Let A = {x \(\in \) N| 1 < x < 4}, B = {x \(\in \) W| 0 ≤ x < 2) and C = {x \(\in \) N| x < 3} Then verify that
(i) A x (B U C) = (A x B) U (A x C)
(ii) A x (B ∩ C) = (A x B) ∩ (A x C)

If B x A = {(-2,3), (-2,4),(0,3), (0,4),(3,3),(3,4)} find A and B.

If A = {5,6}, B = {4,5,6}, C = {5,6,7}, Show that A x A = (B x B) ∩ (C x C)

Let A = {x \(\in \) W| x < 2}, B = {x \(\in \) N| 1 < x ≤ 4} and C = (3,5). Verify that
A x (B U C) = (A x B) U (A x C)

Let A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even prime number. Verify that
(A ∩ B) x C = (A x C) ∩ (B x C)

Represent each of the given relations by (a) an arrow diagram, (b) a graph and (c) a set in roster form, wherever possible.
(i) {(x, y)|x = 2y, x \(\in \) {2, 3, 4, 5}, y \(\in \) {1, 2, 3, 4}
(ii) {(x, y)|y = x + 3, x, y are natural numbers < 10}

A company has four categories of employees given by Assistants (A), Clerks (C), Managers (M) and an Executive Officer (E). The company provides Rs.10,000, Rs. 25,000, Rs. 50,000 and Rs.1,00,000 as salaries to the people who work in the categories A, C, M and E respectively. If A₁, A₂, A₃, A₄ and A₅ were Assistants; C₁, C₂, C₃, C₄ were Clerks; M₁, M₂, M₃ were managers and E₁, E₂ were Executive officers and if the relation R is defined by xRy, where x is the salary given to person y, express the relation R through an ordered pair and an arrow diagram.

Given f(x) = 2x - x², find
(i) f (1)
(ii) f (x + 1)
(iii) f (x) + f (1)

Given the function f:x ⟶ x²- 5x + 6, evaluate
i) f( -1)
ii) f (2a)
iii) f (2)
iv) f (x - 1)

A graph representing the function f (x) is given in Fig it is clear that f (9) = 2.
(i) Find the following values of the function
(a) f(0)
(b) f(7)
(c) f(2)
(d) f(10)
(ii) For what value of x is f (x) = 1?
(iii) Describe the following (i) Domain (ii) Range.
(iv) What is the image of 6 under f ?

An open box is to be made from a square piece of material, 24 cm on a side, by cutting equal squares from the corners and turning up the sides as shown Fig. Express the volume V of the box as a function of x.

The data in the adjacent table depicts the length of a person forehand and her corresponding height. Based on this data, a student finds a relationship between the height (y) and the forehand length(x) as y = ax + b,  where a, b are constants.
(i) Check if this relation is a function.
(ii) Find a and b.
(iii) Find the height of a woman whose forehand length is 40 cm.
(iv) Find the length of forehand of a woman if her height is 53.3 inches.

Length ‘x’ of forehand (in cm)	Height 'y' (in inches)
35	56
45	65
50	69.5
55	74

Let A = {1,2,3,4} and B = { 2, 5, 8, 11,14} be two sets. Let f: A ⟶ B be a function given by f(x) = 3x − 1. Represent this function
(i) by arrow diagram
(ii) in a table form
(iii) as a set of ordered pairs
(iv) in a graphical form

Forensic scientists can determine the height (in cms) of a person based on the length of their thigh bone. They usually do so using the function h(b) = 2.47b + 54.10 where b is the length of the thigh bone.
(i) Check if the function h is one – one or not
(ii) Also find the height of a person if the length of his thigh bone is 50 cm.
(iii) Find the length of the thigh bone if the height of a person is 147.96 cm.

If the function f: R⟶ R defined by 
\(f(x)=\left\{\begin{array}{l} 2 x+7, x<-2 \\ x^{2}-2,-2 \leq x<3 \\ 3 x-2, x \geq 3 \end{array}\right.\)
(i) f( 4)
(ii) f( -2)
(iii) f(4) + 2f(1)
(iv) \(\frac { f(1)-3f(4) }{ f(-3) } \)