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

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 = `[(x, 5, 2),(2, y, 3),(1, 1, z)]`, xyz = 80, 3x + 2y + 10z = 20, ten A adj. A = `[(81, 0, 0),(0, 81, 0),(0, 0, 81)]`

If A = `[(0, 1, 3),(1, 2, x),(2, 3, 1)]`, A^–1 = `[(1/2, -4, 5/2),(-1/2, 3, -3/2),(1/2, y, 1/2)]` then x = 1, y = ^–1.

If A and B are invertible matrices, then which of the following is not correct?

(A³)^–1 = (A^–1)³, where A is a square matrix and |A| ≠ 0.

`("aA")^-1 = 1/"a"  "A"^-1`, where a is any real number and A is a square matrix.

|A^–1| ≠ |A|^–1, where A is non-singular matrix.

|adj. A| = |A|², where A is a square matrix of order two.

Show that the function f(x) = 4x³ – 18x² + 27x – 7 has neither maxima nor minima.