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RD Sharma solutions for Class 12 Mathematics chapter 3 - Binary Operations

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Chapter 3: Binary Operations

Ex. 3.1Ex. 3.2Ex. 3.3Ex. 3.4Ex. 3.50Ex. 3.5Ex. 3.6Ex. 3.7

Chapter 3: Binary Operations Exercise 3.1 solutions [Pages 4 - 5]

Ex. 3.1 | Q 1.1 | Page 4

Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = ab for all a, b ∈ N.

Ex. 3.1 | Q 1.2 | Page 4

Determine whether the following operation define a binary operation on the given set or not : 'O' on Z defined by a O b = ab for all a, b ∈ Z.

Ex. 3.1 | Q 1.3 | Page 4

Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = a + b - 2 for all a, b ∈ N

Ex. 3.1 | Q 1.4 | Page 4

Determine whether the following operation define a binary operation on the given set or not : '×6' on S = {1, 2, 3, 4, 5} defined by

a ×6 b = Remainder when ab is divided by 6.

Ex. 3.1 | Q 1.5 | Page 4

Determine whether the following operation define a binary operation on the given set or not :

\[' +_6 ' \text{on S} = \left\{ 0, 1, 2, 3, 4, 5 \right\} \text{defined by}\] 
\[a +_6 b = \begin{cases}a + b & ,\text{ if a} + b < 6 \\ a + b - 6 & , \text{if a} + b \geq 6\end{cases}\]

Ex. 3.1 | Q 1.6 | Page 4

Determine whether the following operation define a binary operation on the given set or not : '⊙' on N defined by a ⊙ b= ab + ba for all a, b ∈ N

Ex. 3.1 | Q 1.7 | Page 4

Determine whether the following operation define a binary operation on the given set or not :

\[' * ' \text{on Q defined by } a * b = \frac{a - 1}{b + 1} \text{for all a, b} \in Q .\]

Ex. 3.1 | Q 2.1 | Page 4

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
On Z+, defined * by a * b = a − b

Here, Z+ denotes the set of all non-negative integers.

Ex. 3.1 | Q 2.2 | Page 4

Determine whether or not the definition of *given below gives a binary operation. In the event that * is not a binary operation give justification of this.

On Z+, defined * by a * b = ab

Here, Z+ denotes the set of all non-negative integers.

Ex. 3.1 | Q 2.3 | Page 4

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.

On R, define by a*b = ab2

Here, Z+ denotes the set of all non-negative integers.

Ex. 3.1 | Q 2.4 | Page 4

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this. 

On Z+ define * by a * b = |a − b|

Here, Z+ denotes the set of all non-negative integers.

Ex. 3.1 | Q 2.5 | Page 4

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.

 On Z+, define * by a * b = a

Here, Z+ denotes the set of all non-negative integers.

Ex. 3.1 | Q 2.6 | Page 4

Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.

On R, define * by a * b = a + 4b2

Here, Z+ denotes the set of all non-negative integers.

Ex. 3.1 | Q 3 | Page 4

Let * be a binary operation on the set I of integers, defined by a * b = 2a + b − 3. Find the value of 3 * 4.

Ex. 3.1 | Q 4 | Page 4

Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.

Ex. 3.1 | Q 5 | Page 5

Let S = {abc}. Find the total number of binary operations on S.

Ex. 3.1 | Q 6 | Page 5

Find the total number of binary operations on {ab}.

Ex. 3.1 | Q 7 | Page 5

Let S be the set of all rational numbers of the form \[\frac{m}{n}\] , where m ∈ Z and n = 1, 2, 3. Prove that * on S defined by a * b = ab is not a binary operation.

Ex. 3.1 | Q 8 | Page 5

Prove that the operation * on the set

\[M = \left\{ \begin{bmatrix}a & 0 \\ 0 & b\end{bmatrix}; a, b \in R - \left\{ 0 \right\} \right\}\] defined by A * B = AB is a binary operation.

Ex. 3.1 | Q 9 | Page 5

The binary operation * : R × R → R is defined as a * b = 2a + b. Find (2 * 3) * 4.

Ex. 3.1 | Q 10 | Page 5

Let * be a binary operation on N given by a * b = LCM (a, b) for all a, b ∈ N. Find 5 * 7.

Chapter 3: Binary Operations Exercise 3.2 solutions [Pages 12 - 13]

Ex. 3.2 | Q 1.1 | Page 12

Let '*' be a binary operation on N defined by
a * b = 1.c.m. (a, b) for all a, b ∈ N
Find 2 * 4, 3 * 5, 1 * 6.

Ex. 3.2 | Q 1.2 | Page 12

Let '*' be a binary operation on N defined by a * b = 1.c.m. (a, b) for all a, b ∈ N

Check the commutativity and associativity of '*' on N.

Ex. 3.2 | Q 2.1 | Page 12

Determine which of the following binary operation is associative and which is commutative : * on N defined by a * b = 1 for all a, b ∈ N ?

Ex. 3.2 | Q 2.2 | Page 12

Determine which of the following binary operations are associative and which are commutative : * on Q defined by \[a * b = \frac{a + b}{2} \text{ for all a, b } \in Q\] ?

Ex. 3.2 | Q 3 | Page 12

Let A be any set containing more than one element. Let '*' be a binary operation on A defined by a * b = b for all a, b ∈ A Is '*' commutative or associative on A ?

Ex. 3.2 | Q 4.01 | Page 12

Check the commutativity and associativity of the following binary operation '*'. on Z defined by a * b = a + b + ab for all ab ∈ Z ?

Ex. 3.2 | Q 4.02 | Page 12

Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2ab for all a, b ∈ N ?

Ex. 3.2 | Q 4.03 | Page 12

Check the commutativity and associativity of the following binary operations '*'. on Q defined by a * b = a − b for all a, b ∈ Q ?

Ex. 3.2 | Q 4.04 | Page 12

Check the commutativity and associativity of the following binary operations '⊙' on Q defined by a ⊙ b = a2 + b2 for all a, b ∈ Q ?

Ex. 3.2 | Q 4.05 | Page 12

Check the commutativity and associativity of the following binary operation 'o' on Q defined by \[\text{a o b }= \frac{ab}{2}\] for all a, b ∈ Q ?

Ex. 3.2 | Q 4.06 | Page 12

Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = ab2 for all ab ∈ Q ?

Ex. 3.2 | Q 4.07 | Page 12

Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = a + ab for all ab ∈ Q ?

Ex. 3.2 | Q 4.08 | Page 12

Check the commutativity and associativity of the following binary operation  '*' on R defined by a * b = a + b − 7 for all ab ∈ R ?

Ex. 3.2 | Q 4.09 | Page 12

Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = (a − b)2 for all ab ∈ Q ?

Ex. 3.2 | Q 4.1 | Page 12

 Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = ab + 1 for all a, b ∈ Q ?

Ex. 3.2 | Q 4.11 | Page 12

Check the commutativity and associativity of the following binary operation '*' on N, defined by a * b = ab for all ab ∈ N ?

Ex. 3.2 | Q 4.12 | Page 12

Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a − b for all ab ∈ Z ?

Ex. 3.2 | Q 4.13 | Page 12

Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] for all ab ∈ Q ?

Ex. 3.2 | Q 4.14 | Page 12

Check the commutativity and associativity of the following binary operation  '*' on Z defined by a * b = a + b − ab for all a, b ∈ Z ?

Ex. 3.2 | Q 4.15 | Page 12

Check the commutativity and associativity of the following binary operation '*' on N defined by a * b = gcd(a, b) for all a, b ∈ N ?

Ex. 3.2 | Q 5 | Page 12

If the binary operation o is defined by aob = a + b − ab on the set Q − {−1} of all rational numbers other than 1, shown that o is commutative on Q − [1].

Ex. 3.2 | Q 6 | Page 12

Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?

Ex. 3.2 | Q 7 | Page 12

On the set Z of integers a binary operation * is defined by a * b = ab + 1 for all a , b ∈ Z. Prove that * is not associative on Z.

Ex. 3.2 | Q 8 | Page 12

Let S be the set of all real numbers except −1 and let '*' be an operation defined by a * b = a + b + ab for all ab ∈ S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.

Ex. 3.2 | Q 9 | Page 12

On Q, the set of all rational numbers, * is defined by \[a * b = \frac{a - b}{2}\] , shown that * is no associative ?

Ex. 3.2 | Q 10 | Page 12

On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b − 4. Prove that * is neither commutative nor associative on Z.

Ex. 3.2 | Q 11 | Page 12

On the set Q of all ration numbers if a binary operation * is defined by \[a * b = \frac{ab}{5}\] , prove that * is associative on Q.

Ex. 3.2 | Q 12 | Page 13

The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.

Ex. 3.2 | Q 13 | Page 13

On Q, the set of all rational numbers a binary operation * is defined by \[a * b = \frac{a + b}{2}\] Show that * is not associative on Q.

Ex. 3.2 | Q 14.1 | Page 13

Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b \[-\] ab, for all a, b \[\in\] S:

Prove that * is a binary operation on S ?

Ex. 3.2 | Q 14.2 | Page 13

Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b \[-\] ab, for all a, b \[\in\] S:

Prove that * is commutative as well as associative ?

Chapter 3: Binary Operations Exercise 3.3 solutions [Page 15]

Ex. 3.3 | Q 1 | Page 15

Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.

Ex. 3.3 | Q 2 | Page 15

Find the identity element in the set of all rational numbers except −1 with respect to *defined by a * b = a + b + ab.

Ex. 3.3 | Q 3 | Page 15

If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.

Ex. 3.3 | Q 4 | Page 15

On the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.

Chapter 3: Binary Operations Exercise 3.4 solutions [Page 25]

Ex. 3.4 | Q 1.1 | Page 25

Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Show that '*' is both commutative and associative ?

Ex. 3.4 | Q 1.2 | Page 25

Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the identity element in Z ?

Ex. 3.4 | Q 1.3 | Page 25

Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the invertible elements in Z ?

Ex. 3.4 | Q 2 | Page 25

Let * be a binary operation on Q0 (set of non-zero rational numbers) defined by \[a * b = \frac{ab}{5} \text{for all a, b} \in Q_0\]

 Show that * is commutative as well as associative. Also, find its identity element if it exists.

Ex. 3.4 | Q 3.1 | Page 25

Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that '*' is both commutative and associative on Q − {−1}.

Ex. 3.4 | Q 3.2 | Page 25

Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Find the identity element in Q − {−1} ?

Ex. 3.4 | Q 3.3 | Page 25

Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element ?

Ex. 3.4 | Q 4.1 | Page 25

Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R :

Show that '⊙' is commutative and associative on A ?

Ex. 3.4 | Q 4.2 | Page 25

Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (ab) ⊙ (cd) = (acbc + d) for all (ab), (cd) ∈ R0 × R :

Find the identity element in A ?

 

Ex. 3.4 | Q 4.3 | Page 25

Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R :

Find the invertible elements in A ?

Ex. 3.4 | Q 5.1 | Page 25

Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by   \[a o b = \frac{ab}{2}, \text{for all a, b} \in Q_0\].

Show that 'o' is both commutative and associate ?

Ex. 3.4 | Q 5.2 | Page 25

Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{ for all a, b } \in Q_0\] :

 Find the identity element in Q0.

Ex. 3.4 | Q 5.3 | Page 25

Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by  \[a o b = \frac{ab}{2}, \text{ for all a, b } \in Q_0\]:

Find the invertible elements of Q0 ?

Ex. 3.4 | Q 6 | Page 25

On R − {1}, a binary operation * is defined by a * b = a + b − ab. Prove that * is commutative and associative. Find the identity element for * on R − {1}. Also, prove that every element of R − {1} is invertible.

Ex. 3.4 | Q 7.1 | Page 25

Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by

(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A

Show that '*' is both commutative and associative on A ?

Ex. 3.4 | Q 7.2 | Page 25

Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by

(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A

Find the identity element in A ?

Ex. 3.4 | Q 7.3 | Page 25

Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by

(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A

Find the invertible element in A ?

Ex. 3.4 | Q 8 | Page 25

Let * be the binary operation on N defined by a * b = HCF of a and b.
Does there exist identity for this binary operation one N ?

Ex. 3.4 | Q 9 | Page 25

Let A  \[=\] R  \[\times\] R and \[*\]  be a binary operation on defined by \[(a, b) * (c, d) = (a + c, b + d) .\] . Show that \[*\] is commutative and associative. Find the binary element for \[*\] on A, if any.

Chapter 3: Binary Operations Exercise 3.50, 3.5 solutions [Pages 33 - 34]

Ex. 3.50 | Q 1 | Page 33

Construct the composition table for ×4 on set S = {0, 1, 2, 3}.

Ex. 3.50 | Q 2 | Page 33

Construct the composition table for +5 on set S = {0, 1, 2, 3, 4}.

Ex. 3.5 | Q 3 | Page 33

Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.

Ex. 3.50 | Q 4 | Page 33

Construct the composition table for ×5 on Z5 = {0, 1, 2, 3, 4}.

Ex. 3.50 | Q 5 | Page 33

For the binary operation ×10 on set S = {1, 3, 7, 9}, find the inverse of 3.

Ex. 3.5 | Q 6 | Page 33

For the binary operation ×7 on the set S = {1, 2, 3, 4, 5, 6}, compute 3−1 ×7 4.

Ex. 3.5 | Q 7 | Page 33

Find the inverse of 5 under multiplication modulo 11 on Z11.

Ex. 3.50 | Q 8 | Page 33

Write the multiplication table for the set of integers modulo 5.

Ex. 3.5 | Q 9.1 | Page 33

Consider the binary operation * defined by the following tables on set S = {a, bcd}.

a b c  d
a a b c d
b b a d c
c c d a b
d d c b a


Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.

Ex. 3.5 | Q 9.2 | Page 33

Consider the binary operation 'o' defined by the following tables on set S = {a, bcd}.

o  a b c d
a a a a a
b a b c d
c a c d b
d a d b c

Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.

Ex. 3.5 | Q 10 | Page 34

Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \[a * b = \begin{cases}a + b & ,\text{ if a  + b} < 6 \\ a + b - 6 & , \text{if a + b} \geq 6\end{cases}\]

Show that 0 is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.

Chapter 3: Binary Operations Exercise 3.6 solutions [Pages 35 - 36]

Ex. 3.6 | Q 1 | Page 35

Write the identity element for the binary operation * on the set R0 of all non-zero real numbers by the rule \[a * b = \frac{ab}{2}\] for all ab ∈ R0.

Ex. 3.6 | Q 2 | Page 35

On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all ab ∈ Z. Write the inverse of 4.

Ex. 3.6 | Q 3 | Page 35

Define a binary operation on a set.

Ex. 3.6 | Q 4 | Page 35

Define a commutative binary operation on a set.

Ex. 3.6 | Q 5 | Page 35

Define an associative binary operation on a set.

Ex. 3.6 | Q 6 | Page 35

Write the total number of binary operations on a set consisting of two elements.

Ex. 3.6 | Q 7 | Page 35

Write the identity element for the binary operation * defined on the set R of all real numbers by the rule

\[a * b = \frac{3ab}{7} \text{ for all a, b} \in R .\] ?

Ex. 3.6 | Q 8 | Page 35

Let * be a binary operation, on the set of all non-zero real numbers, given by \[a * b = \frac{ab}{5} \text { for all a, b } \in R - \left\{ 0 \right\}\]

Write the value of x given by 2 * (x * 5) = 10.

Ex. 3.6 | Q 9 | Page 35

Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ... ,10}.

Ex. 3.6 | Q 10 | Page 35

Define identity element for a binary operation defined on a set.

Ex. 3.6 | Q 11 | Page 36

Write the composition table for the binary operation multiplication modulo 10 (×10) on the set S = {2, 4, 6, 8}.

Ex. 3.6 | Q 12 | Page 36

For the binary operation multiplication modulo 10 (×10) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.

Ex. 3.6 | Q 13 | Page 36

For the binary operation multiplication modulo 5 (×5) defined on the set S = {1, 2, 3, 4}. Write the value of \[\left( 3 \times_5 4^{- 1} \right)^{- 1}.\] 

Ex. 3.6 | Q 14 | Page 36

Write the composition table for the binary operation ×5 (multiplication modulo 5) on the set S = {0, 1, 2, 3, 4}.

Ex. 3.6 | Q 15 | Page 36

A binary operation * is defined on the set R of all real numbers by the rule \[a * b = \sqrt{  a^2 + b^2} \text{for all a, b } \in R .\]

Write the identity element for * on R.

Ex. 3.6 | Q 16 | Page 36

Let +6 (addition modulo 6) be a binary operation on S = {0, 1, 2, 3, 4, 5}. Write the value of \[2 +_6 4^{- 1} +_6 3^{- 1} .\]

Ex. 3.6 | Q 17 | Page 36

Let * be a binary operation defined by a * b = 3a + 4b − 2. Find 4 * 5.

Ex. 3.6 | Q 18 | Page 36

If the binary operation * on the set Z of integers is defined by a * b = a + 3b2, find the value of 2 * 4.

Ex. 3.6 | Q 19 | Page 36

Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.

Ex. 3.6 | Q 20 | Page 36

Let * be a binary operation on set of integers I, defined by a * b = 2a + b − 3. Find the value of 3 * 4.

Chapter 3: Binary Operations Exercise 3.7 solutions [Pages 36 - 39]

Ex. 3.7 | Q 1 | Page 36

If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .

  • (42 + 52) + 32

  • (4 + 5)2 + 32

  • 412 + 32

  • (4 + 5 + 3)2

Ex. 3.7 | Q 2 | Page 36

If a * b denote the bigger among a and b and if a ⋅ b = (a * b) + 3, then 4.7 = __________ .

  • 14

  • 31

  • 10

  • 8

Ex. 3.7 | Q 3 | Page 37

On the power set P of a non-empty set A, we define an operation ∆ by

\[X ∆ Y = \left( \overline{X} \cap Y \right) \cup \left( X \cap \overline{Y} \right)\]

Then which are of the following statements is true about ∆.

  • commutative and associative without an identity

  • commutative but not associative with an identity

  • associative but not commutative without an identity

  • associative and commutative with an identity

Ex. 3.7 | Q 4 | Page 37

If the binary operation * on Z is defined by a * b = a2 − b2 + ab + 4, then value of (2 * 3) * 4 is ____________ .

  • 233

  • 33

  • 55

  • -55

Ex. 3.7 | Q 5 | Page 37

Mark the correct alternative in the following question:-

For the binary operation * on Z defined by a * b = a + b + 1, the identity element is ________________ .

  • 0

  • -1

  • 1

  • 2

Ex. 3.7 | Q 6 | Page 37

If a binary operation * is defined on the set Z of integers as a * b = 3a − b, then the value of (2 * 3) * 4 is ___________ .

  • 2

  • 3

  • 4

  • 5

Ex. 3.7 | Q 7 | Page 37

Q+ denote the set of all positive rational numbers. If the binary operation a ⊙ on Q+ is defined as \[a \odot = \frac{ab}{2}\] ,then the inverse of 3 is __________ .

  • `4/3`

  • 2

  • `1/3`

  • `2/3`

Ex. 3.7 | Q 8 | Page 37

If G is the set of all matrices of the form

\[\begin{bmatrix}x & x \\ x & x\end{bmatrix}, \text{where x } \in R - \left\{ 0 \right\}\] then the identity element with respect to the multiplication of matrices as binary operation, is ______________ .

  • \[\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\]

  • \[\begin{bmatrix}- 1/2 & - 1/2 \\ - 1/2 & - 1/2\end{bmatrix}\]

  • \[\begin{bmatrix}1/2 & 1/1 \\ 1/2 & 1/2\end{bmatrix}\]

  • \[\begin{bmatrix}- 1 & - 1 \\ - 1 & - 1\end{bmatrix}\]

Ex. 3.7 | Q 9 | Page 37

Q+ is the set of all positive rational numbers with the binary operation * defined by \[a * b = \frac{ab}{2}\] for all ab ∈ Q+. The inverse of an element a ∈ Q+ is ______________ .

  • a

  • `1/a`

  • `2/a`

  • `4/a`

Ex. 3.7 | Q 10 | Page 37

If the binary operation ⊙ is defined on the set Q+ of all positive rational numbers by \[a \odot b = \frac{ab}{4} . \text{ Then }, 3 \odot \left( \frac{1}{5} \odot \frac{1}{2} \right)\] is equal to __________ .

  • `3/160`

  • `5/160`

  • `3/10`

  • `3/40`

Ex. 3.7 | Q 11 | Page 37

Let * be a binary operation defined on set Q − {1} by the rule a * b = a + b − ab. Then, the identify element for * is ____________ .

  • 1

  • `(a-1)/a`

  • `a/(a-1)`

  • 0

Ex. 3.7 | Q 12 | Page 37

Which of the following is true ?

  • * defined by \[a * b = \frac{a + b}{2}\] is a binary operation on Z .

  • * defined by \[a * b = \frac{a + b}{2}\]  is a binary operation on Q .

  • all binary commutative operations are associative.

  • subtraction is a binary operation on N.

Ex. 3.7 | Q 13 | Page 38

The binary operation * defined on N by a * b = a + b + ab for all a, b N is ________________ .

  • commutative only

  • associative only

  • commutative and associative both

  • none of these

Ex. 3.7 | Q 14 | Page 38

The binary operation * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is equal to ______________ .

  • 20

  • 40

  • 400

  • 445

Ex. 3.7 | Q 15 | Page 38

Let * be a binary operation on R defined by a * b = ab + 1. Then, * is _________________ .

  • commutative but not associative

  • associative but not commutative

  • neither commutative nor associative

  • both commutative and associative

Ex. 3.7 | Q 16 | Page 38

Subtraction of integers is ___________________ .

  • commutative but no associative

  • commutative and associative

  • associative but not commutative

  • neither commutative nor associative

Ex. 3.7 | Q 17 | Page 38

The law a + b = b + a is called _________________ .

  • closure law

  • associative law

  • commutative law

  • distributive law

Ex. 3.7 | Q 18 | Page 38

An operation * is defined on the set Z of non-zero integers by \[a * b = \frac{a}{b}\]  for all ab ∈ Z. Then the property satisfied is _______________ .

  • closure

  • commutative

  • associative

  • none of these

Ex. 3.7 | Q 19 | Page 38

On Z an operation * is defined by a * b = a2 + b2 for all a, b ∈ Z. The operation * on Z is _______________ .

  • commutative and associative

  • associative but not commutative

  • not associative

  • not a binary operation

Ex. 3.7 | Q 20 | Page 38

A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is ________________ .

  • commutative

  • associative

  • not commutative

  • commutative and associative

Ex. 3.7 | Q 21 | Page 38

Let * be a binary operation on Q+ defined by \[a * b = \frac{ab}{100} \text{ for all a, b } \in Q^+\] The inverse of 0.1 is _________________ .

  • 105

  • 104

  • 106

  • none of these

Ex. 3.7 | Q 22 | Page 38

Let * be a binary operation on N defined by a * b = a + b + 10 for all ab ∈ N. The identity element for * in N is _____________ .

  • −10

  • 0

  • 10

  • non-existent

Ex. 3.7 | Q 23 | Page 38

Consider the binary operation * defined on Q − {1} by the rule
a * b = a + b − ab for all a, b ∈ Q − {1}
The identity element in Q − {1} is _______________ .

  • 0

  • 1

  • `1/2`

  • -1

Ex. 3.7 | Q 24 | Page 38

For the binary operation * defined on R − {1} by the rule a * b = a + b + ab for all a, b ∈ R − {1}, the inverse of a is ________________ .

  • \[- a\]

  • \[- \frac{a}{a + 1}\]

  • \[\frac{1}{a}\]

  • \[a^2\]

Ex. 3.7 | Q 25 | Page 38

For the multiplication of matrices as a binary operation on the set of all matrices of the form \[\begin{bmatrix}a & b \\ - b & a\end{bmatrix}\] a, b ∈ R the inverse of \[\begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix}\] is ___________________ .

  • \[\begin{bmatrix}- 2 & 3 \\ - 3 & - 2\end{bmatrix}\]

  • \[\begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix}\]

  • \[\begin{bmatrix}2/13 & - 3/13 \\ 3/13 & 2/13\end{bmatrix}\]

  • \[\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]

Ex. 3.7 | Q 26 | Page 38

On the set Q+ of all positive rational numbers a binary operation * is defined by \[a * b = \frac{ab}{2} \text{ for all, a, b }\in Q^+\]. The inverse of 8 is _________ .

  • `1/8`

  • `1/2`

  • 2

  • 4

Ex. 3.7 | Q 27 | Page 39

Let * be a binary operation defined on Q+ by the rule

\[a * b = \frac{ab}{3} \text{ for all a, b } \in Q^+\] The inverse of 4 * 6 is ___________ .

  • `9/8`

  • `2/3`

  • `3/2`

  • none of these

Ex. 3.7 | Q 28 | Page 39

The number of binary operation that can be defined on a set of 2 elements is _________ .

  • 8

  • 4

  • 16

  • 64

Ex. 3.7 | Q 29 | Page 39

The number of commutative binary operations that can be defined on a set of 2 elements is ____________ .

  • 8

  • 6

  • 4

  • 2

Chapter 3: Binary Operations

Ex. 3.1Ex. 3.2Ex. 3.3Ex. 3.4Ex. 3.50Ex. 3.5Ex. 3.6Ex. 3.7

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

RD Sharma solutions for Class 12 Mathematics chapter 3 - Binary Operations

RD Sharma solutions for Class 12 Maths chapter 3 (Binary Operations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 3 Binary Operations are Types of Relations, Types of Functions, Composition of Functions and Invertible Function, Inverse of a Function, Concept of Binary Operations, Introduction of Relations and Functions.

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