Mathematics · 2023 · Set 65/3/1

CBSE Class 12 Mathematics 2023 — Set 65/3/1

CBSE Class XII Board 2023 · Set 65/3/1

Real board examination

Sets

This paper has 2 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 2023. Every question below is solved the concept-first way. Sample papers are labelled honestly — never shown as a past exam.

Total marks: 80
Questions: 38
Duration: 180 min
Sections: 5

The marks / questions / duration above are the official exam pattern. We currently have 38 of this paper’s questions (100% of the full paper), with 38 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	MCQ/Assertion-Reason	20	1	20
BSection B	Very Short Answer	5	2	10
CSection C	Short Answer	6	3	18
DSection D	Long Answer	4	5	20
ESection E	Case study	3	4	12
Total		38		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 2023 · Set 65/3/1

Series/Set: 65/3/1Roll No. ________

Time Allowed: 3 hoursMaximum Marks: 80

General Instructions

This question paper contains 38 questions divided into 5 sections — A, B, C, D, E.
Section A comprises 20 questions of 1 mark each (MCQ/Assertion-Reason).
Section B comprises 5 questions of 2 marks each (Very Short Answer).
Section C comprises 6 questions of 3 marks each (Short Answer).
Section D comprises 4 questions of 5 marks each (Long Answer).
Section E comprises 3 questions of 4 marks each (Case study).

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

Section A

MCQ/Assertion-Reason · 1 mark each · 20 of 20 shown

Q1.

A = \begin{bmatrix} 1 & 4 & x \\ z & 2 & y \\ -3 & -1 & 3 \end{bmatrix}

is a symmetric matrix, then the value of

x + y + z

is :

(a) $10$
(b) $6$
(c) $8$
(d) $0$

[1]

Q2.

A \cdot (\text{adj } A) = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}

, then the value of

|A| + |\text{adj } A|

is equal to :

(a) $12$
(b) $9$
(c) $3$
(d) $27$

[1]

Q3.

A

and

B

are skew-symmetric matrices of same order.

AB

is symmetric, if :

(a) $AB = O$
(b) $AB = -BA$
(c) $AB = BA$
(d) $BA = O$

[1]

Q4.

For what value of

x \in \left[0, \frac{\pi}{2}\right]

, is

A + A' = \sqrt{3}\, I

, where

A = \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix}

(a) $\frac{\pi}{3}$
(b) $\frac{\pi}{6}$
(c) $0$
(d) $\frac{\pi}{2}$

[1]

Q5.

Let

A

be the area of a triangle having vertices

(x_1, y_1)

(x_2, y_2)

and

(x_3, y_3)

. Which of the following is correct ?

(a) $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = \pm A$
(b) $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = \pm 2A$
(c) $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = \pm \frac{A}{2}$
(d) $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}^2 = A^2$

[1]

Q6.

\int 2^{x+2}\, dx

is equal to :

(a) $2^{x+2} + C$
(b) $2^{x+2} \log 2 + C$
(c) $\frac{2^{x+2}}{\log 2} + C$
(d) $2 \cdot \frac{2^x}{\log 2} + C$

[1]

Page 1 of 6

Q7.

∫ (2 cos 2x - 1)/(1 + 2 sin x) dx is equal to : (a) x - 2 cos x + C (b) x + 2 cos x + C (c) -x - 2 cos x + C (d) -x + 2 cos x + C

[1]

Q8.

The solution of the differential equation (dx)/(x) + (dy)/(y) = 0 is : (a) (1)/(x) + (1)/(y) = C (b) log x - log y = C (c) xy = C (d) x + y = C

[1]

Q9.

What is the product of the order and degree of the differential equation (d² y)/(dx²) sin y + ((dy)/(dx))³ cos y = √(y) ? (a) 3 (b) 2 (c) 6 (d) not defined

[1]

Q10.

If a vector makes an angle of (π)/(4) with the positive directions of both x-axis and y-axis, then the angle which it makes with positive z-axis is : (a) (π)/(4) (b) (3π)/(4) (c) (π)/(2) (d) 0

[1]

Q11.

veca and vecb are two non-zero vectors such that the projection of veca on vecb is 0. The angle between veca and vecb is : (a) (π)/(2) (b) π (c) (π)/(4) (d) 0

[1]

Q12.

In triangle ABC, vecAB = hati + hatj + 2hatk and vecAC = 3hati - hatj + 4hatk. If D is mid-point of BC, then vector vecAD is equal to : (a) 4hati + 6hatk (b) 2hati - 2hatj + 2hatk (c) hati - hatj + hatk (d) 2hati + 3hatk

[1]

Q13.

The value of λ for which the angle between the lines vecr = hati + hatj + hatk + p(2hati + hatj + 2hatk) and vecr = (1+q)hati + (1+qλ)hatj + (1+q)hatk is (π)/(2) is : (a) -4 (b) 4 (c) 2 (d) -2

[1]

Q14.

If P(A ∩ B) = (1)/(8) and P(barA) = (3)/(4), then P((B)/(A)) is equal to : (a) (1)/(2) (b) (1)/(3) (c) (1)/(6) (d) (2)/(3)

[1]

Q15.

The value of k for which function f(x) = x², x ≥ 0 kx, x < 0 is differentiable at x = 0 is : (a) 1 (b) 2 (c) any real number (d) 0

[1]

Page 2 of 6

Q16.

If y = (cos x - sin x)/(cos x + sin x), then (dy)/(dx) is : (a) -sec²((π)/(4) - x) (b) sec²((π)/(4) - x) (c) log|sec((π)/(4) - x)| (d) -log|sec((π)/(4) - x)|

[1]

Q17.

The number of feasible solutions of the linear programming problem given as Maximize z = 15x + 30y subject to constraints : 3x + y ≤ 12, x + 2y ≤ 10, x ≥ 0, y ≥ 0 is (a) 1 (b) 2 (c) 3 (d) infinite

[1]

Q18.

The feasible region of a linear programming problem is shown in the figure below (a shaded region bounded by the lines x + 2y = 4 and x + y = 3). Which of the following are the possible constraints ? (a) x + 2y ≥ 4, x + y ≤ 3, x ≥ 0, y ≥ 0 (b) x + 2y ≤ 4, x + y ≤ 3, x ≥ 0, y ≥ 0 (c) x + 2y ≥ 4, x + y ≥ 3, x ≥ 0, y ≥ 0 (d) x + 2y ≥ 4, x + y ≥ 3, x ≤ 0, y ≤ 0

[1]

Q19.

(Assertion-Reason) Assertion (A) : Range of [sin⁻¹ x + 2 cos⁻¹ x] is [0, π]. Reason (R) : Principal value branch of sin⁻¹ x has range [-(π)/(2), (π)/(2)]. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (c) Assertion (A) is true and Reason (R) is false. (d) Assertion (A) is false and Reason (R) is true.

[1]

Q20.

(Assertion-Reason) Assertion (A) : A line through the points (4, 7, 8) and (2, 3, 4) is parallel to a line through the points (-1, -2, 1) and (1, 2, 5). Reason (R) : Lines vecr = veca₁ + λ vecb₁ and vecr = veca₂ + μ vecb₂ are parallel if vecb₁ · vecb₂ = 0. (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (c) Assertion (A) is true and Reason (R) is false. (d) Assertion (A) is false and Reason (R) is true.

[1]

Section B

Very Short Answer · 2 marks each · 5 of 5 shown

Q1.

If vecr = 3hati - 2hatj + 6hatk, find the value of (vecr × hatj) · (vecr × hatk) - 12.

[2]

Q2.

If the angle between the lines (x-5)/(α) = (y+2)/(-5) = fracz + (24)/(5)β and (x)/(1) = (y)/(0) = (z)/(1) is (π)/(4), find the relation between α and β.

[2]

Page 3 of 6

Q3.

If f(x) = a(tan x - cot x), where a > 0, then find whether f(x) is increasing or decreasing function in its domain.

[2]

Q4.

(a) Evaluate : 3 sin⁻¹(frac1√(2)) + 2 cos⁻¹(frac√(3)2) + cos⁻¹(0) OR (b) Draw the graph of f(x) = sin⁻¹ x, x ∈ [-frac1√(2), frac1√(2)]. Also, write range of f(x).

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

[2]

Q5.

(a) If y = x(1)/(x), then find (dy)/(dx) at x = 1. OR (b) If x = a sin 2t, y = a(cos 2t + log tan t), then find (dy)/(dx).

[2]

Section C

Short Answer · 3 marks each · 6 of 6 shown

Q1.

(a) Find the general solution of the differential equation : (d)/(dx)(xy²) = 2y(1 + x²) OR (b) Solve the following differential equation : xe(y)/(x) - y + x(dy)/(dx) = 0

[3]

Q2.

Evaluate : ∫₁³ frac√(4-x)√(x) + √(4-x) dx

[3]

Q3.

Evaluate : ∫₁e frac1√(4x² - (x log x)²) dx

[3]

Q4.

(a) Find : ∫ (cos x)/(sin 3x) dx OR (b) Find : ∫ x² log(x² + 1) dx

[3]

Page 4 of 6

Q5.

Determine graphically the minimum value of the following objective function : z = 500x + 400y subject to constraints x + y ≤ 200, x ≥ 20, y ≥ 4x, y ≥ 0.

[3]

Q6.

(a) A pair of dice is thrown simultaneously. If X denotes the absolute difference of numbers obtained on the pair of dice, then find the probability distribution of X. OR (b) There are two coins. One of them is a biased coin such that P(head) : P(tail) is 1 : 3 and the other coin is a fair coin. A coin is selected at random and tossed once. If the coin showed head, then find the probability that it is a biased coin.

[3]

Section D

Long Answer · 5 marks each · 4 of 4 shown

Q1.

Show that a function f : mathbbR → mathbbR defined as f(x) = (5x - 3)/(4) is both one-one and onto.

[5]

Q2.

The area of the region bounded by the line y = mx (m > 0), the curve x² + y² = 4 and the x-axis in the first quadrant is (π)/(2) units. Using integration, find the value of m.

[5]

Q3.

(a) If A = 1 0 2 \0 2 1 \2 0 3 , then show that A³ - 6A² + 7A + 2I = O. OR (b) If A = 3 2 \5 -7 , then find A⁻¹ and use it to solve the following system of equations : 3x + 5y = 11, 2x - 7y = -3.

[5]

Page 5 of 6

Q4.

(a) Find the value of b so that the lines (x-1)/(2) = (y-b)/(3) = (z-3)/(4) and (x-4)/(5) = (y-1)/(2) = z are intersecting lines. Also, find the point of intersection of these given lines. OR (b) Find the equations of all the sides of the parallelogram ABCD whose vertices are A(4, 7, 8), B(2, 3, 4), C(-1, -2, 1) and D(1, 2, 5). Also, find the coordinates of the foot of the perpendicular from A to CD.

[5]

Section E

Case study · 4 marks each · 3 of 3 shown

Q1.

Case Study - 1 : An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16 vertices. The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the bottom face and the following table give the probability distribution of X : X : 1, 2, 3, 4, 5, 6, 7, 8 P(X) : p,2p,2p,p,2p,p²,2p²,7p² + p Based on the above information, answer the following questions : (i) Find the value of p. [1 mark] (ii) Find P(X > 6). [1 mark] (iii) (a) Find P(X = 3m), where m is a natural number. [2 marks] OR (iii) (b) Find the mean E(X). [2 marks]

⚠ This question is not in the current syllabus — Out-of-syllabus: random-variable / probability-distribution / mean-variance / binomial content — removed from the rationalized NCERT Class-12 Ch13 syllabus. Retained for reference, flagged out of core syllabus.

[4]

Q2.

Case Study - 2 : In order to set up a rain water harvesting system, a tank to collect rain water is to be dug. The tank should have a square base and a capacity of 250 m³. The cost of land is Rs 5,000 per square metre and cost of digging increases with depth and for the whole tank, it is Rs 40,000 h², where h is the depth of the tank in metres. x is the side of the square base of the tank in metres. Based on the above information, answer the following questions : (i) Find the total cost C of digging the tank in terms of x. [1 mark] (ii) Find (dC)/(dx). [1 mark] (iii) (a) Find the value of x for which cost C is minimum. [2 marks] OR (iii) (b) Check whether the cost function C(x) expressed in terms of x is increasing or not, where x > 0. [2 marks]

[4]

Q3.

Case Study - 3 : A volleyball player serves the ball which takes a parabolic path given by the equation h(t) = -(7)/(2)t² + (13)/(2)t + 1, where h(t) is the height of ball at any time t (in seconds), (t ≥ 0). Based on the above information, answer the following questions : (i) Is h(t) a continuous function ? Justify. [2 marks] (ii) Find the time at which the height of the ball is maximum. [2 marks]

[4]

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