NCERT Solutions for Exercise 1.4 Class 12 Maths Chapter 1

NCERT Solutions for Exercise 1.4 Class 12 Maths Chapter 1 - Relations and Functions

Edited By Ramraj Saini | Updated on Dec 03, 2023 01:35 PM IST | #CBSE Class 12th

NCERT Solutions For Class 12 Maths Chapter 1 Exercise 1.4

NCERT Solutions for Exercise 1.4 Class 12 Maths Chapter 1 Relations and Functions are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. NCERT Solutions for class 12 maths chapter 1 exercise 1.4 talks about commutative, associative, binary operations etc. Exercise 1.4 Class 12 Maths has questions which includes finding whether a relation follows binary operation or not. After analysis from previous year questions it is clear that NCERT Solutions for class 12 maths chapter 1 exercise 1.4 plays an important role to score well in CBSE class 12 board exam. Also it has a significant contribution in competitive exams like JEE main. Hence it is recommended for students to solve all the questions of Class 12th maths chapter 1 exercise 1.4 to score well in their examination.

Latest: JEE Main high scoring chapters | JEE Main 10 year's papers

This Story also Contains

NCERT Solutions For Class 12 Maths Chapter 1 Exercise 1.4
Assess NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4
NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions: Exercise 1.4
Key Features Of NCERT Solutions for Exercise 1.4 Class 12 Maths Chapter 1

12th class Maths exercise 1.4 answers are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.

PTE Registrations 2024

Apply

Assess NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4

Download PDF

NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions: Exercise 1.4

Question:1(i) Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

(i) On $Z^+$ , define ∗ by $a * b = a - b$

Answer:

(i) On $Z^+$ , define ∗ by $a * b = a - b$

It is not a binary operation as the image of $(1,2)$ under * is $1\ast 2=1-2$ $=-1 \notin Z^{+}$ .

Question:1(ii) Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

(ii) On $Z^+$ , define ∗ by $a * b = ab$

Answer:

(ii) On $Z^+$ , define ∗ by $a * b = ab$

We can observe that for $a,b \in Z^+$ ,there is a unique element ab in $Z^+$ .

This means * carries each pair $(a,b)$ to a unique element $a * b = ab$ in $Z^+$ .

Therefore,* is a binary operation.

Question:1(iii) Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

(iii) On $R$ , define ∗ by $a * b = ab^2$

Answer:

(iii) On $R$ , define ∗ by $a * b = ab^2$

We can observe that for $a,b \in R$ ,there is a unique element $ab^{2}$ in $R$ .

This means * carries each pair $(a,b)$ to a unique element $a * b = ab^{2}$ in $R$ .

Therefore,* is a binary operation.

Question:1(iv) Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

(iv) On $Z^+$ , define ∗ by $a * b = | a - b |$

Answer:

(iv) On $Z^+$ , define ∗ by $a * b = | a - b |$

We can observe that for $a,b \in Z^+$ ,there is a unique element $| a - b |$ in $Z^+$ .

This means * carries each pair $(a,b)$ to a unique element $a * b = | a - b |$ in $Z^+$ .

Therefore,* is a binary operation.

Question:1(v) Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this

(v) On $Z^+$ , define ∗ by $a * b = a$

Answer:

(v) On $Z^+$ , define ∗ by $a * b = a$

* carries each pair $(a,b)$ to a unique element $a * b = a$ in $Z^+$ .

Therefore,* is a binary operation.

Question:2(i) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(i)On $Z$ , define $a * b = a-b$

Answer:

a*b=a-b

b*a=b-a

$a*b\neq b*a$

so * is not commutative

(a*b)*c=(a-b)-c

a*(b*c)=a-(b-c)=a-b+c

(a*b)*c not equal to a*(b*c), so * is not associative

Question:2(ii) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(ii) On $Q$ , define $a * b = ab + 1$

Answer:

(ii) On $Q$ , define $a * b = ab + 1$

ab = ba for all $a,b \in Q$

ab+1 = ba + 1 for all $a,b \in Q$

$\Rightarrow$ $a\ast b=b\ast a$ for $a,b \in Q$

$(1*2)*3 = (1\times 2+1) * 3 = 3 * 3 = 3\times 3+1 = 10$

$1*(2*3) = 1 * (2\times 3+1) = 1 * 7 = 1\times 7+1 = 8$

$\therefore$ $(1\ast 2)\ast 3\neq 1\ast (2\ast 3)$ $;$ where $1,2,3 \in Q$

$\therefore$ operation * is not associative.

Question:2(iii) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(iii) On $Q$ , define $a * b = \frac{ab}{2}$

Answer:

(iii) On $Q$ , define $a * b = \frac{ab}{2}$

ab = ba for all $a,b \in Q$

$\frac{ab}{2}=\frac{ba}{2}$ for all $a,b \in Q$

$\Rightarrow$ $a\ast b=b\ast a$ for $a,b \in Q$

$\therefore$ operation * is commutative.

$(a*b)*c = \frac{ab}{2}*c = \frac{(\frac{ab}{2})c}{2} = \frac{abc}{4}$

$a*(b*c) = a*\frac{bc}{2} = \frac{a(\frac{bc}{2})}{2} = \frac{abc}{4}$

$\therefore$ $(a*b)*c=a*(b*c)$ $;$

$\therefore$ operation * is associative.

Question:2(iv) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(iv) On $Z^+$ , define $a * b = 2^{ab}$

Answer:

(iv) On $Z^+$ , define $a * b = 2^{ab}$

ab = ba for all $a,b \in Z^{+}$

2ab = 2ba for all $a,b \in Z^{+}$

$\Rightarrow$ $a\ast b=b\ast a$ for $a,b \in Z^{+}$

$\therefore$ the operation is commutative.

$(1*2)*3 = 2^{1\times 2} * 3 = 4 * 3 = 2^{4\times 3} = 2^{12}$

$1*(2*3) = 1 * 2^{2\times 3} = 1 * 64 = 2^{1\times 64}=2^{64}$

$\therefore$ $(1\ast 2)\ast 3\neq 1\ast (2\ast 3)$ $;$ where $1,2,3 \in Z^{+}$

$\therefore$ operation * is not associative.

Question:2(v) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(v) On $Z^+$ , define $a * b = a^b$

Answer:

(v) On $Z^+$ , define $a * b = a^b$

$1\ast 2 = 1^{2}= 1$ and $2\ast 1 = 2^{1}= 2$

$\Rightarrow$ $1\ast 2\neq 2\ast 1$ for $1,2 \in Z^{+}$

$\therefore$ the operation is not commutative.

$(2\ast 3)\ast 4 = 2^{3} \ast 4 = 8 \ast 4 = 8^{ 4}=2^{12}$

$2\ast (3\ast 4) = 2 \ast 3^{ 4} = 2 \ast 81 = 2^{81}$

$\therefore$ $(2\ast 3)\ast 4\neq 2\ast (3\ast 4)$ $;$ where $2,3,4 \in Z^{+}$

$\therefore$ operation * is not associative.

Question:2(vi) For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(vi) On $R - \{-1 \}$ , define $a * b = \frac{a}{b +1}$

Answer:

(iv) On $R - \{-1 \}$ , define $a * b = \frac{a}{b +1}$

$1\ast 2 = \frac{1}{2+1}=\frac{1}{3}$ and $2\ast 1 = \frac{2}{2+1}= \frac{2}{3}$

$\Rightarrow$ $1\ast 2\neq 2\ast 1$ for $1,2 \in R - \{-1 \}$

$\therefore$ the operation is not commutative.

$(1\ast 2)\ast 3 = (\frac{1}{2+1}) \ast 3 = \frac{1}{3} \ast 3 = \frac{\frac{1}{3}}{3+1}= \frac{1}{12}$

$1\ast (2\ast 3) = 1 \ast (\frac{2}{3+1}) = 1 \ast \frac{2}{4} = 1 \ast \frac{1}{2} = \frac{1}{\frac{1}{2}+1} = \frac{2}{3}$

$\therefore$ $(1\ast 2)\ast 3\neq 1\ast (2\ast 3)$ $;$ where $1,2,3 \in R - \{-1 \}$

$\therefore$ operation * is not associative.

Question:3 Consider the binary operation $\wedge$ on the set $\{1, 2, 3, 4, 5\}$ defined by $a \wedge b = min \{a, b\}$ . Write the operation table of the operation $\wedge$ .

Answer:

$\{1, 2, 3, 4, 5\}$

$a \wedge b = min \{a, b\}$ for $a,b \in \{1, 2, 3, 4, 5\}$

The operation table of the operation $\wedge$ is given by :

$\wedge$	1	2	3	4	5
1	1	1	1	1	1
2	1	2	2	2	2
3	1	2	3	3	3
4	1	2	3	4	4
5	1	2	3	4	5

Question:4(i) Consider a binary operation ∗ on the set $\{1, 2, 3, 4, 5\}$ given by the following multiplication table (Table 1.2).

(i) Compute $(2 * 3) * 4$ and $2 * (3 * 4)$

(Hint: use the following table)

1653387717508

Answer:

(i)

1653387755767

$(2 * 3) * 4 = 1*4 =1$

$2 * (3 * 4) = 2*1=1$

Question:4(ii) Consider a binary operation ∗ on the set $\{1, 2, 3, 4, 5\}$ given by the following multiplication table (Table 1.2).

(ii) Is ∗ commutative?

(Hint: use the following table)

1653387844990

Answer:

(ii)

1653387854531

For every $a,b \in\{1, 2, 3, 4, 5\}$ , we have $a*b = b*a$ . Hence it is commutative.

Question:4(iii) Consider a binary operation ∗ on the set { $\{1, 2, 3, 4, 5\}$ given by the following multiplication table (Table 1.2).

(iii) Compute (2 ∗ 3) ∗ (4 ∗ 5).

(Hint: use the following table)

1653387903438

Answer:

(iii) (2 ∗ 3) ∗ (4 ∗ 5).

1653387914428

from the above table

$(2*3)*(4*5)=1*1=1$

Question:5 Let ∗′ be the binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by $a *' b = H.C.F. \;of\;a\;and\;b$ . Is the operation ∗′ same as the operation ∗ defined
in Exercise 4 above? Justify your answer.

Answer:

$a *' b = H.C.F. \;of\;a\;and\;b$ for $a,b \in \{1, 2, 3, 4, 5\}$

The operation table is as shown below:

$\ast$	1	2	3	4	5
1	1	1	1	1	1
2	1	2	1	2	1
3	1	1	3	1	1
4	1	2	1	4	1
5	1	1	1	1	5

The operation ∗′ same as the operation ∗ defined in Exercise 4 above.

Question:6(i) let ∗ be the binary operation on N given by . Find

(i) 5 ∗ 7, 20 ∗ 16

Answer:

a*b=LCM of a and b

(i) 5 ∗ 7, 20 ∗ 16

$5*7 = L.C.M \, of\, 5\, \, and \, 7=35$

$20*16 = L.C.M \, of\, 20\, \, and \, 16 =80$

Question:6(ii) Let ∗ be the binary operation on N given by $a * b = L.C.M. \;of \;a\; and \;b$ . Find

(ii) Is ∗ commutative?

Answer:

$a * b = L.C.M. \;of \;a\; and \;b$

(ii) $L.C.M. \;of \;a\; and \;b = L.C.M. \;of \;b\; and \;a$ for all $a,b \in N$

$\therefore \, \, \, \, \, \, \, a*b = b*a$

Hence, it is commutative.

Question:6(iii) Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

(iii) Is ∗ associative?

Answer:

a $*$ b = L.C.M. of a and b

(iii) $a,b,c \in N$

$(a*b)*c = (L.C.M \, of\, a\, and \, b)*c= L.C.M\, of \, a,b\, and\, c$

$a*(b*c) = a*(L.C.M \, of\, b\, and \, c)= L.C.M\, of \, a,b\, and\, c$

$\therefore \, \, \, \, \, \, \, \, \, (a*b)*c=a*(b*c)$

Hence, the operation is associative.

Question:6(iv) Let ∗ be the binary operation on N given by $a* b = L.C.M.\; of \;a\; and \;b$ . Find

(iv) the identity of ∗ in N

Answer:

$a* b = L.C.M.\; of \;a\; and \;b$

(iv) the identity of ∗ in N

We know that $L.C.M.\; of \;a\; and \;1 = a = L.C.M.\; of 1\, \, and \;a\;$

$\therefore$ $a*1=a=1*a$ for $a \in N$

Hence, 1 is the identity of ∗ in N.

Question 6(v) Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

(v) Which elements of N are invertible for the operation ∗?

Answer:

$a* b = L.C.M.\; of \;a\; and \;b$

An element a is invertible in N

if $a*b=e=b*a$

Here a is inverse of b.

a*b=1=b*a

a*b=L.C.M. od a and b

a=b=1

So 1 is the only invertible element of N

Question:7 Is ∗ defined on the set $\{1, 2, 3, 4, 5\}$ by $a * b = L.C.M. \;of \;a\; and \;b$ a binary operation? Justify your answer.

Answer:

$a * b = L.C.M. \;of \;a\; and \;b$

A = $\{1, 2, 3, 4, 5\}$

Operation table is as shown below:

$*$	1	2	3	4	5
1	1	2	3	4	5
2	2	2	6	4	10
3	3	6	3	12	15
4	4	4	12	4	20
5	5	10	15	20	5

From the table, we can observe that

$2*3=3*2=6 \notin A$

$2*5=5*2=10 \notin A$

$3*4=4*3=12 \notin A$

$3*5=5*3=15 \notin A$

$4*5=5*4=20 \notin A$

Hence, the operation is not a binary operation.

Question:8 Let ∗ be the binary operation on N defined by a ∗ b = H.C.F. of a and b. Is ∗ commutative? Is ∗ associative? Does there exist identity for this binary
operation on N?

Answer:

a ∗ b = H.C.F. of a and b for all $a,b \in A$

H.C.F. of a and b = H.C.F of b and a for all $a,b \in A$

$\therefore \, \, \, \, a*b=b*a$

Hence, operation ∗ is commutative.

For $a,b,c \in N$ ,

$(a*b)*c = (H.C.F \, of\, a\, and\, b)*c= H.C.F\, of \, a,b,c.$

$a*(b*c )= a*(H.C.F \, of\, b\, and\, c)= H.C.F\, of \, a,b,c.$

$\therefore$ $(a*b)*c=a*(b*c)$

Hence, ∗ is associative.

An element $c \in N$ will be identity for operation * if $a*c=a= c*a$ for $a \in N$ .

Hence, the operation * does not have any identity in N.

Question:9(i) Let ∗ be a binary operation on the set Q of rational numbers as follows:

(i) $a * b = a - b$ Find which of the binary operations are commutative and which are associative.

Answer:

On the set Q ,the operation * is defined as $a * b = a - b$ .It is observed that:

$\frac{1}{2}*\frac{1}{3}= \frac{1}{2}-\frac{1}{3}=\frac{1}{6}$

$\frac{1}{3}*\frac{1}{2}= \frac{1}{3}-\frac{1}{2}=\frac{-1}{6}$

$\therefore$ $\frac{1}{2}*\frac{1}{3}\neq \frac{1}{3}*\frac{1}{2}$ here $\frac{1}{2},\frac{1}{3} \in Q$

Hence, the * operation is not commutative.

It can be observed that

$(\frac{1}{2}*\frac{1}{3})*\frac{1}{4} = \left ( \frac{1}{2}-\frac{1}{3}\right )*\frac{1}{4}=\frac{1}{6}*\frac{1}{4}=\frac{1}{6}-\frac{1}{4}=\frac{-1}{12}$

$\frac{1}{2}*(\frac{1}{3}*\frac{1}{4})= \frac{1}{2}*\left ( \frac{1}{3} - \frac{1}{4}\right ) = \frac{1}{2}*\frac{1}{12} = \left ( \frac{1}{2} - \frac{1}{12} \right ) = \frac{5}{12}$

$\left ( \frac{1}{2}*\frac{1}{3} \right )*\frac{1}{4}\neq \frac{1}{2}*(\frac{1}{3}*\frac{1}{4})$ for all $\frac{1}{2},\frac{1}{3}, \frac{1}{4} \in Q$

The operation * is not associative.

Question:9(ii) Let ∗ be a binary operation on the set Q of rational numbers as follows:

(ii) $a*b = a^2 + b^2$ Find which of the binary operations are commutative and which are associative.

Answer:

On the set Q ,the operation * is defines as $a*b = a^2 + b^2$ .It is observed that:

For $a,b \in Q$

$a*b=a^{2}+b^{2}= b^{2}+a^{2}=b*a$

$\therefore$ $a*b=b*a$

Hence, the * operation is commutative.

It can be observed that

$(1*2)*3 =(1^{2}+2^{2})*3 = 5*3 = 5^{2}+3^{2} = 25+9 =34$

$1*(2*3) =1*(2^{2}+3^{2}) = 1*13 = 1^{2}+13^{2} = 1+169 =170$

$(1*2)*3 \neq 1*(2*3)$ for all $1,2,3 \in Q$

The operation * is not associative.

Question:9(iii) Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iii) $a * b = a + ab$ Find which of the binary operations are commutative and which are associative.

Answer:

On the set Q ,the operation * is defines as $a * b = a + ab$ .It is observed that:

For $a,b \in Q$

$1 * 2 = 1+1\times 2 =1 + 2 = 3$

$2 * 1= 2+2\times 1 =2 + 2 = 4$

$\therefore$ $1*2\neq 2*1$ for $1,2 \in Q$

Hence, the * operation is not commutative.

It can be observed that

$1*(2*3) =1*(2+3\times 2) = 1*8 = 1+1\times 8 = 1+8 =9$

$(1*2)*3 \neq 1*(2*3)$ for all $1,2,3 \in Q$

The operation * is not associative.

Question:9(iv) Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iv) $a * b = (a-b)^2$ Find which of the binary operations are commutative and which are associative.

Answer:

On the set Q ,the operation * is defined as $a * b = (a-b)^2$ .It is observed that:

For $a,b \in Q$

$a * b = (a-b)^2$

$b* a = (b-a)^2 = \left [ -\left ( a-b \right ) \right ]^{2} = (a-b)^{2}$

$\therefore$ $a*b = b* a$ for $a,b \in Q$

Hence, the * operation is commutative.

It can be observed that

$(1*2)*3 =(1-2)^{2}*3 = 1*3 =(1-3)^{2}= (-2)^{2} =4$

$1*(2*3) =1*(2-3)^{2} = 1*1 =(1-1)^{2} = 0^{2} =0$

$(1*2)*3 \neq 1*(2*3)$ for all $1,2,3 \in Q$

The operation * is not associative.

Question:9(v) Let ∗ be a binary operation on the set Q of rational numbers as follows:

(v) $a * b = \frac{ab}{4}$ Find which of the binary operations are commutative and which are associative.

Answer:

On the set Q ,the operation * is defines as $a * b = \frac{ab}{4}$ .It is observed that:

For $a,b \in Q$

$a * b = \frac{ab}{4}$

$b* a = \frac{ba}{4}$

$\therefore$ $a*b = b* a$ for $a,b \in Q$

Hence, the * operation is commutative.

It can be observed that

$(a*b)*c =(\frac{ab}{4})*c = \frac{\frac{ab}{4}c}{4}=\frac{abc}{16}$

$a*(b*c) =a*(\frac{bc}{4}) = \frac{\frac{bc}{4}a}{4}=\frac{abc}{16}$

$(a*b)*c = a*(b*c)$ for all $a,b,c \in Q$

The operation * is associative.

Question:9(vi) Let ∗ be a binary operation on the set Q of rational numbers as follows:

(vi) $a* b = ab^2$ Find which of the binary operations are commutative and which are associative.

Answer:

On the set Q ,the operation * is defines as $a* b = ab^2$ .It is observed that:

For $a,b \in Q$

$1* 2 = 1\times 2^2=1\times 4=4$

$2* 1 = 2\times 1^2=2\times 1=2$

$\therefore$ $1*2\neq 2*1$ for $1,2 \in Q$

Hence, the * operation is not commutative.

It can be observed that

$(1*2)*3 = (1\times 2^{2})*3 = 4*3 = 4\times 3^{2}=4\times 9=36$

$1*(2*3) = 1*(2\times 3^{2}) = 1*18 = 1\times 18^{2}=1\times 324=324$

$(1*2)*3\neq 1*(2*3)$ for all $1,2,3 \in Q$

The operation * is not associative.

Question:10 Find which of the operations given above has identity.

Answer:

An element $p \in Q$ will be identity element for operation *

if $a*p = a = p*a$ for all $a \in Q$

$(v) a * b = \frac{ab}{4}$

$a * p = \frac{ap}{4}$

$p * a = \frac{pa}{4}$

$a*p = a = p*a$ when $p=4$ .

Hence, $(v) a * b = \frac{ab}{4}$ has identity as 4.

However, there is no such element $p \in Q$ which satisfies above condition for all rest five operations.

Hence, only (v) operations have identity.

Question:11 Let $A = N \times N$ and ∗ be the binary operation on A defined by $(a, b) * (c, d) = (a + c, b + d)$ Show that ∗ is commutative and associative. Find the identity element for ∗ on A, if any.

Answer:

$A = N \times N$ and ∗ be the binary operation on A defined by

$(a, b) * (c, d) = (a + c, b + d)$

Let $(a,b),(c,d) \in A$

Then, $a,b,c,d \in N$

We have

$(a, b) * (c, d) = (a + c, b + d)$

$(c,d)*(a,b) = (c+a,d+b)= (a+c,b+d)$

$\therefore \, \, \, \, \, (a,b)*(c,d)=(c,d)*(a,b)$

Thus it is commutative.

Let $(a,b),(c,d),(e,f) \in A$

Then, $a,b,c,d,e,f \in N$

$[(a, b) * (c, d)]*(e,f)= [(a + c, b + d)]*(e,f) =[(a+c+e),(b+d+f)]$

$(a, b) * [(c, d)*(e,f)]= (a,b)*[(c + e, d + f)] =[(a+c+e),(b+d+f)]$

$\therefore \, \, \, \, \, [(a,b)*(c,d)]*(e,f)=(a, b) * [(c, d)*(e,f)]$

Thus, it is associative.

Let $e= (e1,e2) \in A$ will be a element for operation * if $(a*e)=a=(e*a)$ for all $a= (a1,a2) \in A$ .

i.e. $(a1+e1,a2+e2)= (a1,a2)= (e1+a1,e2+a2)$

This is not possible for any element in A .

Hence, it does not have any identity.

Question:12(i) State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation ∗ on a set N, $a*a = a,\; \forall a\in N$

Answer:

(i) For an arbitrary binary operation ∗ on a set N, $a*a = a,\; \forall a\in N$

An operation * on a set N as $a*b=a+b\, \, \, \, \forall\, \, a,b \in N$

Then , for b=a=2

$2*2= 2+2 = 4\neq 2$

Hence, statement (i) is false.

Question:12(ii) State whether the following statements are true or false. Justify.

(ii) If ∗ is a commutative binary operation on N, then $a *(b*c) =( c*b)*a$

Answer:

(ii) If ∗ is a commutative binary operation on N, then $a *(b*c) =( c*b)*a$

R.H.S $=(c*b)*a$

$=(b*c)*a$ (* is commutative)

$= a*(b*c)$ ( as * is commutative)

= L.H.S

$\therefore$ $a *(b*c) =( c*b)*a$

Hence, statement (ii) is true.

Question:13 Consider a binary operation ∗ on N defined as $a * b = a^3 + b^3$ . Choose the correct answer.

(A) Is ∗ both associative and commutative?
(B) Is ∗ commutative but not associative?
(C) Is ∗ associative but not commutative?
(D) Is ∗ neither commutative nor associative?

Answer:

A binary operation ∗ on N defined as $a * b = a^3 + b^3$ .

For $a,b \in N$

$a * b = a^3 + b^3 = b^{3}+a^{3}=b*a$

Thus, it is commutative.

$(1*2)*3 = (1^{3}+2^{3})*3=9*3 =9^{3}+3^{3}=729+27=756$

$1*(2*3) = 1*(2^{3}+3^{3})=1*35 =1^{3}+35^{3}=1+42875=42876$

$\therefore \, \, \, \, \, (1*2)*3 \neq 1*(2*3)$ where $1,2,3 \in N$

Hence, it is not associative.

Hence, B is the correct option.

More About NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4

The NCERT Class 12 Maths chapter Relations and Functions has a total of 5 exercises including miscellaneous exercise. Exercise 1.4 Class 12 Maths covers solutions to 13 main questions and their sub-questions. Most of the questions are related to concepts of binary operations which includes commutative, associative operations etc. Hence NCERT Solutions for Class 12 Maths chapter 1 exercise 1.4 is recommended for learning these concepts to score well in the exam.

Also Read| NCERT Notes For Class 12 Mathematics Chapter 1

Benefits of NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4

The Class 12 Maths NCERT syllabus chapter 1.4 exercise is provided above in a comprehensive manner which is solved by subject matter experts .
Students are recommended to practice Exercise 1.4 Class 12 Maths to prepare for topics like binary operations which includes commutative, associative operations etc.
These Class 12 Maths chapter 1 exercise 1.4 solutions can be referred by students to revise just before the exam.
NCERT syllabus for Class 12 Maths chapter 1 exercise 1.4 can be used to prepare similar topics of physics also.

JEE Main Highest Scoring Chapters & Topics

Just Study 40% Syllabus and Score upto 100%

Download EBook

Key Features Of NCERT Solutions for Exercise 1.4 Class 12 Maths Chapter 1

Comprehensive Coverage: The solutions encompass all the topics covered in ex 1.4 class 12, ensuring a thorough understanding of the concepts.
Step-by-Step Solutions: In this class 12 maths ex 1.4, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.
Accuracy and Clarity: Solutions for class 12 ex 1.4 are presented accurately and concisely, using simple language to help students grasp the concepts easily.
Conceptual Clarity: In this 12th class maths exercise 1.4 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.
Inclusive Approach: Solutions for ex 1.4 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.
Relevance to Curriculum: The solutions for class 12 maths ex 1.4 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

Also see-

NCERT Solutions Subject Wise

Subject wise NCERT Exemplar solutions

Happy learning!!!

Frequently Asked Question (FAQs)

1. Which concepts are covered in Exercise 1.4 Class 12 Maths?

These concepts are discussed in class 12 ex 1.4. Concepts like proving a binary operation associative, commutative etc. are mentioned in the Exercise 1.4 Class 12 Maths. Practice 12th class maths exercise 1.4 answers to command the cocepts.

2. What are the important topics in chapter relations and functions ?

Definitions of relations and functions, types of relations, types of functions, composition of functions, invertible function and binary operations are the important topics in this chapter.

3. How much weightage is held by the chapter relations and functions for CBSE board exam ?

The weightage of chapter relation and function is more than 5% in the CBSE board examination.

4. How are the NCERT solutions helpful in the board exam ?

As students can assess from previous year questions that many questions are asked directly from NCERT exercise. Hence to score well in the examination, it is required to practice NCERT exercise for Class 12 Maths before the exam.

5. What are relations in Class 12 Maths?

These concepts are discussed in class 12 maths ex 1.4. In maths, relation defines the relationship between sets of values of ordered pairs. Practice these questions to get deeper understanding.

6. Mention the number of questions in Exercise 1.4 Class 12 Maths ?

13 questions are the in Exercise 1.4 Class 12 Maths

7. How many exercises are there in the NCERT Class 12 Maths chapter 1 ?

There are a total of 5 exercises including a miscellaneous exercise in the NCERT Class 12 Maths chapter 1.

Also Read

NCERT Books for Class 12 - Download All Subjects PDFs Here

NCERT Class 12 Physics Chapter 9 Notes Ray Optics and Optical Instruments - Download PDF

NCERT Class 12 Physics Chapter 8 Notes Electromagnetic Waves - Download PDF

NCERT Class 12 Physics Chapter 7 Notes Alternating Current - Download PDF

NCERT Class 12 Physics Chapter 6 Notes Electromagnetic Induction - Download PDF

NCERT Class 12 Physics Chapter 5 Notes Magnetism and Matter - Download PDF

NCERT Class 12 Physics Chapter 4 Notes Moving Charges and Magnetism - Download PDF

NCERT Solutions for Miscellaneous Exercise Chapter 7 Class 12 - Integrals

NCERT Solutions for Exercise 4.1 Class 11 Maths Chapter 4 - Principle of Mathematical Induction

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

NCERT Exemplar Class 12 Physics Solutions Chapter 8 Electromagnetic Waves

Articles

CBSE Sample Papers 2024-25: Download CBSE 10th, 12th Previous Year Papers Here

Sep 05, 2024

UPMSP UP Board 12th Admit Card 2025, Download Admit Card Here

Sep 05, 2024

NIOS Registration 2024 for Class 10th, 12th- Link, Apply before Last Date

Sep 05, 2024

BSEB Bihar Board 12th Exam Date 2025 for Sent UP & Final Exams Soon @biharboardonline.bihar.gov.in

Sep 05, 2024

MPsos Result 2024 December Exam, Check MP State Open School Result @mpsos.nic.in

Sep 05, 2024

CBSE Mid-Term Exam Date 2024-25: Half Yearly Dates Sheet for Class 9, 10, 11 and 12

Sep 05, 2024

CBSE Class 11 English Syllabus 2024-25 Download PDF

Sep 04, 2024

CBSE Single Girl Child Scholarship 2024: Application Process, Eligibility, and Dates

Sep 04, 2024

Upcoming School Exams

National Means Cum-Merit Scholarship

Application Date:01 August,2024 - 16 September,2024

Eligibility Criteria

Application Process

Exam Pattern

International General Knowledge Olympiad

Exam Date:19 September,2024 - 19 September,2024

Exam Pattern

Result

Admit Card

Unified Cyber Olympiad

Exam Date:20 September,2024 - 20 September,2024

Exam Pattern

Admit Card

Result

National Institute of Open Schooling 10th examination

Exam Date:20 September,2024 - 07 October,2024

Application Process

Exam Pattern

Mock Test

National Institute of Open Schooling 12th Examination

Exam Date:20 September,2024 - 07 October,2024

Application Process

Exam Pattern

Admit Card

View All School Exams

Certifications By Top Providers

Full-Stack Web Development

Via Masai School

Introduction to Web Development with HTML CSS JavaScript

Via IBM

Artificial Intelligence Projects

Via Great Learning

Digital Banking Business Model

Via State Bank of India

Introduction to Watson AI

Via IBM

Business Analytics Foundations

Via PW Skills

1113 courses

804 courses

394 courses

222 courses

Harvard University, Cambridge

98 courses

IBM

85 courses

IT & Software

Teaching and Academics

Programming and Development

Health, Fitness and Medicine

Business and Management

Engineering and Architecture

General Management

Financial Management

Environmental Studies

History

Psychology

Artificial Intelligence

Explore Top Universities Across Globe

University of Essex, Colchester

Wivenhoe Park Colchester CO4 3SQ

University College London, London

Gower Street, London, WC1E 6BT

The University of Edinburgh, Edinburgh

Old College, South Bridge, Edinburgh, Post Code EH8 9YL

University of Bristol, Bristol

Beacon House, Queens Road, Bristol, BS8 1QU

University of Delaware, Newark

Newark, DE 19716

University of Nottingham, Nottingham

University Park, Nottingham NG7 2RD

ETH Zurich Scholarship 2024: Requirements and Deadlines

3 min ⋆ Aug 14, 2024 06:08 AM IST

IELTS Exam 2024 - Dates, Registration, Syllabus, Fees, Pattern, Result

11 min ⋆ Jul 30, 2024 12:07 PM IST

TOEFL Syllabus 2024 - Listening, Reading, Speaking and Writing Section

9 min ⋆ Jul 30, 2024 12:07 PM IST

GMAT Exam 2024: Dates, Fees, Registration, Syllabus, Pattern, Result

17 min ⋆ Aug 12, 2024 11:08 AM IST

Engineering in USA for Indian Students: Top Universities, Fees and Scholarships

6 min ⋆ Aug 14, 2024 05:08 AM IST

GMAT Test Centres 2024: List of City Wise Exam Centres in India

6 min ⋆ Jul 22, 2024 05:07 AM IST

Related E-books & Sample Papers

CBSE 12th English Elective Question Paper 2024

57+ Downloads

CBSE 12th Sociology Question Paper 2024

14+ Downloads

CBSE 12th Computer Science Question Paper 2024

18+ Downloads

CBSE 12th Psychology Question Paper 2024

13+ Downloads

CBSE 12th Biology (Series 1-3) Question Paper 2024

15+ Downloads

CBSE 12th Business Studies (Series 1-3) Question Paper 2024

6+ Downloads

Questions related to CBSE Class 12th

Have a question related to CBSE Class 12th ?

i have taken admission in clg after my campartment exam in chemistry along with the neet 2025 preperation but unfortunately i again got campartment after doing all this preparation i got 15 marks only now I can't able to think what should I do next.

You can use them people also used problem

Read Complete Answer

Divyan 06 August,2024

i scored 84 percent in cbse 2 years ago.Am i eligible for medhavi scholarship if give 12th again after 2 drop years and will got 75+ in mpboard

Hi,

The Medhavi National Scholarship Program, under the Human Resources & Development Mission (HRDM), offers financial assistance to meritorious students through a scholarship exam. To be eligible, candidates must be between 16 and 40 years old as of the last date of registration and have at least passed the 10th grade from a recognized board. Higher qualifications, such as 11th/12th grade, graduation, post-graduation, or a diploma, are also acceptable.

To apply, download the Medhavi App from the Google Play Store, sign up, and read the detailed notification about the scholarship exam. Complete the registration within the app, take the exam from home using the app, and receive your results within two days. Following this, upload the necessary documents and bank account details for verification. Upon successful verification, the scholarship amount will be directly transferred to your bank account.

The scholarships are categorized based on the marks obtained in the exam: Type A for those scoring 60% or above, Type B for scores between 50% and 60%, and Type C for scores between 40% and 50%. The cash scholarships range from Rs. 2,000 to Rs. 18,000 per month, depending on the exam and the marks obtained.

Since you already have a 12th-grade qualification with 84%, you meet the eligibility criteria and can apply for the Medhavi Scholarship exam. Preparing well for the exam can increase your chances of receiving a higher scholarship.

Read Complete Answer

Yuvan 01 September,2024

i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

Read Complete Answer

Ashvadha 12 July,2024

Kya mujhe class 12 ka important question milega jo exam mai per year puche jate hai

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

Thank you and wishing you all the best for your bright future.

Read Complete Answer

kumardurgesh1802 10 July,2024

I have completed my 12th with arts cbse. I want to repeat my 12th with pcmb. How to become a private candidate and what are deadline to be able to appear in 2025 exam?

Hello student,

If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:

No school admission needed! Register directly with CBSE. (But if you want to attend the school then you can take admission in any private school of your choice but it will be waste of money)
You have to appear for the 2025 12th board exams.
Registration for class 12th board exam starts around September 2024 (check CBSE website for exact dates).
Aim to register before late October to avoid extra fees.
Schools might not offer classes for private students, so focus on self-study or coaching.

Remember , these are tentative dates based on last year. Keep an eye on the CBSE website ( https://www.cbse.gov.in/ ) for the accurate and official announcement.

I hope this answer helps you. If you have more queries then feel free to share your questions with us, we will be happy to help you.

Good luck with your studies!

Read Complete Answer

kumardurgesh1802 10 July,2024

View All

Stay up-to date with CBSE Class 12th News

Download Careers360 App's

Regular exam updates, QnA, Predictors, College Applications & E-books now on your Mobile

300M+
Students
36,000+
Colleges
550+
Exams
1500+
E-Books
16000+
Certifications

We Appeared in

Central Board of Secondary Education Class 12th Examination

JEE Test Series

Applications Open

Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

Option 1) less than 3	Option 2) more than 3 but less than 6
Option 3) more than 6 but less than 9	Option 4) more than 9

Exams

Top Schools

Products & Resources

NCERT Solutions

Exams

Colleges

Predictors

Resources

Exams

Colleges & Courses

Predictors

Resources

Exams

Colleges

Predictors

Resources

Exams

Colleges

Predictors & E-Books

Resources

Exams

Predictors & Articles

Colleges

Resources

Exams

Colleges

Resources

Exams

Resources

Top Courses & Careers

Colleges

Exams

Colleges

Resources

Quick Links

Exams

Colleges

Resources

Exams

Colleges

Resources

Diploma Colleges

Exams

Resources

Upcoming Events

Other Exams

Exams

Colleges

Top Countries

Student Visas

Exams

Colleges

Upcoming Events

Resources

Engineering Preparation

Medical Preparation

Online Courses

Products

Top Streams

Specializations

Resources

Top Providers

NCERT Solutions for Exercise 1.4 Class 12 Maths Chapter 1 - Relations and Functions

NCERT Solutions For Class 12 Maths Chapter 1 Exercise 1.4

PTE Registrations 2024

Assess NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4

NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions: Exercise 1.4

Key Features Of NCERT Solutions for Exercise 1.4 Class 12 Maths Chapter 1

Also see-

Frequently Asked Question (FAQs)

Also Read

Articles

Upcoming School Exams

Certifications By Top Providers

Explore Top Universities Across Globe

Related E-books & Sample Papers

Questions related to CBSE Class 12th

Popular CBSE Class 12th Questions

Colleges After 12th

Tallentex 2025 - ALLEN's Talent Encouragement Exam