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

Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto.

(Hint: Consider f(x) = x + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`

Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.

Let S = {a, b, c} and T = {1, 2, 3}. Find F⁻¹ of the following functions F from S to T, if it exists.

F = {(a, 3), (b, 2), (c, 1)} 

Let S = {a, b, c} and T = {1, 2, 3}. Find F⁻¹ of the following functions F from S to T, if it exists.

F = {(a, 2), (b, 1), (c, 1)}

Let A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and f, g : A → B be functions defined by f(x) = x² − x, x ∈ A and g(x) = `2|x - 1/2|- 1`, x ∈ A. Are f and g equal?

Justify your answer. (Hint: One may note that two functions f : A → B and g : A → B such that f(a) = g(a) ∀ a ∈ A are called equal functions.)

Let f: R → R be the Signum Function defined as

f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`

and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0, 1]?

Prove the following:

3 sin⁻¹ x = sin⁻¹ (3x − 4x³), `x ∈ [-1/2, 1/2]`

Prove the following: 

3cos⁻¹x = cos⁻¹(4x³ − 3x), `x ∈ [1/2, 1]`

Prove `tan^(-1)  2/11 + tan^(-1)  7/24 = tan^(-1)  1/2`

Prove `2 tan^(-1)  1/2 + tan^(-1)  1/7 = tan^(-1)  31/17`

Write the following function in the simplest form:

`tan^(-1)  (sqrt(1+x^2) -1)/x`, x ≠ 0

Write the function in the simplest form: `tan^(-1)  1/(sqrt(x^2 - 1)), |x| > 1`

Write the following function in the simplest form:

`tan^(-1) (sqrt((1-cos x)/(1 + cos x)))`, 0 < x < π

Write the function in the simplest form:  `tan^(-1)  ((cos x - sin x)/(cos x + sin x)) `,` 0 < x < pi`

Write the following function in the simplest form:

`tan^(-1)  x/(sqrt(a^2 - x^2))`, |x| < a

Write the following function in the simplest form:

`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; (-a)/sqrt3 < x < a/sqrt3`

Find the value of the following:

`tan^-1 [2 cos (2  sin^-1  1/2)]`

Find the value of `cot(tan^(-1) a + cot^(-1) a)`

Find the value of the following:

`tan  1/2 [sin^(-1)  (2x)/(1+ x^2) + cos^(-1)  (1-y^2)/(1+y^2)]`, |x| < 1, y > 0 and xy < 1

if `sin(sin^(-1)  1/5 + cos^(-1) x)  = 1` then find the value of x