# Mathematics Sample 2022-2023 Arts (English Medium) Class 12 Question Paper Solution

Mathematics [Sample]
Marks: 80 CBSE
Arts (English Medium)
Science (English Medium)
Commerce (English Medium)

Date: March 2023
Duration: 3h

General Instructions :

1. This Question paper contains - five sections A, B, C, D, and E. Each section is compulsory. However, there are internal choices in some questions.
2. Section A has 18 MCQs and 02 Assertion-Reason based questions of 1 mark each.
3. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
4. Section C has 6 Short Answer (SA)-type questions of 3 marks each.
5. Section D has 4 Long Answer (LA)-type questions of 5 marks each.
6. Section E has 3 source based/case based/passage based/integrated units of assessment (4 marks each) with sub-parts.

SECTION - A : (Multiple Choice Questions) Each question carries 1 mark
1

If A = [aij] is a skew-symmetric matrix of order n, then ______.

a_(ij) = 1/(a_(ji)) ∀  i, j

a_(ij) ≠ 0  ∀  i, j

a_(ij) = 0, where  i = j

a_(ij) ≠ 0  where  i = j

Concept: Symmetric and Skew Symmetric Matrices
Chapter: [0.03] Matrices
2

If A is a square matrix of order 3, |A′| = −3, then |AA′| = ______.

9

−9

3

−3

Concept: Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method
Chapter: [0.04] Determinants
3

The area of a triangle with vertices A, B, C is given by ______.

|vec("AB") xx vec("AC")|

1/2|vec("AB") xx vec("AC")|

1/4|vec("AC") xx vec("AB")|

1/8|vec("AC") xx vec("AB")|

Concept: Area of a Triangle
Chapter: [0.04] Determinants
4

The value of ‘k’ for which the function f(x) = {{:((1 - cos4x)/(8x^2)",",  if x ≠ 0),(k",",  if x = 0):} is continuous at x = 0 is ______.

0

–1

1

2

Concept: Algebra of Continuous Functions
Chapter: [0.05] Continuity and Differentiability
5

If f'(x) = x + 1/x, then f(x) is ______.

x^2 + log |x| + C

x^2/2 + log |x| + C

x/2 + log |x| + C

x/2 - log |x| + C

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
6

If m and n, respectively, are the order and the degree of the differential equation d/(dx) [((dy)/(dx))]^4 = 0, then m + n = ______.

1

2

3

4

Concept: Order and Degree of a Differential Equation
Chapter: [0.09] Differential Equations
7

The solution set of the inequality 3x + 5y < 4 is ______.

an open half-plane not containing the origin.

an open half-plane containing the origin.

the whole XY-plane not containing the line 3x + 5y = 4.

a closed half-plane containing the origin.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
8

The scalar projection of the vector 3hati - hatj - 2hatk on the vector hati + 2hatj - 3hatk is ______.

7/sqrt(14)

7/14

6/13

7/2

Concept: Product of Two Vectors - Projection of a Vector on a Line
Chapter: [0.1] Vectors
9

The value of int_2^3 x/(x^2 + 1)dx is ______.

log 4

log  3/2

1/2 log2

log  9/4

Concept: Definite Integrals Problems
Chapter: [0.07] Integrals
10

If A, B are non-singular square matrices of the same order, then (AB–1)–1 = ______.

A–1B

A–1B–1

BA–1

AB

Concept: Invertible Matrices
Chapter: [0.03] Matrices
11

The corner points of the shaded unbounded feasible region of an LPP are (0, 4), (0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the objective function Z = 4x + 6y occurs at ______. (0.6, 1.6) only

(3, 0) only

(0.6, 1.6) and (3, 0) only

at every point of the line-segment joining the points (0.6, 1.6) and (3, 0)

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
12

If |(2, 4),(5, 1)| = |(2x, 4),(6, x)|, then the possible value(s) of ‘x’ is/are ______.

3

sqrt(3)

-sqrt(3)

sqrt(3), -sqrt(3)

Concept: Introduction of Determinant
Chapter: [0.04] Determinants
13

If A is a square matrix of order 3 and |A| = 5, then |adj A| = ______.

5

25

125

1/5

Concept: Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method
Chapter: [0.04] Determinants
14

Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6 and P(A' ∩ B') is ______.

0.9

0.18

0.28

0.1

Concept: Independent Events
Chapter: [0.13] Probability
15

The general solution of the differential equation y dx – x dy = 0 is ______.

xy = C

x = Cy2

y = Cx

y = Cx2

Concept: Formation of a Differential Equation Whose General Solution is Given
Chapter: [0.09] Differential Equations
16

If y = sin–1x, then (1 – x2)y2 is equal to ______.

xy1

xy

xy2

x2

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.05] Continuity and Differentiability
17

If two vectors veca and vecb are such that |veca| = 2, |vecb| = 3 and veca.vecb = 4, then |veca - 2vecb| is equal to ______.

sqrt(2)

2sqrt(6)

24

2sqrt(2)

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.1] Vectors
18

P is a point on the line joining the points A(0, 5, −2) and B(3, −1, 2). If the x-coordinate of P is 6, then its z-coordinate is ______.

10

6

–6

–10

Concept: Equation of a Line in Space
Chapter: [0.11] Three - Dimensional Geometry
ASSERTION-REASON BASED QUESTIONS
19

Assertion (A): The domain of the function sec–12x is (-∞, - 1/2] ∪ pi/2, ∞)

Reason (R): sec–1(–2) = - pi/4

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

A is true but R is false.

A is false but R is true.

Concept: Basic Concepts of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
20

Assertion (A): The acute angle between the line barr = hati + hatj + 2hatk  + λ(hati - hatj) and the x-axis is π/4

Reason(R): The acute angle 𝜃 between the lines barr = x_1hati + y_1hatj + z_1hatk  + λ(a_1hati + b_1hatj + c_1hatk) and  barr = x_2hati + y_2hatj + z_2hatk  + μ(a_2hati + b_2hatj + c_2hatk) is given by cosθ = (|a_1a_2 + b_1b_2 + c_1c_2|)/sqrt(a_1^2 + b_1^2 + c_1^2 sqrt(a_2^2 + b_2^2 + c_2^2)

Both A and R are true and R is the correct explanation of A.

Both A and R are true but R is not the correct explanation of A.

A is true but R is false.

A is false but R is true.

Concept: Angle Between Two Lines
Chapter: [0.11] Three - Dimensional Geometry
SECTION - B : This section comprises of very short answer type-questions (VSA) of 2 marks each
21
21.a

Find the value of sin^-1 [sin((13π)/7)]

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
OR
21.b

Prove that the function f is surjective, where f: N → N such that f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if  "n is even"):} Is the function injective? Justify your answer.

Concept: Types of Functions
Chapter: [0.01] Relations and Functions
22

A man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening?

Concept: Rate of Change of Bodies or Quantities
Chapter: [0.06] Applications of Derivatives
23
23.a

If veca = hati - hatj + 7hatk and vecb = 5hati - hatj + λhatk, then find the value of λ so that the vectors veca + vecb and veca - vecb are orthogonal.

Concept: Vectors and Their Types
Chapter: [0.1] Vectors
OR
23.b

Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2

Concept: Basic Concepts of Vector Algebra
Chapter: [0.1] Vectors
24

If ysqrt(1 - x^2) + xsqrt(1 - y^2) = 1, then prove that (dy)/(dx) = - sqrt((1 - y^2)/(1 - x^2))

Concept: Concept of Differentiability
Chapter: [0.05] Continuity and Differentiability
25

Find |vecx| if (vecx - veca).(vecx + veca) = 12, where veca is a unit vector.

Concept: Vectors and Their Types
Chapter: [0.1] Vectors
SECTION - C : This section comprises of short answer type questions (SA) of 3 marks each
26

Find: int (dx)/sqrt(3 - 2x - x^2)

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.07] Integrals
27
27.a

Three friends go for coffee. They decide who will pay the bill, by each tossing a coin and then letting the “odd person” pay. There is no odd person if all three tosses produce the same result. If there is no odd person in the first round, they make a second round of tosses and they continue to do so until there is an odd person. What is the probability that exactly three rounds of tosses are made?

Concept: Variance of a Random Variable
Chapter: [0.13] Probability
OR
27.b

Find the mean number of defective items in a sample of two items drawn one-by-one without replacement from an urn containing 6 items, which include 2 defective items. Assume that the items are identical in shape and size.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.13] Probability
28
28.a

Evaluate: int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)

Concept: Properties of Definite Integrals
Chapter: [0.07] Integrals
OR
28.b

Evaluate the following integral:

$\int\limits_0^4 \left| x - 1 \right| dx$
Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.07] Integrals
29
29.a

Solve the differential equation: y dx + (x – y2)dy = 0

Concept: Formation of a Differential Equation Whose General Solution is Given
Chapter: [0.09] Differential Equations
OR
29.b

Solve the differential equation: xdy – ydx = sqrt(x^2 + y^2)dx

Concept: Solutions of Linear Differential Equation
Chapter: [0.09] Differential Equations
30

Solve the following Linear Programming Problem graphically:

Maximize Z = 400x + 300y subject to x + y ≤ 200, x ≤ 40, x ≥ 20, y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
31

Evaluate the following integral:

$\int\frac{x^3 + x + 1}{x^2 - 1}dx$
Concept: Evaluation of Simple Integrals of the Following Types and Problems
Chapter: [0.07] Integrals
SECTION - D : This section comprises of long answer-type questions (LA) of 5 marks each
32

Make a rough sketch of the region {(x, y): 0 ≤ y ≤ x2, 0 ≤ y ≤ x, 0 ≤ x ≤ 2} and find the area of the region using integration.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.08] Applications of the Integrals
33
33.a

Define the relation R in the set N × N as follows:

For (a, b), (c, d) ∈ N × N, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in N × N.

Concept: Types of Relations
Chapter: [0.01] Relations and Functions
OR
33.b

Given a non-empty set X, define the relation R in P(X) as follows:

For A, B ∈ P(X), (4, B) ∈ R iff A ⊂ B. Prove that R is reflexive, transitive and not symmetric.

Concept: Types of Relations
Chapter: [0.01] Relations and Functions
34
34.a

An insect is crawling along the line barr = 6hati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk) and another insect is crawling along the line barr = - 4hati - hatk + μ(3hati - 2hatj - 2hatk). At what points on the lines should they reach so that the distance between them s the shortest? Find the shortest possible distance between them.

Concept: Shortest Distance Between Two Lines
Chapter: [0.11] Three - Dimensional Geometry
OR
34.b

The equations of motion of a rocket are:
x = 2t,y = –4t, z = 4t, where the time t is given in seconds, and the coordinates of a ‘moving point in km. What is the path of the rocket? At what distances will the rocket be from the starting point O(0, 0, 0) and from the following line in 10 seconds? vecr = 20hati - 10hatj + 40hatk + μ(10hati - 20hatj + 10hatk)

Concept: Distance of a Point from a Plane
Chapter: [0.11] Three - Dimensional Geometry
35

If A = [(2, -3, 5),(3, 2, -4),(1, 1, -2)], find A–1. Use A–1 to solve the following system of equations 2x − 3y + 5z = 11, 3x + 2y – 4z = –5, x + y – 2z = –3

Concept: Inverse of Matrix - Inverse of a Square Matrix by the Adjoint Method
Chapter: [0.04] Determinants
SECTION - E
36 | This section comprises 3 case-study/passage-based questions of 4 marks each with two sub-parts. First, two case study questions have three sub-parts (i), (ii), and (iii) of marks 1, 1, 2 respectively. The third case study question has two sub-parts of 2 marks each. The temperature of a person during an intestinal illness is given by f(x) = 0.1x2 + mx + 98.6, 0 ≤ x ≤ 12, m being a constant, where f(x) is the temperature in °F at x days.
1. Is the function differentiable in the interval (0, 12)? Justify your answer.
2. If 6 is the critical point of the function, then find the value of the constant m.
3. Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
Concept: Maxima and Minima
Chapter: [0.06] Applications of Derivatives
37 In an elliptical sport field the authority wants to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of x^2/a^2 + y^2/b^2 = 1.
1. If the length and the breadth of the rectangular field be 2x and 2y respectively, then find the area function in terms of x.
2. Find the critical point of the function.
3. Use First derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
OR
Use Second Derivative Test to find the length 2x and width 2y of the soccer field (in terms of a and b) that maximize its area.
Concept: Maxima and Minima
Chapter: [0.06] Applications of Derivatives
38 There are two antiaircraft guns, named as A and B. The probabilities that the shell fired from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an airplane at the same time.
1. What is the probability that the shell fired from exactly one of them hit the plane?
2. If it is known that the shell fired from exactly one of them hit the plane, then what is the probability that it was fired from B?
Concept: Properties of Conditional Probability
Chapter: [0.13] Probability

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