figure. The output voltage $\left(\mathrm{V}_{0}\right)$ is :

The gravitational potential at a point above the surface of earth is $-5.12 \times 10^{7} \mathrm{~J} / \mathrm{kg}$ and the acceleration due to gravity at that point is $6.4 \mathrm{~m} / \mathrm{s}^{2}$. Assume that the mean radius of earth to be 6400 km . The height of this point above the earth's surface is :

An electric toaster has resistance of $60 \Omega$ at room temperature $\left(27^{\circ} \mathrm{C}\right)$. The toaster is connected to a 220 V supply. If the current flowing through it reaches 2.75 A , the temperature attained by toaster is around : (if $\alpha=2 \times 10^{-4} /{ }^{\circ} \mathrm{C}$ )

A Zener diode of breakdown voltage 10 V is used as a voltage regulator as shown in the figure. The current through the Zener diode is

The electrostatic potential due to an electric dipole at a distance ' $r$ ' varies as :

A spherical body of mass 100 g is dropped from a height of 10 m from the ground. After hitting the ground, the body rebounds to a height of 5 m . The impulse of force imparted by the ground to the body is given by : (given $\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^{2}$ )

A particle of mass $m$ projected with a velocity ' $u$ ' making an angle of $30^{\circ}$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height h is :

At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at $47^{\circ} \mathrm{C}$ ?

A series L,R circuit connected with an ac source $E=(25 \sin 1000 t) V$ has a power factor of $\frac{1}{\sqrt{2}}$. If the source of emf is changed to $\mathrm{E}=(20 \sin 2000$ t) V , the new power factor of the circuit will be :

The horizontal component of earth's magnetic field at a place is $3.5 \times 10^{-5} \mathrm{~T}$. A very long straight conductor carrying current of $\sqrt{2} A$ in the direction from South east to North West is placed. The force per unit length experienced by the conductor is $\_\_\_\_\_\_\_$ $\times 10^{-6} \mathrm{~N} / \mathrm{m}$.

Two cells are connected in opposition as shown. Cell $\mathrm{E}_{1}$ is of 8 Vemf and $2 \Omega$ internal resistance; the cell $E_{2}$ is of 2 V emf and $4 \Omega$ internal resistance. The terminal potential difference of cell $\mathrm{E}_{2}$ is:

A electron of hydrogen atom on an excited state is having energy $\mathrm{E}_{\mathrm{n}}=-0.85 \mathrm{eV}$. The maximum number of allowed transitions to lower energy level is $\_\_\_\_\_\_\_$ .

Each of three blocks $\mathrm{P}, \mathrm{Q}$ and R shown in figure has a mass of 3 kg . Each of the wire A and B has cross-sectional area $0.005 \mathrm{~cm}^{2}$ and Young's modulus $2 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$. Neglecting friction, the longitudinal strain on wire $B$ is $\_\_\_\_\_\_\_$ $\times 10^{-4}$ $\left(\right.$ Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ )

The distance between object and its two times magnified real image as produced by a convex lens is 45 cm . The focal length of the lens used is $\_\_\_\_\_\_$ cm .

The displacement and the increase in the velocity of a moving particle in the time interval of t to $(\mathrm{t}+1) \mathrm{s}$ are 125 m and $50 \mathrm{~m} / \mathrm{s}$, respectively. The distance travelled by the particle in $(t+2)^{\mathrm{th}} \mathrm{s}$ is $\_\_\_\_\_\_\_$ m .

A capacitor of capacitance $C$ and potential V has energy E. It is connected to another capacitor of capacitance 2 C and potential 2 V . Then the loss of energy is $\frac{x}{3} E$, where x is

Consider a Disc of mass 5 kg , radius 2 m , rotating with angular velocity of $10 \mathrm{rad} / \mathrm{s}$ about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is $\_\_\_\_\_\_$ J.

In a closed organ pipe, the frequency of fundamental note is 30 Hz . A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to 110 Hz . If the organ pipe has a cross-sectional area of $2 \mathrm{~cm}^{2}$, the amount of water poured in the organ tube is $\_\_\_\_\_\_$ g. (Take speed of sound in air is $330 \mathrm{~m} / \mathrm{s}$ )

A ceiling fan having 3 blades of length 80 cm each is rotating with an angular velocity of 1200 rpm . The magnetic field of earth in that region is 0.5 G and angle of dip is $30^{\circ}$. The emf induced across the blades is $N \pi \times 10^{-5} \mathrm{~V}$. The value of N is $\_\_\_\_\_\_\_$ .

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

Statement-I: The gas liberated on warming a salt with dil $\mathrm{H}_{2} \mathrm{SO}_{4}$, turns a piece of paper dipped in lead acetate into black, it is a confirmatory test for sulphide ion.

Statement-II: In statement-I the colour of paper turns black because of formation of lead sulphite.

In the light of the above statements, choose the most appropriate answer from the options given below:

This reduction reaction is known as:

Sugar which does not give reddish brown precipitate with Fehling's reagent is:

Given below are the two statements, one is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A): There is a considerable increase in covalent radius from N to P . However from As to Bi only a small increase in covalent radius is observed.

Reason (R): Covalent and ionic radii in a particular oxidation state increases down the group.

In the light of the above statement, choose the most appropriate answer from the options given below:

Diamagnetic Lanthanoid ions are:

Aluminium chloride in acidified aqueous solution forms an ion having geometry

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

Statement-I: The orbitals having same energy are called as degenerate orbitals.

Statement-II: In hydrogen atom, 3p and 3d orbitals are not degenerate orbitals.

In the light of the above statements, choose the most appropriate answer from the options given

Given below are two statement one is labeled as Assertion (A) and the other is labeled as Reason (R) .

Assertion (A): $\mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CH}_{2}-\mathrm{Cl}$ is an example of allyl halide

Reason (R): Allyl halides are the compounds in which the halogen atom is attached to $\mathrm{sp}^{2}$ hybridised carbon atom.

In the light of the two above statements, choose the most appropriate answer from the options given below:

What happens to freezing point of benzene when small quantity of napthalene is added to benzene?

Match List-I with List-II

Choose the correct answer from the options given below:

The Lassiagne's extract is boiled with dil $\mathrm{HNO}_{3}$ before testing for halogens because,

Choose the correct Statements from the following:

(A) Ethane-1 2-diamine is a chelating ligand.

(B) Metallic aluminium is produced by elecrtrolysis of aluminium oxide in presence of cryolite.

(C) Cyanide ion is used as ligand for leaching of silver.

(D) Phosphine act as a ligand in Wilkinson catalyst.

(E) The stability constants of $\mathrm{Ca}^{2+}$ and $\mathrm{Mg}^{2+}$ are similar with EDTA complexes.

Choose the correct answer from the options given below:

The rate of first order reaction is $0.04 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$ at 10 minutes and $0.03 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}$ at 20 minutes after

initiation. Half life of the reaction is $\qquad$ minutes. (Given $\log 2=0.3010, \log 3=0.4771$ )

The pH at which $\mathrm{Mg}(\mathrm{OH})_{2}\left[\mathrm{~K}_{\mathrm{sp}}=1 \times 10^{-11}\right]$

begins to precipitate from a solution containing 0.10 M $\mathrm{Mg}^{2+}$ ions is $\qquad$

An ideal gas undergoes a cyclic transformation starting from the point A and coming back to the same point by tracing the path $\mathrm{A} \rightarrow \mathrm{B} \rightarrow \mathrm{C} \rightarrow \mathrm{A}$ as shown in the diagram. The total work done in the process is $\qquad$ J.

If the IUPAC name of an element is "Unununnium", then the element belongs to the $n^{\mathrm{th}}$ group of the periodic table. Find the value of $n$.

The total number of molecular orbitals formed from 2 s and 2 p atomic orbitals of a diatomic molecule

On a thin layer chromatographic plate, an organic compound moved by 3.5 cm , while the solvent moved by 5 cm . The retardation factor of the organic compound is $\_\_\_\_$ $\times 10^{-1}$

The compound formed by the reaction of ethanal with semicarbazide contains $\_\_\_\_$ number of nitrogen atoms.

0.05 cm thick coating of silver is deposited on a plate of $0.05 \mathrm{~m}^{2}$ area.

The number of silver atoms deposited on plate are $\qquad$ $\times 10^{23} .($ At mass Ag $\left.=108, \mathrm{~d}=7.9 \mathrm{~g} \mathrm{~cm}^{-3}\right)$

$2 \mathrm{MnO}_{4}^{-}+\mathrm{bI}^{-}+\mathrm{cH}_{2} \mathrm{O} \rightarrow \mathrm{xI}_{2}+\mathrm{yMnO}_{2}+\mathrm{zOH}^{-}$

If the above equation is balanced with integer coefficients, the value of $z$ is $\qquad$

The mass of sodium acetate $\mathrm{(CH_3COONa)}$ required to prepare $250\ \mathrm{mL}$ of $0.35\ \mathrm{M}$ aqueous solution is $\_\_\_\_$ $\ \mathrm{g}$.

[Given: molar mass of $\mathrm{CH_3COONa} = 82.02\ \mathrm{g\ mol^{-1}}$]