#### Chapters

Chapter 2 - Relations and Functions

Chapter 3 - Trigonometric Functions

Chapter 4 - Principle of Mathematical Induction

Chapter 5 - Complex Numbers and Quadratic Equations

Chapter 6 - Linear Inequalities

Chapter 7 - Permutations and Combinations

Chapter 8 - Binomial Theorem

Chapter 9 - Sequences and Series

Chapter 10 - Straight Lines

Chapter 11 - Conic Sections

Chapter 12 - Introduction to Three Dimensional Geometry

Chapter 13 - Limits and Derivatives

Chapter 14 - Mathematical Reasoning

Chapter 15 - Statistics

Chapter 16 - Probability

## Chapter 2 - Relations and Functions

#### Pages 33 - 34

If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A × B)?

if `(x/3 + 1, y - 2/3) = (5/3, 1/3)` find the values of *x* and *y*.

If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.

State whether the following statement are true or false. If the statement is false, rewrite the given statement correctly.

If P = {*m*, *n*} and Q = {*n*, *m*}, then P × Q = {(*m*, *n*), (*n*, *m*)}.

State whether the following statement are true or false. If the statement is false, rewrite the given statement correctly.

If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (*x*, *y*) such that *x* ∈ A and *y* ∈ B.

State whether the following statement are true or false. If the statement is false, rewrite the given statement correctly.

If A = {1, 2}, B = {3, 4}, then A × (B ∩ Φ) = Φ.

If A = {–1, 1}, find A × A × A.

If A × B = {(*a*, *x*), (*a*, *y*), (*b*, *x*), (*b*, *y*)}. Find A and B.

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that A × (B ∩ C) = (A × B) ∩ (A × C)

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that A × C is a subset of B × D

Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.

Let A and B be two sets such that *n*(A) = 3 and *n* (B) = 2. If (*x*, 1), (*y*, 2), (*z*, 1) are in A × B, find A and B, where *x*, *y* and *z* are distinct elements.

The Cartesian product A × A has 9 elements among which are found (–1, 0) and (0, 1). Find the set A and the remaining elements of A × A.

#### Pages 35 - 36

Let A = {1, 2, 3, … , 14}. Define a relation R from A to A by R = {(*x*, *y*): 3*x* – *y* = 0, where *x*, *y* ∈ A}. Write down its domain, codomain and range.

Define a relation R on the set **N** of natural numbers by R = {(*x*, *y*): *y* = *x* + 5, *x* is a natural number less than 4; *x*, *y* ∈ **N**}. Depict this relationship using roster form. Write down the domain and the range.

A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(*x*, *y*): the difference between *x* and *y* is odd; *x* ∈ A, *y *∈ B}. Write R in roster form.

The given figure shows a relationship between the sets P and Q. write this relation

(i) in set-builder form (ii) in roster form.

What is its domain and range?

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(*a*, *b*): *a*, *b* ∈ A, *b* is exactly divisible by *a*}.

(i) Write R in roster form

(ii) Find the domain of R

(iii) Find the range of R.

Determine the domain and range of the relation R defined by R = {(*x*, *x* + 5): *x* ∈ {0, 1, 2, 3, 4, 5}}.

Write the relation R = {(*x*, *x*^{3}): *x *is a prime number less than 10} in roster form.

Let A = {*x*, *y*, z} and B = {1, 2}. Find the number of relations from A to B.

Let R be the relation on **Z** defined by R = {(*a*, *b*): *a*, *b* ∈ **Z**, *a *– *b* is an integer}. Find the domain and range of R.

#### Page 44

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

(iii) {(1, 3), (1, 5), (2, 5)}

Find the domain and range of the given real function:

*f*(*x*) = –|*x*|

Find the domain and range of the following real function:

f(x) = `sqrt(9 - x^2)`

A function *f* is defined by *f*(*x*) = 2*x* – 5. Write down the values of

(i) *f*(0), (ii) *f*(7), (iii) *f*(–3)

The function ‘*t*’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by `t(C) = "9C"/5 + 32`

Find

(i) t(0)

(ii) t(28)

(iii) t(–10)

(iv) The value of C, when t(C) = 212.

Find the range of each of the following functions.

*f*(*x*) = 2 – 3*x*, *x* ∈ **R**, *x* > 0.

Find the range of each of the following functions.

*f*(*x*) = *x*, *x* is a real number

Find the range of each of the following functions.

*f*(*x*) = *x*^{2} + 2, *x*, is a real number.

#### Pages 46 - 47

The relation *f* is defined by f(x) = `{(x^2,0<=x<=3),(3x,3<=x<=10):}`

The relation* g* is defined by g(x) = `{(x^2, 0 <= x <= 2),(3x,2<= x <= 10):}`

Show that *f* is a function and* g *is not a function.

If *f*(*x*) = *x*^{2}, find `(f(1.1) - f(1))/((1.1 - 1))`

Find the domain of the function f(x) = `(x^2 + 2x + 1)/(x^2 - 8x + 12)`

Find the domain and the range of the real function *f* defined by `f(x)=sqrt((x-1))`

Find the domain and the range of the real function *f* defined by *f* (*x*) = |*x* – 1|.

Let `f = {(x, x^2/(1+x^2)):x ∈ R}` be a function from **R** into **R**. Determine the range of *f*.

Let *f*, *g*: **R** → **R** be defined, respectively by *f*(*x*) = *x *+ 1, *g*(*x*) = 2*x* – 3. Find *f* + *g*, *f* – *g* and `f/g`

Let *f *= {(1, 1), (2, 3), (0, –1), (–1, –3)} be a function from **Z** to **Z** defined by *f*(*x*) = *ax* + *b*, for some integers *a*, *b*. Determine *a*, *b*.

Let R be a relation from **N** to **N** defined by R = {(*a*, *b*): *a*, *b* ∈ **N** and *a* = *b*^{2}}. Are the following true?

(i) (*a*, *a*) ∈ R, for all* a *∈ **N**

(ii) (*a*, *b*) ∈ R, implies (*b*, *a*) ∈ R

(iii) (*a*, *b*) ∈ R, (*b*, *c*) ∈ R implies (*a*, *c*) ∈ R.

Justify your answer in each case.

Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and *f *= {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?

(i) *f* is a relation from A to B (ii) *f* is a function from A to B.

Justify your answer in each case.

Let *f* be the subset of **Z** × **Z** defined by *f *= {(*ab*, *a* + *b*): *a*, *b* ∈ **Z**}. Is *f* a function from **Z** to **Z**: justify your answer.

Let A = {9, 10, 11, 12, 13} and let *f*: A → **N** be defined by *f*(*n*) = the highest prime factor of *n*. Find the range of *f*.