Mathematics · 2019

CBSE Class 12 Mathematics 2019 — Previous-Year Question Paper

CBSE Class XII Board 2019 · Set 65/1/1

Real board examination⚠ Old pattern · pre-2020 syllabus

Sets

About the 2019 exam: The 2019 exam used CBSE's older pattern — 29 questions for 100 marks, all subjective (no MCQs). All three series (65/1/1, 65/2/1, 65/3/1) are here in full, each verified against the official CBSE paper. Topics since removed from the syllabus are badged and filterable above.

This paper has 9 questions on a topic removed in CBSE’s 2023-24 syllabus update (each marked Not in syllabus). It’s kept for historical accuracy — the exam really asked it that year — but isn’t in the current syllabus and doesn’t count toward a concept’s importance. Switch to to focus on what’s still examinable.

About this paper

The real Class-12 board examination held in 2019. Every question below is solved the concept-first way. Sample papers are labelled honestly — never shown as a past exam.

Total marks: 100
Questions: 29
Duration: 180 min
Sections: 4

The marks / questions / duration above are the official exam pattern. We currently have 29 of this paper’s questions (100% of the full paper), with 29 fully solved. Questions we couldn’t yet extract or verify are held — never shown as complete.

Sections & marks

Section	Type	Questions	Marks each	Total
ASection A	Very short answer	4	1	4
BSection B	Short answer	8	2	16
CSection C	Long answer I	11	4	44
DSection D	Long answer II	6	6	36
Total		29		100

The question paper

The questions we hold for this paper, laid out by section. Solutions are on the Answers tab.

Board Examination

Mathematics

CBSE Class XII Board 2019 · Set 65/1/1

Series/Set: 65/1/1Roll No. ________

Time Allowed: 3 hoursMaximum Marks: 100

General Instructions

This question paper contains 29 questions divided into 4 sections — A, B, C, D.
Section A comprises 4 questions of 1 mark each (Very short answer).
Section B comprises 8 questions of 2 marks each (Short answer).
Section C comprises 11 questions of 4 marks each (Long answer I).
Section D comprises 6 questions of 6 marks each (Long answer II).

Above is the official exam pattern. The questions printed below are those we currently hold for this paper.

Section A

Very short answer · 1 mark each · 4 of 4 shown

Q1.

If A and B are square matrices of the same order 3, such that

|A| = 2

and

AB = 2I

, write the value of

|B|

[1]

Q2.

f(x) = x + 1

, find

\frac{d}{dx}(fof)(x)

⚠ This question is not in the current syllabus — Composite functions (removed 2023-24)

[1]

Q3.

Find the order and the degree of the differential equation

x^2 \frac{d^2y}{dx^2} = \left\{ 1 + \left(\frac{dy}{dx}\right)^2 \right\}^4

[1]

Q4.

If a line makes angles

90^\circ, 135^\circ, 45^\circ

with the

x

y

and

z

axes respectively, find its direction cosines.

(OR)

Find the vector equation of a line which passes through the point

(3, 4, 5)

and is parallel to the vector

2\hat{i} + 2\hat{j} - 3\hat{k}

[1]

Section B

Short answer · 2 marks each · 8 of 8 shown

Q1.

Examine whether the operation

*

defined on

R

a * b = ab + 1

(i) a binary or not,
(ii) if a binary operation, is it associative or not?

[2]

Page 1 of 6

Q2.

Find a matrix A such that 2A - 3B + 5C = O, where B = -2 2 0 \3 1 4 and C = 2 0 -2 \7 1 6 .

[2]

Q3.

Find: ∫ fracsec² x√(tan² x + 4)dx.

[2]

Q4.

Find: ∫ √(1 - sin 2x)dx, (π)/(4) < x < (π)/(2). OR Find: ∫ sin⁻¹(2x)dx.

[2]

Q5.

Form the differential equation representing the family of curves y = e2x(a + bx), where 'a' and 'b' are arbitrary constants.

⚠ This question is not in the current syllabus — Formation of a differential equation from a family of curves (removed 2023-24)

[2]

Q6.

If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is √(3). OR If veca = 2hati + 3hatj + hatk, vecb = hati - 2hatj + hatk and vecc = -3hati + hatj + 2hatk, find [vecavecbvecc].

⚠ This question is not in the current syllabus — Scalar triple product of vectors (removed 2023-24)

[2]

Q7.

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event 'number obtained is even' and B be the event 'number obtained is red'. Find if A and B are independent events.

[2]

Q8.

A die is thrown 6 times. If 'getting an odd number' is a success, what is the probability of (i) 5 successes? (ii) at least 5 successes? (iii) at most 5 successes? OR The random variable X has probability distribution P(X) of the following form, where k is some number: P(X = x) = k, if x = 0 \2k, if x = 1 \3k, if x = 2 \0, otherwise (a) Determine the value of 'k'. (b) Find P(X < 2), P(X ≥ 2), P(X ≤ 2).

⚠ This question is not in the current syllabus — Binomial distribution / Bernoulli trials (removed 2023-24)

[2]

Page 2 of 6

Section C

Long answer I · 4 marks each · 11 of 11 shown

Q1.

Prove that ∫₀a f(x) dx = ∫₀a f(a-x) dx and hence evaluate ∫₀π/2 (x)/(sin x + cos x) dx

[4]

Q2.

If the lines (x - 1)/(-3) = (y - 2)/(2λ) = (z - 3)/(2) and (x - 1)/(3λ) = (y - 1)/(2) = (z - 6)/(-5) are perpendicular, find the value of λ. Hence find whether the lines are intersecting or not.

[4]

Q3.

Show that the relation R on R defined as R = \(a, b) : a ≤ b\, is reflexive and transitive but not symmetric. OR Prove that the function f : N → N, defined by f(x) = x² + x + 1 is one-one but not onto. Find the inverse of f : N → S, where S is range of f.

⚠ This question is not in the current syllabus — Composite functions and inverse of a function (removed 2023-24)

[4]

Q4.

Solve: tan⁻¹ 4x + tan⁻¹ 6x = (π)/(4).

⚠ This question is not in the current syllabus — Elementary properties of inverse trigonometric functions (removed 2023-24)

[4]

Q5.

Using properties of determinants, prove that a² + 2a 2a + 1 1 \2a + 1 a + 2 1 \3 3 1 = (a - 1)³.

⚠ This question is not in the current syllabus — Properties of determinants used to prove identities (removed 2023-24)

[4]

Page 3 of 6

Q6.

If log(x² + y²) = 2tan⁻¹((y)/(x)), show that (dy)/(dx) = (x + y)/(x - y). OR If xy - yx = ab, find (dy)/(dx).

[4]

Q7.

If y = (sin⁻¹ x)², prove that (1 - x²)(d²y)/(dx²) - x(dy)/(dx) - 2 = 0.

[4]

Q8.

Find the equation of tangent to the curve y = √(3x - 2) which is parallel to the line 4x - 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact.

[4]

Q9.

Find: ∫ (3x + 5)/(x² + 3x - 18)dx.

[4]

Q10.

Solve the differential equation: xdy - ydx = √(x² + y²)dx, given that y = 0 when x = 1. OR Solve the differential equation: (1 + x²)(dy)/(dx) + 2xy - 4x² = 0, subject to the initial condition y(0) = 0.

[4]

Q11.

If hati + hatj + hatk, 2hati + 5hatj, 3hati + 2hatj - 3hatk and hati - 6hatj - hatk respectively are the position vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Find whether vecAB and vecCD are collinear or not.

[4]

Page 4 of 6

Section D

Long answer II · 6 marks each · 6 of 6 shown

Q1.

If A = 1 1 1 \1 0 2 \3 1 1 , find A⁻¹. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12. OR Find the inverse of the following matrix using elementary operations: A = 1 2 -2 \-1 3 0 \0 -2 1 .

⚠ This question is not in the current syllabus — Elementary operations for inverse matrix (removed 2023-24)

[6]

Q2.

A tank with rectangular base and rectangular sides, open at the top, is to be constructed so that its depth is 2 m and volume is 8 m³. If building of tank costs ₹70 per square metre for the base and ₹45 per square metre for sides, what is the cost of least expensive tank?

[6]

Q3.

Using integration, find the area of triangle ABC, whose vertices are A(2, 5), B(4, 7) and C(6, 2). OR Find the area, lying above the x-axis and included between the circle x² + y² = 8x and the parabola y² = 4x.

⚠ This question is not in the current syllabus — Area between two curves (removed 2023-24)

[6]

Q4.

Find the vector and Cartesian equations of the plane passing through the points (2, 2, -1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above. OR Find the vector equation of the plane that contains the line vecr = (hati + hatj) + λ(hati + 2hatj - hatk) and the point (-1, 3, -4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.

[6]

Page 5 of 6

Q5.

A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, whereas the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced. What is the probability that it was produced by A?

[6]

Q6.

A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours of work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer's profit on an item of model A is ₹15 and on an item of model B is ₹10. How many items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.

[6]

Page 6 of 6