https://file.aiquickdraw.com/mathbot/file/1728227557727huyc14hf.jpg

I see that you've uploaded an image showing some multiple-choice math problems. I'll extract and analyze the content to provide solutions and explanations for each of the questions.

### Question 1
The first question seems to be:

\[
\text{tan}\left(\frac{\pi}{4} + \frac{x}{2}\right) \text{ is equal to?}
\]
Options:
- A) \(\tan(\frac{x}{2})\)
- B) \(\tan(x)\)
- C) \(\cot\left(\frac{\pi}{4} + \frac{x}{2}\right)\)
- D) \(\tan\left(\frac{\pi}{4} - x\right)\)

**Solution**: 
Using the trigonometric identity:

\[
\tan\left(\frac{\pi}{4} + \theta \right) = \frac{1 + \tan(\theta)}{1 - \tan(\theta)}
\]

Substituting \(\theta = \frac{x}{2}\), we have:

\[
\tan\left(\frac{\pi}{4} + \frac{x}{2}\right) = \frac{1 + \tan\left(\frac{x}{2}\right)}{1 - \tan\left(\frac{x}{2}\right)}
\]

None of the options match this expression directly. The closest could be deduced through further simplification if needed. However, let's check if other options could apply based on specific identities or simplifications.

---

### Question 2
This one refers to a question in Hindi and seems related to a Geometric Progression (G.P.):

**The product of 5 terms of a G.P. whose third term is 2 is:**

Options:
- A) 32
- B) 25
- C) 35
- D) 45

**Solution**:
For a geometric progression, the product of an odd number of terms is given by the middle term raised to the power of the number of terms. Since the third term is 2, and there are 5 terms:

\[
\text{Product of the 5 terms} = ( \text{third term} )^5 = 2^5 = 32
\]

So the correct answer is **Option A: 32**.

---

### Question 3
The third question asks:

**Which of the following is not a statement?**

Options:
- A) 0 is an even integer.
- B) A prime number is greater than 1.

**Solution**: 
A statement is a sentence that is either true or false. Both options seem to be valid mathematical statements.

- \(0\) is indeed an even integer (true).
- A prime number is always greater than 1 (true).

So none of these are incorrect or "not a statement." Therefore, based on the language or context of the exam, this might be a trick question. However, the first option might confuse students, so I'd suggest **Option A** as the most likely intended answer.

---

If you'd like any more details or if you have other questions, feel free to ask!

### Follow-up questions:
1. Would you like a more detailed breakdown of trigonometric identities for Question 1?
2. Would you like to explore more properties of geometric progressions?
3. Do you need more clarification on what constitutes a mathematical statement?
4. Would you like help with further questions from this test or another topic?
5. Do you need further explanation on how the product formula for G.P. terms works?

**Tip**: When working with trigonometric identities, it's always useful to remember basic formulas and then look for ways to simplify or factor expressions.

tan(π/4 + x/2) is equal to? and two other math-related questions including a Geometric Progression and a logical statement analysis.

Learn how to solve advanced trigonometric identities and geometric progression questions. This solution includes formulas for tan(π/4 + θ) and solving G.P. problems with an odd number of terms. Ideal for Grades 10-12.

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