Question

Assertion :If ${\cos }^2\frac{\pi }{8}$ is a root of the equation $x^2+ax+b=0$, where $a,bϵQ$, then ordered pair $(a,b)$ is $[-1,\frac{1}{8}]$. Reason: If $a+mb=0$ and m is irrational, then $a,b=0$.

• 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

Since, ${\cos }^2\frac{\pi }{8}$ is the root of the equation $x^2+ax+b=0$.
$\therefore {\left({\cos }^2\frac{\pi }{8}\right)}^2+a{\cos }^2\frac{\pi }{8}+b=0$
$\Rightarrow {\left(\frac{1+\cos \frac{\pi }{4}}{2}\right)}^2+a\left(\frac{1+\cos \frac{\pi }{4}}{2}\right)+b=0\{\because \cos 2\theta =2{\cos }^2\theta -1\}$
$\Rightarrow \frac{{(\sqrt{2}+1)}^2}{8}+\frac{a(\sqrt{2}+1)}{2\sqrt{2}}+b=0$
$\Rightarrow (\frac{3}{8}+b+\frac{a}{2})+\sqrt{2}\left(\frac{a+1}{4}\right)=0$
Since, $\sqrt{2}$ is irrational
$\therefore \frac{a+1}{4}=\frac{3}{8}+b+\frac{a}{2}=0$
$\Rightarrow (a,b)=(-1,\frac{1}{8})$
Thus the statement is correct and the reason is correct explanation for the assertion.