Question

Assertion :There exists no A.P. whose three terms are $\sqrt{3},\sqrt{5}$ and $\sqrt{7}$. Reason: If $t_p,t_q$ and $t_r$ are three distinct terms of an A.P., then $\frac{t_r-t_p}{t_q-t_p}$ is a rational number.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Suppose $\sqrt{3},\sqrt{5}$ and $\sqrt{7}$ are the pth, qth and rth terms of an A.P. whose common difference is d, then
$t_r-t_p=(r-p)d$

and $t_q-t_p=(q-p)d$

$\Rightarrow \frac{t_r-t_p}{t_q-t_p}=\frac{r-p}{q-p}$ which is rational numbers

$\Rightarrow \frac{\sqrt{7}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ is a rational numbers.

$\iff \frac{(\sqrt{7}-\sqrt{3})(\sqrt{5}+\sqrt{3})}{5-3}$ is rational

$\iff \frac{(\sqrt{35}-\sqrt{15})-\sqrt{21}-3}{2}$ is rational

$\iff \sqrt{35}-\sqrt{15}+\sqrt{21}$ is rational, say r.

Now,
$\sqrt{35}-\sqrt{15}+\sqrt{21}=r$

$\Rightarrow \sqrt{15}-\sqrt{21}=\sqrt{35}-r$

Squaring both sides, we get
$15+21-\left(2\right)\left(6\right)\sqrt{35}=35+r^2-2r\sqrt{35}$
$\Rightarrow \sqrt{35}=\frac{1-r^2}{12+2r}$
$\Rightarrow \sqrt{35}$ is rational
This is a contradiction
Hence $\sqrt{3},\sqrt{5}$ and $\sqrt{7}$ cannot be three terms of an A.P