JEE (Advanced) 2024 Paper 1 Study Guide

JEE (Advanced) 2024 Paper 1: General Instructions and Marking Schemes

Mathematics Section Structure and Evaluation

Section 1 (Maximum Marks: 12) contains 4 questions. Each question is a single correct choice (MCQ). The marking scheme is as follows:

• Full Marks: +3 if only the correct option is chosen.
• Zero Marks: 0 if unanswered.
• Negative Marks: −1 in all other cases.

Section 2 (Maximum Marks: 12) contains 3 questions. Each question has one or more correct options. Marking:

• Full Marks: +4 if all correct options are chosen.
• Partial Marks: +3 if four options are correct but only three chosen.
• Partial Marks: +2 if three or more correct but only two correct chosen.
• Partial Marks: +1 if two or more correct but only one correct chosen.
• Zero Marks: 0 if unanswered.
• Negative Marks: −2 in all other cases.

Section 3 (Maximum Marks: 24) contains 6 questions where the answer is a non-negative integer. Marking:

• Full Marks: +4 if the correct integer is entered.
• Zero Marks: 0 in all other cases.

Section 4 (Maximum Marks: 12) contains 4 matching list sets. Each set is a multiple-choice question. Marking:

• Full Marks: +3 if correct combination is chosen.
• Zero Marks: 0 if unanswered.
• Negative Marks: −1 in all other cases.

Mathematics: Questions and Details

Section 1: Single Correct Option (Q.1 – Q.4)

Q.1 Functional Limit Question Let $f(x)$ be a continuously differentiable function on $(0, \infty)$ with $f(1) = 2$. The function satisfies the equation: $\lim_{t \to x} \frac{t^{10}f(x) - x^{10}f(t)}{t^9 - x^9} = f(x) - −x^{10}$ for each $x > 0$. Find $f(x)$. Options: (A) $\frac{31}{11}x^{10} - \frac{9}{11}x$ (B) $\frac{9}{11}x^{10} + \frac{13}{11}x$ (C) $-\frac{9}{11}x^{10} + \frac{31}{11}x$ (D) $\frac{13}{11}x^{10} + \frac{9}{11}x$

Q.2 Probability Analysis A student takes a quiz of true-false questions. He either knows the answer or guesses. If he knows it, he is always correct. The probability of correct answering, given he guessed, is $\frac{1}{2}$. The probability that he guessed, given his answer was correct, is $\frac{1}{6}$. Find the probability that the student knows the answer of a randomly chosen question. Options: (A) $\frac{1}{12}$ (B) $\frac{1}{7}$ (C) $\frac{5}{7}$ (D) $\frac{5}{12}$

Q.3 Trigonometry Problem Given $\frac{\pi}{2} < x < \pi$ and $\cot(x) = -\frac{5}{11}$. Evaluate the expression: $\frac{\sin(x) + \cos(x) - (\sin^6(x) + \cos^6(x))^{11}}{\sin^2(x) - (\sin^6(x) + \cos^6(x))}$ Options: (A) $1 - 2 \times 3^{-11}$ (B) $1 + 2 \times 3^{-11}$ (C) $1 + 3 \times 2^{-11}$ (D) $1 - 3 \times 2^{-11}$

Q.4 Ellipse and Geometry Ellipse equation: $\frac{x^2}{9} + \frac{y^2}{4} = 1$. A point $S(p, q)$ is in the first quadrant outside the ellipse ($\frac{p^2}{9} + \frac{q^2}{4} > 1$). Two tangents from $S$ meet the ellipse: one at the minor axis endpoint and the other at point $T$ in the fourth quadrant. The vertex with positive x-coordinate is $R$, and center is $O$. Area of $\triangle ORT = \frac{3}{2}$. Find $p$ and $q$. Options: (A) $q = 2, p = 3\sqrt{3}$ (B) $q = 2, p = 4\sqrt{3}$ (C) $q = 1, p = 5\sqrt{3}$ (D) $q = 1, p = 6\sqrt{3}$

Section 3: Integer Type (Q.8 – Q.11)

Q.8 Logarithmic Equation Given $a = 3\sqrt{2}$ and $b = 6^{1/6} \times 5^{1/6}$. For $x, y \in \mathbb{R}$, equations are: $\log_a(x^5 + y^3) = \sqrt{18 + 4\sqrt{5}}$$\log_b(x^2 - y) = 1080$ Find the value of $4x + 5y$.

Q.9 Polynomial Roots$f(x) = x^4 + ax^3 + bx^2 + c$ has real coefficients and $f(1) = -9$. If $i\sqrt{3}$ is a root of $x^3 + ax^2 + bx = 0$, and $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $f(x) = 0$, find \alpha_1^2 + \alpha_2^2 + ̑\alpha_3^2 + \alpha_4^2.

Q.10 Determinant Counting Set $S$ contains matrices with entries $a, b, c, d, e \in \{0, 1\}$ and $|A| \in \{-1, 1\}$. The matrix template is: $\begin{pmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{pmatrix}$ Find the number of elements in $S$.

Q.11 Combinatorics A group of 9 students ($s_1$ to $s_9$) is divided into 3 teams: $X$ (size 2), $Y$ (size 3), and $Z$ (size 4). Constraint: $s_1$ cannot be in team $X$, and $s_2$ cannot be in team $Y$. Find the number of ways to form teams.

Physics Section Structure and Evaluation

The structure for Physics Paper 1 (Marks, Sections, Evaluation) matches the Mathematics structure: Section 1 (Single Correct, +3/-1), Section 2 (Multi-correct, +4/Partial/-2), Section 3 (Integer, +4/0), and Section 4 (Matching, +3/-1).

Physics: Questions and Details

Section 1: Single Correct Option (Q.1 – Q.4)

Q.1 Dimensional Analysis A dimensionless quantity is formed as $e^{\alpha} \varepsilon_0^{\beta} h^{\gamma} c^{\delta}$. If $n$ is a non-zero integer, determine the tuple $(\alpha, \beta, \gamma, \delta)$. Constants: $e$ (electronic charge), $\varepsilon_0$ (permittivity), $h$ (Planck's constant), $c$ (speed of light). Options: (A) $(2n, -n, -n, -n)$ (B) $(n, -n, -2n, -n)$ (C) $(n, -n, -n, -2n)$ (D) $(2n, -n, -2n, -2n)$

Q.2 Magnetostatics An infinite wire on the z-axis carries current $I$ in the $+z$ direction. Calculate $\int \mathbf{B} \cdot d\mathbf{l}$ along a straight line from $(-\sqrt{3}a, a, 0)$ to $(a, a, 0)$. Options: (A) $7\mu_0 I / 24$ (B) $7\mu_0 I / 12$ (C) $\mu_0 I / 8$ (D) $\mu_0 I / 6$

Q.3 Oscillations on a Hoop Two beads (charge $q$, mass $m$) are on a horizontal circular hoop (radius $R$). One bead is glued. The other oscillates. Find the square of the angular frequency $\omega^2$ for small oscillations. Options: (A) $q^2 / (4\pi\varepsilon_0 R^3 m)$ (B) $q^2 / (32\pi\varepsilon_0 R^3 m)$ (C) $q^2 / (8\pi\varepsilon_0 R^3 m)$ (D) $\frac{q^2}{16\pi\varepsilon_0 R^3 m}$

Q.4 Dynamics Mass $5\,\text{kg}$ moves along x-axis with force $F = (-20x + 10)\,\text{N}$. At $t = 0\,\text{s}$, it is at rest at $x = 1\,\text{m}$. Find position and momentum at $t = \frac{\pi}{4}\,\text{s}$. Options: (A) $-0.5\,\text{m}, 5\,\text{kg}\,\text{m/s}$ (B) $0.5\,\text{m}, 0\,\text{kg}\,\text{m/s}$ (C) $0.5\,\text{m}, -5\,\text{kg}\,\text{m/s}$ (D) $-1\,\text{m}, 5\,\text{kg}\,\text{m/s}$

Section 3: Integer Type (Q.8 – Q.13)

Q.8 Calorimetry Specific heat $C = kT$. Find the heat required to raise $1\,\text{kg}$ of substance from $-73\,^{\circ}\text{C}$ to $27\,^{\circ}\text{C}$ in terms of $n k$. Find $n$.

Q.9 Conservation of Angular Momentum Disc 1 (Mass $M$, Radius $R$) is on a vertical axis. A motor (negligible mass) is at its circumference. Disc 2 (Mass $M$, Radius $R/2$) is on the motor shaft. Motor turns smaller disc at $\omega$. If large disc rotates at $\omega/n$, find $n$.

Q.11 Doppler Effect Source frequency is $240\,\text{Hz}$. Case 1: Observer and source move toward each other at speed $v$, frequency heard is $288\,\text{Hz}$. Case 2: They move away at speed $v$, frequency heard is $n\,\text{Hz}$. Find $n$.

Q.12 Hydrodynamics Two tanks at height $H$ are filled with water to height $h$. Tank 1 flows from hole at bottom. Tank 2 has a pipe of length $H$ to ground. Ratio of time to empty $t_1/t_2$ if $H = (16/9)h$.

Section 4: Matching Lists (Q.14 – Q.17)

Q.14 Thermodynamics (Cyclic Process) Monoatomic ideal gas (1 mole) undergoes cyclic process $J \rightarrow K \rightarrow L \rightarrow M \rightarrow J$. List-I properties: Work done, Change in internal energy ($JK$ and $MJ$), Heat given ($KL$). List-II values: $RT_0 - 4RT_0 \ln(2)$, $0$, $3RT_0$, $-2RT_0 \ln(2)$, $-3RT_0 \ln(2)$.

Chemistry Section Structure and Evaluation

The structure for Chemistry Paper 1 matches Mathematics and Physics: Section 1 (Single Correct, +3/-1), Section 2 (Multi-correct, +4/Partial/-2), Section 3 (Integer, +4/0), and Section 4 (Matching, +3/-1).

Chemistry: Questions and Details

Section 1: Single Correct Option (Q.1 – Q.4)

Q.1 Ideal Gas and RMS Velocity Vessel contains $10\,\text{g}$ of gas $X$ at $300\,\text{K}$ and $2\,\text{atm}$. Added $80\,\text{g}$ of gas $Y$, pressure becomes $6\,\text{atm}$. Find ratio of RMS velocities of $X$ and $Y$ at $300\,\text{K}$. Options: (A) $2\sqrt{2} : \sqrt{3}$ (B) $2\sqrt{2} : 1$ (C) $1 : 2$ (D) $2 : 1$

Q.2 Inorganic Reactions Disproportionation of aqueous nitrous acid ($HNO_2$) yields: Options: (A) $H_3O^+$, $NO_3^-$, and $NO$ (B) $H_3O^+$, $NO_3^-$, and $NO_2$ (C) $H_3O^+$, $NO^-$, and $NO_2$ (D) $H_3O^+$, $NO_3^-$, and $N_2O$

Q.3 Biomolecules Aspartame is a dipeptide aspartyl phenylalanine methyl ester. Identify the correct structure among four representations (not provided in text, but labeled A-D).

Q.4 Coordination Isomerism Identify Set-I (geometrical isomerism) and Set-II (ionization isomerism). Option (C): Set-I: $[Co(NH_3)_3(NO_2)_3]$ and $[Co(en)_2Cl_2]$; Set-II: $[Co(NH_3)_5Cl]SO_4$ and $[Co(NH_3)_5(SO_4)]Cl$.

Section 3: Integer Type (Q.8 – Q.13)

Q.8 Thermodynamics Expansion of 5 moles of monoatomic gas from $X \rightarrow Y \rightarrow Z$. Find total change in enthalpy ($\Delta H$) in Joules. Data: $C_{v, m} = 12\,J\,K^{-1}\,mol^{-1}$, $R = 8.3\,J\,K^{-1}\,mol^{-1}$.

Q.9 Chemical Kinetics Reaction: $2H_2(g) + 2NO(g) \rightarrow N_2(g) + 2H_2O(g)$. Mechanism given:

1. $2NO \rightleftharpoons N_2O_2$ (fast)
2. $N_2O_2 + H_2 \rightarrow N_2O + H_2O$ (slow)
3. $N_2O + H_2 \rightarrow N_2 + H_2O$ (fast) Find the overall order of the reaction.

Q.11 Isoelectronic Species Find number of species isoelectronic with $Ni(CO)_4$ among: $V(CO)_6$, $Cr(CO)_5$, $Cu(CO)_3$, $Mn(CO)_5$, $Fe(CO)_5$, $[Co(CO)_3]^{3-}$, $[Cr(CO)_4]^{4-}$, $Ir(CO)_3$.

Q.13 Magnetism Total number of diamagnetic species among: $[Mn(NH_3)_6]^{3+}$, $[MnCl_6]^{3-}$, $[FeF_6]^{3-}$, $[CoF_6]^{3-}$, $[Fe(NH_3)_6]^{3+}$, $[Co(en)_3]^{3+}$.

Section 4: Matching Lists (Q.14 – Q.17)

Q.14 Conductometric Titration Match Titrate/Titrant combinations with conductance vs. volume graphs. (P) $KCl$ vs $AgNO_3$ (Q) $AgNO_3$ vs $KCl$ (R) $NaOH$ vs $HCl$ (S) $NaOH$ vs $CH_3COOH$

Q.15 Xenon Chemistry Match Xenon compounds to Geometry/Lone Pairs: (P) $XeF_2$: Trigonal bipyramidal, 3 lone pairs. (Q) $XeF_4$: Octahedral, 2 lone pairs. (R) $XeO_3$: Tetrahedral, 1 lone pair. (S) $XeO_3F_2$: Trigonal bipyramidal, 0 lone pairs.