Commerce (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

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 ______.

Write the domain and range (principle value branch) of the following functions:

f(x) = tan^–1 x.

Find the distance between the lines:

`vecr = (hati + 2hatj - 4hatk) + λ(2hati + 3hatj + 6hatk)`;

`vecr = (3hati + 3hatj - 5hatk) + μ(4hati + 6hatj + 12hatk)`

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let `3/5` be the probability that he knows the answer and `2/5` be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability `1/3`. What is the probability that the student knows the answer, given that he answered it correctly?

Find the value of `tan^-1 [2 cos (2 sin^-1  1/2)] + tan^-1 1`.

The lines `vecr = hati + hatj - hatk + λ(2hati + 3hatj - 6hatk)` and `vecr = 2hati - hatj - hatk + μ(6hati + 9hatj - 18hatk)`; (where λ and μ are scalars) are ______.

Let f(x) be a polynomial function of degree 6 such that `d/dx (f(x))` = (x – 1)³ (x – 3)², then

Assertion (A): f(x) has a minimum at x = 1.

Reason (R): When `d/dx (f(x)) < 0, ∀  x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀  x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.

ASSERTION (A): The relation f : {1, 2, 3, 4} `rightarrow` {x, y, z, p} defined by f = {(1, x), (2, y), (3, z)} is a bijective function.

REASON (R): The function f : {1, 2, 3} `rightarrow` {x, y, z, p} such that f = {(1, x), (2, y), (3, z)} is one-one.