Date: April 2022

Duration: 2h

**General Instructions:**

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

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

Chapter: [0.07] Integrals

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

Chapter: [0.07] Integrals

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

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

Chapter: [0.09] Differential Equations

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

Chapter: [0.1] Vectors

Find the direction cosines of the following line:

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

Chapter: [0.1] Vectors

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.

Chapter: [0.13] Probability

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?

Chapter: [0.13] Probability

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

Chapter: [0.01] Relations and Functions

Find the general solution of the following differential equation:

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

Chapter: [0.09] Differential Equations

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

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

Chapter: [0.09] Differential Equations

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

Chapter: [0.1] Vectors

Find the shortest distance between the following lines:

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

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

Chapter: [0.11] Three - Dimensional Geometry

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)`

Chapter: [0.11] Three - Dimensional Geometry

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

Chapter: [0.07] Integrals

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

Chapter: [0.08] Applications of the Integrals

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

Chapter: [0.08] Applications of the Integrals

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.

Chapter: [0.11] Three - Dimensional Geometry

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.**

- What is the probability that a new policyholder will have an accident within a year of purchasing a policy?
- 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?

Chapter: [0.13] Probability

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