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Mathematics Term 2 2021-2022 CBSE (Science) Class 12 Question Paper Solution

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Mathematics [Term 2]
Marks: 40 Academic Year: 2021-2022
Date: April 2022
Duration: 2h
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General Instructions:

  1. This question paper contains three sections – A, B and C. Each part is compulsory.
  2. Section - A has 6 short answer type (SA1) questions of 2 marks each.
  3. Section – B has 4 short answer type (SA2) questions of 3 marks each.
  4. Section - C has 4 long answer type questions (LA) of 4 marks each.
  5. There is an internal choice in some of the questions.
  6. Q 14 is a case-based problem having 2 sub parts of 2 marks each.

SECTION – A
[2] 1
[2] 1.i

Find: `int logx/(1 + log x)^2 dx`

Concept: Definite Integrals Problems
Chapter: [0.07] Integrals
OR
[2] 1.ii

Find: `int (sin2x)/sqrt(9 - cos^4x) dx`

Concept: Indefinite Integral Problems
Chapter: [0.07] Integrals
[2] 2

Write the sum of the order and the degree of the following differential equation:

`d/(dx) (dy/dx)` = 5

Concept: Order and Degree of a Differential Equation
Chapter: [0.09] Differential Equations
[2] 3

If `hata` and `hatb` are unit vectors, then prove that `|hata + hatb| = 2 cos  theta/2`, where θ is the angle between them.

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.1] Vectors
[2] 4

Find the direction cosines of the following line:

`(3 - x)/(-1) = (2y - 1)/2 = z/4`

Concept: Direction Cosines
Chapter: [0.1] Vectors
[2] 5

A bag contains 1 red and 3 white balls. Find the probability distribution of the number of red balls if 2 balls are drawn at random from the bag one-by-one without replacement.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.13] Probability
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[2] 6

Two cards are drawn at random from a pack of 52 cards one-by-one without replacement. What is the probability of getting first card red and second card Jack?

Concept: Introduction of Probability
Chapter: [0.13] Probability
SECTION – B
[3] 7

Find: `int (x + 1)/((x^2 + 1)x) dx`

Concept: Types of Relations
Chapter: [0.01] Relations and Functions
[3] 8
[3] 8.i

Find the general solution of the following differential equation:

`x (dy)/(dx) = y - xsin(y/x)`

Concept: Formation of a Differential Equation Whose General Solution is Given
Chapter: [0.09] Differential Equations
OR
[3] 8.ii

Find the particular solution of the following differential equation, given that y = 0 when x = `pi/4`.

`(dy)/(dx) + ycotx = 2/(1 + sinx)`

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
[3] 9

If `veca ≠ vec(0), veca.vecb = veca.vecc, veca xx vecb = veca xx vecc`, then show that `vecb = vecc`.

Concept: Vectors and Their Types
Chapter: [0.1] Vectors
[3] 10
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[3] 10.i

Find the shortest distance between the following lines:

`vecr = (hati + hatj - hatk) + s(2hati + hatj + hatk)`

`vecr = (hati + hatj - 2hatk) + t(4hati + 2hatj + 2hatk)`

Concept: Shortest Distance Between Two Lines
Chapter: [0.11] Three - Dimensional Geometry
OR
[3] 10.ii

Find the vector and the cartesian equations of the plane containing the point `hati + 2hatj - hatk` and parallel to the lines `vecr = (hati + 2hatj + 2hatk) + s(2hati - 3hatj + 2hatk)` and `vecr = (3hati + hatj - 2hatk) + t(hati - 3hatj + hatk)`

Concept: Vector and Cartesian Equation of a Plane
Chapter: [0.11] Three - Dimensional Geometry
SECTION – C
[4] 11

Evaluate: `int_(-1)^2 |x^3 - 3x^2 + 2x|dx`

Concept: Definite Integrals Problems
Chapter: [0.07] Integrals
[4] 12
[4] 12.i

Using integration, find the area of the region in the first quadrant enclosed by the line x + y = 2, the parabola y2 = x and the x-axis.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.08] Applications of the Integrals
OR
[4] 12.ii

Using integration, find the area of the region `{(x, y): 0 ≤ y ≤ sqrt(3)x, x^2 + y^2 ≤ 4}`

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.08] Applications of the Integrals
[4] 13

Find the foot of the perpendicular from the point (1, 2, 0) upon the plane x – 3y + 2z = 9. Hence, find the distance of the point (1, 2, 0) from the given plane.

Concept: Distance of a Point from a Plane
Chapter: [0.11] Three - Dimensional Geometry
[4] 14
CASE-BASED/DATA-BASED
An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company’s statistics show that an accident-prone person will have an accident at some time within a fixed one-year period with a probability 0.6, whereas this probability is 0.2 for a person who is not accident prone. The company knows that 20 percent of the population is accident prone.

Based on the given information, answer the following questions.

  1. What is the probability that a new policyholder will have an accident within a year of purchasing a policy?
  2. Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone?
Concept: Bayes’ Theorem
Chapter: [0.13] Probability

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