JEE Main 2024 — 6 April, Shift 2

Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If $P$ is the sum of perimeters and Q is be the sum of areas of all the triangles formed in this process, then:

Let $\mathrm{A}=\{1,2,3,4,5\}$. Let R be a relation on A defined by $x R y$ if and only if $4 x \leq 5 y$. Let $m$ be the number of elements in R and n be the minimum number of elements from $\mathrm{A} \times \mathrm{A}$ that are required to be added to R to make it a symmetric relation. Then $\mathrm{m}+\mathrm{n}$ is equal to:

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:

Suppose the solution of the differential equation $\frac{d y}{d x}=\frac{(2+\alpha) x-\beta y+2}{\beta x-2 \alpha y-(\beta \gamma-4 \alpha)} \quad$

represents a circle passing through origin. Then the radius of this circle is :

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio

$5: 4$, is $a x^{2}+b y^{2}+c x y+d x+e y+170=0$, then the value of $a^{2}+2 b+3 c+4 d+e$ is equal to:

$\quad \lim _{n \rightarrow \infty} \frac{\left(1^{2}-1\right)(n-1)+\left(2^{2}-2\right)(n-2)+\ldots .+\left((n-1)^{2}-(n-1)\right) \cdot 1}{\left(1^{3}+2^{3}+\ldots .+n^{3}\right)-\left(1^{2}+2^{2}+\ldots .+n^{2}\right)}$ is equal to:

Let $0 \leq \mathrm{r} \leq \mathrm{n}$. If ${ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}+1}:{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}:{ }^{\mathrm{n}-1} C_{\mathrm{r}-1}=55: 35: 21$, then $2 n+5 r$ is equal to:

A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to :

If ${{z}_{1}},{{z}_{2}}$ are two distinct complex number such that \left|\frac{{{\text{z}}-{}}2{{\text{z_2}}{}}}{\frac{1}{2}-{{\text{z}}_{1}}{{{\text{\overline{z_2}}}}_{}}} \right|=2 then

If the function $f(x)=\left(\frac{1}{x}\right)^{2 x} ; x>0$ attains the maximum value at $x=\frac{1}{\mathrm{e}}$ then :

Let $\vec{a}=6 \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a is vector such that $|\overrightarrow{\mathrm{c}}| \geq 6, \vec{a} \cdot \overrightarrow{\mathrm{c}}=6|\overrightarrow{\mathrm{c}}|,|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=2 \sqrt{2}$

and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $60^{\circ}$, then $|(\vec{a} \times \vec{b}) \times \vec{c}|$ is equal to:

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{\text {th }}$ position in this arrangement is :

Suppose for a differentiable function $h, h(0)=0$, $h(1)=1$ and $h^{\prime}(0)=h^{\prime}(1)=2$. If $g(x)=h\left(e^{x}\right) e^{h(x)}$, then $\mathrm{g}^{\prime}(0)$ is equal to:

Let $\mathrm{P}(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{Q}(3,-3,1)$ in the line $\frac{x-0}{1}=\frac{y-3}{1}=\frac{z-1}{-1}$ and $R$ be the point $(2,5,-1)$. If the area of the triangle PQR is $\lambda$ and $\lambda^{2}=14 \mathrm{~K}$, then K is equal to:

If $\mathrm{P}(6,1)$ be the orthocentre of the triangle whose vertices are $A(5,-2), B(8,3)$ and $C(h, k)$, then the point C lies on the circle.

Let $f(x)=\frac{1}{7-\sin 5 x}$ be a function defined on $R$.

Then the range of the function $f(x)$ is equal to:

Let $\vec{a}=2 \hat{i}+\hat{j}-\hat{k}, \vec{b}=((\vec{a} \times(\hat{i}+\hat{j})) \times \hat{i}) \times \hat{i}$.

Then the square of the projection of $\vec{a}$ on $\vec{b}$ is :

If the area of the region $\left\{ \left( x,y \right):\frac{a}{{{x}^{2}}}\le y\le \frac{1}{x},1\le x\le 2,0<a<1 \right\}$ is $\left( {{\log }_{e}}2 \right)-\frac{1}{7}$ then the value of $7a-3$ is equal to :

If $\int \frac{1}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x} d x=\frac{1}{12} \tan ^{-1}(3 \tan x)+$ constant, then the maximum value of $\operatorname{asin} x+b \cos x$, is :

If A is a square matrix of order 3 such that $\operatorname{det}(\mathrm{A})=3$ and

$\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$, then $m+\ 2 n$ is equal to

Let [ $t$ ] denote the greatest integer less than or equal to t . Let $\mathrm{f}:[0, \infty) \rightarrow \mathrm{R}$ be a function defined by $f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]$. Let $S$ be the set of all points in the interval $[0,8]$ at which f is not continuous. Then $\sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to $\qquad$

The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $x= \pm \frac{4}{\sqrt{3}}$, respectively. Let the line $y-\sqrt{3} x+\sqrt{3}=0$ touch this hyperbola at $\left(X_0,Y_0\right)$. If m is the product of the focal distances of the point $\left(X_0,Y_0\right)$, then $4 \mathrm{e}^{2}+\mathrm{m}$ is equal to $\qquad$

If $S(x)=(1+x)+2(1+x)^{2}+3(1+x)^{3}+\ldots .$. $+60(1+x)^{60}, x \neq 0$, and

$(60)^{2} S(60)=a(b)^{b}+b$, where $a, b \in N$, then $(a+b)$ equal to $\qquad$

Let [t] denote the largest integer less than or equal to t.

If $\int_{0}^{3}\left(\left[x^{2}\right]+\left[\frac{x^{2}}{2}\right]\right) d x=a+b \sqrt{2}-\sqrt{3}-\sqrt{5}+c \sqrt{6}-\sqrt{7}$, where $a, b, c \in z$, then $a+b+c$

is equal to $\qquad$

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $n-m$ is equal to $\qquad$

In a triangle $\mathrm{ABC}, \mathrm{BC}=7, \mathrm{AC}=8, \mathrm{AB}=\alpha \in \mathrm{N}$ and $\cos \mathrm{A}=\frac{2}{3}$. If $49 \cos (3 \mathrm{C})+42=\frac{\mathrm{m}}{\mathrm{n}}$,

where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to $\qquad$

If the shortest distance between the lines $\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}$ is $\frac{44}{\sqrt{30}}$, then the largest possible value of $|\lambda|$ is equal to $\qquad$

Let $\alpha, \beta$ be roots of $x^{2}+\sqrt{2} x-8=0$. If $\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$, then $\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_{9}}{2 \mathrm{U}_{8}}$ is equal to $\qquad$

If the system of equations

$2 \mathrm{x}+7 \mathrm{y}+\lambda \mathrm{z}=3$

$3 x+2 y+5 z=4$

$x+\mu y+32 z=-1$

has infinitely many solutions, then $(\lambda-\mu)$ is equal to $\qquad$

If the solution $y(x)$ of the given differential equation $\left(e^{y}+1\right) \cos x d x+e^{y} \sin x d y=0$ passes through the point

$\left(\frac{\pi}{2}, 0\right)$, then the value of $e^{y\left(\frac{\pi}{6}\right)}$ is equal to $\qquad$

The longest wavelength associated with Paschen series is : $\left(\right.$ Given $\mathrm{R}_{H}=1.097 \times 10^{7}$ SI unit)

A total of 48 J heat is given to one mole of helium kept in a cylinder. The temperature of helium increases by $2^{\circ} \mathrm{C}$. The work done by the gas is : (Given, $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$.)

In finding out refractive index of glass slab the following observations were made through travelling microscope 50 vernier

scale division $=$ 49 MSD; 20 divisions on main scale in each cm For mark on paper

$\mathrm{MSR}=8.45 \mathrm{~cm}, \mathrm{VC}=26$ For mark on paper seen through slab $\mathrm{MSR}=7.12 \mathrm{~cm}, \mathrm{VC}=41$

For powder particle on the top surface of the glass slab $\mathrm{MSR}=4.05 \mathrm{~cm}, \mathrm{VC}=1$

$(\mathrm{MSR}=$ Main Scale Reading, VC $=$ Vernier Coincidence) Refractive index of the glass slab is:

In the given electromagnetic wave $E_{y}=600 \sin (\omega t-k x) \mathrm{Vm}^{-1}$, intensity of the associated light beam is

(in $\mathrm{W} / \mathrm{m}^{2}$ ); (Given $\epsilon_{0}=$ $9 \times 10^{-12} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}$ )

Assuming the earth to be a sphere of uniform mass density, a body weighed 300 N on the surface of earth. How much it would weigh at R/4 depth under surface of earth?

The acceptor level of a p-type semiconductor is 6 eV . The maximum wavelength of light which can create a hole would be : Given hc $=1242 \mathrm{eV} \mathrm{nm}$.

A car of 800 kg is taking turn on a banked road of radius 300 m and angle of banking $30^{\circ}$. If coefficient of static friction is

0.2 then the maximum speed with which car can negotiate the turn safely: $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}, \sqrt{3}=1.73\right)$

Two identical conducting spheres $P$ and $S$ with charge Q on each, repel each other with a force 16N. A third identical uncharged conducting sphere R is successively brought in contact with the two spheres. The new force of repulsion between P and S is :

In a coil, the current changes form -2 A to +2 A in 0.2 s and induces an emf of 0.1 V . The selfinductance of the coil is :

For the thin convex lens, the radii of curvature are at 15 cm and 30 cm respectively. The focal length the lens is 20 cm . The refractive index of the material is :

Energy of 10 non rigid diatomic molecules at temperature T is :

A body of weight 200 N is suspended form a tree branch thought a chain of mass 10 kg . The branch pulls the chain by a force equal to (if $g=10 \mathrm{~m} / \mathrm{s}^{2}$ ):

When UV light of wavelength 300 nm is incident on the metal surface having work function 2.13 eV , electron emission takes place. The stopping potential is: $($ Given hc $=1240 \mathrm{eV} \mathrm{nm})$

The number of electrons flowing per second in the filament of a 110 W bulb operating at 220 V is : (Given $\mathrm{e}=1.6 \times 10^{-19} \mathrm{C}$ )

When kinetic energy of a body becomes 36 times of its original value, the percentage increase in the momentum of the body will be :

Pressure inside a soap bubble is greater than the pressure outside by an amount : (given : $\mathrm{R}=$ Radius of bubble, $\mathrm{S}=$ Surface tension of bubble)

Match List-I with List-II

List-I (Y vs X)	List-II (Shape of Graph)
(A)	$\text{Y}=$ magnetic susceptibility $\text{X}=$ magnetisingfield	(I)
(B)	$\text{Y}=$ magnetic field$\text{X}=$ distance from centre of a current carryingwire for $\text{x}<\text{a}$ (where $\text{a}=$ radiusof wire)	(II)
(C)	$\text{Y}=$ magnetic field$\text{X}=$ distance from centre of acurrent carrying wire for $\text{x}>\text{a}$ (where $\text{a}=$radius of wire)	(III)
(D)	$\text{Y}=$ magneticfield insidesolenoid $\text{X}=$ distancefrom center	(IV)

Choose the correct answer from the options given below :

In a vernier calliper, when both jaws touch each other, zero of the vernier scale shifts towards left and its $4^{\text {th }}$ division coincides exactly with a certain division on main scale. If 50 vernier scale divisions equal to 49 main scale divisions and zero error in the instrument is 0.04 mm then how many main scale divisions are there in 1 cm ?

Given below are two statements :

Statement (I) : Dimensions of specific heat is $\left[\mathrm{L}^{2} \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]$

Statement (II) : Dimensions of gas constant is $\left[\mathrm{M} \mathrm{L}^{2} \mathrm{~T}^{-1} \mathrm{~K}^{-1}\right]$

A body projected vertically upwards with a certain speed from the top of a tower reaches the ground in $t_{1}$. If it is projected vertically downwards from the same point with the same speed, it reaches the ground in $t_{2}$. Time required to reach the ground, if it is dropped from the top of the tower, is :

In Franck-Hertz experiment, the first dip in the current-voltage graph for hydrogen is observed at 10.2 V. The wavelength of

light emitted by hydrogen atom when excited to the first excitation level is $\qquad$ nm .

(Given hc $=1245 \mathrm{eV} \mathrm{nm}, \mathrm{e}=1.6 \times 10^{-19} \mathrm{C}$ ).

For a given series LCR circuit it is found that maximum current is drawn when value of variable capacitance is 2.5 nF . If resistance of $200 \Omega$ and 100 mH inductor is being used in the given circuit. The frequency of ac source is $\qquad$ $\times 10^{3} \mathrm{~Hz}$ (given $\pi^{2}=10$ )

A particle moves in a straight line so that its displacement $x$ at any time $t$ is given by $x^{2}=1+t^{2}$. Its acceleration at any time t is $\mathrm{x}^{-\mathrm{n}}$ where $\mathrm{n}=$ $\qquad$ .

Three balls of masses $2 \mathrm{~kg}, 4 \mathrm{~kg}$ and 6 kg respectively are arranged at centre of the edges of an equilateral triangle of side 2 m . The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be $\qquad$ $\mathrm{kg} \mathrm{m}^{2}$.

A coil having 100 turns, area of $5 \times 10^{-3} \mathrm{~m}^{2}$, carrying current of 1 mA is placed in uniform magnetic field of 0.20 T such a way that plane of coil is perpendicular to the magnetic field. The work done in turning the coil through $90^{\circ}$ is $\qquad$ $\mu \mathrm{J}$.

In the given figure an ammeter A consists of a $240 \Omega$ coil connected in parallel to a $10 \Omega$ shunt. The reading of the ammeter is $\qquad$ mA .

A wire of cross sectional area A , modulus of elasticity $2 \times 10^{11} \mathrm{Nm}^{-2}$ and length 2 m is stretched between two vertical rigid supports. When a mass of 2 kg is suspended at the middle it sags lower from its original position making angle $\theta=\frac{1}{100}$ radian on the points of support. The value of $A$ is $\qquad$ $\times 10^{-4} \mathrm{~m}^{2}($ consider $\mathrm{x}<<\mathrm{L})$. (given : $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ )

Two coherent monochromatic light beams of intensities $I$ and $4I$ are superimposed. The difference between maximum and minimum possible intensities in the resulting beam is $xI$ . The value of $x$ is $\qquad$ .

Two open organ pipes of length 60 cm and 90 cm resonate at $6^{\text {th }}$ and $5^{\text {th }}$ harmonics respectively. The difference of frequencies for the given modes is $\qquad$ Hz. (Velocity of sound in air $=333 \mathrm{~m} / \mathrm{s}$ )

The correct arrangement for decreasing order of electrophilic substitution for above compounds

Molality $(m)$ of $3\ \mathrm{M}$ aqueous solution of $\mathrm{NaCl}$ is $\_\_\_\_\_$.

[Given: Density of solution $=1.25\ \mathrm{g\ mL^{-1}}$, molar mass of $\mathrm{NaCl}=58.5\ \mathrm{g\ mol^{-1}}$]

The incorrect statements regarding enzymes are:

(A) Enzymes are biocatalysts.

(B) Enzymes are non-specific and can catalyse different kinds of reactions.

(C) Most Enzymes are globular proteins.

(D) Enzyme - oxidase catalyses the hydrolysis of maltose into glucose.

Choose the correct answer from the options given below:

How can an electrochemical cell be converted into an electrolytic cell?

Arrange the following elements in the increasing order of number of unpaired electrons in it.

(A) Sc

(B) Cr

(C) V

(D) Ti

(E) Mn

Choose the correct answer from the options given below:

Match List-I with List-II.

Choose the correct answer from the options given below:

The major products formed:

A and B respectively are:

The incorrect statement regarding the geometrical isomers of 2-butene is:

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

Statement I: $\mathrm{PF}_{5}$ and $\mathrm{BrF}_{5}$ both exhibit $\mathrm{sp}^{3} \mathrm{~d}$ hybridisation.

Statement II: Both $\mathrm{SF}_{6}$ and $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$ exhibit $\mathrm{sp}^{3} \mathrm{~d}^{2}$ hybridisation.

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

The number of ions from the following that are expected to behave as oxidising agent is:

$\mathrm{Sn}^{4+}, \mathrm{Sn}^{2+}, \mathrm{Pb}^{2+}, \mathrm{Tl}^{3+}, \mathrm{Pb}^{4+}, \mathrm{Tl}^{+}$

Identify the product (A) in the following reaction.

The correct statements among the following, for a "chromatography" purification method is:

Evaluate the following statements related to group 14 elements for their correctness.

(A) Covalent radius decreases down the group from C to Pb in a regular manner.

(B) Electronegativity decreases from C to Pb down the group gradually.

(C) Maximum covalence of C is 4 whereas other elements can expand their covalence due to presence of d orbitals.

(D) Heavier elements do not form $p \pi-p \pi$ bonds.

(E) Carbon can exhibit negative oxidation states.

Choose the correct answer from the options given below:

Match List–I (Reaction) with List–II (Type of redox reaction).

Choose the correct answer from the options given below:

The correct IUPAC name of $\left[\mathrm{PtBr}_{2}\left(\mathrm{PMe}_{3}\right)_{2}\right]$ is:

Match List-I with List-II

Choose the correct answer from the option given below:

The ratio $\frac{K_{P}}{K_{C}}$ for the reaction: $\mathrm{CO}_{(\mathrm{g})}+\frac{1}{2} \mathrm{O}_{2(\mathrm{~g})} \rightleftharpoons \mathrm{CO}_{2(\mathrm{~g})}$ is:

An amine ( X ) is prepared by ammonolysis of benzyl chloride. On adding p-toluenesulphonyl chloride to it the solution remains

clear. Molar mass of the amine $(\mathrm{X})$ formed is $\qquad$ $\mathrm{g} \mathrm{mol}^{-1}$.

(Given molar mass in $\mathrm{gmol}^{-1} \mathrm{C}: 12, \mathrm{H}: 1, \mathrm{O}: 16, \mathrm{~N}: 14$ )

Consider the following reactions

The number of protons that do not involve in hydrogen bonding in the product B is $\_\_\_\_\_\_\_\_\_\_\_\_$ .

When ' x ' $\times 10^{-2} \mathrm{~mL}$ methanol (molar mass $=32 \mathrm{~g}$; density $=0.792 \mathrm{~g} / \mathrm{cm}^{3}$ ) is added to 100 mL water $\left(\right.$ density $\left.=1 \mathrm{~g} / \mathrm{cm}^{3}\right)$, the following diagram is obtained.

[Given: Molal freezing point depression constant of water at $273.15 \mathrm{~K}^{-1}$ is $1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$ ]

The ratio of number of oxygen atoms to bromine atoms in the product $Q$ is $\qquad$ $\times 10^{-1}$.

Number of carbocation from the following that are not stabilized by hyperconjugation is.

For the reaction at $298 \mathrm{~K}, 2 \mathrm{~A}+\mathrm{B} \rightarrow \mathrm{C} . \Delta \mathrm{H}$ $=400 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $\Delta \mathrm{S}=0.2 \mathrm{~kJ} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. The reaction will become spontaneous above $\_\_\_\_\_\_\_\_\_$ K

Total number of species from the following with central atom utilising $s \mathrm{p}^{2}$ hybrid orbitals for bonding is…

$\qquad$ $\mathrm{NH}_{3}, \mathrm{SO}_{2}, \mathrm{SiO}_{2}, \mathrm{BeCl}_{2}, \mathrm{C}_{2} \mathrm{H}_{2}, \mathrm{C}_{2} \mathrm{H}_{4}, \mathrm{BCl}_{3}, \mathrm{HCHO}$, $\mathrm{C}_{6} \mathrm{H}_{6}, \mathrm{BF}_{3}, \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Cl}_{2}$

Consider the two different first order reactions given below

$\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}$ (Reaction 1$)$

$\mathrm{P} \rightarrow \mathrm{Q}$ (Reaction 2)

The ratio of the half life of Reaction 1: Reaction 2 is $5: 2$. If $t_{1}$ and $t_{2}$ represent the time taken to complete $2 / 3^{\text {rd }}$ and $4 / 5$ of Reaction 1 and Reaction 2, respectively, then the value of the ratio $\mathrm{t}_{1}: \mathrm{t}_{2}$ is $\qquad$ $\times 10^{-1}$ (nearest integer). [Given: $\log _{10}(3)=0.477$ and $\log _{10}(5)=0.699$ ]

For hydrogen atom, energy of an electron in first excited state is $-3.4\ \mathrm{eV}$. K.E. of the same electron of hydrogen atom is $x\ \mathrm{eV}$. Value of $x$ is \_\_\_\_\_$$\times 10^{-1}\ \mathrm{eV} (nearest integer).

Among $\mathrm{VO}_{2}^{+}, \mathrm{MnO}_{4}^{-}$and $\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}$, the spin-only magnetic moment value of the species with least oxidising ability is.

BM (Nearest integer). (Given atomic member $\mathrm{V}=23, \mathrm{Mn}=25, \mathrm{Cr}=24$ )