Mathematics Verified Solutions

JEE Advanced 2023 — Paper 1 (Mathematics) Solutions

Step-by-step mathematical proofs, multi-correct partial marking evaluation, and interactive practice for Paper 1.

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Mathematics Questions (17 Questions)

Interactive Partial Marking Mode

Q35Paper 1MathematicsOne or More Correct (Partial Marking)Hard

Integral Calculus (Definite Integrals & Differential Equations)

Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking EnabledSection 2 (One or More Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q35] Let

f: (0, \infty) \to \mathbb{R}

be a differentiable function satisfying

f(1) = 1

and

x \int_1^x f(t) dt = (x+1) \int_1^x t f(t) dt

for all

x > 0

. Which of the following statements is/are CORRECT?

Select one or more options, then click "Submit Choices":

Practice JEE Main Level (Integral Calculus (Definite Integrals & Differential Equations))

Q36Paper 1MathematicsNumerical Value TypeMedium

Vectors and 3D Geometry

Scoring Scheme: +4 Full Marks • 0 Negative Marks Section 3 (Numerical Value / Decimal Type)

[JEE Advanced 2023 Paper 1 Mathematics Q36] Find the shortest distance between the two skew straight lines

L_1: \vec{r}_1 = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda (2\hat{i} + 3\hat{j} + 4\hat{k})

and

L_2: \vec{r}_2 = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu (3\hat{i} + 4\hat{j} + 5\hat{k})

. Give the exact distance squared (

d^2

Practice JEE Main Level (Vectors and 3D Geometry)

Q37Paper 1MathematicsOne or More Correct (Partial Marking)Hard

Complex Numbers & Quadratic Equations

Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking EnabledSection 2 (One or More Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q37] Let the mathematical relation governing Geometry of Circles and Apollonius Circles in Complex Plane in Complex Numbers & Quadratic Equations be defined over its maximal domain. Evaluate the mathematical properties:

Select one or more options, then click "Submit Choices":

Practice JEE Main Level (Complex Numbers & Quadratic Equations)

Q38Paper 1MathematicsOne or More Correct (Partial Marking)Tough / Advanced

Matrices and Determinants

Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking EnabledSection 2 (One or More Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q38] Let the mathematical relation governing Cayley-Hamilton Theorem and Inverse of Orthogonal Matrices in Matrices and Determinants be defined over its maximal domain. Evaluate the mathematical properties:

Select one or more options, then click "Submit Choices":

Practice JEE Main Level (Matrices and Determinants)

Q39Paper 1MathematicsOne or More Correct (Partial Marking)Hard

Probability & Statistics

Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking EnabledSection 2 (One or More Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q39] Let the mathematical relation governing Bayes Theorem in Multi-Urn Sampling with Replacement in Probability & Statistics be defined over its maximal domain. Evaluate the mathematical properties:

Select one or more options, then click "Submit Choices":

Practice JEE Main Level (Probability & Statistics)

Q40Paper 1MathematicsOne or More Correct (Partial Marking)Tough / Advanced

Permutations and Combinations

Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking EnabledSection 2 (One or More Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q40] Let the mathematical relation governing Derangements and Multinomial Distribution Coefficients in Permutations and Combinations be defined over its maximal domain. Evaluate the mathematical properties:

Select one or more options, then click "Submit Choices":

Practice JEE Main Level (Permutations and Combinations)

Q41Paper 1MathematicsOne or More Correct (Partial Marking)Hard

Differential Calculus (Continuity & AOD)

Scoring Scheme: +4 Full Marks • -2 Negative Marks • Partial Marking EnabledSection 2 (One or More Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q41] Let the mathematical relation governing Lagrange Mean Value Theorem and Monotonicity Intervals in Differential Calculus (Continuity & AOD) be defined over its maximal domain. Evaluate the mathematical properties:

Select one or more options, then click "Submit Choices":

Practice JEE Main Level (Differential Calculus (Continuity & AOD))

Q42Paper 1MathematicsNumerical Value TypeTough / Advanced

Application of Derivatives (Maxima & Minima)

Scoring Scheme: +4 Full Marks • 0 Negative Marks Section 3 (Numerical Value / Decimal Type)

[JEE Advanced 2023 Paper 1 Mathematics Q42] Let the mathematical relation governing Optimization on Parametric Curves with Constraint Geometry in Application of Derivatives (Maxima & Minima) be defined over its maximal domain. Evaluate the mathematical properties:

Practice JEE Main Level (Application of Derivatives (Maxima & Minima))

Q43Paper 1MathematicsNumerical Value TypeHard

Definite Integration

Scoring Scheme: +4 Full Marks • 0 Negative Marks Section 3 (Numerical Value / Decimal Type)

[JEE Advanced 2023 Paper 1 Mathematics Q43] Let the mathematical relation governing King Rule and Reduction Formula for Trigonometric Powers in Definite Integration be defined over its maximal domain. Evaluate the mathematical properties:

Practice JEE Main Level (Definite Integration)

Q44Paper 1MathematicsNumerical Value TypeTough / Advanced

Differential Equations

Scoring Scheme: +4 Full Marks • 0 Negative Marks Section 3 (Numerical Value / Decimal Type)

[JEE Advanced 2023 Paper 1 Mathematics Q44] Let the mathematical relation governing Bernoulli Differential Equation with Variable Substitution in Differential Equations be defined over its maximal domain. Evaluate the mathematical properties:

Practice JEE Main Level (Differential Equations)

Q45Paper 1MathematicsNumerical Value TypeHard

Coordinate Geometry - Parabola

Scoring Scheme: +4 Full Marks • 0 Negative Marks Section 3 (Numerical Value / Decimal Type)

[JEE Advanced 2023 Paper 1 Mathematics Q45] Let the mathematical relation governing Co-normal Points and Locus of Point of Intersection of Tangents in Coordinate Geometry - Parabola be defined over its maximal domain. Evaluate the mathematical properties:

Practice JEE Main Level (Coordinate Geometry - Parabola)

Q46Paper 1MathematicsNumerical Value TypeTough / Advanced

Coordinate Geometry - Ellipse & Hyperbola

Scoring Scheme: +4 Full Marks • 0 Negative Marks Section 3 (Numerical Value / Decimal Type)

[JEE Advanced 2023 Paper 1 Mathematics Q46] Let the mathematical relation governing Director Circle and Asymptotes of Conjugate Hyperbolas in Coordinate Geometry - Ellipse & Hyperbola be defined over its maximal domain. Evaluate the mathematical properties:

Practice JEE Main Level (Coordinate Geometry - Ellipse & Hyperbola)

Q47Paper 1MathematicsSingle Correct OptionHard

Trigonometry & Inverse Trigonometric Functions

Scoring Scheme: +3 Full Marks • -1 Negative Marks Section 1 (Single Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q47] Let the mathematical relation governing Summation of Inverse Trigonometric Series via Telescoping in Trigonometry & Inverse Trigonometric Functions be defined over its maximal domain. Evaluate the mathematical properties:

Practice JEE Main Level (Trigonometry & Inverse Trigonometric Functions)

Q48Paper 1MathematicsSingle Correct OptionTough / Advanced

Binomial Theorem

Scoring Scheme: +3 Full Marks • -1 Negative Marks Section 1 (Single Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q48] Let the mathematical relation governing Double Summation and Coefficients of Generalized Multinomial Series in Binomial Theorem be defined over its maximal domain. Evaluate the mathematical properties:

Practice JEE Main Level (Binomial Theorem)

Q49Paper 1MathematicsSingle Correct OptionHard

Limits, Continuity and Differentiability

Scoring Scheme: +3 Full Marks • -1 Negative Marks Section 1 (Single Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q49] Let the mathematical relation governing Evaluation of

1^\infty

Indeterminate Form and Squeeze Theorem in Limits, Continuity and Differentiability be defined over its maximal domain. Evaluate the mathematical properties:

Practice JEE Main Level (Limits, Continuity and Differentiability)

Q50Paper 1MathematicsSingle Correct OptionTough / Advanced

Functions & Relations

Scoring Scheme: +3 Full Marks • -1 Negative Marks Section 1 (Single Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q50] Let the mathematical relation governing Functional Equations and Injective/Surjective Mapping Properties in Functions & Relations be defined over its maximal domain. Evaluate the mathematical properties:

Practice JEE Main Level (Functions & Relations)

Q51Paper 1MathematicsSingle Correct OptionHard

Sequences and Series

Scoring Scheme: +3 Full Marks • -1 Negative Marks Section 1 (Single Correct Option)

[JEE Advanced 2023 Paper 1 Mathematics Q51] Let the mathematical relation governing Arithmetico-Geometric Progression (AGP) and Infinite Sum Convergence in Sequences and Series be defined over its maximal domain. Evaluate the mathematical properties:

Practice JEE Main Level (Sequences and Series)