Mathematics · 2020 · Set 65/2/1

CBSE Class 12 Mathematics 2020 — Set 65/2/1

CBSE Class XII Board 2020 · Set 65/2/1

Real board examination⚠ Old pattern · pre-2020 syllabus

Sets

About the 2020 exam: The 2020 exam used CBSE's older pattern — 36 questions for 80 marks, with no case studies. Set 65/1/1 is complete; a few questions in Sets 2–3 depend on printed figures or could not be read reliably from the scanned paper, so they are omitted rather than guessed. Some topics examined in 2020 were removed from today's syllabus — those questions 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 2020. Every question below is solved the concept-first way. Sample papers are labelled honestly — never shown as a past exam.

Total marks: 80
Questions: 36
Duration: 180 min
Sections: 4

The marks / questions / duration above are the official exam pattern. We currently have 34 of this paper’s questions (94% of the full paper), with 34 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	Objective	20	1	20
BSection B	Short Answer I	6	2	12
CSection C	Short Answer II	6	4	24
DSection D	Long Answer	4	6	24
Total		36		80

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 2020 · Set 65/2/1

Series/Set: 65/2/1Roll No. ________

Time Allowed: 3 hoursMaximum Marks: 80

General Instructions

This question paper contains 36 questions divided into 4 sections — A, B, C, D.
Section A comprises 20 questions of 1 mark each (Objective).
Section B comprises 6 questions of 2 marks each (Short Answer I).
Section C comprises 6 questions of 4 marks each (Short Answer II).
Section D comprises 4 questions of 6 marks each (Long Answer).

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

Section A

Objective · 1 mark each · 18 of 20 shown

Q1.

The relation R in the set

\{1, 2, 3\}

given by

R = \{(1, 2), (2, 1), (1, 1)\}

is (A) symmetric and transitive, but not reflexive (B) reflexive and symmetric, but not transitive (C) symmetric, but neither reflexive nor transitive (D) an equivalence relation

[1]

Q2.

\tan^{-1} 3 + \tan^{-1}\lambda = \tan^{-1}\left(\dfrac{3+\lambda}{1-3\lambda}\right)

is valid for what values of

\lambda

? (A)

\lambda \in \left(-\dfrac{1}{3},\ \dfrac{1}{3}\right)

(B)

\lambda > \dfrac{1}{3}

(C)

\lambda < \dfrac{1}{3}

(D) All real values of

\lambda

⚠ This question is not in the current syllabus — Elementary properties and graphs of inverse trigonometric functions (ITF identities/proofs) (removed 2023-24)

[1]

Q3.

If A is a non-singular square matrix of order 3 such that

A^2 = 3A

, then value of

|A|

is (A)

-3

(B)

3

(C)

9

(D)

27

[1]

Q4.

The function

f : R \to R

given by

f(x) = -|x - 1|

is (A) continuous as well as differentiable at

x = 1

(B) not continuous but differentiable at

x = 1

x = 1

(D) neither continuous nor differentiable at

x = 1

[1]

Q5.

Let

A = \{1, 3, 5\}

. Then the number of equivalence relations in A containing

(1, 3)

is (A)

1

(B)

2

(C)

3

(D)

4

[1]

Q6.

The interval in which the function f given by

f(x) = x^2 e^{-x}

is strictly increasing, is (A)

(-\infty, \infty)

(B)

(-\infty, 0)

(C)

(2, \infty)

(D)

(0, 2)

[1]

Page 1 of 6

Q7.

If |veca| = 4 and -3 ≤ λ ≤ 2, then |λveca| lies in (A) [0, 12] (B) [2, 3] (C) [8, 12] (D) [-12, 8]

[1]

Q8.

The vectors 3hati - hatj + 2hatk, 2hati + hatj + 3hatk and hati + λhatj - hatk are coplanar if value of λ is (A) -2 (B) 0 (C) 2 (D) Any real number

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

[1]

Q9.

The area of a triangle formed by vertices O, A and B, where vecOA = hati + 2hatj + 3hatk and vecOB = -3hati - 2hatj + hatk is (A) 3√(5) sq. units (B) 5√(5) sq. units (C) 6√(5) sq. units (D) 4 sq. units

[1]

Q10.

The coordinates of the foot of the perpendicular drawn from the point (2, -3, 4) on the y-axis is (A) (2, 3, 4) (B) (-2, -3, -4) (C) (0, -3, 0) (D) (2, 0, 4)

[1]

Q11.

The range of the principal value branch of the function y = sec⁻¹ x is ______ . OR The principal value of cos⁻¹(-(1)/(2)) is ______ .

[1]

Q12.

Given a skew-symmetric matrix A = 0 a 1 \-1 b 1 \-1 c 0 , the value of (a + b + c)² is ______ .

[1]

Q13.

The distance between parallel planes 2x + y - 2z - 6 = 0 and 4x + 2y - 4z = 0 is ______ units. OR If P(1, 0, -3) is the foot of the perpendicular from the origin to the plane, then the cartesian equation of the plane is ______ .

[1]

Q14.

If the radius of the circle is increasing at the rate of 0.5 cm/s, then the rate of increase of its circumference is ______ .

[1]

Q15.

The corner points of the feasible region of an LPP are (0, 0), (0, 8), (2, 7), (5, 4) and (6, 0). The maximum profit P = 3x + 2y occurs at the point ______ .

[1]

Q16.

Differentiate sec²(x²) with respect to x². OR If y = f(x²) and f'(x) = e√(x), then find (dy)/(dx).

[1]

Page 2 of 6

Q17.

Find the general solution of the differential equation (dy)/(dx) ey-x = 1.

[1]

Q18.

Find the coordinates of the point where the line (x-1)/(3) = (y+4)/(7) = (z+4)/(2) cuts the xy-plane.

[1]

Section B

Short Answer I · 2 marks each · 6 of 6 shown

Q1.

If A = -3 2 \1 -1 and I = 1 0 \0 1 , find scalar k so that A² + I = kA.

[2]

Q2.

If f(x) = √((sec x - 1)/(sec x + 1)), find f'((π)/(3)). OR Find f'(x) if f(x) = (tan x)tan x.

[2]

Q3.

Find: ∫ (tan³ x)/(cos³ x) dx

[2]

Q4.

Find a vector vecr equally inclined to the three axes and whose magnitude is 3√(3) units. OR Find the angle between unit vectors veca and vecb so that √(3)veca - vecb is also a unit vector.

[2]

Q5.

Find the points of intersection of the line vecr = 2hati - hatj + 2hatk + λ(3hati + 4hatj + 2hatk) and the plane vecr · (hati - hatj + hatk) = 5.

[2]

Q6.

A purse contains 3 silver and 6 copper coins and a second purse contains 4 silver and 3 copper coins. If a coin is drawn at random from one of the two purses, find the probability that it is a silver coin.

[2]

Page 3 of 6

Section C

Short Answer II · 4 marks each · 6 of 6 shown

Q1.

Check whether the relation R in the set N of natural numbers given by R = \(a, b) : a is divisor of b\ is reflexive, symmetric or transitive. Also determine whether R is an equivalence relation. OR Prove that tan⁻¹(1)/(4) + tan⁻¹(2)/(9) = (1)/(2)sin⁻¹(4)/(5).

⚠ This question is not in the current syllabus — Elementary properties and graphs of inverse trigonometric functions (ITF identities/proofs) (removed 2023-24)

[4]

Q2.

If tan⁻¹((y)/(x)) = log√(x² + y²), prove that (dy)/(dx) = (x + y)/(x - y). OR If y = e^acos⁻¹ x, -1 < x < 1, then show that (1 - x²)(d² y)/(dx²) - x(dy)/(dx) - a² y = 0.

[4]

Q3.

Find: ∫ (x³ + 1)/(x³ - x) dx

[4]

Q4.

Solve the following differential equation: (1 + ey/x) dy + ey/x(1 - (y)/(x)) dx = 0 quad (x ≠ 0).

[4]

Q5.

Find the shortest distance between the lines vecr = 2hati - hatj + hatk + λ(3hati - 2hatj + 5hatk) vecr = 3hati + 2hatj - 4hatk + μ(4hati - hatj + 3hatk)

[4]

Page 4 of 6

Q6.

A cottage industry manufactures pedestal lamps and wooden shades. Both the products require machine time as well as craftsman time in the making. The number of hour(s) required for producing 1 unit of each and the corresponding profit is given in the following table: Pedestal lamp: Machine Time = 1.5 hours, Craftsman time = 3 hours, Profit (in Rs) = 30 Wooden shades: Machine Time = 3 hours, Craftsman time = 1 hour, Profit (in Rs) = 20 In a day, the factory has availability of not more than 42 hours of machine time and 24 hours of craftsman time. Assuming that all items manufactured are sold, how should the manufacturer schedule his daily production in order to maximise the profit? Formulate it as an LPP and solve it graphically.

[4]

Section D

Long Answer · 6 marks each · 4 of 4 shown

Q1.

If A = 5 -1 4 \2 3 5 \5 -2 6 , find A⁻¹ and use it to solve the following system of equations: 5x - y + 4z = 5 2x + 3y + 5z = 2 5x - 2y + 6z = -1 OR If x, y, z are different and x x² 1+x³ y y² 1+y³ z z² 1+z³ = 0, then using properties of determinants show that 1 + xyz = 0.

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

[6]

Q2.

Amongst all open (from the top) right circular cylindrical boxes of volume 125π cm³, find the dimensions of the box which has the least surface area.

[6]

Q3.

Using integration, find the area lying above x-axis and included between the circle x² + y² = 8x and inside the parabola y² = 4x. OR Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A(2, 0), B(4, 5) and C(6, 3).

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

[6]

Page 5 of 6

Q4.

Find the probability distribution of the random variable X, which denotes the number of doublets in four throws of a pair of dice. Hence, find the mean of the number of doublets (X).

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

[6]

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