JEE Advanced 2024 — Paper 1

Let $f(x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f(1)=2$ and $\lim _{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^{9}-x^{9}}=1$ for each $x>0$. Then, for all $x>0, f(x)$ is equal to

A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guess it, is $\frac{1}{2}$. Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is $\frac{1}{6}$. Then the probability that the student knows the answer of a randomly chosen question is

Let $\frac{\pi}{2}<x<\pi$ be such that $\cot x=\frac{-5}{\sqrt{11}}$. Then $\left(\sin \frac{11 x}{2}\right)(\sin 6 x-\cos 6 x)+\left(\cos \frac{11 x}{2}\right)(\sin 6 x+\cos 6 x)$ is equal to

Consider the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$. Let $S(p, q)$ be a point in the first quadrant such that $\frac{p^{2}}{9}+\frac{q^{2}}{4}>1$. Two tangents are drawn from $S$ to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point T in the fourth quadrant. Let R be the vertex of the ellipse with positive $x$-coordinate and $O$ be the centre of the ellipse. If the area of the triangle $\triangle \mathrm{ORT}$ is $\frac{3}{2}$, then which of the following options is correct ?

Let $S=\{a+b \sqrt{2}: a, b \in Z\}, T_{1}=\left\{(-1+\sqrt{2})^{n}: n \in Z\right\}$, and $T_{2}=\left\{(1+\sqrt{2})^{n}: n \in N\right\}$. Then which of the following statements is(are) TRUE ?

Let $R^{2}$ denote $R \times R$. Let $S=\left\{(a, b, c): a, b, c \in R\right.$ and $x^{2}+2 b x y+y^{2}>0$ for all $\left.(x, y) \in R^{2}-\{(0,0)\}\right\}$ Then which of the following statements is (are) TRUE?

Let $R^{3}$ denote the three dimensional space. Take two points $P=(1,2,3)$ and $Q=(4,2,7)$. Let $\operatorname{dist}(X, Y)$

denote the distance between two points $X$ and $Y$ in $R^{3}$

Let $S=\left\{X \in R^{3}:(\operatorname{dist}(X, P))^{2}-(\operatorname{dist}(X, Q))^{2}=50\right\}$ and $T=\left\{Y \in R^{3}:(\operatorname{dist}(Y, Q))^{2}-(\operatorname{dist}(Y, P))^{2}=50\right\}$.

Then which of the following statements is(are) TRUE ?

Let $a=3 \sqrt{2}$ and $b=\frac{1}{5^{1 / 6} \sqrt{6}}$. If $x, y \in R$ are such that $3 x+2 y=\log _{a}(18)^{5 / 4} \text { and } 2 x-y=\log _{b}(\sqrt{1080})$ then $4 x+5 y$ is equal to

Let $f(x)=x^{4}+a x^{3}+b x^{2}+c$ be a polynomial with real coefficients such that $f(1)=-9$.

Suppose that $i \sqrt{3}$ is a root of the equation $4 x^{3}+3 a x^{2}+2 b x=0$, where $i=\sqrt{-1}$.

If $\alpha_{1}, \alpha_{2}, \alpha_{3}$, and $\alpha_{4}$ are all the roots of the equation $f(x)=0$, then $\left|\alpha_{1}\right|^{2}+\left|\alpha_{2}\right|^{2}+\left|\alpha_{3}\right|^{2}+\left|\alpha_{4}\right|^{2}$ is equal to $\qquad$

Let $S=\left\{A\left(\begin{array}{lll}0 & 1 & c\\ 1 & a & d\\ 1 & b & e\end{array}\right): a, b, c, d, e \in\{0,1\}\right.$ and $\left.|A| \in\{-1,1\}\right\}$, where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is $\_\_\_\_\_$ .

A group of 9 students $s_{1}, s_{2}, \ldots \ldots . s_{9}$ is to be divided to form three teams $X, Y$ and $Z$ of sizes 2, 3 and 4 respectively. Suppose that $\mathrm{s}_{1}$ cannot be selected for the team X , and $\mathrm{s}_{2}$ cannot be selected for team Y . Then the number of ways to form such teams, is $\_\_\_\_\_$ .

Let $\overrightarrow{\mathrm{OP}}=\frac{\alpha-1}{\alpha} \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{OQ}}=\hat{\mathrm{i}}+\frac{\beta-1}{\beta} \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{OR}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\frac{1}{2} \hat{\mathrm{k}}$ be three vectors, where $\alpha, \beta \in$ $\mathrm{R}-\{0\}$ and O denotes the origin. If $(\overrightarrow{\mathrm{OP}} \times \overrightarrow{\mathrm{OQ}}) \cdot \overrightarrow{\mathrm{OR}}=0$ and the point $(\alpha, \beta, 2)$ lies on the plane $3 \mathrm{x}+3 \mathrm{y}-\mathrm{z}+l=0$, then the value of $l$ is $\_\_\_\_\_$ .

Let $X$ be a random variable, and let $P(X=x)$ denote the probability that $X$ takes the values $x$. Suppose that the points $(x, P(X=x)), x=0,1,2,3,4$, lie on a fixed straight line in the $x y$-plane, and $P(X=x)=0$ for all $X \in R-\{0,1,2,3,4\}$. If the mean of $X$ is $\frac{5}{2}$, and the variance of $X$ is $\alpha$, then the value of $24 \alpha$ is $\_\_\_\_\_$ .

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^{2}+x-1=0$. Consider the set $T=\{1, \alpha, \beta\}$. For a $3 \times 3$ matrix $M=\left(a_{i j}\right)_{3 \times 3}$, define $R_{i}=a_{i 1}+a_{i 2}+a_{i 3}$ and $C_{j}=a_{1 j}+a_{2 j}+a_{3 j}$ for $i=1,2$, 3 and $\mathrm{j}=1,2,3$.

Match each entry in List-I to the correct entries in List-II.

Let the straight line $y=2 x$ touch a circle with centre $(0, \alpha), \alpha>0$, and radius $r$ at a point $A_{1}$. Let $B_{1}$ be the point on the circle such the line segment $A_{1} B_{1}$ is a diameter of the circle. Let $\alpha+r=5+\sqrt{5}$.

Match each entry in List-I to the correct entries in List-II.

The correct option is:

Let $\gamma \in R$ be such that the lines $L_{1}: \frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}$ and $L_{2}: \frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}$ intersect. Let $R_{1}$ be the point of intersection of $L_{1}$ and $L_{2}$. Let $O=(0,0,0)$, and $\hat{n}$ denote a unit normal vector to the plane containing both the lines $L_{1}$ and $L_{2}$.

Match each entry in List-I to the correct entries in List-II.

The correct option is:

Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be functions defined by

Let $a, b, c, d \in R$. Define the function $h: R \rightarrow R$ by

Match each entry in List-I to the correct entries in List-II.

The correct option is:

A dimensionless quantity is constructed in terms of electronic charge e, permittivity of free space $\varepsilon_{0}$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^{\alpha} \varepsilon_{0}^{\beta} h^{\gamma} c^{\delta}$ and n is a non-zero integer, then $(\alpha, \beta, \gamma, \delta)$ is given by

An infinitely long wire, located on the z-axis, carries a current I along the +z-direction and produces the magnetic field $\overrightarrow{\mathrm{B}}$. The magnitude of the line integral $\int \overrightarrow{\mathrm{B}} . \overrightarrow{\mathrm{d} l}$ along a straight line from the point $(-\sqrt{3} a, a, 0)$ to $(a, a, 0)$ is given by[0pt] [ $\mu_{0}$ is the magnetic permeability of free space.]

Two beads, each with charge $q$ and mass $m$, are on a horizontal, frictionless, non-conducting, circular hoop of radius $R$. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by [ $\varepsilon_{0}$ is the permittivity of free space.]

A block of mass 5 kg moves along the $x$-direction subject to the force $F=(-20 x+10) \mathrm{N}$, with the value of $x$ in metre. At time $t=0 \mathrm{~s}$, it is at rest at position $x=1 \mathrm{~m}$. The position and momentum of the block at $t=(\pi / 4) \mathrm{s}$ are

A particle of mass $m$ is moving in a circular orbit under the influence of the central force $F(r)=-k r$, corresponding to the potential energy $V(r)=k r^{2} / 2$, where $k$ is a positive force constant and $r$ is the radial distance from the origin. According to the Bohr's quantization rule, the angular

momentum of the particle is given by $\mathrm{L}=\mathrm{n} \hbar$, where $\hbar=\mathrm{h} /(2 \pi)$, h is the Planck's constant, and n a positive

integer. If $v$ and $E$ are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?

Two uniform strings of mass per unit length $\mu$ and $4 \mu$, and length $L$ and $2 L$, respectively, are joined at point $O$, and tied at two fixed ends $P$ and $Q$, as shown in the figure. The strings are under a uniform tension $T$. If we define the frequency $v_{0}=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}$,

which of the following statement(s) is(are) correct?

A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is 9 cm , while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index $n$ up to the level QPR. If the image of a point object O at a height of h (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct?

The specific heat capacity of a substance is temperature dependent and is given by the formula C $=\mathrm{kT}$, where k is a constant of suitable dimensions in SI units,and T is the absolute temperature. If the heat required to raise the temperature of 1 kg of the substance from $-73^{\circ} \mathrm{C}$ to $27^{\circ} \mathrm{C}$ is nk , the value of $n$ is $\_\_\_\_\_$ . [Given: $0 \mathrm{~K}=-273^{\circ} \mathrm{C}$ ]

A disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. $A$ battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass $M$ and radius $R / 2$ is fixed to the motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed $\omega$. If the angular speed at which the large disc rotates is $\omega / n$, then the value of $n$ is $\_\_\_\_\_$ .

A point source $S$ emits unpolarized light uniformly in all directions. At two points $A$ and $B$, the ratio $r=I_{A} / I_{B}$ of the intensities of light is 2 .If a set of two polaroids having $45^{\circ}$ angle between their pass-axes is placed just before point $B$, then the new value of $r$ will be $\qquad$

A source (S) of sound has frequency 240 Hz . When the observer ( O ) and the source move towards each other at a speed $v$ with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be 288 Hz . However, when the observer and the source move away from each other at the same speed v with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be nHz .

The value of n is $\qquad$ .

Two large, identical water tanks, 1 and 2, kept on the top of a building of height H , are filled with water up to height $h$ in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2, and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are $t_{1}$ and $t_{2}$, respectively. If $H=\left(\frac{16}{9}\right) \mathrm{h}$, then the ratio $t_{1} / t_{2}$ is $\qquad$

A thin uniform rod of length $L$ and certain mass is kept on a frictionless horizontal table with a massless string of length $L$ fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point $O$. If a horizontal impulse $P$ is imparted to the rod at a distance $x=L / n$ from the mid-point of the rod (see figure), then the rod and string revolve together around the point $O$, with the rod remaining aligned with the string. In such a case, the value of $n$ is $\qquad$ .

One mole of a monatomic ideal gas undergoes the cyclic process $\mathrm{J} \rightarrow \mathrm{K} \rightarrow \mathrm{L} \rightarrow \mathrm{M} \rightarrow \mathrm{J}$, as shown in the P-T diagram. Match the quantities mentioned in List-I with their values in List-II and choose the correct option. [ $R$ is the gas constant.]

A light ray is incident on the surface of a sphere of refractive index n at an angle of incidence $\theta_{0}$. The ray partially refracts into the sphere with angle of refraction $\phi_{0}$ and then partly reflects from the back surface. The reflected ray then emerges out of the sphere after a partial refraction. The total angle of deviation of the emergent ray with respect to the incident ray is $\alpha$. Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

The circuit shown in the figure contains an inductor $L$, a capacitor $C_{0}$, a resistor $R_{0}$ and an ideal battery. The circuit also contains two keys $\mathrm{K}_{1}$ and $\mathrm{K}_{2}$. Initially, both the keys are open and there is no charge on the capacitor. At an instant, key $\mathrm{K}_{1}$ is closed and immediately after this the current in $R_{0}$ is found to be $I_{1}$. After a long time, the current attains a steady state value $I_{2}$. Thereafter, $K_{2}$ is closed and simultaneously $\mathrm{K}_{1}$ is opened and the voltage across $\mathrm{C}_{0}$ oscillates with amplitude $\mathrm{V}_{0}$ and angular frequency $\omega_{0}$ Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

A closed vessel contains 10 g of an ideal gas $\mathbf{X}$ at 300 K , which exerts 2 atm pressure. At the same temperature, 80 g of another ideal gas $\mathbf{Y}$ is added to it and the pressure becomes 6 atm . The ratio of root mean square velocities of $X$ and $Y$ at 300 K is

At room temperature, disproportionation of aqueous in situ generated $\mathrm{HNO_2}$ gives the species:

Among the following, the correct statement(s) for electrons in an atom is(are)

The option(s) in which at least three molecules follow Octet Rule is(are)

Consider the following volume - temperature (V - T) diagram for the expansion of 5 moles of an ideal monoatomic gas. Consider only P - V work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence $\mathbf{X} \rightarrow \mathbf{Y} \rightarrow \mathbf{Z}$ is $\_\_\_\_\_$ -. [Use the given data: Molar heat capacity of the gas for the given temperature range, $\mathrm{C}_{\mathrm{v}, \mathrm{m}}=12 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and gas constant, $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ ]

Consider the following reaction,

Which follows the mechanism given below? $2 \mathrm{NO}(\mathrm{g}) \underset{\mathrm{K}_{-1}}{\stackrel{\mathrm{~K}_{1}}{\rightleftharpoons}} \mathrm{~N}_{2} \mathrm{O}_{2}(\mathrm{~g})$ (fast equilibrium) $\mathrm{N}_{2} \mathrm{O}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g}) \xrightarrow{\mathrm{k}_{2}} \mathrm{~N}_{2} \mathrm{O}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g})$ (slow reaction) $\mathrm{N}_{2} \mathrm{O}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{~g}) \xrightarrow{\mathrm{K}_{3}} \mathrm{~N}_{2}(\mathrm{~g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g})$ (fast reaction) The order of the reaction is $\_\_\_\_\_$ .

Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. NaOH solution, gives $\mathbf{P}$ and $\mathbf{Q}$. Compound $\mathbf{P}$ does not give Tollen's test, whereas $\mathbf{Q}$ on acidification gives positive Tollen's test. Treatment of $\mathbf{P}$ with excess cyclohexanone in the presence of catalytic amount of p-tolunesulfonic acid (PTSA) gives product $\mathbf{R}$. Sum of the number of methylene groups $\left(-\mathrm{CH}_{2}-\right)$ and oxygen atoms in $\mathbf{R}$ is $\qquad$ .

Among $\mathrm{V}(\mathrm{CO})_{6}, \mathrm{Cr}(\mathrm{CO})_{5}, \mathrm{Mn}(\mathrm{CO})_{5}, \mathrm{Fe}(\mathrm{CO})_{5},\left[\mathrm{Co}(\mathrm{CO})_{3}\right]^{3-},\left[\mathrm{Cr}(\mathrm{CO})_{4}\right]^{4-}$ and $\operatorname{Ir}(\mathrm{CO})_{3}$, the total number of species isoelectronic with $\mathrm{Ni}(\mathrm{CO})_{4}$ is $\_\_\_\_\_$ [Given, atomic number: $\mathrm{V}=23, \mathrm{Cr}=24, \mathrm{Mn}=2 \overline{5, \mathrm{Fe}=26, \mathrm{Co}=27}, \mathrm{Ni}=28, \mathrm{Cu}=29$, $\mathrm{Ir}=77$ ]

In the following reaction sequence, major product $\mathbf{P}$ is formed. Glycerol reacts completely with excess $\mathbf{P}$ in the presence of an acid catalyst to form $\mathbf{Q}$. Reaction of $\mathbf{Q}$ with excess NaOH followed by the treatment with $\mathrm{CaCl}_{2}$ yields Ca - soap $\mathbf{R}$, quantitatively. Starting with one mole of $\mathbf{Q}$, the amount of $\mathbf{R}$ produced in gram is $\_\_\_\_\_$ [Given, atomic weight: $\mathrm{H}=1, \mathrm{C}=12, \mathrm{~N}=14, \mathrm{O}=16, \mathrm{Na}=23, \mathrm{Cl}=35, \mathrm{Ca}=40$ ]

Among the following complexes, the total number of diamagnetic species is $\_\_\_\_\_$ . $\left[\mathrm{Mn}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+},\left[\mathrm{MnCl}_{6}\right]^{3-},\left[\mathrm{FeF}_{6}\right]^{3-},\left[\mathrm{CoF}_{6}\right]^{3-},\left[\mathrm{Fe}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$ and $\left[\mathrm{Co}(\mathrm{en})_{3}\right]^{3+}$ [Given, atomic number: $\mathrm{Mn}=25, \mathrm{Fe}=26, \mathrm{Co}=27$; en $=\mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{2}$ ]

Based on VSEPR model, match the xenon compounds given in List-I with the corresponding geometries and the number of lone pairs on xenon given in List-II and choose the correct option