Question

Assertion :$co{s}^{7}x+si{n}^{4}x=1$ has only two non-zero solutions in the interval $-\pi <x<\pi$ Reason: $co{s}^{5}x+co{s}^{2}x-2=0$ is possible only when $cosx=1$

• 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 D Assertion is incorrect but Reason is correct

statement 1.
$co{s}^{7}x+si{n}^{4}x=1$
$\Rightarrow co{s}^{7}x=1-si{n}^{4}x=(1-si{n}^{2}x)(1+si{n}^{2}x)=co{s}^{2}x(2-co{s}^{2}x)$
$\Rightarrow co{s}^{2}x(co{s}^{5}x+co{s}^{2}x-2)=0$
$cosx=0\Rightarrow x=-\frac{\pi }{2},\frac{\pi }{2}$
or, $co{s}^{5}x+co{s}^{2}x=2$ we know $-1\le cosx\le 1$
thus $cosx=1$ is the only solution $\Rightarrow x=0$
Hence number of solution is 3.
Therefore statement 1 is incorrect.
statement 2.
$co{s}^{5}x+co{s}^{2}x=2$
we know $-1\le cosx\le 1$
Hence this equation is only solvable if $cosx=1$
therefore statement 2 is correct.