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 |
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 that
a * 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.
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