Question

Assertion :The largest and smallest value of $f\left(\theta \right)$ differs by $\frac{1}{\sqrt{2}}$ Reason: $a\sin x+b\cos x+c\in [c-\sqrt{a^2+b^2},c+\sqrt{a^2+b^2}]\forall x\in R$ where $a,b,c\in R$

• A. Assertion is true, Reason is also true; Reason is the correct explanation of Assertion.
• B. Assertion is true, Reason is also true; Reason is not the correct explanation of Assertion
• C. Assertion is true, Reason is false.
• D. Assertion is false, Reason is true.

Answer 1

The correct option is A Assertion is true, Reason is also true; Reason is the correct explanation of Assertion.

$f\left(\theta \right)=\frac{4{\sin }^2\theta \cos \theta -\cos \theta +\sin \theta }{4(\cos \theta +\sin \theta )}$

$=\frac{2\sin \theta \left(\right(\sin \theta +\cos \theta {)}^2-1)-(\cos \theta +\sin \theta )+2\sin \theta }{4(\cos \theta +\sin \theta )}$

$=\frac{1}{2}\sin \theta (\sin \theta +\cos \theta )-\frac{1}{4}$

$=\frac{1}{4}(\sin 2\theta -\cos 2\theta ϵ[-\frac{\sqrt{2}}{4},\frac{\sqrt{2}}{4}]$

Hence, $\left(a\right)$ is correct answer