__TN SCERT-10th Maths Guide Book with Answers -Chapter 1 – __

__Relations and Functions (Full chapter)__

__Chapter 1 Relations and Functions Ex 1.1__

__Chapter 1 Relations and Functions Ex 1.1__

**Question 1.**

**Find A × B, A × A and B × A**

**(i) A = {2,-2,3} and B = {1,-4}**

**(ii) A = B = {p,q]**

**(iii) A= {m,n} ; B = (Φ)**

**Solution:**

(i) A = {2,-2,3}, B = {1,-4}

A × B = {(2, 1), (2, -4), (-2, 1), (-2, -4), (3,1) , (3,-4)}

A × A = {(2, 2), (2,-2), (2, 3), (-2, 2), (-2, -2), (-2, 3), (3, 2), (3, -2), (3,3) }

B × A = {(1, 2), (1, -2), (1, 3), (-4, 2), (-4, -2), (-4,3)}

(ii) A = B = {(p,q)]

A × B = {(p, p), {p, q), (q, p), (q, q)}

A × A = {(p, p), (p, q), (q, p), (q, q)}

B × A = {(p,p), {p, q), (q,p), (q, q)}

(iii) A = {m,n} × Φ

A × B = { }

A × A = {(m,m), (m,n), (n, m), (n, n)}

B × A = { }

**Question 2.**

**Let A = {1, 2, 3} and B = {x | x is a prime number less than 10}. Find A × B and B × A.**

**Solution:**

A = {1, 2, 3}, B = {2,3, 5,7}

A × B = {(1,2), (1,3), (1,5), (1,7), (2,2), (2.3) , (2,5), (2,7), (3,2), (3,3), (3, 5), (3,7)}

B × A = {(2,1), (2,2), (2,3), (3,1), (3,2), (3.3) , (5,1), (5,2), (5,3), (7,1), (7,2) , (7, 3)}

**Question 3.**

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

**Solution:**

B × A ={(-2,3), (-2,4), (0,3), (0,4), (3,3), (3,4)}

A = {3, 4), B = { -2, 0, 3}

**Question 4.**

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

**Solution:**

A = {5,6}, B = {4,5,6},C = {5,6, 7}

A × A = {(5, 5), (5, 6), (6, 5), (6, 6)} …(1)

B × B = {(4, 4), (4, 5), (4, 6), (5, 4),

(5,5), (5,6), (6,4), (6,5), (6,6)} …(2)

C × C = {(5,5), (5,6), (5,7), (6,5), (6,6),

(6, 7), (7, 5), (7, 6), (7, 7)} …(3)

(B × B) ∩ (C × C) = {(5, 5), (5,6), (6, 5), (6,6)} …(4)

(1) = (4)

A × A = (B × B) ∩ (C × C)

It is proved.

**Question 5.**

**Given A ={1, 2, 3}, B = {2, 3, 5}, C = {3, 4} and D = {1, 3, 5}, check if (A ∩ C) x (B ∩ D) = (A × B) ∩ (C × D) is true?**

**Solution:**

LHS = {(A∩C) × (B∩D)

A ∩C = {3}

B ∩D = {3,5}

(A ∩ C) × (B ∩ D) = {(3, 3) (3, 5)} …(1)

RHS = (A × B) ∩ (C × D)

A × B = {(1,2), (1,3), (1,5), (2,2), (2,3), (2, 5), (3, 2), (3,3), (3,5)}

C × D = {(3,1), (3,3), (3,5), (4,1), (4, 3), (4,5)}

(A × B) ∩ (C × D) = {(3, 3), (3, 5)} …(2)

∴ (1) = (2) ∴ It is true.

**Question 6.**

**Let A = {x ∈ W | x < 2}, B = {x ∈ N |1 < x < 4} and C = {3, 5}. Verify that**

**(i) A × (B ∪ C) = (A × B) ∪ (A × C)**

**(ii) A × (B ∩ C) = (A × B) ∩ (A × C)**

**(iii) (A ∪ B) × C = (A × C) ∪ (B × C)**

**(i) A × (B ∪ C) = (A × B) ∪ (A × C)**

**Solution:**

A = {x ∈ W|x < 2} = {0,1}

B = {x ∈ N |1 < x < 4} = {2,3,4}

C = {3,5}

LHS =A × (B ∪ C)

B ∪ C = {2,3,4} ∪ {3,5}

= {2, 3, 4, 5}

A × (B ∪ C) = {(0, 2), (0, 3), (0,4), (0, 5), (1.2) , (1,3), (1,4),(1,5)} …(1)

RHS = (A × B) ∪ (A × C)

(A × B) = {(0,2), (0,3), (0,4), (1,2), (1,3), (1,4)}

(A × C) = {(0,3), (0,5), (1,3), (1,5)}

(A × B) ∪ (A × C)= {(0, 2), (0, 3), (0,4), (1, 2), (1.3), (1,4), (0, 5), (1,5)} ….(2)

(1) = (2), LHS = RHS

Hence it is proved.

(ii) A × (B ∩ C) = (A × B) ∩ (A × C)

LHS = A × (B ∩ C)

(B ∩ C) = {3}

A × (B ∩ C) = {(0, 3), (1, 3)} …(1)

RHS = (A × B) ∩ (A × C)

(A × B) = {(0,2),(0,3),(0,4),(1,2), (1,3),(1,4)}

(A × C) = {(0,3), (0,5), (1,3), (1,5)}

(A × B) ∩ (A × C) = {(0, 3), (1, 3)} …(2)

(1) = (2) ⇒ LHS = RHS.

Hence it is verified.

(iii) (A ∪ B) × C = (A × C) ∪ (B × C)

LHS = (A ∪ B) × C

A ∪ B = {0,1,2,3,4}

(A ∪ B) × C = {(0,3), (0,5), (1,3), (1,5), (2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} (1)

RHS = (A × C) ∪ (B × C)

(A × C) = {(0,3), (0,5), (1,3), (1,5)}

(B × C) = {(2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)}

(A × C) ∪ (B × C) = {(0, 3), (0, 5), (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} …(2)

(1) = (2)

∴ LHS = RHS. Hence it is verified.

**Question 7.**

**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**

**(i) (A ∩ B) × c = (A × C) ∩ (B × C)**

**(ii) A × (B – C ) = (A × B) – (A × C)**

**A = {1,2, 3, 4, 5, 6, 7}**

**B = {2, 3, 5, 7}**

**C = {2}**

**Solution:**

(i)(A ∩ B) × C = (A × c) ∩ (B × C)

LHS = (A ∩ B) × C

A ∩ B = {2, 3, 5, 7}

(A ∩ B) × C = {(2, 2), (3, 2), (5, 2), (7, 2)} …(1)

RHS = (A × C) ∩ (B × C)

(A × C) = {(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), (7,2)}

(B × C) = {2,2), (3,2), (5,2), (7,2)}

(A × C) ∩ (B × C) = {(2,2), (3,2), (5,2), (7,2)} …(2)

(1) = (2)

∴ LHS = RHS. Hence it is verified.

(ii) A × (B – C) = (A × B) – (A × C)

LHS = A × (B – C)

(B – C) = {3,5,7}

A × (B – C) = {(1,3), (1, 5), (1,7), (2,3), (2,5), (2.7) , (3,3), (3,5), (3,7), (4,3), (4,5), (4,7), (5,3), (5,5), (5,7), (6,3) , (6,5), (6,7), (7,3), (7,5), (7.7)} …(1)

RHS = (A × B) – (A × C)

(A × B) = {(1,2), (1,3), (1,5), (1,7),

(2, 2), (2, 3), (2, 5), (2, 7),

(3, 2), (3, 3), (3, 5), (3, 7),

(4, 2), (4, 3), (4, 5), (4, 7),

(5, 2), (5, 3), (5, 5), (5, 7),

(6, 2), (6, 3), (6, 5), (6, 7),

(7, 2), (7, 3), (7, 5), (7,7)}

(A × C) = {(1,2), (2,2),(3,2),(4,2), (5,2), (6,2), (7,2)}

(A × B) – (A × C) = {(1, 3), (1, 5), (1, 7), (2, 3), (2, 5), (2, 7), (3, 3), (3, 5), (3, 7), (4, 3), (4, 5), (4, 7), (5, 3), (5, 5), (5, 7), (6, 3), (6, 5), (6, 7), (7, 3), (7, 5), (7,7) } …(2)

(1) = (2) ⇒ LHS = RHS.

Hence it is verified.

__Chapter 1 Relations and Functions Ex 1.2__

### Question 1.

Let A = {1,2,3,7} and B = {3,0,-1,7}, which of the following are relation from A to B ?

(i) R1 = {(2,1), (7,1)}

(ii) R2 = {(-1,1)}

(iii) R3 = {(2,-1), (7,7), (1,3)}

(iv) R4 = {(7,-1), (0,3), (3,3), (0,7)}

(i) A = {1,2, 3,7}, B = {3, 0,-1, 7}

Solution:

### R1 = {(2,1), (7,1)}

### It is not a relation there is no element as 1 in B.

### (ii) R2 = {(-1, 1)}

It is not [∵ -1 ∉ A, 1 ∉ B]

(iii) R3 = {(2,-1), (7, 7), (1,3)}

It is a relation.

R4 = {(7,-1), (0, 3), (3, 3), (0, 7)}

It is also not a relation. [∵ 0 ∉ A]

### Question 2.

Let A = {1, 2, 3, 4,…,45} and R be the relation defined as “is square of ” on A. Write R as a subset of A × A. Also, find the domain and range of R.

Solution:

### A = {1, 2, 3, 4, . . . 45}, A × A = {(1, 1), (2, 2) ….. (45,45)}

R – is square of’

R = {(1,1), (2,4), (3, 9), (4, 16), (5,25), (6,36)}

R ⊂ (A × A)

Domain of R = {1, 2, 3, 4, 5, 6}

Range of R = {1,4, 9, 16, 25, 36}

### Question 3.

A Relation R is given by the set {(x, y) /y = x + 3, x ∈ {0, 1, 2, 3, 4, 5}}. Determine its domain and range.

Solution:

### x = {0,1,2,3,4,5}

y = x + 3

### ⇒ y = {3, 4, 5, 6, 7, 8}

R = {(x,y)}

= {(0, 3),(1, 4),(2, 5),(3, 6), (4, 7), (5, 8)}

Domain of R = {0, 1, 2, 3, 4, 5}

Range of R = {3, 4, 5, 6, 7, 8}

### Question 4.

Represent each of the given relation by (a) an arrow diagram, (b) a graph and (c) a set in roster form, wherever possible.

(i) {(x,y)|x = 2y,x ∈ {2,3,4,5},y ∈ {1, 2,3,4)

(ii) {(x, y)y = x + 3, x, y are natural numbers <10}

Solution:

### (i){(x,y)|x = 2y,x ∈ {2,3,4,5},y ∈ {1,2,3,4}} R = (x = 2y)

2 = 2 × 1 = 2

4 = 2 × 2 = 4

### (c) {(2,1), (4, 2)}

(ii) {(x, v)[y = x + 3, x,+ are natural numbers <10}

### x = {1,2, 3, 4, 5, 6, 7, 8,9} R = (y = x + 3)

### y = {1,2, 3, 4,5,6, 7, 8,9}

R = {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}

### (c) R = {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}

### Question 5.

A company has four categories of employees given by Assistants (A), Clerks (C), Managers (M) and an Executive Officer (E). The company provide ₹10,000, ₹25,000, ₹50,000 and ₹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 As 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.

### Solution:

### A – Assistants → A1, A2, A3, A4, A5

C – Clerks → C1, C2, C3, C4

D – Managers → M1, M2, M3

E – Executive officer → E1, E2

(a) R = {(10,000, A1), (10,000, A2), (10,000, A3),

(10,000, A4), (10,000, A5), (25,000, C1),

(25,000, C2), (25,000, C3), (25,000, C4),

(50,000, M1), (50,000, M2), (50,000, M3),

(1,00,000, E1), (1,00,000, E2)}

__Chapter 1 Relations and Functions Ex 1.3__

**Question 1.**

**Let f = {(x,y)|x,y ∈ N and y = 2x} be a relation on N. Find the domain, co-domain and range. Is this relation a function?**

**Solution:**

**Question 2.**

**Let X = {3, 4, 6, 8}. Determine whether the relation R = {(x,fx))|x ∈ X, f(x) = x2 + 1} is a function from X to N ?**

**Solution:**

**Question 3.**

**Given the function f: x → x2 – 5x + 6, evaluate**

**(i) f(-1)**

**(ii) A(2a)**

**(iii) f(2)**

**(iv) f(x – 1)**

**Solution:**

**Question 4.**

**A graph representing the function f(x) is given in figure 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?**

**Solution:**

**Question 5.**

**Let f(x) = 2x + 5. If x ≠ 0 then find**

**Solution:**

**Question 6.**

**A function fis defined by f(x) = 2x – 3**

**(i) find f(0)+f(1)2**

**(ii) find x such that f(x) = 0.**

**(iii) find x such that f(x) = x.**

**(iv) find x such that f(x) = f(1 – x).**

**Solution:**

**Question 7.**

**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 in figure. Express the volume V of the box as a function of x.**

**Solution:**

**Question 8.**

**A function f is defined bv f(x) = 3 – 2x . Find x such that f(x2) = (f(x))2.**

**Solution:**

**Question 9.**

**A plane is flying at a speed of 500 km per hour. Express the distance d travelled by the plane as function of time t in hours.**

**Solution:**

**Question 10.**

**The data in the adjacent table depicts the length of a woman’s 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.**

**Solution:**

__Chapter 1 Relations and Functions Ex 1.4__

__Chapter 1 Relations and Functions Ex 1.4__

**Question 1.**

**Determine whether the graph given below represent functions. Give reason for your answers concerning each graph.**

**Solution:**

**Question 2.**

**Let f :A → B be a function defined by f(x) = x2 – 1, Where A = {2,4,6,10,12},**

**B = {0,1, 2, 4, 5, 9}. Represent/by**

**(i) set of ordered pairs;**

**(ii) a table;**

**(iii) an arrow diagram;**

**(iv) a graph**

**Solution:**

**Question 3.**

**Represent the function f = {(1, 2),(2, 2),(3, 2), (4,3), (5,4)} through**

**(i) an arrow diagram**

**(ii) a table form**

**(iii) a graph**

**Solution:**

**Question 4.**

**Show that the function f : N → N defined by f{x) = 2x – 1 is one – one but not onto.**

**Solution:**

**Question 5.**

**Show that the function f: N → N defined by f (m) = m2 + m + 3 is one – one function.**

**Solution:**

**Question 6.**

**Let A = {1, 2,3,4) and B = N. Letf: A → B be**

**defined by f(x) = x3 then,**

**(i) find the range off**

**(ii) identify the tpe of function**

**Solution:**

**Question 7.**

**In each of the following cases state whether the function is bijective or not. Justify your answer.**

**(i) f: R → R defined by f(x) = 2x + 1**

**(ii) f: R → R defined by f(x) = 3 – 4×2**

**Solution:**

**Question 8.**

**Let A = {-1, 1} and B = {0, 2}. If the function f: A → B defined by f(x) = ax + b is an onto function? Find a and b.**

**Solution:**

**Question 9.**

**If the function f is defined by**

**(i) f(3)**

**(ii) f(0)**

**(iii) f(-1.5)**

**(iv) f(2) + f(-2)**

**Solution:**

**Question 10.**

**A function f: [-5,9] → R is defined as follows:**

**Solution:**

**Question 11**

**The distance S an object travels under the influence of gravity in time t seconds is given by S(t) = 12 gt2 + at + b where, (g is the 2 acceleration due to gravity), a, b are constants. Check if the function S (t)is one-one.**

**Solution:**

**Question 12.**

**The function ‘t’ which maps temperature in Celsius (C) into temperature in Fahrenheit (F) is defined by ty(C)= F where F = 95 C +32 . Find,**

**(i) t(0)**

**(ii) t(28)**

**(iii) t(-10)**

**(iv) the value of C when t(C) = 212**

**(v) the temperature when the Celsius value is equal to the Farenheit value.**

**Solution:**

__Chapter 1 Relations and Functions Ex 1.5__

**Question 1**

**Using the functions f and g given below, find fog and gof. Check whether fog = gof.**

**(i) f(x) = x – 6, g(x) = x2**

**(ii) f(x) = 2x, g(x) = 2×2– 1**

**(iii) f(x) = x+63g(x) = 3 – x**

**(iv) f(x) = 3 + x, g(x) = x – 4**

**(v) f(x) = 4×2– 1,g(x) = 1 + x**

**Solution:**

**Question 2.**

**Find the value of k, such that fog = gof**

**(i) f(x) = 3x + 2, g(x) = 6x – k**

**(ii) f(x) = 2x – k, g(x) = 4x + 5**

**Solution:**

**Question 3.**

**if f(x) = 2x – 1, g(x) = x+12, show that fog = gof = x**

**Solution:**

**Question 4.**

**(i) If f (x) = x2 – 1, g(x) = x – 2 find a, if gof(a) = 1.**

**(ii) Find k, if f(k) = 2k – 1 and fof (k) = 5.**

**Solution:**

**Question 5.**

**Let A,B,C ⊂ N and a function f: A → B be defined by f(x) = 2x + 1 and g : B → C be defined by g(x) = x2. Find the range of fog and gof**

**Solution:**

**Question 6.**

**If f(x) = x2 – 1. Find (i)f(x) = x2 – 1, (ii)fofof**

**Solution:**

**Question 7.**

**If f: R → R and g : R → R are defined by f(x) = x5 and g(x) = x4 then check if f,g are one-one and fog is one-one?**

**Solution:**

**Question 8.**

**Consider the functions f(x), g(x), h(x) as given below. Show that (fog)oh = fo(goh) in each case.**

**(i) f(x) = x – 1, g(x) = 3x + 1 and h(x) = x2**

**(ii) f(x) = x2, g(x) = 2x and h(x) = x + 4**

**(iii) f(x) = x – 4, g(x) = x2 and h(x) = 3x – 5**

**Solution:**

**Question 9.**

**Let f ={(-1, 3),(0, -1),(2, -9)} be a linear function from Z into Z . Find f(x).**

**Solution:**

**Question 10.**

**In electrical circuit theory, a circuit C(t) is called a linear circuit if it satisfies the superposition principle given by C(at1 + bt2) = aC(t1) + bC(t2), where a,b are constants. Show that the circuit C(t) = 31 is linear.**

**Solution:**

__Chapter 1 Relations and Functions Ex 1.6__

**Question 1.**

**If n(A × B) = 6 and A = {1, 3} then n(B) is**

**(1) 1**

**(2) 2**

**(3) 3**

**(4) 6**

**Question 2.**

**A = {a, b, p}, B = {2, 3}, C = {p, q, r, s} then**

**n[(A ∪ C) × B] is**

**(1) 8**

**(2) 20**

**(3) 12**

**(4) 16**

**Question 3.**

**If A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5,6,7,8} then state which of the following statement is true.**

**(1) (A × C) ⊂ (B × D)**

**(2) (B × D) ⊂ (A × C)**

**(3) (A × B) ⊂ (A × D)**

**(4) (D × A) ⊂ (B × A)**

**Question 4.**

**If there are 1024 relations from a set A = {1, 2, 3, 4, 5} to a set B, then the number of elements in B is**

**(1) 3**

**(2) 2**

**(3) 4**

**(4) 6**

**Question 5.**

**The range of the relation R = {(x, x2)|x is a prime number less than 13} is**

**(1) {2,3,5,7}**

**(2) {2,3,5,7,11}**

**(3) {4,9,25,49,121}**

**(4) {1,4,9,25,49,121}**

**Question 6.**

**If the ordered pairs (a + 2,4) and (5,2a+b)are equal then (a,b) is**

**(1) (2,-2)**

**(2) (5,1)**

**(3) (2,3)**

**(4) (3,-2)**

**Question 7.**

**Let n(A) = m and n(B) = n then the total number of non-empty relations that can be defined from A to B is**

**(1) mn**

**(2) nm**

**(3) 2mn-1**

**(4) 2mn**

**Question 8.**

**If {(a,8),(6,b)}represents an identity function, then the value of a and b are respectively**

**(1) (8,6)**

**(2) (8,8)**

**(3) (6,8)**

**(4) (6,6)**

**Question 9.**

**Let A = {1,2,3,4} and B = {4,8,9,10}. A function f : A → B given by f = {(1,4),(2,8),(3,9),(4,10)} is a**

**(1) Many-one function**

**(2) Identity function**

**(3) One-to-one function**

**(4) Into function**

**Question 10.**

**If f(x) = 2×2 and g (x) = 13x, Then fog is**

**Queston 11.**

**If f: A → B is a bijective function and if n(B) = 7, then n(A) is equal to**

**(1) 7**

**(2) 49**

**(3) 1**

**(4) 14**

**Question 12.**

**Let f and g be two functions given by f = {(0,1), (2,0), (3, -4), (4,2), (5,7)} g = {(0,2), (1, 0), (2, 4), (-4, 2), (7,0)} then the range of fog is**

**(1) {0,2,3,4,5}**

**(2) {-4,1,0,2,7}**

**(3) {1,2,3,4,5}**

**(4) {0,1,2}**

**Question 13.**

**Let f(x) = 1+x2−−−−−√ then**

**(1) f(xy) = f(x),f(y)**

**(2) f(xy) > f(x),f(y)**

**(3) f(xy) < f(x).f(y)**

**(4) None of these**

**Question 14.**

**If g = {(1, 1),(2, 3),(3, 5),(4, 7)} is a function given by g(x) = ∝x + β then the values of ∝ and β are**

**(1) (-1,2)**

**(2) (2,-1)**

**(3) (-1,-2)**

**(4) (1,2)**

**Question 15.**

**f(x) = (x + 1)3 – (x – 1)3 represents a function which is**

**(1) linear**

**(2) cubic**

**(3) reciprocal**

**(4) quadratic**