Question

Assertion :${(\sqrt{2}-1)}^n$ can be expressed as $\sqrt{N}-\sqrt{N-1}$ for $\forall N>1$ and $n\in N$. Reason: ${(\sqrt{2}-1)}^n$ can be written in the form $\alpha +\beta \sqrt{2}\forall ,\alpha ,\beta$ are integers & n is a positive integer.

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion,
• B. Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
• C. Assertion is true but Reason is false,
• D. Assertion is false but Reason is true.

Answer 1

The correct option is B Both Assertion & Reason are individually true but Reason is not the correct (proper) explanation of Assertion,

The assertion is true since
${(\sqrt{2}-1)}^2=3-2\sqrt{2}=\sqrt{9}-\sqrt{8}{(\sqrt{2}-1)}^3=5\sqrt{2}-7=\sqrt{50}-\sqrt{49}$
The reason is also true since any integral power of $(\sqrt{2}-1)$ will have rational and irrational part.
The irrational part will have only one surd $\sqrt{2}$.
Thus ${(\sqrt{2}-1)}^n=A+B\sqrt{2}$ where A and B are integers.
But the reason does not explain the assertion.