JEE Main 2024 — 8 April, Shift 2

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

equal lo :

Let $\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \quad \vec{b}=2 \hat{i}+3 \hat{j}-5 \hat{k}$ and $\overrightarrow{\mathrm{c}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}$ be three vectors. Let $\overrightarrow{\mathrm{r}}$ be a unit vector along

$\vec{b}+\vec{c}$. If $\vec{r} \cdot \vec{a}=3$, then $3 \lambda$ is equal to :

If $\alpha \neq a, \beta \neq b, \gamma \neq c$ and $\left| \begin{matrix}\alpha & b & c \\a & \beta & c \\a & b & \gamma \\\end{matrix} \right|=0$, then $\frac{\mathrm{a}}{\alpha-\mathrm{a}}+\frac{\mathrm{b}}{\beta-\mathrm{b}}+\frac{\gamma}{\gamma-\mathrm{c}}$ is equal to :

In an increasing geometric progression ol positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ 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 :-

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 :

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

If the system of equations $x+4 y-z=\lambda$, $7 x+9 y+\mu z=-3,5 x+y+2 z=-1$ has infinitely many solutions,

then $(2 \mu .+3 \lambda)$ is equal to :

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

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

$\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to :

Let $y=y(x)$ be the solution curve of the differential equation secy $\frac{d y}{d x}+2 x \sin y=x^{3} \cos y$ $y(1)=0$.

Then $y(\sqrt{3})$ is equal to $:$

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

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

If the function $f(x)=2 x^{3}-9 a x^{2}+12 a^{2} x+1, a>0$ has a local maximum at $\mathrm{x}=\alpha$ and a local minimum

$x=\alpha^{2}$, then $\alpha$ and $\alpha^{2}$ are the roots of the equation :

Let $\int_{\alpha}^{\log _{e} 4} \frac{\mathrm{dx}}{\sqrt{\mathrm{e}^{\mathrm{x}}-1}}=\frac{\pi}{6}$. Then $\mathrm{e}^{\alpha}$ and $\mathrm{e}^{-\alpha}$ are the roots of the equation :

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

For\ a,b>0,\ \text{let}

be a continuous function at $x=0$. Then $\frac{b}{a}$ is equal to

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

Let A be the region enclosed by the parabola $y^{2}=2 x$ and the line $x=24$. Then the maximum area of

the rectangle inscribed in the region A is $\qquad$

If $\alpha=\lim _{x \rightarrow 0^{+}}\left(\frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$ and $\beta=\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}$ are the roots of the quadratic equation

$\mathrm{ax}^{2}+\mathrm{bx}-\sqrt{\mathrm{e}}=0$, then 12 $\log _{e}(a+b)$ is equal to $\qquad$ .

Let $S$ be the focus of the hyperbola $\frac{x^{2}}{3}-\frac{y^{2}}{5}=1$, on the positive x -axis. Let C be the circle with its centre at $A(\sqrt{6}, \sqrt{5})$ 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 $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$. Then $2 \alpha+\beta+\gamma$ is equal to $\qquad$

An arithmetic progression is written in the following way

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

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

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

Let $\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{N}$ and $\mathrm{a}<\mathrm{b}<\mathrm{c}$. 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 $\frac{136}{5}$, respectively. Then $2 \mathrm{a}+\mathrm{b}-\mathrm{c}$ is equal to $\_\_\_\_\_$

Lei $\alpha|x|=|y| e^{x y-\beta}, \alpha, \beta \in N$ be the solution of the differential equation

$x d y-y d x+x y(x d y+y d x)=0$, $y(1)=2$. Then $\alpha+\beta$ is equal to

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

$\alpha+\beta+20 \mathrm{AB}$ is $\qquad$

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 : (Neglect any energy losses during collision with the ground)

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

If $\epsilon_{0}$ is the permittivity of free space and $E$ is the electric field, then $\in_{0} E^{2}$ has the dimensions :

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

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

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 option given below:

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

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 $\frac{\mathrm{a}}{2}$ and 2 a from axis of the wire is :

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

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 $\qquad$ to $\qquad$ times of its initial length if the water is to be boiled in 15 minutes.

A capacitor has air as dielectric medium and two conducting plates of area $12 \mathrm{~cm}^{2}$ and they are 0.6 cm apart. When a slab

of dielectric having area $12 \mathrm{~cm}^{2}$ and 0.6 cm thickness is inserted between the plates, one of the conducting plates has to

be moved by 0.2 cm to keep the capacitance same as in previous case. The dielectric constant of the slab is :

$\left(\right.$ Given $\left.\epsilon_{0}=8.834 \times 10^{-12} \mathrm{~F} / \mathrm{m}\right)$

A given object takes $n$ times the time to slide down $45^{\circ}$ rough inclined plane as it takes the time to slide down an identical perfectly smooth $45^{\circ}$ inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane 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 V across resistance. Inductance of the coil 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 the 4 linear scale divisions are clearly visible while 60 divisions on circular scale coincide with the reference line. The diameter of the wire is :

A proton and an electron have the same de Broglie wavelength. If $K_{p}$ and $K_{e}$ be the kinetic energies of proton and electron respectively. Then choose the correct relation :

Least count of a vernier caliper is $\frac{1}{20 \mathrm{~N}} \mathrm{~cm}$. 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 :

If $M_{o}$ is the mass of isotope ${ }_{5}^{12} B, M_{p}$ and $M_{n}$ are the masses of proton and neutron, then nuclear binding energy of isotope is :

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

The coercivity of a magnet is $5 \times 10^{3} \mathrm{~A} / \mathrm{m}$. 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 $\qquad$ A.

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

If the net electric field at point P along Y axis is zero, then the ratio of $\left|\frac{\mathrm{q}_{2}}{\mathrm{q}_{3}}\right|$ is $\frac{8}{5 \sqrt{\mathrm{x}}}$, where $\mathrm{x}=$ $\qquad$

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 $\qquad$ $\Omega$.

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

Two slits are 1 mm apart and the screen is located 1 m away from the slits. A light 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 $\ldots \ldots \ldots . \times 10^{-4} \mathrm{~m}$.

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 $x=0.04 \mathrm{~m}$ the object has kinetic energy 0.5 J and potential energy 0.4 J . The amplitude of oscillation is $\qquad$ cm .

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 $\qquad$ cm .

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 2 M thrown at a velocity $\frac{v}{2}$ from the top of the tower of height 4 H will touch the ground at a distance of $\qquad$ .m.

A circular table is rotating with an angular velocity of $\omega \mathrm{rad} / \mathrm{s}$ 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 surface 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 $\mathrm{x} \sqrt{2} \omega \mathrm{~m} / \mathrm{s}$, where the value of x is $\qquad$

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 :

For a reaction $A \xrightarrow{\mathrm{~K}_{1}} \mathrm{~B} \xrightarrow{\mathrm{~K}_{2}} \mathrm{C}$ If the rate of formation of $B$ is set to be zero then the concentration of B is given by :

When $\psi_{\mathrm{A}}$ and $\Psi_{\mathrm{B}}$ are the wave functions of atomic orbitals, then $\sigma^{*}$ is represented by :

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

The shape of carbocation is :

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

Statement (I) : $\mathrm{S}_{\mathrm{N}} 2$ reactions are 'stereospecific', indicating that they result in the formation only one stereo-isomers as the product.

Statement (II) : $\mathrm{S}_{\mathrm{N}} 1$ 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 :

Match List-I with List-II. List-I (Reactions) List-II (Products)

Choose the correct answer from the options given below :

Match List-I with List-II.

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 $\mathrm{p} \pi-\mathrm{p} \pi$ 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 IUPAC name of following hydrocarbon $(\mathrm{X})$ is :

The equilibrium $\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-} \rightleftharpoons 2 \mathrm{CrO}_{4}^{2-}$ is shifted to the right in :

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

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 $\mathrm{H}_{2} \mathrm{CO}_{3} / \mathrm{HCO}_{3}^{\ominus}$ concentrations.

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 :

$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{COOH}, \mathrm{CH}_{3} \mathrm{COOH}, \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2} \mathrm{COOH}$, $\mathrm{HCOOH}$

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

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

[$\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NH}_{2} ;\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)_{2} \mathrm{NH} ;\left(\mathrm{C}_{6} \mathrm{H}_{5}\right)_{3} \mathrm{~N}$]

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

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

The emf of cell

$\text{T}1\left| \underset{\left( 0.01\text{M} \right)}{\mathop{\text{T}{{1}^{+}}}}\, \right|\left| \text{C}{{\text{u}}^{2+}}\underset{\left( 0.01M \right)}{\mathop{~}}\, \right|\text{Cu}$

is 0.83 V at 298 K . It could be increased 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 non-metals 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:

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

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 reaction; $\frac{1}{2} \mathrm{H}_{2(\mathrm{~g})}+\mathrm{AgCl}_{(\mathrm{s})} \rightarrow \mathrm{H}_{(\mathrm{aq})}^{+}+\mathrm{Cl}_{(\mathrm{aq})}^{-}+\mathrm{Ag}_{(\mathrm{s})}$ occurs in which of the following galvanic cell :

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

Statement (I) : Fusion of $\mathrm{MnO}_{2}$ with KOH and an oxidising agent gives dark green $\mathrm{K}_{2} \mathrm{MnO}_{4}$.

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.

Choose the correct answer from the options given below :

$\Delta_{\text {vap }} \mathrm{H}^{\ominus}$ for water is $+40.49 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at 1 bar and $100^{\circ} \mathrm{C}$. Change in internal energy for this vapourisation under same condition is $\qquad$ kJ $\operatorname{mol}^{-1}$. (Integer answer) (Given $\mathrm{R}=8.3 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$ )

Number of molecules having bond order 2 from the following molecule is $\qquad$

$\mathrm{C}_{2}, \mathrm{O}_{2}, \mathrm{Be}_{2}, \mathrm{Li}_{2}, \mathrm{Ne}_{2}, \mathrm{~N}_{2}, \mathrm{He}_{2}$

Total number of optically active compounds from the following is

The total number of carbon atoms present in tyrosine, an amino acid, is $\qquad$

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

Molality of an aqueous solution of urea is $4.44\ m$. Mole fraction of urea in solution is $X \times 10^{-3}$. The value of $X$ is $\_\_\_\_\_$ (integer answer).

Total number of unpaired electrons in the complex ion $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$ and $\left[\mathrm{NiCl}_{4}\right]^{2-}$ is

Wavenumber for a radiation having $5800\ \mathrm{\AA}$ wavelength is $x \times 10^{4}\ \mathrm{cm^{-1}}$. The value of $x$ is:

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]