JEE Main 2024 — 4 April, Shift 1

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function given by $f(x)=\begin{cases}\frac{1-\cos 2x}{x^2}, & x<0,\\\alpha, & x=0,\\\frac{\beta\sqrt{1-\cos x}}{x}, & x>0.\end{cases}$ where $\beta \in \mathrm{R}$ f is continues at $\mathrm{x}=0$, then $\alpha^2+\beta^2$ is equal to :

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is :

The vertices of a triangle are $\mathrm{A}(-1,3), \mathrm{B}(-2,2)$ and $\mathrm{C}(3,-1)$. A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

If the solution $y=y(x)$ of the differential equation $\left(x^{4}+2 x^{3}+3 x^{2}+2 x+2\right) d y-\left(2 x^{2}+2 x+3\right) d x=0$ satisfies $y(-1)=-\frac{\pi}{4}$, then $y(0)$ is equal to :

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^{2}-3 x+8}{2 x^{2}+3 x+8}$ be $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$. Then $m+n$ is equal to:

One of the points of intersection of the curves $y=1+3 x-2 x^{2}$ and $y=\frac{1}{x}$ is $\left(\frac{1}{2}, 2\right)$.

Let the area of the region enclosed by these curves be $\frac{1}{24}(\ell \sqrt{5}+\mathrm{m})-\mathrm{n} \log _{\mathrm{e}}(1+\sqrt{5})$, where $\ell, \mathrm{m}, \mathrm{n} \in$ N .

Then $\ell+\mathrm{m}+\mathrm{n}$ is equal to

If the system of equations

$x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0$,$x+(\cos \alpha) y+(\sin \alpha) z=0$,$x+(\sin \alpha) y-(\cos \alpha) z=0$

has a non-trivial solution, then $\alpha \in\left(0, \frac{\pi}{2}\right)$ is equal to :

There are 5 points $\mathrm{P}_{1}, \mathrm{P}_{2}, \mathrm{P}_{3}, \mathrm{P}_{4}, \mathrm{P}_{5}$ on the side AB , excluding A and B , of a triangle ABC . Similarly there are 6 points $\mathrm{P}_{6}, \mathrm{P}_{7}, \ldots, \mathrm{P}_{11}$ on the side BC and 7 points $\mathrm{P}_{12}, \mathrm{P}_{13}, \ldots, \mathrm{P}_{18}$ on the side CA of the triangle. The number of triangles, that can be formed using the points $\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{18}$ as vertices, is :

Let $f\left( \text{x} \right)=\left\{ \begin{matrix}-2, & -2\le \text{x}\le 0 \\\text{x}-2, & 0<\text{x}\le 2 \\\end{matrix} \right.$ and $\text{h}\left( \text{x} \right)=\text{f}\left(\left| \text{x}\right| \right)+\left| \text{f}\left( \text{x} \right) \right|$.Then $\int _{-2}^{2}\text{ }\!\!~\!\!\text{ h}\left( \text{x} \right)\text{dx}$ is equal to :

The sum of all rational terms in the expansion of $\left(2^\frac{1}{5}+5^\frac{1}{3}\right)^{15}\\$ is equal to:

Let a unit vector which makes an angle of $60^{\circ}$ with $2 \hat{i}+2 \hat{j}-\hat{k}$ and an angle of $45^{\circ}$ with $\hat{i}-\hat{k}$ be $\vec{C}$. Then $\overrightarrow{\mathrm{C}}+\left(-\frac{1}{2} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{\sqrt{2}}{3} \hat{\mathrm{k}}\right)$ is :

Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $7^{\text {th }}, 8^{\text {th }}$ and $13^{\text {th }}$ terms of an A.P. If the $5^{\text {th }}$ term of the G.P. is the $\mathrm{n}^{\text {th }}$ term of the A.P., then $n$ is equal to

Let $a, b \in R$. Let the mean and the variance of 6 observations $-3,4,7,-6, \mathrm{a}, \mathrm{b}$ be 2 and 23 , respectively. The mean deviation about the mean of these 6 observations is :

If 2 and 6 are the roots of the equation $a x^{2}+b x+1=0$, then the quadratic equation, whose roots are $\frac{1}{2 a+b}$ and $\frac{1}{6 a+b}$, is :

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^{2}+|z|=0, z \in C$. Then $4\left(\alpha^{2}+\beta^{2}\right)$ is equal to :

Let the point, on the line passing through the points $\mathrm{P}(1,-2,3)$ and $\mathrm{Q}(5,-4,7)$, farther from the origin and at a distance of 9 units from the point $P$, be $(\alpha, \beta, \gamma)$. Then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is equal to :

A square is inscribed in the circle $x^{2}+y^{2}-10 x-6 y+30=0$. One side of this square is parallel to $y=x+3$.

If $\left(x_{i}, y_{i}\right)$ are the vertices of the square, then $\sum\left(x_{i}^{2}+y_{i}^{2}\right)$ is equal to :

If the domain of the function $\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{e}\left(\frac{3 x^{2}-8 x+5}{x^{2}-3 x-10}\right)$ is $(\alpha, \beta]$, then $3 \alpha+10 \beta$ is equal to :

Let $f(\mathrm{x})=\mathrm{x}^{5}+2 \mathrm{e}^{\mathrm{x} / 4}$ for all $\mathrm{x} \in$ R. Consider a function $g(x)$ such that (gof) $(x)=x$ for all $x \in R$. Then the value of $8 \mathrm{~g}^{\prime}(2)$ is :

Let \text{A}=\left[ \begin{array}{*{35}{l}}1 & 2 & \alpha \\1 & 0 & 1 \\0 & 1 & 2 \\\end{array} \right].If $\operatorname{det}\left(\operatorname{adj}\left(2 \mathrm{~A}-\mathrm{A}^{\mathrm{T}}\right) \cdot \operatorname{adj}\left(\mathrm{A}-2 \mathrm{~A}^{\mathrm{T}}\right)\right)=2^{8}$, then $(\operatorname{det}(\mathrm{A}))^{2}$ is equal to :

If $\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{m \sqrt{5}}{n(2 n)^{2 / 3}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $8 \mathrm{~m}+12 \mathrm{n}$ is equal to $\qquad$

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to $\qquad$

Let the solution $\mathrm{y}=\mathrm{y}(\mathrm{x})$ of the differential equation $\frac{d y}{d x}-y=1+4 \sin x$ satisfy $y(\pi)=1$. Then $\mathrm{y}\left(\frac{\pi}{2}\right)+10$ is equal to $\qquad$

If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3 \sqrt{5}} \mathrm{k} \quad$

and $\quad \int_{0}^{\mathrm{k}}\left[\mathrm{x}^{2}\right] \mathrm{dx}=\alpha-\sqrt{\alpha}, \quad$ where $\quad[\mathrm{x}]$ denotes the greatest integer function, then $6 \alpha^{3}$ is equal to $\qquad$

Let A be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is -3 . If the points $(x, y)$ satisfying $A^{2}+x A+y I=0$ lie on a hyperbola, whose transverse axis is parallel to the x -axis, eccentricity is e and the length of the latus rectum is $\ell$, then $\mathrm{e}^{4}+\ell^{4}$ is equal to $\qquad$

Let $\text{a}=1+\frac{{{~}^{2}}{{\text{C}}_{2}}}{3!}+\frac{{{~}^{3}}{{\text{C}}_{2}}}{4!}+\frac{{{~}^{4}}{{\text{C}}_{2}}}{5!}+\ldots$

$\text{b}=1+\frac{{{~}^{1}}{{\text{C}}_{0}}+{{~}^{1}}{{\text{C}}_{1}}}{1!}+\frac{{{~}^{2}}{{\text{C}}_{0}}+{{~}^{2}}{{\text{C}}_{1}}+{{~}^{2}}{{\text{C}}_{2}}}{2!}+\frac{{{~}^{3}}{{\text{C}}_{0}}+{{~}^{3}}{{\text{C}}_{1}}+{{~}^{3}}{{\text{C}}_{2}}+{{~}^{3}}{{\text{C}}_{3}}}{3!}+\ldots$

Then $\frac{2b}{{{a}^{2}}}$ is equal to $\text{ }\!\!~\!\!\text{ }$

Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that \text{A}\left[ \begin{array}{*{35}{l}}1 \\1 \\1 \\\end{array} \right]=3\left[ \begin{array}{*{35}{l}}1 \\1 \\1 \\\end{array} \right]. Then the maximum value of $\operatorname{det}(\mathrm{A})$ is $\qquad$

Let the length of the focal chord $P Q$ of the parabola $y^{2}=12 x$ be 15 units. If the distance of $P Q$ from the origin is $p$, then $10 p^{2}$ is equal to $\qquad$

Let $ABC$ be a triangle of area $15\sqrt{2}$. Let the vectors representing its sides be:

Then the square of the length of the largest side of the triangle $ABC$ is :

If $\int_{0}^{\frac{\pi}{4}} \frac{\sin ^{2} x}{1+\sin x \cos x} d x=\frac{1}{a} \log _{e}\left(\frac{a}{3}\right)+\frac{\pi}{b \sqrt{3}}$, where $a$, $\mathrm{b} \in \mathrm{N}$, then $\mathrm{a}+\mathrm{b}$ is equal to $\qquad$

An electron is projected with uniform velocity along the axis inside a current carrying long solenoid. Then :

The electric field in an electromagnetic wave is given by $\vec{E}=\hat{\mathrm{i}} 40 \cos \omega\left(t-\frac{z}{c}\right) N C^{-1} . \quad$

The magnetic field induction of this wave is (in SI unit):

Which of the following nuclear fragments corresponding to nuclear fission between neutron $\left({ }_{0}^{1} \mathrm{n}\right)$ and uranium isotope $\left({ }_{92}^{235} \mathrm{U}\right)$ is correct:

In an experiment to measure focal length (f) of convex lens, the least counts of the measuring scales for the position of object ( $u$ ) and for the position of image ( $v$ ) are $\Delta u$ and $\Delta v$, respectively. The error in the measurement of the focal length of the convex lens will be :

Given below are two statements :

Statement I : When speed of liquid is zero everywhere, pressure difference at any two points depends on equation $\mathrm{P}_{1}-\mathrm{P}_{2}=\rho g\left(\mathrm{~h}_{2}-\mathrm{h}_{1}\right)$

Statement II : In ventury tube shown, $2 \mathrm{gh}=v_{1}^{2}-v_{2}^{2}$

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

The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are $8 \Omega$ and $10 \Omega$ respectively. After inserting in a hot bath of temperature $400^{\circ} \mathrm{C}$, the resistance of platinum wire is :

A metal wire of uniform mass density having length $L$ and mass $M$ is bent to form a semicircular arc and a particle of mass $m$ is placed at the centre of the arc. The gravitational force on the particle by the wire is:

On celcius scale the temperature of body increases by $40^{\circ} \mathrm{C}$. The increase in temperature on Fahrenheit scale is:

An effective power of a combination of 5 identical convex lenses which are kept in contact along the principal axis is 25 D. Focal length of each of the convex lens is :

A wooden block, initially at rest on the ground, is pushed by a force which increases linearly with time $t$. Which of the following curve best describes acceleration of the block with time :

If a rubber ball falls from a height $h$ and rebounds upto the height of $h / 2$. The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are :

The equation of stationary wave is : $y=2 a \sin \left(\frac{2 \pi n t}{\lambda}\right) \cos \left(\frac{2 \pi x}{\lambda}\right)$. Which of the following is NOT correct?

A body travels $102.5 \mathrm{~m}\; {\text {in }}\; \mathrm{n}^{\text {th }}$ second and 115.0 m in $(\mathrm{n}+2)^{\mathrm{th}}$ second. The acceleration is :

To measure the internal resistance of a battery, potentiometer is used. For $\mathrm{R}=10 \Omega$, the balance point is observed at $\ell=500 \mathrm{~cm}$ and for $\mathrm{R}=1 \Omega$ the balance point is observed at $\ell=400 \mathrm{~cm}$. The internal resistance of the battery is approximately:

An infinitely long positively charged straight thread has a linear charge density $\lambda \mathrm{Cm}^{-1}$. An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is :

The value of net resistance of the network as shown in the given figure is :

The co-ordinates of a particle moving in $x-y$ plane are given by : $x=2+4 t, y=3 t+8 t^{2}$

The motion of the particle is :

In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to :

A. pure inductor.

B. pure capacitor.

C. pure resistor.

D. combination of an inductor and capacitor.

Choose the correct answer from the options given below:

An infinite plane sheet of charge having uniform surface charge density $+\sigma_{\mathrm{s}} \mathrm{C} / \mathrm{m}^{2}$ is placed on $\mathrm{x}-\mathrm{y}$ plane. Another infinitely long line charge having uniform linear charge density $+\lambda_{\mathrm{e}} \mathrm{C} / \mathrm{m}$ is placed at $\mathrm{z}=4 \mathrm{~m}$ plane and parallel to y -axis. If the magnitude values $\left|\sigma_{s}\right|=2\left|\lambda_{e}\right|$ then at point $(0,0,2)$, the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $\pi \sqrt{\mathrm{n}}: 1$. The value of $n$ is $\qquad$ .

A hydrogen atom changes its state from $n=3$ to $\mathrm{n}=2$. Due to recoil, the percentage change in the wave length

of emitted light is approximately $1 \times 10^{-\mathrm{n}}$. The value of n is $\qquad$ . [Given Rhc $=13.6 \mathrm{eV}, \mathrm{hc}=1242 \mathrm{eV} \mathrm{nm}$,

$\mathrm{h}=6.6 \times 10^{-34} \mathrm{~J}$ s, mass of the hydrogen atom $\left.=1.6 \times 10^{-27} \mathrm{~kg}\right]$

The magnetic field existing in a region is given by $\vec{B}=0.2(1+2 x) \hat{k} T$. A square loop of edge 50 cm carrying 0.5 A current is placed in $\mathrm{x}-\mathrm{y}$ plane with its edges parallel to the $x-y$ axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is $\qquad$ mN .

A alternating current at any instant is given by $\mathrm{i}=\left[6+\sqrt{56} \sin \left(100 \pi \mathrm{t}+\frac{\pi}{3}\right)\right]$ A. The rms value of the current

is $\qquad$ A.

Twelve wires each having resistance $2 \Omega$ are joined to form a cube. A battery of 6 V emf is joined across point a and c . The voltage difference between $e$ and $f$ is $\qquad$ V.

A soap bubble is blown to a diameter of 7 cm . 36960 erg of work is done in blowing it further. If surface tension of soap solution is 40 dyne/cm then the new radius is $\qquad$ cm. Take $:\left(\pi=\frac{22}{7}\right)$

Two wavelengths $\lambda_{1}$ and $\lambda_{2}$ are used in Young's double slit experiment $\lambda_{1}=450 \mathrm{~nm}$ and $\lambda_{2}=650 \mathrm{~nm}$. The minimum order of fringe produced by $\lambda_{2}$ which overlaps with the fringe produced by $\lambda_{1}$ is $n$. The value of $n$ is $\qquad$

An elastic spring under tension of 3 N has a length a. Its length is $b$ under tension 2 N . For its length $(3 a-2 b)$, the value of tension will be $\qquad$ N

Two forces $\overrightarrow{\mathrm{F}}_{1}$ and $\overrightarrow{\mathrm{F}}_{2}$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\vec{F}_{1}$ and $\vec{F}_{2}$ is $\cos ^{-1}\left(\frac{1}{n}\right)$. The value of $|n|$ is $\qquad$ .

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed $v$. The sphere and the cylinder reaches upto maximum heights $h_{1}$ and $h_{2}$, respectively, above the initial level. The ratio $h_{1}: h_{2}$ is $\frac{n}{10}$. The value of $n$ is $\qquad$ .

What pressure (bar) of $\mathrm{H}_{2}$ would be required to make emf of hydrogen electrode zero in pure water at $25^{\circ} \mathrm{C}$ ?

The correct sequence of ligands in the order of decreasing field strength is :

What will be the decreasing order of basic strength of the following conjugate bases ?

${ }^{-} \mathrm{OH}, \mathrm{RO}, \mathrm{CH}_{3} \mathrm{CO} \bar{O}, \mathrm{C} \overline{1}$

In the precipitation of the iron group (III) in qualitative analysis, ammonium chloride is added before adding ammonium hydroxide to :

One of the commonly used electrode is calomel electrode. Under which of the following categories calomel electrode comes?

Number of complexes from the following with even number of unpaired "d" electrons is $\qquad$ .

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

[Given atomic numbers: $\mathrm{V}=23, \mathrm{Cr}=24, \mathrm{Fe}=26$, $\mathrm{Ni}=28, \mathrm{Cu}=29]$

Which one of the following molecules has maximum dipole moment ?

Which among the following is incorrect statement?

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

Statement I : Acidity of $\alpha$-hydrogens of aldehydes and ketones is responsible for Aldol reaction.

Statement II : Reaction between benzaldehyde and ethanal will NOT give Cross - Aldol product.

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

Which of the following nitrogen containing compound does not give Lassaigne's test?

The element which shows only one oxidation state other than its elemental form is :

Number of elements from the following that can not form compounds with valencies which match with their respective group valencies is $\qquad$ . B, C, N, S, O, F, P, Al, Si

The molarity $(M)$ of an aqueous solution containing $5.85\ \mathrm{g}$ of $\mathrm{NaCl}$ in $500\ \mathrm{mL}$ water is:

(Given : Molar mass of $\mathrm{Na} = 23 \mathrm{gmol}^{-1}$ and $\mathrm{Cl} = 35.5 \mathrm{gmol}^{-1}$ )

Identify the correct set of reagents or reaction conditions ' X ' and ' Y ' in the following set of transformation.

The correct order of first ionization enthalpy values of the following elements is:

(A) $\mathrm{O}$ (B) $\mathrm{N}$ (C) $\mathrm{Be}$ (D) $\mathrm{F}$ (E) $\mathrm{B}$

Choose the correct answer from the options given below:

The magnitude of enthalpy of formation of $\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)$ from ethylene by addition of hydrogen where the bondenergies of $\mathrm{C}-\mathrm{H}, \mathrm{C}-\mathrm{C}, \mathrm{H}-\mathrm{H}$ are $414 \mathrm{~kJ}, 347 \mathrm{~kJ}$, 615 kJ and 435 kJ respectively is - $\qquad$ kJ

The number of correct reaction(s) among the following is $\qquad$

$\mathrm{X} g$ of ethylamine is subjected to reaction with $\mathrm{NaNO}_{2} / \mathrm{HCl}$ followed by water and evolved dinitrogen gas which occupied $2.24\ \mathrm{L}$ volume at STP. The value of $X$ is $\_\_\_\_\_$ $\times 10^{-1} \mathrm{~g}$.

The de-Broglie wavelength of an electron in the $4^{\text {th }}$ orbit is $\_\_\_\_ \pi a_0$.

$(a_0 =$ Bohr radius$)$

Only 2 mL of $\mathrm{KMnO}_{4}$ solution of unknown molarity is required to reach the end point of a titration of 20 mL of oxalic acid ( 2 M ) in acidic medium. The molarity of $\mathrm{KMnO}_{4}$ solution should be $\qquad$ M.

Consider the following reaction $\mathrm{MnO}_{2}+\mathrm{KOH}+\mathrm{O}_{2} \rightarrow \mathrm{~A}+\mathrm{H}_{2} \mathrm{O}$. Product ' $A$ ' in neutral or acidic medium

disproportionate to give products ' B ' and ' C ' along with water. The sum of spin-only magnetic moment values of B and C

is $\qquad$ BM. (nearest integer) (Given atomic number of Mn is 25 )

Consider the following transformation involving first order elementary reaction in each step at constant temperature as

shown below. $\mathrm{A}+\mathrm{B} \underset{\text { Step } 3}{\stackrel{\text { Step } 1}{\rightleftharpoons}} \mathrm{C} \xrightarrow{\text { Step 2 }} \mathrm{P}$ Some details of the above reaction are listed below.

if the overall rate constant of the above transformation $\left( \text{k} \right)$ is given as $\text{k}=\frac{{{\text{k}}_{1}}{{\text{k}}_{2}}}{{{\text{k}}_{3}}}$ and the overall activation energy

$\left( {{\text{E}}_{\text{a}}} \right)$ is $400\text{ }\!\!~\!\!\text{ kJ }\!\!~\!\!\text{ mo}{{\text{l}}^{-1}}$, then the value of $\text{E}{{\text{a}}_{3}}$ is $\text{ }\!\!~\!\!\text{ }$ $\text{kJmo}{{\text{l}}^{-1}}$ (nearest integer)