JEE Main 2024 — 8 April, Shift 1

The value of $\mathrm{k} \in \mathbb{N}$ for which the integral $I_{n}=\int_{0}^{1}\left(1-x^{k}\right)^{n} d x, n \in \mathbb{N}$, satisfies $147 I_{20}=148 I_{21}$ is:

The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot(8)^{x}+48=0$ is :

Let the circles $C_{1}:(x-\alpha)^{2}+(y-\beta)^{2}=r_{1}^{2}$ and $C_{2}:(x-8)^{2}+\left(y-\frac{15}{2}\right)^{2}=r_{2}^{2}$ touch each other

externally at the point $(6,6)$. If the point $(6,6)$ divides the line segment joining the centres of the circles $\mathrm{C}_{1}$

and $\mathrm{C}_{2}$ internally in the ratio $2: 1$, then $(\alpha+\beta)+4\left(r_{1}^{2}+r_{2}^{2}\right)$ equals

Let $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ be a point in the first octant, whose projection in the xy-plane is the point Q . Let $\mathrm{OP}=\gamma$; the angle between OQ and the positive $x$-axis be $\theta$; and the angle between OP and the positive $z$-axis be $\phi$, where O is the origin. Then the distance of P from the x -axis is :

The number of critical points of the function $f(x)=(x-2)^{2 / 3}(2 x+1)$ is :

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $\mathrm{e}^{-\mathrm{a}}+4 \mathrm{a}^{2}+\mathrm{a}-1$. Then the differential equation, whose general solution is $\mathrm{y}=\mathrm{c}_{1} f(\mathrm{x})+\mathrm{c}_{2}$, where $\mathrm{c}_{1}$ and $\mathrm{c}_{2}$ are arbitrary constants, is :

Let $f(x)=4 \cos ^{3} x+3 \sqrt{3} \cos ^{2} x-10$. The number of points of local maxima of $f$ in interval $(0,2 \pi)$ is:

Let A=\left[ \begin{array}{*{35}{l}}2 & a & 0 \\1 & 3 & 1 \\0 & 5 & b \\\end{array} \right].If $A^{3}=4 A^{2}-A-21 I$, where I is the identity matrix of order $3 \times 3$, then $2 a+3 b$ is equal to :

If the shortest distance between the lines

$\mathrm{L}_{1}: \overrightarrow{\mathrm{r}}=(2+\lambda) \hat{\mathrm{i}}+(1-3 \lambda) \hat{\mathrm{j}}+(3+4 \lambda) \hat{\mathrm{k}}, \lambda \in \mathbb{R}$

$\mathrm{L}_{2}: \overrightarrow{\mathrm{r}}=2(1+\mu) \hat{\mathrm{i}}+3(1+\mu) \hat{\mathrm{j}}+(5+\mu) \hat{\mathrm{k}}, \mu \in \mathbb{R}$ is $\frac{\mathrm{m}}{\sqrt{\mathrm{n}}}$,

where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then the value of $\mathrm{m}+\mathrm{n}$ equals.

Let the sum of two positive integers be 24 . If the probability, that their product is not less than $\frac{3}{4}$ times their greatest positive product, is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $n-m$ equals :

If $\text{sin}x=-\frac{3}{5}$, where $\pi <x<\frac{3\pi }{2}$, then $80\left( \text{ta}{{\text{n}}^{2}}x-\text{cos}x \right)$ is equal to :

Let $I(x)=\int \frac{6}{\sin ^{2} x(1-\cot x)^{2}} d x$. If $I(0)=3$, then $\mathrm{I}\left(\frac{\pi}{12}\right)$ is equal to :

The equations of two sides AB and AC of $a$ triangle $A B C$ are $4 x+y=14$ and $3 x-2 y=5$, respectively. The point $\left(2,-\frac{4}{3}\right)$ divides the third side BC internally in the ratio $2: 1$. The equation of the side BC is :

Let $[\mathrm{t}]$ be the greatest integer less than or equal to t. Let A be the set of al prime factors of 2310 and $f: A \rightarrow \mathbb{Z}$

be the function $f(x)=\left[\log _{2}\left(x^{2}+\left[\frac{x^{3}}{5}\right]\right)\right]$. The number of one-to-one functions from A to the range of $f$ is :

Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{lm}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is :

If the set $R=\{(a, b) ; a+5 b=42, a, b \in \mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^{m}\left(1+i^{n!}\right)=x+i y$,

where $I=\sqrt{-1}$, then the value of $m+x+y$ is :

For the function $f(x)=(\cos x)-x+1, x \in \mathbb{R}$, between the following two statements

Statement I : $f(x)=0$ for only one value of $x$ is $[0, \pi]$.

Statement II : $f(x)$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right]$

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

The set of all $\alpha$, for which the vector $\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k} \quad$ and$\quad \vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k} \quad$ are inclined at an obtuse angle for all $t \in \mathbb{R}$ is :

Let $y=y(x)$ be the solution of the differential equation $\left(1+y^{2}\right) e^{\tan x} d x+\cos ^{2} x\left(1+e^{2 \tan x}\right) d y=0$,

$y(0)=1$. Then $y\left(\frac{\pi}{4}\right)$ is equal to :

Let $H: \frac{-x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4 \sqrt{3}$. Suppose the point $(\alpha, 6), \alpha>0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$, then $\alpha^{2}+\beta$ is equal to :

Let $A=\left[ \begin{matrix}2 & -1 \\1 & 1 \\\end{matrix} \right]$. If the sum of the diagonal elements of $A^{13}$ is $3^{n}$, then $n$ is equal to $\_\_\_\_$

If the orthocentre of the triangle formed by the lines $2 x+3 y-1=0, x+2 y-1=0$ and $\mathrm{ax}+\mathrm{by}-1=0$,

is the centroid of another triangle, whose circumecentre and orthocentre respectively are $(3,4)$ and $(-6,-8)$,

then the value of $|a-b|$ is

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables X and Y respectively denote the number of blue and Yellow balls. If $\overline{\mathrm{X}}$ and $\overline{\mathrm{Y}}$ are the means of $X$ and $Y$ respectively, then $7 \bar{X}+4 \bar{Y}$ is equal to $\qquad$ .

The number of 3-digit numbers, formed using the digits $2,3,4,5$ and 7 , when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to $\qquad$

Let the positive integers be written in the form

If the $\mathrm{k}^{\text {th }}$ row contains exactly k numbers for every natural number k , then the row in which the number 5310 will be, is $\qquad$

If the range of $f(\theta)=\frac{\sin ^{4} \theta+3 \cos ^{2} \theta}{\sin ^{4} \theta+\cos ^{2} \theta}, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac{\alpha}{\beta}$, is equal to $\qquad$

Let $\alpha=\sum_{r=0}^{n}\left(4 r^{2}+2 r+1\right)^{n} C_{r}$ and $\beta=\left(\sum_{\mathrm{r}=0}^{\mathrm{n}} \frac{{ }^{\mathrm{n}} C_{\mathrm{r}}}{\mathrm{r}+1}\right)+\frac{1}{\mathrm{n}+1}$. If $140<\frac{2 \alpha}{\beta}<281$,

then the value of $n$ is $\qquad$

Let $\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}, \vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k} \quad$ and $\overrightarrow{\mathrm{c}}=17 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ be three given vectros. If $\overrightarrow{\mathrm{r}}$ is a vector

such that $\vec{r} \times \vec{a}=(\vec{b}+\vec{c}) \times \vec{a}$ and $\vec{r} .(\vec{b}-\vec{c})=0$, then $\frac{|593 \overrightarrow{\mathrm{r}}+67 \overrightarrow{\mathrm{a}}|^{2}}{(593)^{2}}$ is equal to $\qquad$

Let the area of the region enclosed by the curve $\mathrm{y}=\min \{\sin \mathrm{x}, \cos \mathrm{x}\}$ and the x -axis between $\mathrm{x}=-\pi$ to $x=\pi$ be $A$. Then $A^{2}$ is equal to $\qquad$

The value of $\lim _{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots . . \sqrt[10]{\cos 10 x}}{x^{2}}\right)$ is $\qquad$

Three bodies $\mathrm{A}, \mathrm{B}$ and C have equal kinetic energies and their masses are $400 \mathrm{~g}, 1.2 \mathrm{~kg}$ and 1.6 kg respectively. The ratio of their linear momenta is :

Average force exerted on a non-reflecting surface at normal incidence is $2.4 \times 10^{-4} \mathrm{~N}$. If $360 \mathrm{~W} / \mathrm{cm}^{2}$ is the light energy flux during span of 1 hour 30 minutes. Then the area of the surface is:

A proton and an electron are associated with same de-Broglie wavelength. The ratio of their kinetic energies is:

(Assume $\mathrm{h}=6.63 \times 10^{-34} \mathrm{~J} \mathrm{~s}, \mathrm{~m}_{\mathrm{e}}=9.0 \times 10^{-31} \mathrm{~kg}$ and $\mathrm{m}_{\mathrm{p}}=1836$ times $\mathrm{m}_{\mathrm{e}}$ )

A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature $\left(27^{\circ} \mathrm{C}\right)$. The ratio of specific heat of gases at constant volume respectively is:

In an expression $\mathrm{a} \times 10^{\mathrm{b}}$ :

Binding energy of a certain nucleus is $18 \times 10^{8} \mathrm{~J}$. How much is the difference between total mass of all the nucleons and nuclear mass of the given nucleus:

Paramagnetic substances:

A. align themselves along the directions of external magnetic field.

B. attract strongly towards external magnetic field.

C. has susceptibility little more than zero.

D. move from a region of strong magnetic field to weak magnetic field.

Choose the most appropriate answer from the options given below:

A clock has $75 \mathrm{~cm}, 60 \mathrm{~cm}$ long second hand and minute hand respectively. In 30 minutes duration the tip of second hand will travel x distance more than the tip of minute hand. The value of $x$ in meter is nearly (Take $\pi=3.14$ ) :

Young's modulus is determined by the equation given by $Y=49000 \frac{\mathrm{~m}}{\ell} \frac{\text { dyne }}{\mathrm{cm}^{2}}$ where $M$ is the mass and $\ell$ is the extension of wire used in the experiment. Now error in Young modules(Y) is estimated by taking data from $\mathrm{M}-\ell$ plot in graph paper. The smallest scale divisions are 5 g and 0.02 cm along load axis and extension axis respectively. If the value of M and $\ell$ are 500 g and 2 cm respectively then percentage error of Y is :

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in $\mathrm{P}-\mathrm{V}$ diagram. The relation between the ratio $\frac{V_{a}}{V_{d}}$ and the $\operatorname{ratio} \frac{\mathrm{V}_{\mathrm{b}}}{\mathrm{V}_{\mathrm{c}}}$ is: \end{enumerate}

Two planets $A$ and $B$ having masses $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ move around the sun in circular orbits of $r_{1}$ and $r_{2}$ radii respectively. If angular momentum of A is L and that of $B$ is $3 L$, the ratio of time period $\left(\frac{T_{A}}{T_{B}}\right)$ is:

A LCR circuit is at resonance for a capacitor C , inductance $L$ and resistance $R$. Now the value of resistance is halved keeping all other parameters same. The current amplitude at resonance will be now:

Two charged conducting spheres of radii $a$ and $b$ are connected to each other by a conducting wire. The ratio of charges of the two spheres respectively is:

Correct Bernoulli's equation is (symbols have their usual meaning) :

A player caught a cricket ball of mass 150 g moving at a speed of $20 \mathrm{~m} / \mathrm{s}$. If the catching process is completed in 0.1 s , the magnitude of force exerted by the ball on the hand of the player is:

A stationary particle breaks into two parts of masses $\mathrm{m}_{\mathrm{A}}$ and $\mathrm{m}_{\mathrm{B}}$ which move with velocities $\mathrm{v}_{\mathrm{A}}$ and $\mathrm{v}_{B}$ respectively. The ratio of their kinetic energies $\left(\mathrm{K}_{\mathrm{B}}: \mathrm{K}_{\mathrm{A}}\right)$ is :

Critical angle of incidence for a pair of optical media is $45^{\circ}$. The refractive indices of first and second media are in the ratio:

The diameter of a sphere is measured using a vernier caliper whose 9 divisions of main scale are equal to 10 divisions of vernier scale. The shortest division on the main scale is equal to 1 mm . The main scale reading is 2 cm and second division of vernier scale coincides with a division on main scale. If mass of the sphere is 8.635 g , the density of the sphere is:

A uniform thin metal plate of mass 10 kg with dimensions is shown. The ratio of x and y coordinates of center of mass of plate in $\frac{n}{9}$.The value of $n$ is $\qquad$ .

An electron with kinetic energy 5 eV enters a region of uniform magnetic field of $3 \mu \mathrm{~T}$ perpendicular to its direction. An electric field E is applied perpendicular to the direction of velocity and magnetic field. The value of E, so that electron moves along the same path, is $\qquad$ $\mathrm{NC}^{-1}$. (Given, mass of electron $=9 \times 10^{-31} \mathrm{~kg}$, electric charge $=1.6 \times 10^{-19} \mathrm{C}$ )

Resistance of a wire at $0^{\circ} \mathrm{C}, 100^{\circ} \mathrm{C}$ and $\mathrm{t}{ }^{\circ} \mathrm{C}$ is found to be $10 \Omega, 10.2 \Omega$ and $10.95 \Omega$ respectively. The temperature $t$ in Kelvin scale is $\qquad$ .

An electric field, $\overrightarrow{\mathrm{E}}=\frac{2 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}}{\sqrt{6}}$ passes through the surface of $4 \mathrm{~m}^{2}$ area having unit vector $\hat{n}=\left(\frac{2 \hat{i}+\hat{j}+\hat{k}}{\sqrt{6}}\right)$. The electric flux for that surface is $\qquad$ V m.

A liquid column of height 0.04 cm balances excess pressure of soap bubble of certain radius. If density of liquid is $8 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$ and surface tension of soap solution is $0.28 \mathrm{Nm}^{-1}$, then diameter of the soap bubble is $\qquad$ cm . (if $\mathrm{g}=10 \mathrm{~ms}^{-2}$ )

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is $\left(\frac{a-1}{a}\right)$ then the value of $a$ is $\qquad$ .

Three vectors $\overrightarrow{\mathrm{OP}}, \overrightarrow{\mathrm{OQ}}$ and $\overrightarrow{\mathrm{OR}}$ each of magnitude A are acting as shown in figure. The resultant of the three vectors is $A \sqrt{x}$. The value of $x$ is $\qquad$

A parallel beam of monochromatic light of wavelength 600 nm passes through single slit of 0.4 mm width. Angular divergence corresponding to second order minima would be $\qquad$ $\times 10^{-3} \mathrm{rad}$.

In an alpha particle scattering experiment distance of closest approach for the $\alpha$ particle is $4.5 \times 10^{-14} \mathrm{~m}$. If target nucleus has atomic number 80, then maximum velocity of $\alpha$-particle is $\qquad$ $\times 10^{5}$ $\mathrm{m} / \mathrm{s}$ approximately. $\left(\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9}\right.$ SI unit, mass of $\alpha$ particle $=$ $\left.6.72 \times 10^{-27} \mathrm{~kg}\right)$

Given below are two statements:

Statement I :

IUPAC name of Compound A is 4-chloro-1, 3-dinitrobenzene:

Statement II:

IUPAC name of Compound B is 4-ethyl-2-methylaniline. In the light of the above statements, choose the most appropriate answer from the options given below:

Combustion of glucose $\mathrm{(C_6H_{12}O_6)}$ produces $\mathrm{CO_2}$ and water. The amount of oxygen (in g) required for the complete combustion of $900\ \mathrm{g}$ of glucose is:

[Given: Molar mass of glucose = $180\ \mathrm{g\ mol^{-1}}$]

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

Assertion (A): The stability order of +1 oxidation state of Ga , In and Tl is $\mathrm{Ga}<\mathrm{In}<\mathrm{Tl}$.

Reason (R): The inert pair effect stabilizes the lower oxidation state down the group.

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

Match List I with List-II

Choose the correct answer from the option below :

Match List I and with List II

Choose the correct answer from the option below :

For the given hypothetical reactions, the equilibrium constants are as follows:

$\mathrm{X} \rightleftharpoons \mathrm{Y} ; \mathrm{K}_{1}=1.0$

$\mathrm{Y} \rightleftharpoons \mathrm{Z} ; \mathrm{K}_{2}=2.0$

$\mathrm{Z} \rightleftharpoons \mathrm{W} ; \mathrm{K}_{3}=4.0$

The equilibrium constant for the reaction $\mathrm{X} \rightleftharpoons \mathrm{W}$ is $\_ \_\_\_\_\_$.

Thiosulphate reacts differently with iodine and bromine in the reaction given below :

$2 \mathrm{~S}_{2} \mathrm{O}_{3}^{2-}+\mathrm{I}_{2} \rightarrow \mathrm{~S}_{4} \mathrm{O}_{6}^{2-}+2 \mathrm{I}^{-}$

$\mathrm{S}_{2} \mathrm{O}_{3}^{2-}+5 \mathrm{Br}_{2}+5 \mathrm{H}_{2} \mathrm{O} \rightarrow 2 \mathrm{SO}_{4}^{2-}+4 \mathrm{Br}^{-}+10 \mathrm{H}^{+}$

Which of the following statement justifies the above dual behaviour of thiosulphate?

An octahedral complex with the formula $\mathrm{CoCl}_{3} \mathrm{nNH}_{3}$ upon reaction with excess of $\mathrm{AgNO}_{3}$ solution given 2 moles of AgCl . Consider the oxidation state of Co in the complex is ' $x$ '. The value of " $x+n$ " is $\qquad$

The incorrect statement regarding the given structure is

Among the following halogens $\mathrm{F}_{2}, \mathrm{Cl}_{2}, \mathrm{Br}_{2}$ and $\mathrm{I}_{2}$ Which can undergo disproportionation reaction?

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

Statement I : $\mathrm{N}\left(\mathrm{CH}_{3}\right)_{3}$ and $\mathrm{P}\left(\mathrm{CH}_{3}\right)_{3}$ can act as ligands to form transition metal complexes.

Statement II: As $\mathrm{N}$ and $\mathrm{P}$are from same group, the nature of bonding of $\mathrm{N}\left(\mathrm{CH}_{3}\right)_{3}$ and $\mathrm{P}\left(\mathrm{CH}_{3}\right)_{3}$ is always same with transition metals.

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

Match List I with List II.

Choose the correct answer from the options given below

Iron (III) catalyses the reaction between iodide and persulphate ions, in which

A. $\mathrm{Fe}^{3+}$ oxidises the iodide ion

B. $\mathrm{Fe}^{3+}$ oxidises the persulphate ion

C. $\mathrm{Fe}^{2+}$ reduces the iodide ion

D. $\mathrm{Fe}^{2+}$ reduces the persulphate ion

Choose the most appropriate answer from the options given below:

Match List I with List II

Choose the correct answer from the options given below :

Number of complexes with even number of electrons in $\mathrm{t}_{2 \mathrm{~g}}$ orbitals is -

$\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+},\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+},\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}$,$\left[\mathrm{Cu}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+},\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$

Identify the product $(\mathrm{P })$ in the following reaction:

Consider the figure provided.

1 mol of an ideal gas is kept in a cylinder, fitted with a piston, at the position A , at $18^{\circ} \mathrm{C}$. If the piston is moved to position B ,

keeping the temperature unchanged, then ' $x$ ' L atm work is done in this reversible process. $\mathrm{x}=$ $\qquad$

L atm. (nearest integer) [Given : Absolute temperature $={ }^{\circ} \mathrm{C}+$ 273.15, $\left.\mathrm{R}=0.08206{\mathrm{~L} \mathrm{~atm} \mathrm{~mol}^{-1}}^{-1} \mathrm{~K}^{-1}\right]$

Number of amine compounds from the following giving solids which are soluble in NaOH upon reaction with Hinsberg's reagent is $\qquad$

The 'spin only' magnetic moment value of $\mathrm{MO}_{4}{ }^{2-}$ is $\qquad$ BM. (Where M is a metal having least metallic radii. among

$\mathrm{Sc}, \mathrm{Ti}, \mathrm{V}, \mathrm{Cr}, \mathrm{Mn}$ and Zn ). (Given atomic number : $\mathrm{Sc}=21, \mathrm{Ti}=22, \mathrm{~V}=23$, $\mathrm{Cr}=24, \mathrm{Mn}=25$ and

$\mathrm{Zn}=30$ )

If 279 g of aniline is reacted with one equivalent of benzenediazonium chloride, the mximum amount of aniline yellow formed will be $\qquad$ g. (nearest integer) (consider complete conversion)

Consider the following reaction $\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}$ The time taken for A to become $1 / 4^{\text {th }}$ of its initial concentration is twice the time taken to become $1 / 2$ of the same. Also, when the change of concentration of B is plotted against time, the resulting graph gives a straight line with a negative slope and a positive intercept on the concentration axis. The overall order of the reaction is $\qquad$