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

Let f: [0, 1] → [0, 1] be defined by f(x) = `{{:(x",",  "if"  x  "is rational"),(1 - x",",  "if"  x  "is irrational"):}`. Then (f o f) x is ______.

Let f: N → R be the function defined by f(x) = `(2x - 1)/2` and g: Q → R be another function defined by g(x) = x + 2. Then (g o f) `3/2` is ______.

Let f = {(1, 2), (3, 5), (4, 1) and g = {(2, 3), (5, 1), (1, 3)}. Then g o f = ______ and f o g = ______.

Let f: R → R be the function defined by f(x) = sin (3x+2) ∀ x ∈ R. Then f is invertible.

The composition of functions is commutative.

The composition of functions is associative.

Every function is invertible.

If `|(2x, 5),(8, x)| = |(6, 5),(8, 3)|`, then find x

Prove that (A^–1)′ = (A′)^–1, where A is an invertible matrix.

Show that if the determinant ∆ = `|(3, -2, sin3theta),(-7, 8, cos2theta),(-11, 14, 2)|` = 0, then sinθ = 0 or `1/2`.

If `|(2x, 5),(8, x)| = |(6, -2),(7, 3)|`, then value of x is ______.

Verify the following using the concept of integration as an antiderivative

`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`

Evaluate the following:

`int x^2/(1 - x^4) "d"x` put x² = t

Evaluate the following:

`int (x^2"d"x)/(x^4 - x^2 - 12)`

Evaluate the following:

`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`

Evaluate the following:

`int_"0"^pi  (x"d"x)/(1 + sin x)`

Evaluate the following:

`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`

Evaluate the following:

`int "e"^(-3x) cos^3x  "d"x`

Evaluate the following:

`int sqrt(tanx)  "d"x`  (Hint: Put tanx = t²)

If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.