Question

Assertion :Statement 1: If $f\left(\theta \right)=(\sin \theta +cosec\theta {)}^2+(\cos \theta +\sec \theta {)}^2$, then the minimum value of $f\left(\theta \right)$ is 9. Reason: Statement 2: Maximum value of $\sin 2\theta$ is 1.

• A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1
• B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1
• C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
• D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE

Answer 1

The correct option is A Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1

$f\left(x\right)={\sin }^2\theta +cose{c}^2\theta +2+{\cos }^2\theta +{\sec }^2\theta +2$

$f\left(x\right)=5+cose{c}^2\theta +{\sec }^2\theta$

$f^{\prime }\left(x\right)=0=-2cose{c}^2\theta \cot \theta +2se{c}^2\theta \tan \theta$

$2cose{c}^2\theta \cot \theta =2{\sec }^2\theta \tan \theta$

${\tan }^4\theta =1$

$\theta =\frac{\pi }{4}$

$f\left(x\right)=5+cose{c}^2\theta +se{c}^2\theta =5+2+2=9$

Reason:

consider, $\sin 2\theta =2\cos \theta \sin \theta$

Max value of x for both sin and cos together is ${45}^o$ which is $1/\sqrt{2}$

$\therefore \sin 2\theta =2\times \frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}=1$

Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1