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JEE Advanced 2019 — 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 2019 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":

Q36Paper 1MathematicsOne or More Correct (Partial Marking)Medium

Vectors and 3D Geometry

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

[JEE Advanced 2019 Paper 1 Mathematics Q36] Find the square of the shortest distance (

d^2

) between the two skew 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})

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

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 2019 Paper 1 Mathematics Q37] Let

z

be a complex number satisfying

|z - 3 - 4i| = 2

. Which of the following statements is/are CORRECT?

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

Q38Paper 1MathematicsOne or More Correct (Partial Marking)Medium

Matrices and Determinants

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

[JEE Advanced 2019 Paper 1 Mathematics Q38] Let

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

. Which of the following statements regarding matrix

A

is/are CORRECT?

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

Q39Paper 1MathematicsOne or More Correct (Partial Marking)Medium

Probability & Statistics

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

[JEE Advanced 2019 Paper 1 Mathematics Q39] An urn

U_1

contains 3 white and 2 black balls, while urn

U_2

contains 1 white and 4 black balls. A fair coin is tossed; if heads appears, a ball is drawn from

U_1

, otherwise from

U_2

. Given that the drawn ball is WHITE, which of the following statements is/are CORRECT?

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

Q40Paper 1MathematicsOne or More Correct (Partial Marking)Hard

Permutations and Combinations

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

[JEE Advanced 2019 Paper 1 Mathematics Q40] Let

D_n

denote the number of derangements of

n

distinct objects (permutations where no object appears in its original position). Consider 5 letters addressed to 5 different recipients placed randomly into 5 corresponding envelopes. Which of the following statements is/are CORRECT?

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

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 2019 Paper 1 Mathematics Q41] Let

f: [0, 2] \to \mathbb{R}

be a continuous function on

[0, 2]

and twice differentiable on

(0, 2)

such that

f(0) = 0

f(1) = 1

, and

f(2) = 1

. Which of the following statements is/are necessarily TRUE?

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

Q42Paper 1MathematicsNumerical Value TypeMedium

Definite Integration

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

[JEE Advanced 2019 Paper 1 Mathematics Q42] Evaluate the definite integral

I = \int_0^{\pi/2} \frac{\sin^{2025} x}{\sin^{2025} x + \cos^{2025} x} dx

. Calculate the exact value of

\frac{4 I}{\pi}

Q43Paper 1MathematicsNumerical Value TypeMedium

Binomial Theorem

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

[JEE Advanced 2019 Paper 1 Mathematics Q43] For

n = 5

, evaluate the double summation

S = \sum_{0 \le i < j \le 5} \binom{5}{i} \binom{5}{j}

. What is the value of

S

Q44Paper 1MathematicsNumerical Value TypeMedium

Coordinate Geometry - Parabola

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

[JEE Advanced 2019 Paper 1 Mathematics Q44] From a point

P(h, k)

, three normal lines are drawn to the parabola

y^2 = 4x

. If the sum of the ordinates of the three co-normal points is

S_y

, find the value of

S_y

Q45Paper 1MathematicsNumerical Value TypeMedium

Trigonometry & Inverse Trigonometric Functions

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

[JEE Advanced 2019 Paper 1 Mathematics Q45] Evaluate the infinite sum

S = \sum_{k=1}^\infty \tan^{-1}\left(\frac{1}{1 + k + k^2}\right)

. Calculate the numerical value of

\frac{4 S}{\pi}

Q46Paper 1MathematicsNumerical Value TypeMedium

Limits, Continuity and Differentiability

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

[JEE Advanced 2019 Paper 1 Mathematics Q46] Evaluate the limit

L = \lim_{x \to 0} (\cos x)^{1/x^2}

. If

L = e^{-k}

, find the value of

k

Q47Paper 1MathematicsSingle Correct OptionMedium

Integral Calculus (Area Bounded by Curves)

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

[JEE Advanced 2019 Paper 1 Mathematics Q47] The area of the region bounded by the parabola

y = x^2

and the line

y = 2x

is:

Q48Paper 1MathematicsSingle Correct OptionMedium

Differential Equations

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

[JEE Advanced 2019 Paper 1 Mathematics Q48] The integrating factor for the first order linear differential equation

\frac{dy}{dx} + \frac{2}{x} y = x^3

(for

x > 0

) is:

Q49Paper 1MathematicsSingle Correct OptionMedium

Functions & Relations

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

[JEE Advanced 2019 Paper 1 Mathematics Q49] If

f(x)

is a polynomial function satisfying

f(x) f(1/x) = f(x) + f(1/x)

for all

x \ne 0

, and

f(3) = 28

, then

f(2)

equals:

Q50Paper 1MathematicsSingle Correct OptionMedium

Coordinate Geometry - Ellipse & Hyperbola

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

[JEE Advanced 2019 Paper 1 Mathematics Q50] The locus of the point of intersection of two perpendicular tangents to the ellipse

\frac{x^2}{16} + \frac{y^2}{9} = 1

is a concentric circle (director circle) of radius:

Q51Paper 1MathematicsSingle Correct OptionMedium

Sequences and Series

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

[JEE Advanced 2019 Paper 1 Mathematics Q51] Find the sum of the infinite Arithmetico-Geometric Progression (AGP):

S = 1 + \frac{2}{2} + \frac{3}{4} + \frac{4}{8} + \frac{5}{16} + \dots