Question

Assertion :Statement 1:The minimum value of ${27}^{\cos 2x}{81}^{\sin 2x}$ is $\frac{1}{243}$ Reason: Statement 2:The minimum value of $a\cos \theta +b\sin \theta$ is $-\sqrt{a^2+b^2}$

• 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

${27}^{\cos \left(2x\right)}\times {81}^{\sin \left(2x\right)}=3^{3\cos \left(2x\right)+4\sin \left(2x\right)}$
Now the minimum value of $a\cos \theta +b\sin \theta$ is $-\sqrt{a^2+b^2}$ and thus, the minimum value becomes $-\sqrt{9+16}=-5$
Hence, minimum value becomes $3^{-5}=\frac{1}{243}$

Both the statements are true and statement 2 is the correct explanation of statement 1