Question

If $A=4sin\theta+cos^2\theta$, then which of the following is not true?

(a) Maximum value of $A$ is 5

(b) Minimum value of $A$ is -4

(c) Maximum value of $A$ occurs when $sin\theta=\dfrac{1}{2}$

(d) Minimum value of $A$ occurs when $sin\theta=1$​

Answer 1

i hope this will help you

Answer 2

We know the following :-

$\boxed{\red{\bold{\sf{-1\leqslant sin\theta \leqslant 1}}}}$

Also,

$\boxed{\red{\bold{\sf{0 \leqslant cos^{2}\theta \leqslant 1}}}}$

$\rule{200}{2}$

Now,

We can Thus say that :-

$\boxed{- 4 \leqslant 4 \sin \theta \leqslant 4}$

And,

$\boxed{0 \leqslant { \cos}^{2} \theta \leqslant 1}$

Adding the two we can say :-

$- 4 + 0 \leqslant 4 \sin \theta + { \cos}^{2} \theta\leqslant 4 + 1 \\ \\ \\ - 4 \leqslant 4 \sin \theta + { \cos}^{2} \theta\leqslant 5$

Thus,

Maximum value of A is 5

Minimum Value of A is (-4)

Hence,

Option A and Option B are correct.

$\rule{200}{2}$

Maximum value of A occurs when 4 sin theta = 4

This means that,

$\sin \theta = 1$

Hence,

Option C is incorrect.

$\rule{200}{2}$

Minimum value of A occurs when Maximum value of sin theta is equal to 1.

Thus,

For minimum value of A, the value of sin theta must be equal to (-1)

Hence,

Option D is also incorrect.

It would have been correct when it states that :-

Minimum value of A occurs when maximum value of sin theta is 1.

$\rule{200}{2}$