Mathematics · 2019 · Set 65/3/1

CBSE Class 12 Mathematics 2019 — Set 65/3/1

CBSE Class XII Board 2019 · Set 65/3/1

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 8 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 65/3/1

Series/Set: 65/3/1Roll 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.

A

is a square matrix of order 3 with

|A| = 4

, then write the value of

|-2A|

[1]

Q2.

y = \sin^{-1} x + \cos^{-1} x

, find

\dfrac{dy}{dx}

[1]

Q3.

Write the order and degree of the differential equation

\left(\dfrac{d^4y}{dx^4}\right)^2 = \left[x + \left(\dfrac{dy}{dx}\right)^2\right]^3

[1]

Q4.

If a line has the direction ratios

-18, 12, -4

, then what are its direction cosines?

(OR)

Find the Cartesian equation of the line which passes through the point

(-2, 4, -5)

and is parallel to the line

\dfrac{x+3}{3} = \dfrac{4-y}{5} = \dfrac{z+8}{6}

[1]

Section B

Short answer · 2 marks each · 8 of 8 shown

Q1.

*

is defined on the set

R

of all real numbers by

a * b = \sqrt{a^2 + b^2}

, find the identity element, if it exists, in

R

with respect to

*

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

[2]

Page 1 of 6

Q2.

If A = 0 2 \3 -4 and kA = 0 3a \2b 24 , then find the values of k, a and b.

[2]

Q3.

Find ∫ dfracsin x - cos x√(1 + sin 2x) dx, 0 < x < (π)/(2).

[2]

Q4.

Find ∫ (sin(x-a))/(sin(x+a)) dx. OR Find ∫ (log x)² dx.

[2]

Q5.

Form the differential equation representing the family of curves y² = m(a² - x²) by eliminating the arbitrary constants m and a.

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

[2]

Q6.

Find a unit vector perpendicular to both the vectors veca and vecb, where veca = hati - 7hatj + 7hatk and vecb = 3hati - 2hatj + 2hatk. OR Show that the vectors hati - 2hatj + 3hatk, -2hati + 3hatj - 4hatk and hati - 3hatj + 5hatk are coplanar.

[2]

Q7.

Mother, father and son line up at random for a family photo. If A and B are two events given by A = Son on one end, B = Father in the middle, find P(B/A).

[2]

Q8.

Let X be a random variable which assumes values x₁, x₂, x₃, x₄ such that 2P(X = x₁) = 3P(X = x₂) = P(X = x₃) = 5P(X = x₄). Find the probability distribution of X. OR A coin is tossed 5 times. Find the probability of getting (i) at least 4 heads, and (ii) at most 4 heads.

⚠ 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.

Using properties of determinants, show that 3a -a+b -a+c \-b+a 3b -b+c \-c+a -c+b 3c = 3(a+b+c)(ab+bc+ca)

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

[4]

Q2.

Show that the relation R on the set Z of all integers, given by R = \(a, b) : 2 divides (a - b)\, is an equivalence relation. OR If f(x) = (4x+3)/(6x-4), x ≠ (2)/(3), show that (f ° f)(x) = x for all x ≠ (2)/(3). Also, find the inverse of f.

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

[4]

Q3.

If tan⁻¹ x - cot⁻¹ x = tan⁻¹(dfrac1√(3)), x > 0, then find the value of x and hence find the value of sec⁻¹((2)/(x)).

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

[4]

Q4.

If sin y = x sin(a + y), prove that (dy)/(dx) = (sin²(a+y))/(sin a). OR If (sin x)y = x + y, find (dy)/(dx).

[4]

Q5.

If y = (sec⁻¹ x)², x > 0, show that x²(x² - 1)(d²y)/(dx²) + (2x³ - x)(dy)/(dx) - 2 = 0.

[4]

Page 3 of 6

Q6.

Find the equation of the tangent and the normal to the curve y = (x-7)/((x-2)(x-3)) at the point where it cuts the x-axis.

[4]

Q7.

Find: ∫ (sin 2x)/((sin² x + 1)(sin² x + 3)) dx.

[4]

Q8.

Prove that ∫ab f(x) dx = ∫ab f(a + b - x) dx and hence evaluate ∫π/6π/3 dfracdx1 + √(tan x).

[4]

Q9.

Solve the differential equation: (dy)/(dx) = (x + y)/(x - y). OR Solve the differential equation: (1 + x²) dy + 2xy dx = cot x dx.

[4]

Q10.

[4]

Q11.

Find the value of λ for which the following lines are perpendicular to each other: (x-5)/(5λ+2) = (2-y)/(5) = (1-z)/(-1); (x)/(1) = dfracy + (1)/(2)2λ = (z-1)/(3); hence, find whether the lines intersect or not.

[4]

Page 4 of 6

Section D

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

Q1.

Show that for the matrix A = 1 1 1 \1 2 -3 \2 -1 3 , A³ - 6A² + 5A + 11I = 0. Hence, find A⁻¹. OR Using matrix method, solve the following system of equations : 3x - 2y + 3z = 8 2x + y - z = 1 4x - 3y + 2z = 4

[6]

Q2.

Find the vector and cartesian equations of the plane passing through the points having position vectors hati + hatj - 2hatk, 2hati - hatj + hatk and hati + 2hatj + hatk. Write the equation of a plane passing through a point (2, 3, 7) and parallel to the plane obtained above. Hence, find the distance between the two parallel planes. OR Find the equation of the line passing through (2, -1, 2) and (5, 3, 4) and of the plane passing through (2, 0, 3), (1, 1, 5) and (3, 2, 4). Also, find their point of intersection.

⚠ This question is not in the current syllabus — Out-of-syllabus: plane content (equation of a plane / plane through intersection of lines) — removed from the rationalized NCERT Class-12 Ch11 syllabus. Retained for reference, flagged out of core syllabus.

[6]

Q3.

Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.

[6]

Q4.

Find the area of the triangle whose vertices are (-1, 1), (0, 5) and (3, 2), using integration. OR Find the area of the region bounded by the curves (x-1)² + y² = 1 and x² + y² = 1, using integration.

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

[6]

Page 5 of 6

Q5.

There are two boxes I and II. Box I contains 3 red and 6 black balls. Box II contains 5 red and 5 black balls. One of the two boxes, box I and box II, is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from box II.

[6]

Q6.

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours and 20 minutes available for cutting and 4 hours available for assembling. The profit is Rs. 50 each for type A and Rs. 60 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit? Formulate the above LPP and solve it graphically and also find the maximum profit.

[6]

Page 6 of 6