Mathematics · 2020 · Set 65/1/1

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

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

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 7 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 36 of this paper’s questions (100% of the full paper), with 36 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 65/1/1

Series/Set: 65/1/1Roll 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 · 20 of 20 shown

Q1.

If A is a square matrix of order 3 and

|A| = 5

, then the value of

|2A'|

is (A)

-10

(B)

10

(C)

-40

(D)

40

[1]

Q2.

If A is a square matrix such that

A^2 = A

, then

(I - A)^3 + A

is equal to (A)

I

(B)

0

(C)

I - A

(D)

I + A

[1]

Q3.

The principal value of

\tan^{-1} \left(\tan \frac{3\pi}{5}\right)

is (A)

\frac{2\pi}{5}

(B)

-\frac{2\pi}{5}

(C)

\frac{3\pi}{5}

(D)

-\frac{3\pi}{5}

[1]

Q4.

If the projection of

\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}

\vec{b} = 2\hat{i} + \lambda\hat{k}

is zero, then the value of

\lambda

is (A)

0

(B)

1

(C)

-\frac{2}{3}

(D)

-\frac{3}{2}

[1]

Q5.

The vector equation of the line passing through the point

(-1, 5, 4)

and perpendicular to the plane

z = 0

is (A)

\vec{r} = -\hat{i} + 5\hat{j} + 4\hat{k} + \lambda(\hat{i} + \hat{j})

(B)

\vec{r} = -\hat{i} + 5\hat{j} + (4 + \lambda)\hat{k}

(C)

\vec{r} = \hat{i} - 5\hat{j} - 4\hat{k} + \lambda\hat{k}

(D)

\vec{r} = \lambda\hat{k}

[1]

Q6.

The number of arbitrary constants in the particular solution of a differential equation of second order is (are) (A)

0

(B)

1

(C)

2

(D)

3

[1]

Page 1 of 6

Q7.

∫-(π)/(4)(π)/(4) sec² x dx is equal to (A) -1 (B) 0 (C) 1 (D) 2

[1]

Q8.

The length of the perpendicular drawn from the point (4, -7, 3) to the y-axis will be (A) 3 units (B) 4 units (C) 5 units (D) 7 units

[1]

Q9.

If A and B are two independent events, where P (A) = (1)/(3) and P (B) = (1)/(4), then P(B'|A) is equal to (A) (1)/(4) (B) (1)/(3) (C) (3)/(4) (D) 1

[1]

Q10.

The corner points of the feasible region determined by a system of linear inequalities are (0, 0), (4, 0), (2, 4) and (0, 5). If the maximum value of z = ax + by, where a, b > 0, occurs at both points (2, 4) and (4, 0), then (A) a = 2b (B) 2a = b (C) a = b (D) 3a = b Fill in the blanks for all questions from question number 11 to 15.

[1]

Q11.

If for all a₁, a₂ ∈ A, (a₁, a₂) ∈ R implies (a₂, a₁) ∈ R, then the relation R defined on set A is called a _____ relation.

[1]

Q12.

The greatest integer function f(x) = [x], defined for 0 < x < 2, is not differentiable at x = \_\_\_\_\_\_\_\_\_\_.

[1]

Q13.

If the order of matrix A is 3 × 2, then the order of matrix A' will be _____. OR A square matrix A will be a skew-symmetric matrix, if _____.

[1]

Q14.

The equation of the normal to the parabola y² = 8x at its origin is ____. OR The radius of a circle is increasing uniformly at the rate of 3 cm/s. At the instant when the radius of the circle is 2 cm, the area of the circle is increasing at the rate of ____ cm²/s.

[1]

Q15.

The position vectors of two points A and B are respectively vecOA = 2hati - hatj - hatk and vecOB = 2hati - hatj + 2hatk. If point P divides the line segment AB in the ratio 2:1, then its position vector is ____. Questions number 16 to 20 are Very Short Answer Type Questions.

[1]

Q16.

If A = 2 0 0 \-1 2 3 \3 3 5 , then find A(adj A).

[1]

Page 2 of 6

Q17.

Evaluate: ∫ x⁴ log x dx OR Evaluate: ∫ frac2x√[3]x²+1 dx

[1]

Q18.

Two cards are drawn successively and without replacement from a well-shuffled deck of 52 cards. Find the probability that one card is red and the other is black.

[1]

Q19.

Find : ∫ fracdx√(9-4x²)

[1]

Q20.

Evaluate : ∫₁³ |2x - 1| dx

[1]

Section B

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

Q1.

Prove that: sin⁻¹(2x√(1-x²)) = 2cos⁻¹x, frac1√(2) ≤ x ≤ 1. OR Consider f: R_+ → (7, ∞) defined by f(x) = 16x² + 24x + 7. Find the inverse of the function f.

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

[2]

Q2.

Find the points on the curve y = x³ - 3x² - 4x at which the tangents are parallel to the line 4x + y - 3 = 0.

[2]

Q3.

Find a unit vector perpendicular to each of the vectors veca = 5hati + 6hatj - 2hatk and vecb = 7hati + 6hatj + 2hatk. OR Find the volume of the parallelepiped whose adjacent edges are 2veca, -vecb and 3vecc, where veca = hati - hatj + 2hatk, vecb = 3hati + 4hatj - 5hatk and vecc = 2hati - hatj + 3hatk.

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

[2]

Q4.

Find the value of k for which the lines x - 2 = 2y + 1 = -z + 1 and x = -y = kz are perpendicular to each other.

[2]

Page 3 of 6

Q5.

If x = at², y = 2at, then find (d²y)/(dx²).

[2]

Q6.

The probability of finding a green signal on a busy crossing X is 30%. What is the probability of finding a green signal on X on two consecutive days out of three?

[2]

Section C

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

Q1.

Let N be the set of natural numbers. A relation R on N × N is defined by "(a, b) R (c, d) if and only if ad = bc, for all a, b, c, d ∈ N". Show that R is an equivalence relation.

[4]

Q2.

If y = ex^2 cos x + (cos x)x, then find (dy)/(dx).

[4]

Q3.

Find the general solution of the differential equation y ey dx = (y³ + 2x ey) dy. OR Find the particular solution of the differential equation x (dy)/(dx) = y - x tan ((y)/(x)), given that x = 1 and y = (π)/(4).

⚠ This question is not in the current syllabus — Linear differential equation of the form dx/dy + Px = Q (removed 2023-24)

[4]

Q4.

A furniture dealer invests his money in tables or chairs or both. He has ₹ 50,000 to invest and has space to store at most 35 items. A chair costs ₹ 1,000 and a table costs ₹ 2,000. This dealer earns a profit of ₹ 150 by selling a chair and ₹ 250 by selling a table. Formulate a Linear Programming Problem and solve the problem graphically to maximize the profit.

[4]

Page 4 of 6

Q5.

Two bags I and II are given. Bag I contains 3 red and 5 black balls, while Bag II contains 4 red and 3 black balls. A ball is transferred at random from Bag I to Bag II and then a ball is drawn at random from Bag II. If the ball drawn is black, find the probability that the transferred ball is black. OR An urn contains 5 red, 2 white and 3 black balls. Three balls are drawn at random from this urn one-by-one without replacement. Find the probability distribution of the number of white balls drawn. Also, find the mean and variance of the number of white balls drawn.

⚠ This question is not in the current syllabus — Variance of a random variable (removed 2023-24)

[4]

Q6.

Find : ∫ sec³ x dx

[4]

Section D

Long Answer · 6 marks each · 4 of 4 shown

Q1.

If A = 1 2 -3 \3 2 -2 \2 -1 1 , then find A⁻¹ and using it, solve the following system of equations : x + 2y - 3z = 6 3x + 2y - 2z = 3 2x - y + z = 2 OR Using properties of determinants, prove that (b+c)² a² bc \(c+a)² b² ca \(a+b)² c² ab = (a-b)(b-c)(c-a)(a+b+c)(a²+b²+c²)

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

[6]

Q2.

Using integration, find the area of the region bounded by the triangle whose vertices are (2, -2), (4, 5) and (6, 2).

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

[6]

Q3.

Show that the height of the right circular cylinder of maximum volume inscribed in a right circular cone of radius r and height h is one-third of the height of the cone and the maximum volume of the cylinder is (4)/(9) of the volume of the cone.

[6]

Page 5 of 6

Q4.

Find the equation of the plane which contains the point A(2, 1, -1) and is perpendicular to the line of intersection of the planes 2x + y - z = 3 and x + 2y + z = 2. Also, find the angle between the obtained plane and y-axis. OR Find the distance of the point P(-2, -4, 7) from the point Q which is the intersection of the line vecr = (3hati - 2hatj + 6hatk) + λ(2hati - hatj + 2hatk) and the plane vecr · (hati - hatj + hatk) = 6. Also, write the vector equation of the line PQ.

⚠ This question is not in the current syllabus — Angle between a line and a plane (removed 2023-24)

[6]

Page 6 of 6