Commerce (English Medium)
Science (English Medium)
Arts (English Medium)
Academic Year: 2021-2022
Date & Time: 7th June 2022, 10:30 am
Duration: 2h
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General Instructions :
- This question paper contains three Sections - A, Band C.
- Each section is compulsory.
- Section - A has 6 short answer type-I questions of 2 marks each.
- Section - B has 4 short answer type-II questions of 3 marks each.
- Section - C has 4 long answer type questions of 4 marks each.
- There is an internal choice in some questions.
- Question 14 is a case study based question with two sub-parts of 2 marks each.
The Cartesian equation of a line AB is: `(2x - 1)/2 = (y + 2)/2 = (z - 3)/3`. Find the direction cosines of a line parallel to line AB.
Chapter: [0.11] Three - Dimensional Geometry
Find the general solution of the following differential equation:
`(dy)/(dx) = e^(x-y) + x^2e^-y`
Chapter: [0.09] Differential Equations
Let X be a random variable which assumes values x1 , x2, x3 , x4 such that 2P (X = x1) = 3P (X = x2) = P (X = x3) = 5P (X = x4). Find the probability distribution of X.
Chapter: [0.13] Probability
If `veca = hati + hatj + hatk, veca.vecb` = 1 and `veca xx vecb = hatj - hatk`, then find `|vecb|`.
Chapter: [0.1] Vectors
If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.
Chapter: [0.11] Three - Dimensional Geometry
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Events A and Bare such that P(A) = `1/2`, P(B) = `7/12` and `P(barA ∪ barB) = 1/4`. Find whether the events A and B are independent or not.
Chapter: [0.13] Probability
A box B1 contains 1 white ball and 3 red balls. Another box B2 contains 2 white balls and 3 red balls. If one ball is drawn at random from each of the boxes B1 and B2, then find the probability that the two balls drawn are of the same colour.
Chapter: [0.13] Probability
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Chapter: [0.07] Integrals
If `veca` and `vecb` are two vectors such that `|veca + vecb| = |vecb|`, then prove that `(veca + 2vecb)` is perpendicular to `veca`.
Chapter: [0.1] Vectors
If `veca` and `vecb` are unit vectors and θ is the angle between them, then prove that `sin θ/2 = 1/2 |veca - vecb|`.
Chapter: [0.1] Vectors
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Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Chapter: [0.07] Integrals
Find the distance of the point (2, 3, 4) measured along the line `(x - 4)/3 = (y + 5)/6 = (z + 1)/2` from the plane 3x + 2y + 2z + 5 = 0.
Chapter: [0.11] Three - Dimensional Geometry [0.11] Three - Dimensional Geometry
Three persons A, B and C apply for a job a manager in a private company. Chances of their selection are in the ratio 1:2:4. The probability that A, B and C can introduce chances to increase the profits of a company are 0.8, 0.5 and 0.3 respectively. If increase in the profit does not take place, find the probability that it is due to the appointment of A.
Chapter: [0.13] Probability
Find the area bounded by the curve y = |x – 1| and y = 1, using integration.
Chapter: [0.08] Applications of the Integrals
There are two boxes, namely box-I and box-II. Box-I contains 3 red and 6 black balls. Box-II contains 5 red and 5 black balls. One of the two boxes, is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from box-II.
Chapter: [0.13] Probability
Read the following passage and answer the questions given below.
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Two motorcycles A and B are running at the speed more than the allowed speed on the roads represented by the lines `vecr = λ(hati + 2hatj - hatk)` and `vecr = (3hati + 3hatj) + μ(2hati + hatj + hatk)` respectively.
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Based on the above information, answer the following questions:
- Find the shortest distance between the given lines.
- Find the point at which the motorcycles may collide.
Chapter: [0.11] Three - Dimensional Geometry
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