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

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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.

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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.

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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.

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NCERT Solutions Subject Wise

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Frequently Asked Questions (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 Class 12 Maths Notes - Download Chapter wise Free PDFs

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NCERT Solutions for Miscellaneous Exercise Chapter 7 Class 12 - Integrals

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Questions related to CBSE Class 12th

Have a question related to CBSE Class 12th ?

quartely question paper for mathematics cbse

Hello there! Thanks for reaching out to us at Careers360.

Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.

Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!

Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.

If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.

Let me know if you need any other tips for your math prep. Good luck with your studies!

Read Complete Answer

Jefferson 25 September,2024

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.

It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.

Possible steps:

Re-evaluate Your Study Strategies:
- Identify Weak Areas: Pinpoint the specific topics or concepts that caused difficulties.
- Seek Clarification: Reach out to teachers, tutors, or online resources for additional explanations.
- Practice Regularly: Consistent practice is key to mastering chemistry.
Consider Professional Help:
- Tutoring: A tutor can provide personalized guidance and support.
- Counseling: If you're feeling overwhelmed or unsure about your path, counseling can help.
Explore Alternative Options:
- Retake the Exam: If you're confident in your ability to improve, consider retaking the chemistry compartment exam.
- Change Course: If you're not interested in pursuing chemistry further, explore other academic options that align with your interests.
Focus on NEET 2025 Preparation:
- Stay Dedicated: Continue your NEET preparation with renewed determination.
- Utilize Resources: Make use of study materials, online courses, and mock tests.
Seek Support:
- Talk to Friends and Family: Sharing your feelings can provide comfort and encouragement.
- Join Study Groups: Collaborating with peers can create a supportive learning environment.

Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.

I hope this information helps you.

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Kanishka kaushikii 17 September,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,

Qualifications:
Age: As of the last registration date, you must be between the ages of 16 and 40.
Qualification: You must have graduated from an accredited board or at least passed the tenth grade. Higher qualifications are also accepted, such as a diploma, postgraduate degree, graduation, or 11th or 12th grade.
How to Apply:
Get the Medhavi app by visiting the Google Play Store.
Register: In the app, create an account.
Examine Notification: Examine the comprehensive notification on the scholarship examination.
Sign up to Take the Test: Finish the app's registration process.
Examine: The Medhavi app allows you to take the exam from the comfort of your home.
Get Results: In just two days, the results are made public.
Verification of Documents: Provide the required paperwork and bank account information for validation.
Get Scholarship: Following a successful verification process, the scholarship will be given. You need to have at least passed the 10th grade/matriculation scholarship amount will be transferred directly to your bank account.

Scholarship Details:

Type A: For candidates scoring 60% or above in the exam.

Type B: For candidates scoring between 50% and 60%.

Type C: For candidates scoring between 40% and 50%.

Cash Scholarship:

Scholarships can range from Rs. 2,000 to Rs. 18,000 per month, depending on the marks obtained and the type of scholarship exam (SAKSHAM, SWABHIMAN, SAMADHAN, etc.).

Since you already have a 12th grade qualification with 84%, you meet the qualification criteria and are eligible to apply for the Medhavi Scholarship exam. Make sure to prepare well for the exam to maximize your chances of receiving a higher scholarship.

Hope you find this useful!

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Yuvan 30 August,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.

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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.

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kumardurgesh1802 10 July,2024

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Central Board of Secondary Education Class 12th Examination

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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

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

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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

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