Q) Prove that 2 + 3√5 is an irrational number given that √5 is an irrational number.

(Q 21 A – 30/4/2 – CBSE 2026 Question Paper)

Ans:

Step 1: Let’s start by considering (2 + 3√5) is a rational number (by the method of contradiction)

If (2 + 3√5)  is a rational number, then it can be expressed in the form of , where p and q are integers and q ≠  0.

∴ (2 + 3√5)  =

∴ 3 √5   = – 2

∴ 3 √5   =

∴ √5   = ……..(i)

Step 2: Since p and q are integers, so is also a rational number.

Since, in above equation (i), LHS = RHS.

Therefore, if RHS is rational, then LHS is also rational.

Therefore √5 is a rational number.

Step 3: But it is given that √5 is an irrational number – this contradicts our finding

It means that our assumption that “(2 + 3√5) is a rational number” is wrong.

Therefore, it is confirmed that (2 + 3√5) is an irrational number.

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