JEE Advanced 26 May 2024 Paper 1 question paper with solutions

Question 1

Maths · Continuity and Differentiability · Single correct

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

1. $\frac{31}{11x}$ - $\frac{9}{11}$x^{10}
2. $\frac{9}{11x}$ + $\frac{13}{11}$x^{10}
3. -$\frac{9}{11x}$ + $\frac{31}{11}$x^{10}
4. $\frac{13}{11x}$ + $\frac{9}{11}$x^{10}

Answer: (b)

Question 2

Maths · Probability · Single correct

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 guessed 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

1. $\frac{1}{12}$
2. $\frac{1}{7}$
3. $\frac{5}{7}$
4. $\frac{5}{12}$

Answer: (c)

Question 3

Maths · Trigonometric Functions · Single correct

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

1. $\frac{\sqrt{11} - 1}{2\sqrt{3}}$
2. $\frac{\sqrt{11} + 1}{2\sqrt{3}}$
3. $\frac{\sqrt{11} + 1}{3\sqrt{2}}$
4. $\frac{\sqrt{11} - 1}{3\sqrt{2}}$

Answer: (b)

Question 4

Maths · Conic Sections · Single correct

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 center of the ellipse. If the area of the triangle $\triangle ORT$ is $\frac{3}{2}$, then which of the following options is correct?

1. $q = 2, p = 3\sqrt{3}$
2. $q = 2, p = 4\sqrt{3}$
3. $q = 1, p = 5\sqrt{3}$
4. $q = 1, p = 6\sqrt{3}$

Answer: (a)

Question 5

Maths · Complex Numbers and Quadratic Equations · Multiple correct

Let $S = \{ a + b \sqrt{2} : a, b \in \mathbb{Z} \}$, $T_1 = \{ (-1 + \sqrt{2})^n : n \in \mathbb{N} \}$ and $T_2 = \{ (1 + \sqrt{2})^n : n \in \mathbb{N} \}$. Then which of the following statements is (are) TRUE?

1. $\mathbb{Z} \cup T_1 \cup T_2 \subset S$
2. $T_1 \cap \left( 0, \frac{1}{2024} \right) = \phi$, where $\phi$ denotes the empty set
3. $T_2 \cap (2024, \infty) \neq \phi$
4. For any given $a, b \in \mathbb{Z}$, $\cos \left( \pi (a + b \sqrt{2}) \right) + i \sin \left( \pi (a + b \sqrt{2}) \right) \in \mathbb{Z}$ if and only if $b = 0$, where $i = \sqrt{-1}$

Answer: (a), (c), (d)

Question 6

Maths · Determinants · Multiple correct

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

1. $\left(2, \frac{7}{2}, 6\right) \in S$
2. If $\left(3, b, \frac{1}{12}\right) \in S$, then $|2b| < 1$
3. For any given $(a, b, c) \in S$, the system of linear equations $ax + by = 1$, $bx + cy = -1$ has a unique solution.
4. For any given $(a, b, c) \in S$, the system of linear equations $(a+1)x + by = 0$, $bx + (c+1)y = 0$ has a unique solution.

Answer: (b), (c), (d)

Question 7

Maths · Three Dimensional Geometry · Multiple correct

Let $\mathbb{R}^3$ denote the three-dimensional space. Take two points $P = (1, 2, 3)$ and $Q = (4, 2, 7)$. Let $dist(X, Y)$ denote the distance between two points $X$ and $Y$ in $\mathbb{R}^3$. Let $$S = \left\{ X \in \mathbb{R}^3 : (dist(X, P))^2 - (dist(X, Q))^2 = 50 \right\}$$ and $$T = \left\{ Y \in \mathbb{R}^3 : (dist(Y, Q))^2 - (dist(Y, P))^2 = 50 \right\}.$$ Then which of the following statements is (are) TRUE?

1. There is a triangle whose area is 1 and all of whose vertices are from $S$.
2. There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $LM$ is also in $T$.
3. There are infinitely many rectangles of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.
4. There is a square of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.

Answer: (a), (b), (c), (d)

Question 8

Maths · Determinants · Fill in the blank

Let $a = 3\sqrt{2}$ and $b = \frac{1}{5^{\frac{1}{6}} \sqrt{6}}$. If $x, y \in \mathbb{R}$ are such that $$3x + 2y = \log_a \left(18^{\frac{5}{4}}\right)$$ and $$2x - y = \log_b \left(\sqrt{1080}\right),$$ then $4x + 5y$ is equal to ________.

Answer: 8

Question 9

Maths · Complex Numbers and Quadratic Equations · Numerical

Let $f(x) = x^4 + ax^3 + bx^2 + c$ be a polynomial with real coefficients such that $f(1) = -9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^3 + 3ax^2 + 2bx = 0$, 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 $|\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2$ is equal to .

Answer: 20

Question 10

Maths · Determinants · Numerical

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

Answer: 16

Question 11

Maths · Permutations and Combinations · Numerical

A group of 9 students, $s_1, s_2, \ldots, s_9$, is to be divided to form three teams $X$, $Y$ and $Z$ of sizes 2, 3, and 4, respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is______.

Answer: 665

Question 12

Maths · Vector Algebra · Numerical

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

Answer: 5

Question 13

Maths · Probability (Advanced) · Numerical

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

Answer: 42

Question 14

Maths · Matrices · Single correct

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 $M = (a_{ij})_{3 \times 3}$, define $R_i = a_{i1} + a_{i2} + a_{i3}$ and $C_j = a_{1j} + a_{2j} + a_{3j}$ for $i = 1, 2, 3$ and $j = 1, 2, 3$. Match each entry in List-I to the correct entry in List-II.

1. → (4) (Q) → (2) (R) → (5) (S) → (1)
2. → (2) (Q) → (4) (R) → (1) (S) → (5)
3. → (2) (Q) → (4) (R) → (3) (S) → (5)
4. → (1) (Q) → (5) (R) → (3) (S) → (4)

Answer: (c)

Question 15

Maths · Conic Sections · Single correct

Let the straight line $y = 2x$ touch a circle with center $(0, \alpha)$, $\alpha > 0$, and radius $r$ at a point $A_1$. Let $B_1$ be the point on the circle such that the line segment $A_1B_1$ is a diameter of the circle. Let $\alpha + r = 5 + \sqrt{5}$. Match each entry in List-I to the correct entry in List-II.

1. → (4) (Q) → (2) (R) → (1) (S) → (3)
2. → (2) (Q) → (4) (R) → (1) (S) → (3)
3. → (4) (Q) → (2) (R) → (5) (S) → (3)
4. → (2) (Q) → (4) (R) → (3) (S) → (5)

Answer: (c)

Question 16

Maths · Three Dimensional Geometry · Single correct

Let $\gamma \in \mathbb{R}$ 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 entry in List-II.

1. → (3) (Q) → (4) (R) → (1) (S) → (2)
2. → (5) (Q) → (4) (R) → (1) (S) → (2)
3. → (3) (Q) → (4) (R) → (1) (S) → (5)
4. → (3) (Q) → (1) (R) → (4) (S) → (5)

Answer: (c)

Question 17

Maths · Relations and Functions · Single correct

Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ be functions defined by $$f(x) = \begin{cases} x \left| x \right| \sin \left( \frac{1}{x} \right), & x \neq 0, \\ 0, & x = 0, \end{cases}$$ and $$g(x) = \begin{cases} 1 - 2x, & 0 \leq x \leq \frac{1}{2}, \\ 0, & otherwise \end{cases}$$ Let $a, b, c, d \in \mathbb{R}$. Define the function $h : \mathbb{R} \to \mathbb{R}$ by $$h(x) = af(x) + b \left( g(x) + g \left( \frac{1}{2} - x \right) \right) + c(x - g(x)) + d \ g(x), \ x \in \mathbb{R}$$ Match each entry in List-I to the correct entry in List-II.

1. → (4) (Q) → (3) (R) → (1) (S) → (2)
2. → (5) (Q) → (2) (R) → (4) (S) → (3)
3. → (5) (Q) → (3) (R) → (2) (S) → (4)
4. → (4) (Q) → (2) (R) → (1) (S) → (3)

Answer: (c)

Question 18

Physics · Physical World, Units and Measurements · Single correct

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

1. $(2n, -n, -n, -n)$
2. $(n, -n, -2n, -n)$
3. $(n, -n, -n, -2n)$
4. $(2n, -n, -2n, -2n)$

Answer: (a)

Question 19

Physics · Moving Charges and Magnetism · Single correct

An infinitely long wire, located on the z-axis, carries a current $I$ along the $+z$-direction and produces the magnetic field $\vec{B}$. The magnitude of the line integral $\int \vec{B} \cdot d\vec{l}$ along a straight line from the point $(-\sqrt{3}a, a, 0)$ to $(a, a, 0)$ is given by [$\mu$_0 is the magnetic permeability of free space.]

1. 7$\mu$_0 I / 24
2. 7$\mu$_0 I / 12
3. $\mu$_0 I / 8
4. $\mu$_0 I / 6

Answer: (a)

Question 20

Physics · Oscillations · Single correct

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.]

1. $q^2 / (4\pi\varepsilon_0 R^3 m)$
2. $q^2 / (32\pi\varepsilon_0 R^3 m)$
3. $q^2 / (8\pi\varepsilon_0 R^3 m)$
4. $q^2 / (16\pi\varepsilon_0 R^3 m)$

Answer: (b)

Question 21

Physics · Laws of Motion · Single correct

A block of mass 5 kg moves along the x-direction subject to the force $F = (-20x + 10) \mathrm{\, N}$, 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

1. -0.5 m, 5 kg m/s
2. 0.5 m, 0 kg m/s
3. 0.5 m, -5 kg m/s
4. -1 m, 5 kg m/s

Answer: (c)

Question 22

Physics · Atoms · Multiple correct

A particle of mass $m$ is moving in a circular orbit under the influence of the central force $F(r) = -kr$, corresponding to the potential energy $V(r) = kr^2/2$, 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 $L = nh$, where $h = h/(2\pi)$, $h$ 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?

1. $r^2 = nh \sqrt{\frac{1}{mk}}$
2. $v^2 = nh \sqrt{\frac{k}{m^3}}$
3. $\frac{L}{mr^2} = \sqrt{\frac{k}{m}}$
4. $E = \frac{nh}{2} \sqrt{\frac{k}{m}}$

Answer: (a), (b), (c)

Question 23

Physics · Waves · Multiple correct

Two uniform strings of mass per unit length $\mu$ and $4\mu$, and length $L$ and $2L$, 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 $v_0 = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$, which of the following statement(s) is(are) correct?

1. With a node at $O$, the minimum frequency of vibration of the composite string is $v_0$
2. With an antinode at $O$, the minimum frequency of vibration of the composite string is $2v_0$
3. When the composite string vibrates at the minimum frequency with a node at $O$, it has 6 nodes, including the end nodes
4. No vibrational mode with an antinode at $O$ is possible for the composite string

Answer: (a), (c), (d)

Question 24

Physics · Ray Optics and Optical Instruments · Multiple 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?

1. For $n = 1.42$, $h = 50 \, \mathrm{cm}$
2. For $n = 1.35$, $h = 36 \, \mathrm{cm}$
3. For $n = 1.45$, $h = 65 \, \mathrm{cm}$
4. For $n = 1.48$, $h = 85 \, \mathrm{cm}$

Answer: (a), (b)

Question 25

Physics · Thermal Properties of Matter · Numerical

The specific heat capacity of a substance is temperature dependent and is given by the formula $C = kT$, 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: $0 \, \mathrm{K} = -273^\circ \mathrm{C}$.]

Answer: 25000

Question 26

Physics · System of Particles and Rotational Motion · Fill in the blank

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 _____.

Answer: 12

Question 27

Physics · Wave Optics · Numerical

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

Answer: 8

Question 28

Physics · Waves · Numerical

A source (S) of sound has frequency 240 $\,$ $\mathrm{Hz}$. 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 $\,$ $\mathrm{Hz}$. 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 $n$ $\,$ $\mathrm{Hz}$. The value of $n$ is .

Answer: 200

Question 29

Physics · Mechanical Properties of Fluids · Numerical

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 the tanks 1 and 2 through the holes, the times taken to empty the tanks are $t_1$ and $t_2$, respectively. If $H = \left( \frac{16}{9} \right) h$, then the ratio $t_1/t_2$ is _______.

Answer: 3

Question 30

Physics · System of Particles and Rotational Motion · Fill in the blank

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 ______.

Answer: 18

Question 31

Physics · Thermodynamics · Single correct

One mole of a monatomic ideal gas undergoes the cyclic process $J \rightarrow K \rightarrow L \rightarrow M \rightarrow J$, 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.]

1. P $\rightarrow$ 1; Q $\rightarrow$ 3; R $\rightarrow$ 5; S $\rightarrow$ 4
2. P $\rightarrow$ 4; Q $\rightarrow$ 3; R $\rightarrow$ 5; S $\rightarrow$ 2
3. P $\rightarrow$ 4; Q $\rightarrow$ 1; R $\rightarrow$ 2; S $\rightarrow$ 2
4. P $\rightarrow$ 2; Q $\rightarrow$ 5; R $\rightarrow$ 3; S $\rightarrow$ 4

Answer: (b)

Question 32

Physics · Electrostatic Potential and Capacitance · Single correct

Four identical thin, square metal sheets, $S_1$, $S_2$, $S_3$, and $S_4$, each of side $a$ are kept parallel to each other with equal distance $d(<<a)$ between them, as shown in the figure. Let $C_0 = \varepsilon_0 \frac{a^2}{d}$, where $\varepsilon_0$ is the permittivity of free space. Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

1. P $\rightarrow$ 3; Q $\rightarrow$ 2; R $\rightarrow$ 4; S $\rightarrow$ 5
2. P $\rightarrow$ 2; Q $\rightarrow$ 3; R $\rightarrow$ 2; S $\rightarrow$ 1
3. P $\rightarrow$ 3; Q $\rightarrow$ 2; R $\rightarrow$ 4; S $\rightarrow$ 1
4. P $\rightarrow$ 3; Q $\rightarrow$ 2; R $\rightarrow$ 2; S $\rightarrow$ 5

Answer: (c)

Question 33

Physics · Ray Optics and Optical Instruments · Single correct

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.

1. P → 5; Q → 2; R → 1; S → 4
2. P → 5; Q → 1; R → 2; S → 4
3. P → 3; Q → 2; R → 1; S → 4
4. P → 3; Q → 1; R → 2; S → 5

Answer: (a)

Question 34

Physics · Alternating Current · Single correct

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 $K_1$ and $K_2$. Initially, both the keys are open and there is no charge on the capacitor. At an instant, key $K_1$ 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 $K_1$ is opened and the voltage across $C_0$ oscillates with amplitude $V_0$ and angular frequency $\omega_0$. Match the quantities mentioned in List-I with their values in List-II and choose the correct option.

1. P $\rightarrow$ 1; Q $\rightarrow$ 3; R $\rightarrow$ 2; S $\rightarrow$ 5
2. P $\rightarrow$ 1; Q $\rightarrow$ 2; R $\rightarrow$ 3; S $\rightarrow$ 5
3. P $\rightarrow$ 1; Q $\rightarrow$ 3; R $\rightarrow$ 2; S $\rightarrow$ 4
4. P $\rightarrow$ 2; Q $\rightarrow$ 5; R $\rightarrow$ 3; S $\rightarrow$ 4

Answer: (a)

Question 35

Chemistry · States of Matter · Single correct

A closed vessel contains 10 g of an ideal gas X at 300 K, which exerts 2 atm pressure. At the same temperature, 80 g of another ideal gas 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

1. 2$\sqrt{2}$ : $\sqrt{3}$
2. 2$\sqrt{2}$ : 1
3. 1 : 2
4. 2 : 1

Answer: (d)

Question 36

Chemistry · The p-Block Elements (Group 15, 16, 17 & 18) · Single correct

At room temperature, disproportionation of an aqueous solution of ${in}$ ${situ}$ generated nitrous acid $(HNO_2)$ gives the species

1. $\mathrm{H_3O^+}$, $\mathrm{NO_3^-}$ and NO
2. $\mathrm{H_3O^+}$, $\mathrm{NO_3^-}$ and NO_2
3. $\mathrm{H_3O^+}$, $\mathrm{NO^-}$ and NO_2
4. $\mathrm{H_3O^+}$, $\mathrm{NO_3^-}$ and N_2O

Answer: (a)

Question 37

Chemistry · Biomolecules · Single correct

Aspartame, an artificial sweetener, is a dipeptide aspartyl phenylalanine methyl ester. The structure of aspartame is

Answer: (b)

Question 38

Chemistry · Co-ordination Compounds · Single correct

Among the following options, select the option in which each complex in Set-I shows geometrical isomerism and the two complexes in Set-II are ionization isomers of each other. $[\mathrm{en} = \mathrm{H_2NCH_2CH_2NH_2}]$

1. \textbf{Set-I:} $[\mathrm{Ni(CO)_4}]$ and $[\mathrm{PdCl_2(PPh_3)_2}]$ \hspace{1.5cm}\textbf{Set-II:} $[\mathrm{Co(NH_3)_5Cl}]\mathrm{SO_4}$ and $[\mathrm{Co(NH_3)_5(SO_4)}]\mathrm{Cl}$
2. \textbf{Set-I:} $[\mathrm{Co(en)(NH_3)_2Cl_2}]$ and $[\mathrm{PdCl_2(PPh_3)_2}]$ \hspace{1.5cm}\textbf{Set-II:} $[\mathrm{Co(NH_3)_6}][\mathrm{Cr(CN)_6}]$ and $[\mathrm{Cr(NH_3)_6}][\mathrm{Co(CN)_6}]$
3. \textbf{Set-I:} $[\mathrm{Co(NH_3)_3(NO_2)_3}]$ and $[\mathrm{Co(en)_2Cl_2}]$ \hspace{1.5cm}\textbf{Set-II:} $[\mathrm{Co(NH_3)_5Cl}]\mathrm{SO_4}$ and $[\mathrm{Co(NH_3)_5(SO_4)}]\mathrm{Cl}$
4. \textbf{Set-I:} $[\mathrm{Cr(NH_3)_5Cl}]\mathrm{Cl_2}$ and $[\mathrm{Co(en)(NH_3)_2Cl_2}]$ \hspace{1.5cm}\textbf{Set-II:} $[\mathrm{Cr(H_2O)_6}]\mathrm{Cl_3}$ and $[\mathrm{Cr(H_2O)_5Cl}]\mathrm{Cl_2}\cdot\mathrm{H_2O}$

Answer: (c)

Question 39

Chemistry · Structure of Atom · Multiple correct

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

1. Uncertainty principle rules out the existence of definite paths for electrons.
2. The energy of an electron in 2s orbital of an atom is lower than the energy of an electron that is infinitely far away from the nucleus.
3. According to Bohr's model, the most negative energy value for an electron is given by $n = 1$, which corresponds to the most stable orbit.
4. According to Bohr's model, the magnitude of velocity of electrons increases with increase in values of $n$.

Answer: (a), (b), (c)

Question 40

Chemistry · Haloalkanes and Haloarenes · Multiple correct

Reaction of $\textit{iso}$-propylbenzene with $\mathrm{O}_2$ followed by the treatment with $\mathrm{H}_3\mathrm{O}^+$ forms phenol and a by-product $P$. Reaction of $P$ with 3 equivalents of $\mathrm{Cl}_2$ gives compound $Q$. Treatment of $Q$ with $\mathrm{Ca(OH)}_2$ produces compound $R$ and calcium salt $S$. The correct statement(s) regarding $P$, $Q$, $R$ and $S$ is(are)

1. Reaction of $P$ with $R$ in the presence of KOH followed by acidification gives
2. Reaction of $R$ with $\mathrm{O}_2$ in the presence of light gives phosgene gas
3. $Q$ reacts with aqueous NaOH to produce $\mathrm{Cl}_3\mathrm{CCH}_2\mathrm{OH}$ and $\mathrm{Cl}_3\mathrm{CCOONa}$
4. $S$ on heating gives $P$

Answer: (a), (b), (d)

Question 41

Chemistry · Chemical Bonding and Molecular Structure · Multiple correct

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

1. CO_2, C_2H_4, NO and HCl
2. NO_2, O_3, HCl and H_2SO_4
3. BCl_3, NO, NO_2 and H_2SO_4
4. CO_2, BCl_3, O_3 and C_2H_4

Answer: (a), (d)

Question 42

Chemistry · Thermodynamics · Numerical

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

Answer: 8120

Question 43

Chemistry · Chemical Kinetics and Nuclear Chemistry · Fill in the blank

Consider the following reaction, $2\mathrm{H}_2(g) + 2\mathrm{NO}(g) \rightarrow \mathrm{N}_2(g) + 2\mathrm{H}_2\mathrm{O}(g)$ which follows the mechanism given below: $$2\mathrm{NO}(g) \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \mathrm{N}_2\mathrm{O}_2(g)$$ (fast equilibrium) $$\mathrm{N}_2\mathrm{O}_2(g) + \mathrm{H}_2(g) \xrightarrow{k_2} \mathrm{N}_2\mathrm{O}(g) + \mathrm{H}_2\mathrm{O}(g)$$ (slow reaction) $$\mathrm{N}_2\mathrm{O}(g) + \mathrm{H}_2(g) \xrightarrow{k_3} \mathrm{N}_2(g) + \mathrm{H}_2\mathrm{O}(g)$$ (fast reaction) The order of the reaction is_____?

Answer: 3

Question 44

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Numerical

Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. NaOH solution, gives $\textbf{P}$ and $\textbf{Q}$. Compound $\textbf{P}$ does not give Tollens' test, whereas $\textbf{Q}$ on acidification gives positive Tollens' test. Treatment of $\textbf{P}$ with excess cyclohexanone in the presence of catalytic amount of p-toluenesulfonic acid (PTSA) gives product $\textbf{R}$. Sum of the number of methylene groups (–CH$_2$–) and oxygen atoms in $\textbf{R}$ is____.

Answer: 18

Question 45

Chemistry · Co-ordination Compounds · Numerical

Among $\mathrm{V(CO)_6}$, $\mathrm{Cr(CO)_5}$, $\mathrm{Cu(CO)_3}$, $\mathrm{Mn(CO)_5}$, $\mathrm{Fe(CO)_5}$, $[\mathrm{Co(CO)_3}]^{3-}$, $[\mathrm{Cr(CO)_4}]^{4-}$, and $\mathrm{Ir(CO)_3}$, the total number of species isoelectronic with $\mathrm{Ni(CO)_4}$ is ______. [Given atomic number : $\mathrm{V} = 23$, $\mathrm{Cr} = 24$, $\mathrm{Mn} = 25$, $\mathrm{Fe} = 26$, $\mathrm{Co} = 27$, $\mathrm{Ni} = 28$, $\mathrm{Cu} = 29$, $\mathrm{Ir} = 77$]

Answer: 1

Question 46

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Numerical

In the following reaction sequence, the major product $P$ is formed. Glycerol reacts completely with excess $P$ in the presence of an acid catalyst to form $Q$. Reaction of $Q$ with excess NaOH followed by the treatment with CaCl$_2$ yields Ca-soap $R$, quantitatively. Starting with one mole of $Q$, the amount of $R$ produced in gram is _____. [Given, atomic weight: H = 1, C = 12, N = 14, O = 16, Na = 23, Cl = 35, Ca = 40]

Answer: 909

Question 47

Chemistry · Co-ordination Compounds · Numerical

Among the following complexes, the total number of diamagnetic species is . $[\mathrm{Mn(NH_3)_6}]^{3+}$, $[\mathrm{MnCl_6}]^{3-}$, $[\mathrm{FeF_6}]^{3-}$, $[\mathrm{CoF_6}]^{3-}$, $[\mathrm{Fe(NH_3)_6}]^{3+}$ and $[\mathrm{Co(en)_3}]^{3+}$ [Given, atomic number: Mn = 25, Fe = 26, Co = 27; en = $\mathrm{H_2NCH_2CH_2NH_2}$]

Answer: 1

Question 48

Chemistry · Electrochemistry · Single correct

In a conductometric titration, small volume of titrant of higher concentration is added stepwise to a larger volume of titrate of much lower concentration, and the conductance is measured after each addition. The limiting ionic conductivity $(\Lambda_0)$ values (in $\mathrm{mS\ m^2\ mol^{-1}}$) for different ions in aqueous solutions are given below: \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline Ions & $\mathrm{Ag^+}$ & $\mathrm{K^+}$ & $\mathrm{Na^+}$ & $\mathrm{H^+}$ & $\mathrm{NO_3^-}$ & $\mathrm{Cl^-}$ & $\mathrm{SO_4^{2-}}$ & $\mathrm{OH^-}$ & $\mathrm{CH_3COO^-}$ \\ \hline $\Lambda_0$ & 6.2 & 7.4 & 5.0 & 35.0 & 7.2 & 7.6 & 16.0 & 19.9 & 4.1 \\ \hline \end{tabular} For different combinations of titrates and titrants given in \textbf{List-I}, the graphs of `conductance' versus `volume of titrant' are given in \textbf{List-II}. Match each entry in \textbf{List-I} with the appropriate entry in \textbf{List-II} and choose the correct option.

1. P-4, Q-3, R-2, S-5
2. P-2, Q-4, R-3, S-1
3. P-3, Q-4, R-2, S-5
4. P-4, Q-3, R-2, S-1

Answer: (c)

Question 49

Chemistry · Chemical Bonding and Molecular Structure · Single correct

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.

1. P-5, Q-2, R-3, S-1
2. P-5, Q-3, R-2, S-4
3. P-4, Q-3, R-2, S-1
4. P-4, Q-2, R-5, S-3

Answer: (b)

Question 50

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Single correct

List-I contains various reaction sequences and List-II contains the possible products. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

1. P-3, Q-5, R-4, S-1
2. P-3, Q-2, R-4, S-1
3. P-3, Q-5, R-1, S-4
4. P-5, Q-2, R-4, S-1

Answer: (a)

Question 51

Chemistry · Alcohols, Phenols and Ethers · Single correct

List-I contains various reaction sequences and List-II contains different phenolic compounds. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.

1. P-2, Q-3, R-4, S-5
2. P-2, Q-3, R-5, S-1
3. P-3, Q-2, R-4, S-1
4. P-3, Q-2, R-5, S-4

Answer: (c)