Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective.

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

f(x) = `x/2`

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

g(x) = |x|

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

h(x) = x|x|

Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

k(x) = x² 

Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto

If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.

Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.

Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.

Which of the following functions from Z into Z are bijections?

Let f: R → R be the functions defined by f(x) = x³ + 5. Then f^–1(x) is ______.

Let f: R – `{3/5}` → R be defined by f(x) = `(3x + 2)/(5x - 3)`. Then ______.

Let f: `[2, oo)` → R be the function defined by f(x) = x² – 4x + 5, then the range of f is ______.

Let f: R → R be given by f(x) = tan x. Then f^–1(1) is ______.

If f(x) = (4 – (x – 7)³}, then f^–1(x) = ______.

Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.

If A is square matrix such that A² = A, show that (I + A)³ = 7A + I..

If A is a square matrix such that A² = I, then (A – I)³ + (A + I)³ –7A is equal to ______.

Matrix addition is associative as well as commutative.

Matrix multiplication is commutative.