JEE Main 2024 — 27 January, Shift 2

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan ^{-1}(x)+\tan ^{-1}(2 x)=\frac{\pi}{4}$ is :

Consider the function $f:(0,2) \rightarrow R$ defined by $f(x)=\frac{x}{2}+\frac{2}{x}$ and the function $g(x)$ defined by $g(x)=\left\{\begin{array}{cc}\min \{\mathrm{f}(\mathrm{t})\}, & 0<\mathrm{t} \leq \mathrm{x}\; \text { and\; } 0<\mathrm{x} \leq 1\\ \frac{3}{2}+\mathrm{x}, & 1<\mathrm{x}<2\end{array}\right.$. Then

Let the image of the point $(1,0,7)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ be the point $(\alpha, \beta, \gamma)$. Then which one of the following points lies on the line passing through $(\alpha, \beta, \gamma)$ and making angles $\frac{2 \pi}{3}$ and $\frac{3 \pi}{4}$ with $y$-axis and $z$-axis respectively and an acute angle with x -axis?

Let R be the interior region between the lines $3 x-y+1=0$ and $x+2 y-5=0$ containing the origin. The set of all values of $a$, for which the points $\left(a^{2}, a+1\right)$ lie in $R$, is:

The $20^{\text {th }}$ term from the end of the progression $20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4}$ is :-

Let $f: R-\left\{\frac{-1}{2}\right\} \rightarrow R$ and $g: R-\left\{\frac{-5}{2}\right\} \rightarrow R$ be defined as $f(x)=\frac{2 x+3}{2 x+1}$ and $g(x)=\frac{|x|+1}{2 x+5}$. Then the domain of the function fog is :

For $0<a<1$, the value of the integral $\int _{0}^{\pi }\frac{dx}{1-2a\text{cos}x+{{a}^{2}}}$ is :

Let $g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$ and $f^{\prime \prime}(x)>0$ for all $x \in(0,3)$. If $g$ is decreasing in $(0, \alpha)$ and increasing in $(\alpha, 3)$, then $8 \alpha$ is

If $\lim _{x \rightarrow 0} \frac{3+\alpha \sin x+\beta \cos x+\log _{e}(1-x)}{3 \tan ^{2} x}=\frac{1}{3}$, then $2 \alpha-\beta$ is equal to :

If $\alpha, \beta$ are the roots of the equation, $x^{2}-x-1=0$ and $S_{n}=2023 \alpha^{n}+2024 \beta^{n}$, then

Let A and B be two finite sets with m and n elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $\mathrm{P}(\mathrm{m}, \mathrm{n})$ from the point $\mathrm{Q}(-2,-3)$ is

The values of $\alpha$, for which $\left| \begin{array}{l}1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{3}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha + \frac{3}{2}\\1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha + \frac{1}{3}\\2\alpha + 3\,\,\,\,\,\,\,\,\,\,3\alpha + 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\end{array} \right| = 0$ lie in the interval

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is

The integral $\int \frac{\left(x^{8}-x^{2}\right) d x}{\left(x^{12}+3 x^{6}+1\right) \tan ^{-1}\left(x^{3}+\frac{1}{x^{3}}\right)}$ is equal to :

If $2 \tan ^{2} \theta-5 \sec \theta=1$ has exactly 7 solutions in the interval $\left[0, \frac{n \pi}{2}\right]$, for the least value of $n \in N$ then $\sum\limits_{k = 1}^n {\frac{k}{{{2^k}}}}$ is equal to :

The position vectors of the vertices $A, B$ and $C$ of a triangle are $2 \hat{i}-3 \hat{j}+3 \hat{k}, \quad 2 \hat{i}+2 \hat{j}+3 \hat{k}$ and $-\hat{i}+\hat{j}+3 \hat{k}$ respectively. Let $l$ denotes the length of the angle bisector AD of $\angle \mathrm{BAC}$ where D is on the line segment BC , then $2 l^{2}$ equals :

If $y=y(x)$ is the solution curve of the differential equation $\left(x^{2}-4\right) d y-\left(y^{2}-3 y\right) d x=0$, $x>2, y(4)=\frac{3}{2}$ and the slope of the curve is never zero, then the value of $y(10)$ equals :

Let $e_{1}$ be the eccentricity of the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$ and $e_{2}$ be the eccentricity of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b$, which passes through the foci of the hyperbola. If $\mathrm{e}_{1} \mathrm{e}_{2}=1$, then the length of the chord of the ellipse parallel to the x -axis and passing through $(0,2)$ is :

Let $\alpha=\frac{(4!)!}{(4!)^{3!}}$ and $\beta=\frac{(5!)!}{(5!)^{4!}}$. Then :

Let the position vectors of the vertices $\mathrm{A}, \mathrm{B}$ and C of a triangle be $2 \hat{i}+2 \hat{j}+\hat{k}, \quad \hat{i}+2 \hat{j}+2 \hat{k}$ and $2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ respectively. Let $l_{1}, l_{2}$ and $l_{3}$ be the lengths of perpendiculars drawn from the ortho center of the triangle on the sides $\mathrm{AB}, \mathrm{BC}$ and CA respectively, then $l_{1}^{2}+l_{2}^{2}+l_{3}^{2}$ equals :

The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12 . If $\mu$ and $\sigma^{2}$ denote the mean and variance of the correct observations respectively, then $15\left(\mu+\mu^{2}+\sigma^{2}\right)$ is equal to $\_\_\_\_\_\_\_$

If the area of the region $\left\{(x, y): 0 \leq y \leq \min \left\{2 x, 6 x-x^{2}\right\}\right\}$ is $A$, then $12 A$ is equal to....

Let A be a $2 \times 2$ real matrix and I be the identity matrix of order 2. If the roots of the equation $|A-x I|=0$ be -1 and 3 , then the sum of the diagonal elements of the matrix $A^{2}$ is.

If the sum of squares of all real values of $\alpha$, for which the lines $2 x-y+3=0,6 x+3 y+1=0$ and $\alpha x+2 y-2=0$ do not form a triangle is $p$, then the greatest integer less than or equal to p is $\qquad$

The coefficient of $x^{2012}$ in the expansion of $(1-x)^{2008}\left(1+x+x^{2}\right)^{2007}$ is equal to

If the solution curve, of the differential equation $\frac{d y}{d x}=\frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $\tan ^{-1}\left(\frac{y-1}{x-1}\right)-\frac{1}{\beta} \log _{e}\left(\alpha+\left(\frac{y-1}{x-1}\right)^{2}\right)=\log _{e}|x-1|$, then $5 \beta+\alpha$ is equal to

Let $f(x)=\int_{0}^{x} g(t) \log _{e}\left(\frac{1-t}{1+t}\right) d t$, where $g$ is a continuous odd function. If $\int_{-\pi / 2}^{\pi / 2}\left(f(x)+\frac{x^{2} \cos x}{1+e^{x}}\right) d x=\left(\frac{\pi}{\alpha}\right)^{2}-\alpha$, then $\alpha$ is equal to.

Consider a circle $(x-\alpha)^{2}+(y-\beta)^{2}=50$, where $\alpha, \beta>0$. If the circle touches the line $y+x=0$ at the point $P$, whose distance from the origin is $4 \sqrt{2}$ , then $(\alpha+\beta)^{2}$ is equal to.

The lines $\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}$ and $\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}$ intersect at the point $P$. If the distance of P from the line $\frac{\mathrm{x}+1}{2}=\frac{\mathrm{y}-1}{3}=\frac{\mathrm{z}-1}{1}$ is $l$, then $14 l^{2}$ is equal to.

Let the complex numbers $\alpha \,and\,\frac{1}{\alpha }$ lie o the circles ${{\left| z-{{z}_{0}} \right|}^{2}}=4$ and ${{\left| z-{{z}_{0}} \right|}^{2}}=16$ respectively, where ${{z}_{0}}=1+i$. Then the value of 100${{\left| \alpha \right|}^{2}}$ is….

The equation of state of a real gas is given by $\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^{2}}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}$, where $\mathrm{P}, \mathrm{V}$ and T are pressure. volume and temperature respectively and R is the universal gas constant. The dimensions of $\frac{a}{b^{2}}$ is similar to that of :

Wheatstone bridge principle is used to measure the specific resistance $\left(\mathrm{S}_{1}\right)$ of given wire, having length $L$, radius $r$. If $X$ is the resistance of wire, then specific resistance is : $\mathrm{S}_{1}=\mathrm{X}\left(\frac{\pi \mathrm{r}^{2}}{\mathrm{~L}}\right)$. If the length of the wire gets doubled then the value of specific resistance will be :

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

Assertion (A): The angular speed of the moon in its orbit about the earth is more than the angular speed of the earth in its orbit about the sun.

Reason (R): The moon takes less time to move around the earth than the time taken by the earth to move around the sun.

In the light of the above statements, choose the most appropriate 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: The limiting force of static friction depends on the area of contact and independent of materials.

Statement II: The limiting force of kinetic friction is independent of the area of contact and depends on materials.

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

A current of $200 \mu \mathrm{~A}$ deflects the coil of a moving coil galvanometer through $60^{\circ}$. The current to cause deflection through $\frac{\pi}{10}$ radian is :

The atomic mass of ${ }_{6} \mathrm{C}^{12}$ is 12.000000 u and that of ${ }_{6} \mathrm{C}^{13}$ is 13.003354 u . The required energy to remove a neutron from ${ }_{6} \mathrm{C}^{13}$, if mass of neutron is 1.008665 u , will be :

A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle $(\theta)$ of thread deflection in the extreme position will be :

Three voltmeters, all having different internal resistances are joined as shown in figure. When some potential difference is applied across A and B , their readings are $\mathrm{V}_{1}, \mathrm{~V}_{2}$ and $\mathrm{V}_{3}$. Choose the correct option.

The total kinetic energy of 1 mole of oxygen at $27^{\circ} \mathrm{C}$ is : [Use universal gas constant $(\mathrm{R})=8.31 \mathrm{~J} / \mathrm{mole} \mathrm{K]}$

Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R)

Assertion (A): In Vernier calliper if positive zero error exists, then while taking measurements, the reading taken will be more than the actual reading.

Reason (R): The zero error in Vernier Calliper might have happened due to manufacturing defect or due to rough handling.

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

Primary side of a transformer is connected to $230 \mathrm{~V}, 50 \mathrm{~Hz}$ supply. Turns ratio of primary to secondary winding is $10: 1$. Load resistance connected to secondary side is $46 \Omega$. The power consumed in it is :

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of $\frac{C_{p}}{C_{v}}$ for the gas is :

The threshold frequency of a metal with work function 6.63 eV is :

Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R)

Assertion (A): The property of body, by virtue of which it tends to regain its original shape when the external force is removed, is Elasticity.

Reason (R): The restoring force depends upon the bonded inter atomic and inter molecular force of solid.

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

When a polaroid sheet is rotated between two crossed polaroids then the transmitted light intensity will be maximum for a rotation of :

An object is placed in a medium of refractive index 3. An electromagnetic wave of intensity $6 \times 10^{8} \mathrm{~W} / \mathrm{m}^{2}$ falls normally on the object and it is absorbed completely. The radiation pressure on the object would be (speed of light in free space $\left.=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)$ :

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

Assertion (A): Work done by electric field on moving a positive charge on an equipotential surface is always zero.

Reason (R): Electric lines of forces are always perpendicular to equipotential surfaces.

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

A heavy iron bar of weight 12 kg is having its one end on the ground and the other on the shoulder of a man. The rod makes an angle $60^{\circ}$ with the horizontal, the weight experienced by the man is :

A bullet is fired into a fixed target looses one third of its velocity after travelling 4 cm . It penetrates further $\mathrm{D} \times 10^{-3} \mathrm{~m}$ before coming to rest. The value of $D$ is :

The magnetic field at the centre of a wire loop formed by two semicircular wires of radii $\mathrm{R}_{1}=2 \pi \mathrm{~m}$ and $R_{2}=4 \pi \mathrm{~m}$ carrying current $\mathrm{I}=4 \mathrm{~A}$ as per figure given below is $\alpha \times 10^{-7} \mathrm{~T}$. The value of $\alpha$ is (Centre O is common for all segments)

Two charges of $-4 \mu \mathrm{C}$ and $+4 \mu \mathrm{C}$ are placed at the points $\mathrm{A}(1,0,4) \mathrm{m}$ and $\mathrm{B}(2,-1,5) \mathrm{m}$ located in an electric field $\vec{E}=0.20 \hat{\mathrm{i}} \mathrm{V} / \mathrm{cm}$. The magnitude of the torque acting on the dipole is $8 \sqrt{\alpha} \times 10^{-5} \mathrm{Nm}$, Where $\alpha=$ $\_\_\_\_$ .

A closed organ pipe 150 cm long gives 7 beats per second with an open organ pipe of length 350 cm , both vibrating in fundamental mode. The velocity of sound is $\_\_\_\_\_\_\_\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

A body falling under gravity covers two points A and B separated by 80 m in 2 s . The distance of upper point A from the starting point is $m (use x${\rm{g}} = 10{\rm{;m}}{{\rm{s}}^{ - 2}}$ )

The reading of pressure metre attached with a closed pipe is $4.5 \times 10^{4} \mathrm{~N} / \mathrm{m}^{2}$. On opening the valve, water starts flowing and the reading of pressure metre falls to $2.0 \times 10^{4} \mathrm{~N} / \mathrm{m}^{2}$. The velocity of water is found to be $\sqrt{\mathrm{V}} \mathrm{m} / \mathrm{s}$. The value of V is $\qquad$

A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of both bodies are identical and the ratio of their kinetic energies is $\frac{7}{x}$ where $x$ is

A parallel beam of monochromatic light of wavelength 5000\ is incident normally on a single narrow slit of width 0.001 mm . The light is focused by convex lens on screen, placed on its focal plane. The first minima will be formed for the angle of diffraction of (degree).

The electric potential at the surface of an atomic nucleus $\left(\mathrm{z}=50\right.$ ) of radius $9 \times {10^{ - 13}}{\rm{\;cm}}$ is ${\rm{\;}}$ $\times {10^{6{\rm{\;V}}}}$.

If Rydberg's constant is $R$, the longest wavelength of radiation in Paschen series will be $\frac{\alpha}{7 R}$, where $\alpha=$ $\_\_\_\_\_\_\_$ .

A series $L C R$ circuit with $L=\frac{100}{\pi} \mathrm{mH}, \mathrm{C}=\frac{10^{-3}}{\pi} \mathrm{~F}$ and $\mathrm{R}=10 \Omega$, is connected across an ac source of $220 \mathrm{~V}, 50 \mathrm{~Hz}$ supply. The power factor of the circuit would be $\_\_\_\_\_\_\_$ .

Which of the following statements is not correct about rusting of iron?

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

Statement (I) : In the Lanthanoids, the formation of $\mathrm{Ce}^{4+}$ is favoured by its noble gas configuration.

Statement (II) : $\mathrm{Ce}^{4+}$ is a strong oxidant reverting to the common +3 state.

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

Choose the correct option having all the elements with $\mathrm{d}^{10}$ electronic configuration from the following:

Phenolic group can be identified by a positive:

The molecular formula of second homologue in the homologous series of mono carboxylic acids is $\qquad$

The technique used for purification of steam volatile water immiscible substance is:

The final product A , formed in the following reaction sequence is:

Match the List-I with List-II and choose the correct answer from the options given below:

List I (Reaction)	List II (Reagent(s))
(A) $\rightarrow$	(I) $\mathrm{Na}_2 \mathrm{Cr}_2 \mathrm{O}_7, \mathrm{H}_2 \mathrm{SO}_4$
(B) $\rightarrow$	(II) (i) NaOH (ii) $\mathrm{CH}_3 \mathrm{Cl}$
(C) $\rightarrow$	(III) (i) $\mathrm{NaOH}, \mathrm{CHCl}_3$ (ii) NaOH (iii) HCl
(D) $\rightarrow$	(IV) (i) NaOH (ii) $\mathrm{CO}_2$ (iii) HCl

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

Statement (I) : Oxygen being the first member of group 16 exhibits only -2 oxidation state.

Statement (II) : Down the group 16 stability of +4 oxidation state decreases and +6 oxidation state increases.

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

Identify from the following species in which $\mathrm{d}^{2} \mathrm{sp}^{3}$ hybridization is shown by central atom:

The quantity which changes with temperature is:

Which structure of protein remains intact after coagulation of egg white on boiling?

Which of the following cannot function as an oxidising agent?

The incorrect statement regarding conformations of ethane is:

Identity the incorrect pair from the following:

Total number of ions from the following with noble gas configuration is $\qquad$ .

$\mathrm{Sr}^{2+}(\mathrm{Z}=38), \mathrm{Cs}^{+}(\mathrm{Z}=55), \mathrm{La}^{2+}(\mathrm{Z}=57) \mathrm{Pb}^{2+}$ $(\mathrm{Z}=82), \mathrm{Yb}^{2+}(\mathrm{Z}=70)$ and $\mathrm{Fe}^{2+}(\mathrm{Z}=26)$

Time required for completion of $99.9 \%$ of a First order reaction is $\qquad$ times of half life $\left(\mathrm{t}_{1 / 2}\right)$ of the reaction.

The number of non-polar molecules from the following is $\qquad$

$\mathrm{HF}, \mathrm{H}_{2} \mathrm{O}, \mathrm{SO}_{2}, \mathrm{H}_{2}, \mathrm{CO}_{2}, \mathrm{CH}_{4}, \mathrm{NH}_{3}, \mathrm{HCl}, \mathrm{CHCl}_{3}, \mathrm{BF}_{3}$

The Spin only magnetic moment value of square planar complex $\left[\mathrm{Pt}\left(\mathrm{NH}_{3}\right)_{2} \mathrm{Cl}\left(\mathrm{NH}_{2} \mathrm{CH}_{3}\right)\right] \mathrm{Cl}$ is B.M.

(Nearest integer) (Given atomic number for $\mathrm{Pt}=78$ )

For a certain thermochemical reaction $M \rightarrow N$ at $\mathrm{T}=400 \mathrm{~K}, \Delta \mathrm{H}^{\ominus}=77.2 \mathrm{~kJ} \mathrm{~mol}^{-1}, \Delta \mathrm{~S}=122 \mathrm{JK}^{-1}$,

$\log$ equilibrium constant $(\log K)$ is - $\qquad$ $\times 10^{-1}$

Volume of $3\ \mathrm{M}$ $\mathrm{NaOH}$ (formula weight $40\ \mathrm{g\ mol^{-1}}$) which can be prepared from $84\ \mathrm{g}$ of $\mathrm{NaOH}$ is $\_\_\_ \times 10^{-1}\ \mathrm{dm^3}$.

One mole of $\mathrm{PbS}$ is oxidised by $X$ moles of $\mathrm{O_3}$ to form $Y$ moles of $\mathrm{O_2}$. Find the value of $X+Y$.

9.3 g of aniline is subjected to reaction with excess of acetic anhydride to prepare acetanilide. The mass of acetanilide produced if the reaction is 100% completed is $\_\_\_\_\_$ $\times 10^{-1}$ g. (Given molar mass in g mol$^{-1}$ $\mathrm{N}$: $14$, $\mathrm{O}$: $16$, $\mathrm{C}$: $12$, $\mathrm{H}$: $1$)