Question

Assertion: $\sqrt{3}$is an irrational number.
Reason: The sum of rational number and an irrational number is an irrational number.
(a) Both Assertion and Reason are true and Reasom is a correct explanation of Assertion.
(b) Both Assertion and Reason and Reasom are true but Reasom is not a correct explanation of Assertion.
(c) Assertion is true and Reasom is false.
(d) Assertion is false and Reasom is true.

Answer 1

(b) Both Assertion and Reason are true, but Reason is not a correct explanation of Assertion.
$$\begin{gathered} \sqrt{3}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{perfect}\mathrm{square}\mathrm{and}\mathrm{is}\mathrm{irrational}. \\ \mathrm{Reason}:\mathrm{Let}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{a}\mathrm{rational}\mathrm{numbe}ra\mathrm{and}\mathrm{an}\mathrm{irrational}\mathrm{number}\sqrt{b}\mathrm{be}\mathrm{a}\mathrm{rational}\mathrm{number}c. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ a+\sqrt{b}=c \\ \Rightarrow \sqrt{b}=c-a \\ \mathrm{Now},c-a\mathrm{is}\mathrm{rational}\mathrm{because}\mathrm{both}c\mathrm{and}a\mathrm{are}\mathrm{rational},\mathrm{but}\sqrt{b}\mathrm{is}\mathrm{irrational};\mathrm{thus},\mathrm{we}\mathrm{arrive}\mathrm{at}\mathrm{a}\mathrm{contradiction}. \\ \mathrm{Hence},\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{a}\mathrm{rational}\mathrm{number}\mathrm{and}\mathrm{an}\mathrm{irrational}\mathrm{number}\mathrm{is}\mathrm{an}\mathrm{irrational}\mathrm{number}. \\ \mathrm{Thus},\mathrm{Reason}\mathrm{R}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{correct}\mathrm{explanation}. \end{gathered}$$