Mathematics · 2023 · Set 65/2/1

CBSE Class 12 Mathematics 2023 — Set 65/2/1

CBSE Class XII Board 2023 · Set 65/2/1

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 65/2/1

Series/Set: 65/2/1Roll 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} 0 & 1 \\ 0 & 0 \end{bmatrix}

, then

A^{2023}

is equal to:

(a) $\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$
(b) $\begin{bmatrix} 0 & 2023 \\ 0 & 0 \end{bmatrix}$
(c) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
(d) $\begin{bmatrix} 2023 & 0 \\ 0 & 2023 \end{bmatrix}$

[1]

Q2.

\begin{bmatrix} 2 & 0 \\ 5 & 4 \end{bmatrix} = P + Q

, where

P

is a symmetric and

Q

is a skew symmetric matrix, then

Q

is equal to:

(a) $\begin{bmatrix} 2 & \frac{5}{2} \\ \frac{5}{2} & 4 \end{bmatrix}$
(b) $\begin{bmatrix} 0 & -\frac{5}{2} \\ \frac{5}{2} & 0 \end{bmatrix}$
(c) $\begin{bmatrix} 0 & \frac{5}{2} \\ -\frac{5}{2} & 0 \end{bmatrix}$
(d) $\begin{bmatrix} 2 & -\frac{5}{2} \\ \frac{5}{2} & 4 \end{bmatrix}$

[1]

Q3.

\begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & a & 1 \end{bmatrix}

is a non-singular matrix and

a \in A

, then the set

A

is:

(a) $\mathbb{R}$
(b) $\{0\}$
(c) $\{4\}$
(d) $\mathbb{R} - \{4\}$

[1]

Q4.

|A| = |kA|

, where

A

is a square matrix of order 2, then sum of all possible values of

k

is:

(a) 1
(b) $-1$
(c) 2
(d) 0

[1]

Q5.

\frac{d}{dx}[f(x)] = ax + b

and

f(0) = 0

, then

f(x)

is equal to:

(a) $a + b$
(b) $\frac{ax^2}{2} + bx$
(c) $\frac{ax^2}{2} + bx + c$
(d) $b$

[1]

Q6.

Degree of the differential equation

\sin x + \cos\left(\frac{dy}{dx}\right) = y^2

is:

(a) 2
(b) 1
(c) not defined
(d) 0

[1]

Page 1 of 6

Q7.

The integrating factor of the differential equation (1 - y²)(dx)/(dy) + yx = ay, (-1 < y < 1) is: (a) (1)/(y² - 1) (b) frac1√(y² - 1) (c) (1)/(1 - y²) (d) frac1√(1 - y²)

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

[1]

Q8.

Unit vector along vecPQ, where coordinates of P and Q respectively are (2, 1, -1) and (4, 4, -7), is: (a) 2hati + 3hatj - 6hatk (b) -2hati - 3hatj + 6hatk (c) -frac2hati7 - frac3hatj7 + frac6hatk7 (d) frac2hati7 + frac3hatj7 - frac6hatk7

[1]

Q9.

Position vector of the mid-point of line segment AB is 3hati + 2hatj - 3hatk. If the position vector of the point A is 2hati + 3hatj - 4hatk, then the position vector of the point B is: (a) (5)/(2)hati + (5)/(2)hatj - (7)/(2)hatk (b) 4hati + hatj - 2hatk (c) 5hati + 5hatj - 7hatk (d) (1)/(2)hati - (1)/(2)hatj + (1)/(2)hatk

[1]

Q10.

The projection of the vector 2hati + 3hatj on the vector 3hati - 2hatj is: (a) 0 (b) 12 (c) frac12√(13) (d) -frac12√(13)

[1]

Q11.

The equation of the line passing through the point (1, 1, 1) and parallel to the z-axis is: (a) (x)/(1) = (y)/(1) = (z)/(1) (b) (x-1)/(1) = (y-1)/(1) = (z-1)/(1) (c) (x)/(0) = (y)/(0) = (z)/(z-1) (d) (x-1)/(0) = (y-1)/(0) = (z-1)/(1)

[1]

Q12.

A pair of dice is thrown and the sum of the numbers appearing on them is observed to be 9. The probability that the number 4 has appeared on one of the dice is: (a) (1)/(9) (b) (4)/(9) (c) (1)/(18) (d) (1)/(2)

[1]

Q13.

The anti-derivative of (tan x - 1)/(tan x + 1) is: (a) sec²((π)/(4) - x) + c (b) -sec²((π)/(4) - x) + c (c) log|sec((π)/(4) - x)| + c (d) -log|sec((π)/(4) - x)| + c

[1]

Q14.

If (a, b), (c, d) and (e, f) are the vertices of triangle ABC and Δ denotes the area of triangle ABC, then a c e b d f \1 1 1 ² is equal to: (a) 2Δ² (b) 4Δ² (c) 2Δ (d) 4Δ

[1]

Q15.

The function f(x) = x|x| is: (a) continuous and differentiable at x = 0 (b) continuous but not differentiable at x = 0 (c) differentiable but not continuous at x = 0 (d) neither differentiable nor continuous at x = 0

[1]

Page 2 of 6

Q16.

If tan((x + y)/(x - y)) = k, then (dy)/(dx) is equal to: (a) -(y)/(x) (b) (y)/(x) (c) sec²((y)/(x)) (d) -sec²((y)/(x))

[1]

Q17.

The objective function Z = ax + by of an LPP has maximum value 42 at (4, 6) and minimum value 19 at (3, 2). Which of the following is true? (a) a = 9, b = 1 (b) a = 5, b = 2 (c) a = 3, b = 5 (d) a = 5, b = 3

[1]

Q18.

The corner points of the feasible region of a linear programming problem are (0, 4), (8, 0) and ((20)/(3), (4)/(3)). If Z = 30x + 24y is the objective function, then (maximum value of Z − minimum value of Z) is equal to: (a) 40 (b) 144 (c) 120 (d) 136

[1]

Q19.

Assertion (A): The maximum value of (cos⁻¹x)² is π². Reason (R): The range of the principal value branch of cos⁻¹x is [-(π)/(2), (π)/(2)]. (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

[1]

Q20.

Assertion (A): If a line makes angles α, β, γ with the coordinate axes, then sin²α + sin²β + sin²γ = 2. Reason (R): The sum of the squares of the direction cosines of a line is 1. (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

[1]

Section B

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

Q1.

(a) Evaluate sin⁻¹(sin(3π)/(4)) + cos⁻¹(cosπ) + tan⁻¹(1). OR (b) Draw the graph of cos⁻¹x, where x ∈ [-1, 0]. Also, write its range.

[2]

Q2.

A particle moves along the curve 3y = ax³ + 1 such that at a point with x-coordinate 1, y-coordinate is changing twice as fast as x-coordinate. Find the value of a.

[2]

Page 3 of 6

Q3.

If veca, vecb, vecc are three non-zero unequal vectors such that veca · vecb = veca · vecc, then find the angle between veca and vecb - vecc.

[2]

Q4.

Find the coordinates of the points on the line (x)/(1) = (y-1)/(2) = (z+1)/(2) which are at a distance of √(11) units from the origin.

[2]

Q5.

If y = √(ax + b), prove that y(d²y)/(dx²) + ((dy)/(dx))² = 0. OR If f(x) = ax + b, 0 < x ≤ 1 \2x² - x, 1 < x < 2 is a differentiable function in (0, 2), then find the values of a and b.

[2]

Section C

Short Answer · 3 marks each · 6 of 6 shown

Q1.

(a) Evaluate ∫₀π/4 log(1 + tan x) dx. OR (b) Find ∫ fracdx√(sin³ x cos(x - α)).

[3]

Q2.

Find ∫ e^cot⁻¹x((1 - x + x²)/(1 + x²))dx.

[3]

Q3.

Evaluate ∫log√(2)log√(3) frac1(ex + e-x)(ex - e-x) dx.

[3]

Q4.

(a) Find the general solution of the differential equation (xy - x²) dy = y² dx. OR (b) Find the general solution of the differential equation (x² + 1)(dy)/(dx) + 2xy = √(x² + 4).

[3]

Page 4 of 6

Q5.

(a) Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of the number of red balls. Also, find the mean of the random variable. OR (b) A and B throw a die alternately till one of them gets a '6' and wins the game. Find their respective probabilities of winning, if A starts the game first.

⚠ This question is not in the current syllabus — Out-of-syllabus: random-variable / probability-distribution content (part defines X = number of red balls and its distribution) — removed from the rationalized NCERT Class-12 Ch13 syllabus. Retained for reference, flagged out of core.

[3]

Q6.

Solve the following linear programming problem graphically: Minimize Z = 5x + 10y subject to the constraints: x + 2y ≤ 120, x + y ≥ 60, x - 2y ≥ 0, x ≥ 0, y ≥ 0.

[3]

Section D

Long Answer · 5 marks each · 4 of 4 shown

Q1.

(a) If A = -3 -2 -4 \2 1 2 \2 1 3 and B = 1 2 0 \-2 -1 -2 \0 -1 1 , then find AB and use it to solve the following system of equations: x - 2y = 3 2x - y - z = 2 -2y + z = 3 OR (b) If f(α) = cosα -sinα 0 ; sinα cosα 0 \0 0 1 , prove that f(α) · f(-β) = f(α - β).

[5]

Q2.

(a) Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4, 2, -6), Q(5, -3, 1), R(12, 4, 5) and S(11, 9, -2). Use these equations to find the point of intersection of the diagonals. OR (b) A line l passes through the point (-1, 3, -2) and is perpendicular to both the lines (x)/(1) = (y)/(2) = (z)/(3) and (x+2)/(-3) = (y-1)/(2) = (z+1)/(5). Find the vector equation of the line l. Hence, obtain its distance from the origin.

[5]

Q3.

Using integration, find the area of the region bounded by the line y = √(3)x, the curve y = √(4 - x²) and the y-axis in the first quadrant.

[5]

Page 5 of 6

Q4.

A function f : [-4, 4] → [0, 4] is given by f(x) = √(16 - x²). Show that f is an onto function but not a one-one function. Further, find all possible values of a for which f(a) = √(7).

[5]

Section E

Case study · 4 marks each · 3 of 3 shown

Q1.

Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The cylinder bore in the form of a circular cylinder open at the top is to be made from a metal sheet of area 75π cm². Based on the above information, answer the following questions: (i) If the radius of the cylinder is r cm and height is h cm, then write the volume V of the cylinder in terms of radius r. (ii) Find (dV)/(dr). (iii) (a) Find the radius of the cylinder when its volume is maximum. OR (iii) (b) For maximum volume, h > r. State true or false and justify.

[4]

Q2.

Recent studies suggest that roughly 12% of the world population is left handed. Depending upon the parents, the chances of having a left handed child are as follows: A: When both father and mother are left handed: chances of left handed child is 24%. B: When father is right handed and mother is left handed: chances of left handed child is 22%. C: When father is left handed and mother is right handed: chances of left handed child is 17%. D: When both father and mother are right handed: chances of left handed child is 9%. Assuming that P(A) = P(B) = P(C) = P(D) = (1)/(4) and L denotes the event that the child is left handed, answer the following questions: (i) Find P(L mid C). (ii) Find P(barL mid A). (iii) (a) Find P(A mid L). OR (iii) (b) Find the probability that a randomly selected child is left handed, given that exactly one of the parents is left handed.

[4]

Q3.

The use of electric vehicles will curb air pollution in the long run. The use of electric vehicles is increasing every year and the estimated number of electric vehicles in use at any time t is given by the function V: V(t) = (1)/(5)t³ - (5)/(2)t² + 25t - 2, where t represents the time and t = 1, 2, 3, … corresponds to the years 2001, 2002, 2003, … respectively. Based on the above information, answer the following questions: (i) Can the above function be used to estimate the number of vehicles in the year 2000? Justify. (ii) Prove that the function V(t) is an increasing function.

[4]

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