JEE Main 2024 Apr 8 Shift 2 — Question Paper

Let $A = \{2, 3, 6, 8, 9, 11\}$ and $B = \{1, 4, 5, 10, 15\}$. Let $R$ be a relation on $A \times B$ defined by $(a, b)R(c, d)$ if and only if $3ad - 7bc$ is an even integer. Then the relation $R$ is

Let $f(x) = \begin{cases} -a & \text{if } -a \le x \le 0 \\ x + a & \text{if } 0 < x \le a \end{cases}$ where $a > 0$ and $g(x) = (f(|x|) - |f(x)|)/2$.

Then the function $g : [-a, a] \to [-a, a]$ is

If the system of equations $x + 4y - z = \lambda$, $7x + 9y + \mu z = -3$, $5x + y + 2z = -1$ has infinitely many solutions, then $(2\mu + 3\lambda)$ is equal to :

If $\alpha \ne a$, $\beta \ne b$, $\gamma \ne c$ and $\begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0$, then $\dfrac{a}{\alpha - a} + \dfrac{b}{\beta - b} + \dfrac{\gamma}{\gamma - c}$ is equal to :

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :

If the term independent of $x$ in the expansion of $\left( \sqrt{a}x^2 + \dfrac{1}{2x^3} \right)^{10}$ is 105, then $a^2$ is equal to :

The sum of all possible values of $\theta \in [-\pi, 2\pi]$, for which $\dfrac{1 + i\cos\theta}{1 - 2i\cos\theta}$ is purely imaginary, is equal to :

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\dfrac{70}{3}$ and the product of the third and fifth terms is 49. Then the sum of the $4^{\text{th}}$, $6^{\text{th}}$ and $8^{\text{th}}$ terms is equal to :

For $a, b > 0$, let $f(x) = \begin{cases} \dfrac{\tan((a+1)x) + b\tan x}{x}, & x < 0 \\ 3, & x = 0 \\ \dfrac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{a}\, x\sqrt{x}}, & x > 0 \end{cases}$

be a continous function at $x = 0$. Then $\dfrac{b}{a}$ is equal to :

If the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$, $a > 0$ has a local maximum at $x = \alpha$ and a local minimum at $x = \alpha^2$, then $\alpha$ and $\alpha^2$ are the roots of the equation :

Let $\displaystyle\int_{\alpha}^{\log_e 4} \dfrac{\mathrm{d}x}{\sqrt{\mathrm{e}^x - 1}} = \dfrac{\pi}{6}$. Then $\mathrm{e}^{\alpha}$ and $\mathrm{e}^{-\alpha}$ are the roots of the equation :

The area of the region in the first quadrant inside the circle $x^2 + y^2 = 8$ and outside the parabola $y^2 = 2x$ is equal to :

Let $y = y(x)$ be the solution curve of the differential equation $\sec y \dfrac{\mathrm{d}y}{\mathrm{d}x} + 2x\sin y = x^3\cos y$, $y(1) = 0$. Then $y(\sqrt{3})$ is equal to :

If the line segment joining the points $(5, 2)$ and $(2, a)$ subtends an angle $\dfrac{\pi}{4}$ at the origin, then the absolute value of the product of all possible values of $a$ is :

If the image of the point $(-4, 5)$ in the line $x + 2y = 2$ lies on the circle $(x+4)^2 + (y-3)^2 = r^2$, then $r$ is equal to :

If the shortest distance between the lines $\dfrac{x - \lambda}{2} = \dfrac{y - 4}{3} = \dfrac{z - 3}{4}$ and $\dfrac{x - 2}{4} = \dfrac{y - 4}{6} = \dfrac{z - 7}{8}$ is $\dfrac{13}{\sqrt{29}}$, then a value of $\lambda$ is :

Let $\vec{a} = 4\hat{i} - \hat{j} + \hat{k}$, $\vec{b} = 11\hat{i} - \hat{j} + \hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a} + \vec{b}) \times \vec{c} = \vec{c} \times (-2\vec{a} + 3\vec{b})$. If $(2\vec{a} + 3\vec{b}) \cdot \vec{c} = 1670$, then $|\vec{c}|^2$ is equal to :

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}$ and $\vec{c} = 3\hat{i} - \hat{j} + \lambda\hat{k}$ be three vectors. Let $\vec{r}$ be a unit vector along $\vec{b} + \vec{c}$. If $\vec{r} \cdot \vec{a} = 3$, then $3\lambda$ is equal to :

There are three bags X, Y and Z. Bag X contains 5 one-rupee coins and 4 five-rupee coins; Bag Y contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y, is :

If the value of $\dfrac{3\cos 36^\circ + 5\sin 18^\circ}{5\cos 36^\circ - 3\sin 18^\circ}$ is $\dfrac{a\sqrt{5} - b}{c}$, where $a$, $b$, $c$ are natural numbers and $\gcd(a, c) = 1$, then $a + b + c$ is equal to :

The number of distinct real roots of the equation $|x+1|\,|x+3| - 4|x+2| + 5 = 0$, is ________

Let a ray of light passing through the point $(3, 10)$ reflects on the line $2x + y = 6$ and the reflected ray passes through the point $(7, 2)$. If the equation of the incident ray is $ax + by + 1 = 0$, then $a^2 + b^2 + 3ab$ is equal to ________.

An arithmetic progression is written in the following way

$$\begin{array}{ccccccc} & & & 2 & & & \\ & & 5 & & 8 & & \\ & 11 & & 14 & & 17 & \\ 20 & & 23 & & 26 & & 29 \end{array}$$ $$- - - - - - - - - - - - - -$$ $$- - - - - - - - - - - - - - - -$$

The sum of all the terms of the $10^{\text{th}}$ row is ________.

Let A be the region enclosed by the parabola $y^2 = 2x$ and the line $x = 24$. Then the maximum area of the rectangle inscribed in the region A is ________.

If $\alpha = \displaystyle\lim_{x \to 0^+} \left( \dfrac{\mathrm{e}^{\sqrt{\tan x}} - \mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}} \right)$ and $\beta = \displaystyle\lim_{x \to 0} (1 + \sin x)^{\frac{1}{2}\cot x}$ are the roots of the quadratic equation $ax^2 + bx - \sqrt{\mathrm{e}} = 0$, then $12\log_e(a + b)$ is equal to ________.

If $\displaystyle\int \dfrac{1}{\sqrt[5]{(x-1)^4 (x+3)^6}}\, \mathrm{d}x = A\left( \dfrac{\alpha x - 1}{\beta x + 3} \right)^{B} + C$, where C is the constant of integration, then the value of $\alpha + \beta + 20AB$ is ________.

Let $\alpha|x| = |y|\mathrm{e}^{xy - \beta}$, $\alpha, \beta \in \mathbb{N}$ be the solution of the differential equation $x\mathrm{d}y - y\mathrm{d}x + xy(x\mathrm{d}y + y\mathrm{d}x) = 0$, $y(1) = 2$. Then $\alpha + \beta$ is equal to ________

Let S be the focus of the hyperbola $\dfrac{x^2}{3} - \dfrac{y^2}{5} = 1$, on the positive $x$-axis. Let C be the circle with its centre at $\mathrm{A}\left(\sqrt{6}, \sqrt{5}\right)$ and passing through the point S. If O is the origin and SAB is a diameter of C, then the square of the area of the triangle OSB is equal to ________

Let $\mathrm{P}(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{Q}(1, 6, 4)$ in the line $\dfrac{x}{1} = \dfrac{y-1}{2} = \dfrac{z-2}{3}$. Then $2\alpha + \beta + \gamma$ is equal to ________

Let $a, b, c \in \mathbb{N}$ and $a < b < c$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $9, 25, a, b, c$ be 18, 4 and $\dfrac{136}{5}$, respectively. Then $2a + b - c$ is equal to ________

If $\epsilon_0$ is the permittivity of free space and $\mathrm{E}$ is the electric field, then $\epsilon_0 \mathrm{E}^2$ has the dimensions :

The angle of projection for a projectile to have same horizontal range and maximum height is :

A given object takes n times the time to slide down $45°$ rough inclined plane as it takes the time to slide down an identical perfectly smooth $45°$ inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is :

[Figure: a 5 kg block at the top of a frictionless ($\mu = 0$) inclined plane of length 10 m at $30°$; at the bottom, a horizontal surface of length 2 m with $\mu = 0.5$ leads to a spring of $k = 100\ \mathrm{N/m}$ fixed to a wall]

A block is simply released from the top of an inclined plane as shown in the figure above. The maximum compression in the spring when the block hits the spring is :

A thin circular disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity $\omega$. If another disc of same dimensions but of mass $\mathrm{M}/2$ is placed gently on the first disc co-axially, then the new angular velocity of the system is :

Two satellite A and B go round a planet in circular orbits having radii 4R and R respectively. If the speed of A is $3v$, the speed of B will be :

A cube of ice floats partly in water and partly in kerosene oil. The ratio of volume of ice immersed in water to that in kerosene oil (specific gravity of Kerosene oil = 0.8, specific gravity of ice = 0.9) :

[Figure: a container with a layer of kerosene oil floating on water; an ice cube floats at the interface, partly in each liquid]

A diatomic gas $(\gamma = 1.4)$ does $100\ \mathrm{J}$ of work in an isobaric expansion. The heat given to the gas is :

Given below are two statements :

Statement (I) : The mean free path of gas molecules is inversely proportional to square of molecular diameter.

Statement (II) : Average kinetic energy of gas molecules is directly proportional to absolute temperature of gas.

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

A capacitor has air as dielectric medium and two conducting plates of area $12\ \mathrm{cm}^2$ and they are $0.6\ \mathrm{cm}$ apart. When a slab of dielectric having area $12\ \mathrm{cm}^2$ and $0.6\ \mathrm{cm}$ thickness is inserted between the plates, one of the conducting plates has to be moved by $0.2\ \mathrm{cm}$ to keep the capacitance same as in previous case. The dielectric constant of the slab is : (Given $\epsilon_0 = 8.834 \times 10^{-12}\ \mathrm{F/m}$)

Water boils in an electric kettle in 20 minutes after being switched on. Using the same main supply, the length of the heating element should be ________ to ________ times of its initial length if the water is to be boiled in 15 minutes.

A long straight wire of radius $a$ carries a steady current I. The current is uniformly distributed across its cross section. The ratio of the magnetic field at $\frac{a}{2}$ and $2a$ from axis of the wire is :

A coil of negligible resistance is connected in series with $90\ \Omega$ resistor across $120\ \mathrm{V}$, $60\ \mathrm{Hz}$ supply. A voltmeter reads $36\ \mathrm{V}$ across resistance. Inductance of the coil is :

A plane progressive wave is given by $y = 2\cos 2\pi(330t - x)\ \mathrm{m}$. The frequency of the wave is :

The position of the image formed by the combination of lenses is :

[Figure: an object 30 cm to the left of a converging lens $f_1 = 10\ \mathrm{cm}$; a diverging lens $f_2 = -10\ \mathrm{cm}$ is 5 cm to its right; a converging lens $f_3 = 30\ \mathrm{cm}$ is 10 cm further right]

A proton and an electron have the same de Broglie wavelength. If $\mathrm{K_p}$ and $\mathrm{K_e}$ be the kinetic energies of proton and electron respectively, then choose the correct relation :

If $\mathrm{M_o}$ is the mass of isotope $\mathrm{^{12}_{5}B}$, $\mathrm{M_p}$ and $\mathrm{M_n}$ are the masses of proton and neutron, then nuclear binding energy of isotope is :

In a hypothetical fission reaction $$\mathrm{_{92}X^{236} \rightarrow {}_{56}Y^{141} + {}_{36}Z^{92} + 3R}$$ The identity of emitted particles (R) is :

There are 100 divisions on the circular scale of a screw gauge of pitch 1 mm. With no measuring quantity in between the jaws, the zero of the circular scale lies 5 divisions below the reference line. The diameter of a wire is then measured using this screw gauge. It is found that 4 linear scale divisions are clearly visible while 60 divisions on circular scale coincide with the reference line. The diameter of the wire is :

Least count of a vernier caliper is $\frac{1}{20\mathrm{N}}$ cm. The value of one division on the main scale is 1 mm. Then the number of divisions of main scale that coincide with N divisions of vernier scale is :

A body of mass M thrown horizontally with velocity $v$ from the top of the tower of height H touches the ground at a distance of 100 m from the foot of the tower. A body of mass 2M thrown at a velocity $\frac{v}{2}$ from the top of the tower of height 4H will touch the ground at a distance of ________ m.

A circular table is rotating with an angular velocity of $\omega$ rad/s about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of 1 m on the groove. All the surfaces are smooth. If the radius of the table is 3 m, the radial velocity of the ball w.r.t. the table at the time ball leaves the table is $x\sqrt{2}\,\omega$ m/s, where the value of $x$ is __________.

[Figure: a circular table of radius 3 m rotating about a vertical axis with angular velocity $\omega$; a radial groove holds a ball placed 1 m from the centre]

Small water droplets of radius 0.01 mm are formed in the upper atmosphere and falling with a terminal velocity of 10 cm/s. Due to condensation, if 8 such droplets are coalesced and formed a larger drop, the new terminal velocity will be ________ cm/s.

An object of mass 0.2 kg executes simple harmonic motion along x axis with frequency of $\left(\frac{25}{\pi}\right)$ Hz. At the position $x = 0.04$ m the object has kinetic energy 0.5 J and potential energy 0.4 J. The amplitude of oscillation is _________ cm.

If the net electric field at point P along Y axis is zero, then the ratio of $\left|\frac{q_2}{q_3}\right|$ is $\frac{8}{5\sqrt{x}}$, where $x =$ __________.

[Figure: point P lies 4 cm above the base line; charge $+q_2$ is on the base line 2 cm left of the foot of the perpendicular from P, and charge $-q_3$ is 3 cm right of it]

A heater is designed to operate with a power of 1000 W in a 100 V line. It is connected in combination with a resistance of $10\ \Omega$ and a resistance R, to a 100 V mains as shown in figure. For the heater to operate at 62.5 W, the value of R should be _________ $\Omega$.

[Figure: a 100 V source in series with a $10\ \Omega$ resistor feeding points B and C; between B and C the heater and resistance R are connected in parallel]

The coercivity of a magnet is $5 \times 10^3$ A/m. The amount of current required to be passed in a solenoid of length 30 cm and the number of turns 150, so that the magnet gets demagnetised when inside the solenoid is _________ A.

An alternating emf $\mathrm{E} = 110\sqrt{2}\ \sin 100\mathrm{t}$ volt is applied to a capacitor of $2\ \mu\mathrm{F}$, the rms value of current in the circuit is __________ mA.

Two slits are 1 mm apart and the screen is located 1 m away from the slits. A light of wavelength 500 nm is used. The width of each slit to obtain 10 maxima of the double slit pattern within the central maximum of the single slit pattern is ________ $\times 10^{-4}$ m.

A potential divider circuit is connected with a dc source of 20 V, a light emitting diode of glow in voltage 1.8 V and a zener diode of breakdown voltage of 3.2 V. The length (PR) of the resistive wire is 20 cm. The minimum length of PQ to just glow the LED is __________ cm.

[Figure: a 20 V dc source across a resistive wire PR with sliding contact Q; an LED is connected from P and a zener diode from the LED junction back to the tap at Q]

When $\psi_A$ and $\psi_B$ are the wave functions of atomic orbitals, then $\sigma^*$ is represented by :

Given below are two statements :

Statement (I) : A Buffer solution is the mixture of a salt and an acid or a base mixed in any particular quantities.

Statement (II) : Blood is naturally occurring buffer solution whose pH is maintained by $\mathrm{H_2CO_3/HCO_3^{\ominus}}$ concentrations.

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

The emf of cell $\mathrm{Tl}\left|\underset{(0.001\,\mathrm{M})}{\mathrm{Tl}^{+}}\right\|\left.\underset{(0.01\,\mathrm{M})}{\mathrm{Cu}^{2+}}\right|\mathrm{Cu}$ is 0.83 V at 298 K. It could be increased by :

The reaction ;

$$\frac{1}{2}\mathrm{H_{2(g)}} + \mathrm{AgCl_{(s)}} \rightarrow \mathrm{H^+_{(aq)}} + \mathrm{Cl^-_{(aq)}} + \mathrm{Ag_{(s)}}$$

occurs in which of the following galvanic cell :

For a reaction $\mathrm{A} \xrightarrow{K_1} \mathrm{B} \xrightarrow{K_2} \mathrm{C}$

If the rate of formation of B is set to be zero then the concentration of B is given by :

Identify the correct statements about p-block elements and their compounds.

(A) Non metals have higher electronegativity than metals.

(B) Non metals have lower ionisation enthalpy than metals.

(C) Compounds formed between highly reactive nonmetals and highly reactive metals are generally ionic.

(D) The non-metal oxides are generally basic in nature.

(E) The metal oxides are generally acidic or neutral in nature.

Choose the correct answer from the options given below :

Identify the incorrect statements about group 15 elements :

(A) Dinitrogen is a diatomic gas which acts like an inert gas at room temperature.

(B) The common oxidation states of these elements are $-3$, $+3$ and $+5$.

(C) Nitrogen has unique ability to form $\mathrm{p\pi - p\pi}$ multiple bonds.

(D) The stability of $+5$ oxidation states increases down the group.

(E) Nitrogen shows a maximum covalency of 6.

Choose the correct answer from the options given below :

The equilibrium $\mathrm{Cr_2O_7^{2-}} \rightleftharpoons 2\mathrm{CrO_4^{2-}}$ is shifted to the right in :

Given below are two statements :

Statement (I) : Fusion of $\mathrm{MnO_2}$ with KOH and an oxidising agent gives dark green $\mathrm{K_2MnO_4}$.

Statement (II) : Manganate ion on electrolytic oxidation in alkaline medium gives permanganate ion.

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

Match List - I with List - II.

List - I (Complex ion)	List - II (Spin only magnetic moment in B.M.)
(A) $\mathrm{[Cr(NH_3)_6]^{3+}}$	(I) 4.90
(B) $\mathrm{[NiCl_4]^{2-}}$	(II) 3.87
(C) $\mathrm{[CoF_6]^{3-}}$	(III) 0.0
(D) $\mathrm{[Ni(CN)_4]^{2-}}$	(IV) 2.83

Choose the correct answer from the options given below :

In qualitative test for identification of presence of phosphorous, the compound is heated with an oxidising agent. Which is further treated with nitric acid and ammonium molybdate respectively. The yellow coloured precipitate obtained is :

Given below are two statements :

Statement (I) : Kjeldahl method is applicable to estimate nitrogen in pyridine.

Statement (II) : The nitrogen present in pyridine can easily be converted into ammonium sulphate in Kjeldahl method.

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

The shape of carbocation is :

IUPAC name of following hydrocarbon (X) is :

$$\underset{\underset{\mathrm{CH_3}}{|}}{\mathrm{CH_3-CH}}-\mathrm{CH_2-CH_2}-\underset{\underset{\mathrm{CH_3}}{|}}{\mathrm{CH}}-\underset{\underset{\mathrm{CH_3}}{|}}{\mathrm{CH}}-\mathrm{CH_2-CH_3}$$

(X)

Given below are two statements :

Statement (I) : $\mathrm{S_N2}$ reactions are 'stereospecific', indicating that they result in the formation of only one stereo-isomer as the product.

Statement (II) : $\mathrm{S_N1}$ reactions generally result in formation of product as racemic mixtures.

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

The correct sequence of acidic strength of the following aliphatic acids in their decreasing order is :

$\mathrm{CH_3CH_2COOH}$, $\mathrm{CH_3COOH}$, $\mathrm{CH_3CH_2CH_2COOH}$, $\mathrm{HCOOH}$

Match List - I with List - II.

[Figure: List-I shows four benzene-ring reactions and List-II shows four products as structures]

List - I (Reactions) :

(A) Aniline $\xrightarrow{\text{(i) } \mathrm{NaNO_2 + HCl} \text{; (ii) } \mathrm{H_2O}, \text{ warm}}$

(B) Phenol $\xrightarrow{\mathrm{Na_2Cr_2O_7,\ H_2SO_4}}$

(C) Phenol $\xrightarrow{\text{(i) } \mathrm{CHCl_3 + aq\ NaOH} \text{; (ii) } \mathrm{H^+}}$

(D) Phenol $\xrightarrow{\text{(i) } \mathrm{NaOH} \text{; (ii) } \mathrm{CO_2} \text{; (iii) } \mathrm{H^+}}$

List - II (Products) :

(I) 2-hydroxybenzaldehyde (salicylaldehyde)

(II) phenol

(III) 2-hydroxybenzoic acid (salicylic acid)

(IV) benzoquinone (cyclohexa-2,5-diene-1,4-dione)

Choose the correct answer from the options given below :

Which one the following compounds will readily react with dilute NaOH ?

Given below are two statements :

Statement (I) : All the following compounds react with p-toluenesulfonyl chloride.

$\mathrm{C_6H_5NH_2}$ $\quad$ $\mathrm{(C_6H_5)_2NH}$ $\quad$ $\mathrm{(C_6H_5)_3N}$

Statement (II) : Their products in the above reaction are soluble is aqueous NaOH.

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

Match List - I with List - II.

List - I (Test)	List - II (Identification)
(A) Bayer's test	(I) Phenol
(B) Ceric ammonium nitrate test	(II) Aldehyde
(C) Phthalein dye test	(III) Alcoholic-OH group
(D) Schiff's test	(IV) Unsaturation

Choose the correct answer from the options given below :

A solution is prepared by adding 1 mole ethyl alcohol in 9 mole water. The mass percent of solute in the solution is ________ (Integer answer) (Given : Molar mass in $\mathrm{g\ mol^{-1}}$ Ethyl alcohol : 46 water : 18)

Wavenumber for a radiation having 5800 $\mathrm{\AA}$ wavelength is $x \times 10\ \mathrm{cm^{-1}}$. The value of $x$ is ______. (Integer answer)

Number of molecules having bond order 2 from the following molecules is ________.

$\mathrm{C_2,\ O_2,\ Be_2,\ Li_2,\ Ne_2,\ N_2,\ He_2}$

$\Delta_{\mathrm{vap}}\mathrm{H^{\ominus}}$ for water is $+40.79\ \mathrm{kJ\ mol^{-1}}$ at 1 bar and 100 $^{\circ}\mathrm{C}$. Change in internal energy for this vapourisation under same condition is ________ $\mathrm{kJ\ mol^{-1}}$. (Integer answer)

(Given $R = 8.3\ \mathrm{JK^{-1}\ mol^{-1}}$)

Molality of an aqueous solution of urea is 4.44 m. Mole fraction of urea in solution is $x \times 10^{-3}$. Value of $x$ is ________. (Integer answer)

Two moles of benzaldehyde and one mole of acetone under alkaline conditions using aqueous NaOH after heating gives $x$ as the major product. The number of $\pi$ bonds in the product $x$ is ________.

Total number of unpaired electrons in the complex ions $\mathrm{[Co(NH_3)_6]^{3+}}$ and $\mathrm{[NiCl_4]^{2-}}$ is ________.