# Mathematics Set 1 2021-2022 ISC (Science) Class 12 Question Paper Solution

Mathematics [Set 1]
Date: April 2022
Duration: 1h30m

Note :

1.  Candidates are al1owed an additional 10 minutes for only reading the paper.
2. They must NOT start writing during this time.
3. This question paper is divided into 3 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. All working, including rough work, should be done on the same sheet as and adjacent to the rest of the answer.
6. The intended marks for questions or parts of questions are given in brackets [ ].
7. Mathematical tables and graph papers are provided.

SECTION - A
[6] 1 | Choose the correct option for the following questions:
[1] 1.i

If int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c", then the value of k is:

3

2

1

None of the above options

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

int_0^(2"a") "f"("x") "dx" = int_0^"a" "f"("x") "dx" + int_0^"a" "f"("k" - "x") "dx", then the value of k is:

a

2a

Independent of a

None of the above options

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

The degree of the differential equation ("d"^2"y")/("dx"^2) + 3("dy"/"dx")^2 = "x"^2 (("d"^2"y")/("dx"^2))^2 is:

1

2

3

4

Concept: Order and Degree of a Differential Equation
Chapter: [0.034] Differential Equations
[1] 1.iv

Given int "e"^"x" (("x" - 1)/("x"^2)) "dx" = "e"^"x" "f"("x") + "c". Then f(x) satisfying the equation is:

x

x2

1/"x"

None of the above options

Concept: Definite Integrals
Chapter: [0.033] Integrals
[1] 1.v

Two cards are drawn out randomly from a pack of 52 cards one after the other, without replacement. The probability of first card being a king and second card not being a king is:

48/663

24/663

12/663

4/663

Concept: Conditional Probability
Chapter: [0.04] Probability (Section A)
[1] 1.vi

If two balls are drawn from a bag containing 3 white, 4 black and 5 red balls. Then, the probability that the drawn balls are of different colours is:

1/66

3/66

19/66

47/66

Concept: Conditional Probability
Chapter: [0.04] Probability (Section A)
[2] 2
[2] 2.i

Find the following integrals int (x^3 - x^2 + x - 1)/(x - 1) dx

Concept: Integration as an Inverse Process of Differentiation
Chapter: [0.033] Integrals
OR
[2] 2.ii

Evaluate:

∫ log_10 "x dx"

Concept: Some Properties of Indefinite Integral
Chapter: [0.033] Integrals
[2] 3
[2] 3.i

Solve the differential equation:

cosec3 x dy − cosec y dx = 0

Concept: Formation of a Differential Equation Whose General Solution is Given
Chapter: [0.034] Differential Equations
OR
[2] 3.ii

Solve the differential equation:

"dy"/"dx" = 2^(-"y")

Concept: Solutions of Linear Differential Equation
Chapter: [0.034] Differential Equations
[2] 4

Evaluate:

int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"

Concept: Properties of Definite Integrals
Chapter: [0.033] Integrals
[4] 5
[4] 5.i

A bag contains 6 red and 5 blue balls and another bag contains 5 red and 8 blue balls. A ball is drawn from the first bag and without noticing its colour is placed in the second bag. If a ball is drawn from the second bag, then find the probability that the drawn ball is red in colour.

Concept: Conditional Probability
Chapter: [0.04] Probability (Section A)
OR
[4] 5.ii

A bag contains 3 red and 4 white balls and another bag contains 2 red and 3 white balls. If one ball is drawn from the first bag and 2 balls are drawn from the second bag, then find the probability that all three balls are of the same colour.

Concept: Conditional Probability
Chapter: [0.04] Probability (Section A)
[4] 6

Evaluate:

int (1 + sin "x")/(1 - sin "x") "dx"

Concept: Indefinite Integral
Chapter: [0.033] Integrals
[6] 7

In a bolt factory, machines X, Y and Z manufacture 20%, 35% and 45% respectively of the total output. Of their output 8%, 6% and 5% respectively are defective bolts. One bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured in machine Y?

Concept: Bayes’ Theorem
Chapter: [0.04] Probability (Section A)
[6] 8
[6] 8.i

Evaluate:

int "dx"/(sin "x" + sin 2"x")

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

By using the properties of definite integrals, evaluate the integrals

int_0^(pi/4) log (1+ tan x) dx

Concept: Properties of Definite Integrals
Chapter: [0.033] Integrals
SECTION - B
[2] 9 | Choose the correct option for the following questions :
[1] 9.i

The equation of the plane which is parallel to 2x − 3y + z = 0 and which passes through (1, −1, 2) is:

2x − 3y + z − 7 = 0

2x − 3y + z + 7 = 0

2x − 3y + z − 8 = 0

2x − 3y + z + 6 = 0

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

The intercepts made on the coordinate axes by the plane 2x + y − 2z = 3 are:

(-3)/2, -3, (-3)/2

3/2, 3, (-3)/2

3/2, -3, (-3)/2

3/2, 3, 3/2

Concept: Plane - Intercept Form of the Equation of a Plane
Chapter: [0.06] Three - Dimensional Geometry (Section B)
[2] 10

Find the equation of the plane passing through the point (1, 1, 1) and is perpendicular to the line ("x" - 1)/3 = ("y" - 2)/0 = ("z" - 3)/4. Also, find the distance of this plane from the origin.

Concept: Distance of a Point from a Plane
Chapter: [0.06] Three - Dimensional Geometry (Section B)
[4] 11

Using integration, find the area of the region bounded between the line x = 4 and the parabola y2 = 16x.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.07] Application of Integrals (Section B)
SECTION - C
[2] 12
[1] 12.i

If the regression line of x on y is, 9x + 3y − 46 = 0 and y on x is, 3x + 12y − 7 = 0, then the correlation coefficient ‘r’ is equal to:

(-1)/12

1/12

(-1)/(2sqrt3)

1/(2sqrt3)

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

If bar"X" = 40, bar"Y" = 6, σx = 10, σy = 1.5 and r = 0.9 for the two sets of data X and Y, then the regression line of X on Y will be:

x − 6y − 4 = 0

x + 6y − 4 = 0

x − 6y + 4 = 0

x + 6y + 4 = 0

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)
[2] 13

For 5 observations of pairs (x, y) of variables X and Y, the following results are obtained:

∑x = 15, ∑y = 25, ∑x2 = 55, ∑y2 = 135, ∑xy = 83.

Calculate the value of bxy and byx.

Concept: Regression Coefficient of X on Y and Y on X
Chapter: [0.09] Linear Regression (Section C)
[4] 14

A manufacturer wishes to produce two commodities A and B. The number of units of material, labour and equipment needed to produce one unit of each commodity is shown in the table given below. Also shown is the available number of units of each item, material, labour, and equipment.

 Items Commodity A Commodity B Available no. of Units Material 1 2 8 Labour 3 2 12 Equipment 1 1 10

Find the maximum profit if each unit of commodity A earns a profit of ₹ 2 and each unit of B earns a profit of ₹ 3.

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

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