Mathematics Term 2 - Delhi Set 2 2021-2022 Commerce (English Medium) Class 12 Question Paper Solution

Find the vector equation of a line passing through a point with position vector `2hati - hatj + hatk` and parallel to the line joining the points `-hati + 4hatj + hatk` and `-hati + 2hatj + 2hatk`.

Find the general solution of the following differential equation:

`(dy)/(dx) = e^(x-y) + x^2e^-y`

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.

If `veca = hati + hatj + hatk, veca.vecb` = 1 and `veca xx vecb = hatj - hatk`, then find `|vecb|`.

If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.

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.

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.

Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`

If `veca` and `vecb` are two vectors such that `|veca + vecb| = |vecb|`, then prove that `(veca + 2vecb)` is perpendicular to `veca`.

If `veca` and `vecb` are unit vectors and θ is the angle between them, then prove that `sin  θ/2 = 1/2 |veca  - vecb|`.

Let `veca = hati + hatj, vecb = hati - hatj` and `vecc = hati + hatj + hatk`. If `hatn` is a unit vector such that `veca.hatn` = 0 and `vecb.hatn` = 0, then find `|vecc.hatn|`.

If `veca` and `vecb` are unit vectors inclined at an angle 30° to each other, then find the area of the parallelogram with `(veca + 3vecb)` and `(3veca + vecb)` as adjacent sides.

Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`

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.

Find the area bounded by the curve y = |x – 1| and y = 1, using integration.

In a factory, machine A produces 30% of total output, machine B produces 25% and the machine C produces the remaining output. The defective items produced by machines A, B and C are 1%,1.2%, 2% respectively. An item is picked at random from a day's output and found to be defective. Find the probability that it was produced by machine B?

Read the following passage and answer the questions given below.

Based on the above information, answer the following questions:

1. Find the shortest distance between the given lines.
2. Find the point at which the motorcycles may collide.