If `sin^(3) theta + 2 sin^(2) theta + 3 sin theta` expression then
1. Maximum value of this expression is 6 where `theta in R`
2. The value of this expression could not be 0
Which is true in the following option ?
निम्नलिखित व्यंजक से सम्बन्धित कथनों पर विचार कीजिए:
`sin^(3) theta + 2 sin^(2) theta + 3 sin theta`
1. किसी `theta ub R` के लिए इस व्यंजक का अधिकतम मान 6 है।
2. किसी `theta in R` के लिए इस व्यंजक का मान 0 नहीं हो सकता।
उपर्युक्त कथनों में से कौन-सा/ से सही है/हैं?

Options

A. only 1 / केवल 1
B. only 2/ केवल 2
C. both 1 and 2 / दोनों 1 और 2
D. Neither 1 nor 2 / न ही 1 और न ही 2

Explanation

To solve the problem step by step, we need to analyze the expression given and evaluate the statements provided.

### Step 1: Define the Expression
The expression we are analyzing is:
\[ E(\theta) = \sin^3 \theta + 2 \sin^2 \theta + 3 \sin \theta \]

### Step 2: Determine the Range of \(\sin \theta\)
The sine function, \(\sin \theta\), has a range of values from \(-1\) to \(1\). Therefore, we will consider the values of \(E(\theta)\) when \(\sin \theta\) takes on these extreme values.

### Step 3: Find the Maximum Value of the Expression
To find the maximum value of \(E(\theta)\), we can substitute \(\sin \theta = 1\) (the maximum value of sine):
\[ E(90^\circ) = \sin^3(90^\circ) + 2 \sin^2(90^\circ) + 3 \sin(90^\circ) \]
Calculating this gives:
\[ E(90^\circ) = 1^3 + 2 \cdot 1^2 + 3 \cdot 1 = 1 + 2 + 3 = 6 \]
Thus, the maximum value of the expression is \(6\).

### Step 4: Check if the Expression can be Zero
Next, we need to check if the expression can equal \(0\). We can substitute \(\sin \theta = 0\):
\[ E(0^\circ) = \sin^3(0^\circ) + 2 \sin^2(0^\circ) + 3 \sin(0^\circ) \]
Calculating this gives:
\[ E(0^\circ) = 0^3 + 2 \cdot 0^2 + 3 \cdot 0 = 0 + 0 + 0 = 0 \]
Thus, the expression can indeed equal \(0\).

### Conclusion
1. The first statement is true: The maximum value of the expression is \(6\).
2. The second statement is false: The expression can be \(0\).

### Final Answer
The correct option is: **Only statement 1 is true.**

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