CBSE Class 12 Mathematics 2025: question 26(b)

Hints

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Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

1. Reflexive: For any $x\in N$, $x\cdot x=x^2$ is the square of the natural number x. So $(x,x)\in R$ and R is reflexive.

   1 mark

2. Symmetric: If $(x,y)\in R$, then $xy$ is a perfect square. Since $yx=xy$, $yx$ is also a perfect square, so $(y,x)\in R$ and R is symmetric.

   1 mark

3. Transitive: Let $(x,y),(y,z)\in R$, so $xy=a^2$ and $yz=b^2$ for some $a,b\in N$. Then $xy\cdot yz=a^2{b}^2$, so $xz=\frac{a^2{b}^2}{y^2}={\left(\frac{ab}{y}\right)}^2$. Here $xz$ is a natural number and $\frac{ab}{y}$ is a positive rational whose square is a natural number, so $\frac{ab}{y}\in N$. Hence $(x,z)\in R$ and R is transitive.

   ½ mark

4. Since R is reflexive, symmetric and transitive, R is an equivalence relation.

   ½ mark

Answer: Yes, R is an equivalence relation.

Revise the chapter

• Chapter: Relations and Functions (notes, formulas, where marks are lost and every past question)
• Topic: Equivalence Relations and Equivalence Classes