# Mathematics Official 2022-2023 ISC (Commerce) Class 12 Question Paper Solution

Mathematics [Official]
Marks: 70 CISCE
ISC (Commerce)
ISC (Arts)
ISC (Science)

Date & Time: 20th February 2023, 2:00 pm
Duration: 3h
1. Candidates are allowed an additional 15 minutes for only reading the paper.
2. They must NOT start writing during this time.
3. The Question Paper consists of three sections A, B and C.
4. Candidates are required to attempt all questions from Section A and all questions EITHER from Section B OR Section C.
5. Section A: Internal choice has been provided in two questions of two marks each, two questions of four marks each and two questions of six marks each.
6. Section B: Internal choice has been provided in one question of two marks and one question of four marks.
7. Section C: Internal choice has been provided in one question of two marks and one question of four marks.
8. All working, including rough work, should be done on the same sheet as, and adjacent to the rest of the answer.
9. The intended marks for questions or parts of questions are given in brackets []
10. Mathematical tables and graph papers are provided.

SECTION A - 65 MARKS
[10]1 | In subparts (i) to (x), choose the correct options, and in subparts (xi) to (xv), answer the questions as instructed.
[1]1.i

A relation R on (1, 2, 3) is given by R = {(1, 1), (2, 2), (1, 2), (3, 3), (2, 3)}. Then the relation R is ______.

Reflexive

Symmetric

Transitive

Symmetric and Transitive

Concept: Types of Relations
Chapter: [0.01] Relations and Functions (Section A)
[1]1.ii

If A is a square matrix of order 3, then |2A| is equal to ______.

2|A|

4|A|

8|A|

6|A|

Concept: Types of Matrices
Chapter: [0.021] Matrices and Determinants
[1]1.iii

If the following function is continuous at x = 2 then the value of k will be ______.

f(x) = {{:(2x + 1",", if x < 2),(                 k",", if x = 2),(3x - 1",", if x > 2):}

2

3

5

– 1

Concept: Continuous Function of Point
Chapter: [0.031] Continuity, Differentiability and Differentiation
[1]1.iv

An edge of a variable cube is increasing at the rate of 10 cm/sec. How fast will the volume of the cube increase if the edge is 5 cm long?

75 cm3/sec

750 cm3/sec

7500 cm3/sec

1250 cm3/sec

Concept: Rate of Change of Bodies or Quantities
Chapter: [0.032] Applications of Derivatives
[1]1.v

Let $f\left(x\right) = x^3$ be a function with domain {0, 1, 2, 3}. Then domain of $f^{-1}$ is ______.

{3, 2, 1, 0}

{0, −1, −2, −3}

{0, 1, 8, 27}

{0, −1, −8, −27}

Concept: Types of Functions
Chapter: [0.01] Relations and Functions (Section A)
[1]1.vi

For the curve y2 = 2x3 – 7, the slope of the normal at (2, 3) is ______.

4

– 4

1/4

(-1)/4

Concept: Tangents and Normals
Chapter: [0.032] Applications of Derivatives
[1]1.vii

Evaluate: int x/(x^2 + 1)"d"x

2log(x2 + 1) + c

1/2log(x2 + 1) + c

"e"^(x^2 + 1) + "c"

logx + x^2/2 + "c"

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.033] Integrals
[1]1.viii

The derivative of log x with respect to 1/x is ______.

1/x

(-1)/x^3

(-1)/x

– x

Concept: Logarithmic Differentiation
Chapter: [0.031] Continuity, Differentiability and Differentiation
[1]1.ix

The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.

(– ∞, 6)

(6, ∞)

(– 6, 6)

(0, – 6)

Concept: Increasing and Decreasing Functions
Chapter: [0.032] Applications of Derivatives
[1]1.x

Evaluate: int_-1^1 x^17.cos^4x  dx

oo

1

– 1

0

Concept: Properties of Definite Integrals
Chapter: [0.033] Integrals
[1]1.xi

Solve the differential equation:

dy/dx = cosec y

Concept: Solutions of Linear Differential Equation
Chapter: [0.034] Differential Equations
[1]1.xii

For what value of k the matrix [(0, k),(-6, 0)] is a skew symmetric matrix?

Concept: Symmetric and Skew Symmetric Matrices
Chapter: [0.021] Matrices and Determinants
[1]1.xiii

Evaluate:

int_0^1 |2x + 1|dx

Concept: Properties of Definite Integrals
Chapter: [0.033] Integrals
[1]1.xiv

Evaluate:

int (1 + cosx)/(sin^2x)dx

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.033] Integrals
[1]1.xv

A bag contains 19 tickets, numbered from 1 to 19. Two tickets are drawn randomly in succession with replacement. Find the probability that both the tickets drawn are even numbers.

Concept: Multiplication Theorem on Probability
Chapter: [0.04] Probability (Section A)
[2]2
[2]2.i

If f(x) = [4 – (x – 7)3]1/5 is a real invertible function, then find f–1(x).

Concept: Composition of Functions and Invertible Function
Chapter: [0.01] Relations and Functions (Section A)
OR
[2]2.ii

Let A = R – {2} and B = R – {1}. If f: A → B is a function defined by f(x) = (x - 1)/(x - 2) then show that f is a one-one and an onto function.

Concept: Types of Functions
Chapter: [0.01] Relations and Functions (Section A)
[2]3

Evaluate the following determinant without expanding:

|(5, 5, 5),(a, b, c),(b + c, c + a, a + b)|

Concept: Properties of Determinants
Chapter: [0.021] Matrices and Determinants
[2]4

The probability of the event A occurring is 1/3 and of the event B occurring is 1/2. If A and B are independent events, then find the probability of neither A nor B occurring.

Concept: Independent Events
Chapter: [0.04] Probability (Section A)
[2]5

Solve for x:

5tan–1x + 3cot–1x = 2π

Concept: Inverse Trigonometric Functions
Chapter: [0.01] Relations and Functions (Section A)
[2]6
[2]6.i

Evaluate:

$\int \cos^{-1} \left(\sin x \right) \text{dx}$

Concept: Evaluation of Simple Integrals of the Following Types and Problems
Chapter: [0.033] Integrals
OR
[2]6.ii

If int x^5 cos (x^6)"d"x = "k" sin (x^6) + "C", find the value of k.

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.033] Integrals
[4]7

If tan^-1 ((x - 1)/(x + 1)) + tan^-1 ((2x - 1)/(2x + 1)) = tan^-1 (23/36) = then prove that 24x2 – 23x – 12 = 0

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.01] Relations and Functions (Section A)
[4]8

If y = eax. cos bx, then prove that

(d^2y)/(dx^2) - 2ady/dx + (a^2 + b^2)y = 0

Concept: Derivatives of Composite Functions - Chain Rule
Chapter: [0.031] Continuity, Differentiability and Differentiation
[4]9
[4]9.i

In a company, 15% of the employees are graduates and 85% of the employees are non-graduates. As per the annual report of the company, 80% of the graduate employees and 10% of the non-graduate employees are in the Administrative positions. Find the probability that an employee selected at random from those working in administrative positions will be a graduate.

Concept: Bayes’ Theorem
Chapter: [0.04] Probability (Section A)
OR
[4]9.ii
[2]9.ii.a

A Problem in Mathematics is given to the three students A, B and C. Their chances of solving the problem are 1/2, 1/3 and 1/4 respectively. Find the probability that exactly two students will solve the problem.

Concept: Conditional Probability
Chapter: [0.04] Probability (Section A)
[2]9.ii.b

A Problem in Mathematics is given to the three students A, B and C. Their chances of solving the problem are 1/2, 1/3 and 1/4 respectively. Find the probability that at least two of them will solve the problem.

Concept: Conditional Probability
Chapter: [0.04] Probability (Section A)
[4]10
[4]10.i

Solve the differential equation:

(1 + y2) dx = (tan1 y x) dy

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.034] Differential Equations
OR
[4]10.ii

Solve the following differential equation:

(x2 – y2)dx + 2xy dy = 0

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations
Chapter: [0.034] Differential Equations
[6]11

Using the matrix method, solve the following system of linear equations:

2/x + 3/y + 10/z = 4, 4/x - 6/y + 5/z = 1, 6/x + 9/y - 20/z = 2.

Concept: Applications of Determinants and Matrices
Chapter: [0.021] Matrices and Determinants
[6]12
[6]12.i

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is $\cot^{- 1} \left( \sqrt{2} \right)$ .

Concept: Maxima and Minima
Chapter: [0.032] Applications of Derivatives
OR
[6]12.ii

A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.

Concept: Maxima and Minima
Chapter: [0.032] Applications of Derivatives
[6]13
[6]13.i

Evaluate:

int (3"e"^(2x) - 2"e"^x)/("e"^(2x) + 2"e"^x - 8)"d"x

Concept: Indefinite Integral
Chapter: [0.033] Integrals
OR
[6]13.ii

Evaluate:

int 2/((1 - x)(1 + x^2))dx

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.033] Integrals
[6]14

A box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement from the box. Find the probability distribution of the number of unspoiled fruits. Also find the mean of the probability distribution.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.04] Probability (Section A)
SECTION B - 15 MARKS
[5]15 | In subparts (i) and (ii), choose the correct options, and in subparts (iii) to (v), answer the questions as instructed.
[1]15.i

If |veca| = 3, |vecb| = sqrt(2)/3 and veca xx vecb is a unit vector then the angle between veca and vecb will be ______.

π/6

π/4

π/3

π/2

Concept: Vectors and Their Types
Chapter: [0.05] Vectors (Section B)
[1]15.ii

The distance of the point 2hati + hatj - hatk from the plane vecr.(hati - 2hatj + 4hatk) = 9 will be ______.

13

13/sqrt(21)

21

21/sqrt(13)

Concept: Distance of a Point from a Plane
Chapter: [0.06] Three - Dimensional Geometry (Section B)
[1]15.iii

Find the area of the parallelogram whose diagonals are hati - 3hatj + hatk and hati + hatj + hatk.

Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
Chapter: [0.05] Vectors (Section B)
[1]15.iv

Find the equation of the plane passing through the point (2, 4, 6) and making equal intercepts on the coordinate axes.

Concept: Equation of a Plane - Intercept Form of the Equation of a Plane
Chapter: [0.06] Three - Dimensional Geometry (Section B)
[1]15.v

If the two vectors 3hati + αhatj + hatk and 2hati - hatj + 8hatk are perpendicular to each other, then find the value of α.

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.05] Vectors (Section B)
[2]16
[2]16.i

If A(1, 2, – 3) and B(– 1, – 2, 1) are the end points of a vector vec("AB") then find the unit vector in the direction of vec("AB").

Concept: Vectors and Their Types
Chapter: [0.05] Vectors (Section B)
OR
[2]16.ii

If hata is unit vector and (2vecx - 3hata)*(2vecx + 3hata) = 91, find the value of |vecx|.

Concept: Vectors and Their Types
Chapter: [0.05] Vectors (Section B)
[4]17
[4]17.i

Find the equation of the plane passing through the point (1, 1, –1) and perpendicular to the planes x + 2y + 3z = 7 and 2x – 3y + 4z = 0.

Concept: Direction Ratios of the Normal to the Plane.
Chapter: [0.06] Three - Dimensional Geometry (Section B)
OR
[4]17.ii

A line passes through the point (2, – 1, 3) and is perpendicular to the lines vecr = (hati + hatj - hatk) + λ(2hati - 2hatj + hatk) and vecr = (2hati - hatj - 3hatk) + μ(hati + 2hatj + 2hatk) obtain its equation.

Concept: Equation of a Line in Space
Chapter: [0.06] Three - Dimensional Geometry (Section B)
[4]18

Find the area of the region bounded by the curve x2 = 4y and the line x = 4y – 2.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.07] Application of Integrals (Section B)
SECTION C - 15 MARKS
[5]19 | In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
[1]19.i

If the demand function is given by p = 1500 – 2x – x2 then find the marginal revenue when x = 10.

1160

1600

1100

1200

Concept: Application of Calculus in Commerce and Economics in the Marginal Revenue Function and Its Interpretation
Chapter: [0.08] Application of Calculus (Section C)
[1]19.ii

If the two regression coefficients are 0.8 and 0.2, then the value of coefficient of correlation r will be ______.

± 0.4

± 0.16

0.4

0.16

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.09] Linear Regression (Section C)
[1]19.iii

Out of the two regression lines x + 2y – 5 = 0 and 2x + 3y = 8, find the line of regression of y on x.

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
Chapter: [0.09] Linear Regression (Section C)
[1]19.iv

The cost function C(x) = 3x2 – 6x + 5. Find the average cost when x = 2.

Concept: Application of Calculus in Commerce and Economics in the Average Cost
Chapter: [0.08] Application of Calculus (Section C)
[1]19.v

The fixed cost of a product is ₹ 30,000 and its variable cost per unit is ₹ 800. If the demand function is p(x) = 4500 – 100x. Find the break-even values.

Concept: Application of Calculus in Commerce and Economics in the Profit Function and Breakeven Point
Chapter: [0.08] Application of Calculus (Section C)
[2]20
[2]20.i

The total cost function for x units is given by C(x) = sqrt(6x + 5) + 2500. Show that the marginal cost decreases as the output x increases.

Concept: Application of Calculus in Commerce and Economics in the Marginal Cost and Its Interpretation
Chapter: [0.08] Application of Calculus (Section C)
OR
[2]20.ii

The average revenue function is given by AR = 25 - x/4. Find total revenue function and marginal revenue function.

Concept: Application of Calculus in Commerce and Economics in the Marginal Revenue Function and Its Interpretation
Chapter: [0.08] Application of Calculus (Section C)
[4]21

Solve the following Linear Programming Problem graphically.

Maximise Z = 5x + 2y subject to:

x – 2y ≤ 2,

3x + 2y ≤ 12,

– 3x + 2y ≤ 3,

x ≥ 0, y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.1] Linear Programming (Section C)
[4]22
[4]22.i

The following table shows the Mean, the Standard Deviation and the coefficient of correlation of two variables x and y.

 Series x y Mean 8 6 Standard deviation 12 4 Coefficient of correlation 0.6

Calculate:

1. the regression coefficient bxy and byx
2. the probable value of y when x = 20
Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.09] Linear Regression (Section C)
OR
[4]22.ii

An analyst analysed 102 trips of a travel company. He studied the relation between travel expenses (y) and the duration (x) of these trips. He found that the relation between x and y was linear. Given the following data, find the regression equation of y on x.

sumx = 510, sumy = 7140, sumx^2 = 4150, sumy^2 = 740200, sumxy = 54900

Concept: Identification of Regression Equations
Chapter: [0.09] Linear Regression (Section C)

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