Mathematics · 2019 · Set 65/2/1

CBSE Class 12 Mathematics 2019 — Set 65/2/1

CBSE Class XII Board 2019 · Set 65/2/1

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 6 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 65/2/1

Series/Set: 65/2/1Roll 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 is a square matrix satisfying

A'A = I

, write the value of

|A|

[1]

Q2.

y = x|x|

, find

\dfrac{dy}{dx}

for

x < 0

[1]

Q3.

Find the order and degree (if defined) of the differential equation

\dfrac{d^2y}{dx^2} + x\left(\dfrac{dy}{dx}\right)^2 = 2x^2 \log\left(\dfrac{d^2y}{dx^2}\right)

[1]

Q4.

Find the direction cosines of a line which makes equal angles with the coordinate axes.

(OR)

A line passes through the point with position vector

2\hat{i} - \hat{j} + 4\hat{k}

and is in the direction of the vector

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

. Find the equation of the line in cartesian form.

[1]

Section B

Short answer · 2 marks each · 8 of 8 shown

Q1.

Examine whether the operation

*

defined on

\mathbb{R}

, the set of all real numbers, by

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

is a binary operation or not, and if it is a binary operation, find whether it is associative or not.

[2]

Page 1 of 6

Q2.

If A = 4 2 \-1 1 , show that (A - 2I)(A - 3I) = 0.

[2]

Q3.

Find : ∫ (sin³ x + cos³ x)/(sin² x cos² x) dx OR Find : ∫ (x - 3)/((x - 1)³) ex dx

[2]

Q4.

A coin is tossed 5 times. What is the probability of getting (i) 3 heads, (ii) at most 3 heads ? OR Find the probability distribution of X, the number of heads in a simultaneous toss of two coins.

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

[2]

Q5.

Find the differential equation of the family of curves y = Ae2x + Be-2x, 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.

Find: ∫ √(3 - 2x - x²) dx.

[2]

Q7.

If |veca| = 2, |vecb| = 7 and veca × vecb = 3hati + 2hatj + 6hatk, find the angle between veca and vecb. OR Find the volume of a cuboid whose edges are given by -3hati + 7hatj + 5hatk, -5hati + 7hatj - 3hatk and 7hati - 5hatj - 3hatk.

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

[2]

Q8.

If P(not A) = 0.7, P(B) = 0.7 and P(B/A) = 0.5, then find P(A/B).

[2]

Page 2 of 6

Section C

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

Q1.

Check whether the relation R defined on the set A = \1, 2, 3, 4, 5, 6\ as R = \(a, b) : b = a + 1\ is reflexive, symmetric or transitive. OR Let f : N → Y be a function defined as f(x) = 4x + 3, where Y = \y ∈ N : y = 4x + 3, for some x ∈ N\. Show that f is invertible. Find its inverse.

⚠ This question is not in the current syllabus — Inverse of a function (removed 2023-24)

[4]

Q2.

If (x-a)² + (y-b)² = c², for some c > 0, prove that frac[1 + ((dy)/(dx))²]3/2(d²y)/(dx²) is a constant independent of a and b.

[4]

Q3.

Find the equation of the normal to the curve x² = 4y which passes through the point (-1, 4).

[4]

Q4.

Solve the differential equation: x (dy)/(dx) = y - x tan ((y)/(x)) OR Solve the differential equation: (dy)/(dx) = - [ (x + y cos x)/(1 + sin x) ] Solve the differential equation : x (dy)/(dx) = y - x tan ((y)/(x)) OR Solve the differential equation : (dy)/(dx) = - [ (x + y cos x)/(1 + sin x) ]

[4]

Q5.

The scalar product of the vector veca = hati + hatj + hatk with a unit vector along the sum of the vectors vecb = 2hati + 4hatj - 5hatk and vecc = λhati + 2hatj + 3hatk is equal to 1. Find the value of λ and hence find the unit vector along vecb + vecc.

[4]

Page 3 of 6

Q6.

Find the value of sin(cos⁻¹ (4)/(5) + tan⁻¹ (2)/(3)).

[4]

Q7.

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]

Q8.

If x√(1+y) + y√(1+x) = 0 and x ≠ y, prove that (dy)/(dx) = -(1)/((x+1)²). OR If (cos x)y = (sin y)x, find (dy)/(dx).

[4]

Q9.

Find: ∫ (x² + x + 1)/((x+2)(x²+1)) dx.

[4]

Q10.

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

[4]

Q11.

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]

Page 4 of 6

Section D

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

Q1.

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is frac2R√(3). Also find the maximum volume.

[6]

Q2.

Using method of integration, find the area of the triangle whose vertices are (1, 0), (2, 2) and (3, 1). OR Using method of integration, find the area of the region enclosed between two circles x² + y² = 4 and (x - 2)² + y² = 4.

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

[6]

Q3.

A company manufactures two types of items A and B. Each unit of type A requires 3 g of silver and 1 g of gold, while each unit of type B requires 1 g of silver and 2 g of gold. The company can use at most 9 g of silver and 8 g of gold. If a unit of type A fetches a profit of ₹ 40 and a unit of type B fetches a profit of ₹ 50, how many units of each type should the company manufacture to earn maximum profit? Convert the above problem into a Linear Programming Problem and solve it graphically. Also find the maximum profit. A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of type B requires 1 g of silver and 2 g of gold. The company can use at the most 9 g of silver and 8 g of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, find the number of units of each type that the company should produce to maximize profit. Formulate the above LPP and solve it graphically and also find the maximum profit.

[6]

Q4.

If A = 1 3 4 \2 1 2 \5 1 1 , find A⁻¹. Hence solve the system of equations x + 3y + 4z = 8, 2x + y + 2z = 5 and 5x + y + z = 7. OR Find the inverse of the following matrix, using elementary transformations: A = 2 0 -1 \5 1 0 \0 1 3 .

[6]

Page 5 of 6

Q5.

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.

[6]

Q6.

There are three coins. One is a two-headed coin, another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed. If it shows heads, what is the probability that it is the two-headed coin?

[6]

Page 6 of 6