If A = `4 sin theta + cos^(2) theta` , 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 = 1//2` (D) Minimum value of A occurs when `sin theta = 1`.

Options

A. Maximum value of A is 5 .
B. Minimum value of A is `-4`
C. Maximum value of A occurs when `sin theta = 1//2`
D. Minimum value of A occurs when `sin theta = 1`.

Explanation

To solve the problem, we need to analyze the expression \( A = 4 \sin \theta + \cos^2 \theta \) and determine the maximum and minimum values of \( A \). We will also check the truth of the given statements.

### Step 1: Rewrite the expression
We start with the expression:
\[ A = 4 \sin \theta + \cos^2 \theta \]
Using the Pythagorean identity \( \cos^2 \theta = 1 - \sin^2 \theta \), we can rewrite \( A \):
\[ A = 4 \sin \theta + (1 - \sin^2 \theta) \]
This simplifies to:
\[ A = -\sin^2 \theta + 4 \sin \theta + 1 \]

### Step 2: Form a quadratic equation
Now, we can treat \( A \) as a quadratic function in terms of \( x = \sin \theta \):
\[ A = -x^2 + 4x + 1 \]
where \( -1 \leq x \leq 1 \) (since \( \sin \theta \) ranges from -1 to 1).

### Step 3: Find the maximum and minimum values
To find the maximum and minimum values of this quadratic function, we can complete the square:
\[ A = -(x^2 - 4x) + 1 \]
Completing the square for \( x^2 - 4x \):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
Thus,
\[ A = -((x - 2)^2 - 4) + 1 = - (x - 2)^2 + 4 + 1 = - (x - 2)^2 + 5 \]
This shows that the maximum value of \( A \) is 5, which occurs when \( (x - 2)^2 = 0 \) or \( x = 2 \). However, since \( x = \sin \theta \) cannot exceed 1, we need to check the endpoints of the interval \( [-1, 1] \).

### Step 4: Evaluate at the endpoints
1. When \( x = 1 \):
\[ A = -1^2 + 4(1) + 1 = -1 + 4 + 1 = 4 \]
2. When \( x = -1 \):
\[ A = -(-1)^2 + 4(-1) + 1 = -1 - 4 + 1 = -4 \]

### Conclusion
- The maximum value of \( A \) is 4 (not 5).
- The minimum value of \( A \) is -4.

### Step 5: Check the options
(A) Maximum value of A is 5. **(False)**
(B) Minimum value of A is -4. **(True)**
(C) Maximum value of A occurs when \( \sin \theta = \frac{1}{2} \). **(False)**
(D) Minimum value of A occurs when \( \sin \theta = 1 \). **(False)**

### Final Answer
The option that is not true is (A) Maximum value of A is 5.