Here we provide RD Sharma Class 12 Ex 3.4 Solutions Chapter 3 Binary Operations for English medium students, Which will very helpful for every student in their exams. Students can download the latest RD Sharma Class 12 Ex 3.4 Solutions Chapter 3 Binary Operations book pdf download. Now you will get step-by-step solutions to each question.

Textbook | NCERT |

Class | Class 12th |

Subject | Maths |

Chapter | 3 |

Exercise | 3.4 |

Category | RD Sharma Solutions |

Table of Contents

**Question 1. Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I**^{+}.

^{+}.

**Solution:**

Let e be the identity element in I

^{+}with respect to * such thata * e = a = e * a, ∀ a ∈ I

^{+}a * e = a and e * a = a, ∀ a ∈ I

^{+}a + e = a and e + a = a, ∀ a ∈ I

^{+}e = 0, ∀ a ∈ I

^{+}Hence, 0 is the identity element in I

^{+}with respect to *.

**Question 2. Find the identity element in the set of all rational numbers except – 1 with respect to * defined by a * b = a + b + ab**

**Solution:**

Let e be the identity element in I+ with respect to * such that

a * e = a = e * a, ∀ a ∈ Q – {-1}

a * e = a and e * a = a, ∀ a ∈ Q – {-1}

a + e + ae = a and e + a + ea = a, ∀ a ∈ Q – {-1}

e + ae = 0 and e + ea = 0, ∀ a ∈ Q – {-1}

e (1 + a) = 0 and e (1 + a) = 0, ∀ a ∈ Q – {-1}

e = 0, ∀ a ∈ Q – {-1} [because a not equal to -1]

Hence, 0 is the identity element in Q – {-1} with respect to *.

**Question 3. If the binary operation * on the set Z is defined by a*b = a + b – 5, then find the identity element with respect to *.**

**Solution: **

We are given the binary operator * defined on Z as

a*b = a + b – 5 for all a, b ∈ Q

Let e be the identity elements with respect to *

Then, a*e = e*a = a [By identity property]

⇒ a + e – 5 = a

⇒ e = 5

Therefore, the required identity element with respect to * is 5.

**Question 4. On the set Z integers, if the binary operation * is defined by a*b = a + b + 2, then find the identity elements.**

**Solution: **

The binary operator * is defined on Z, and is given by

a*b = a + b +2 for all a, b ∈ Z.

Let a ∈ Z and e ∈ Z be the identity element with respect to *, then

a*e = e*a = a [By identity property]

⇒ a + e + 2 = a

⇒ e = -2 ∈ Z

Therefore, the identity element with respect to * is -2.

I think you got complete solutions for this chapter. If You have any queries regarding this chapter, please comment in the below section our subject teacher will answer you. We tried our best to give complete solutions so you got good marks in your exam.

If these solutions have helped you, you can also share rdsharmasolutions.in to your friends