Arts (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

Find the shortest distance between the following lines:

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

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

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?

If f'(x) = `x + 1/x`, then f(x) is ______.

Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if  "n is even"):}` Is the function injective? Justify your answer.

An insect is crawling along the line `barr = 6hati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)` and another insect is crawling along the line `barr = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`. At what points on the lines should they reach so that the distance between them s the shortest? Find the shortest possible distance between them.

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.

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.

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?

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.

Read the following passage and answer the questions given below.

Based on the above information:

1. Calculate the probability that a randomly chosen seed will germinate.
2. Calculate the probability that the seed is of type A₂, given that a randomly chosen seed germinates.

If `veca, vecb, vecc` are three vectors such that `veca.vecb = veca.vecc` and `veca xx vecb = veca xx vecc, veca ≠ 0`, then show that `vecb = vecc`.

If `|veca`| = 3, `|vecb|` = 5, `|vecc|` = 4 and `veca + vecb + vecc` = `vec0`, then find the value of `(veca.vecb + vecb.vecc + vecc.veca)`.

Find the shortest distance between the following lines:

`vecr = 3hati + 5hatj + 7hatk + λ(hati - 2hatj + hatk)` and `vecr = (-hati - hatj - hatk) + μ(7hati - 6hatj + hatk)`.

Find `int dx/sqrt(sin^3x cos(x - α))`.

Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.

A function f : [– 4, 4] `rightarrow` [0, 4] is given by f(x) = `sqrt(16 - x^2)`. Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = `sqrt(7)`.

Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.

Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.

The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.

`int secx/(secx - tanx)dx` equals ______.