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Please Solve R.D.Sharma Class 12 Chapter 4 Exercise MCQs Question 4 Maths Textbook Solution.

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Answer: The correct option is \text { (B), } \mathrm{B}^{2}=\mathrm{B} \text { and } \mathrm{A}^{2}=\mathrm{A}

Given:A B=A, B A=B


A B=A                                                                                                         ....(1)

BA=B                                                                                                         ....(2)

Now from Equation (2),

\begin{aligned} &B \times(A B)=B \\ &B^{2} A=B \end{aligned}

Again from Equation (2),

\begin{aligned} &\mathrm{B}^{2} \mathrm{~A}=\mathrm{BA} \\ &\mathrm{B}^{2}=\mathrm{B} \end{aligned}

Now from Equation (1),

\begin{aligned} &A \times(B A)=A \\ &A^{2} B=A \end{aligned}

Again from Equation (1),

\begin{aligned} &\mathrm{A}^{2} \mathrm{~B}=\mathrm{AB} \\ &\mathrm{A}^{2}=\mathrm{A} \end{aligned}

So,\mathrm{A}^{2}=\mathrm{A} \text { and } \mathrm{B}^{2}=\mathrm{B}

Hence, the correct option is (B).        

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