Statement-1: If `alpha, beta` are different values of `x` satisfying `a cosx + b sinx =c, (b ne 0)`, then `tan((alpha+beta)/(2))=a/b`
Statement-2: The equation `sin^(2)x -1/sqrt(2)(sqrt(3)+1)sinx + sqrt(3)/4 =0` has two roots in `[0,pi/2]`
Statement-3: The number of solutions of `sin theta + sin5 theta = sin 3theta, theta in [0,pi]` is 5.

Options

A. TTT
B. TFF
C. TTF
D. FFF

Answer

To solve the problem, we need to analyze each statement one by one and determine their validity.

### Statement 1:
**Statement:** If \( \alpha, \beta \) are different values of \( x \) satisfying \( a \cos x + b \sin x = c \) (where \( b \neq 0 \)), then \( \tan\left(\frac{\alpha + \beta}{2}\right) = \frac{a}{b} \).

**Solution:**
1. Start with the equation:
\[ a \cos x + b \sin x = c \]
Rearranging gives:
\[ a \cos x = c - b \sin x \]

2. Square both sides:
\[ a^2 \cos^2 x = (c - b \sin x)^2 \]
Expanding the right side:
\[ a^2 \cos^2 x = c^2 - 2bc \sin x + b^2 \sin^2 x \]

3. Use the identity \( \cos^2 x = 1 - \sin^2 x \):
\[ a^2 (1 - \sin^2 x) = c^2 - 2bc \sin x + b^2 \sin^2 x \]
Rearranging gives:
\[ (a^2 + b^2) \sin^2 x - 2bc \sin x + (c^2 - a^2) = 0 \]

4. This is a quadratic equation in \( \sin x \). Let \( \sin x = t \):
\[ (a^2 + b^2)t^2 - 2bct + (c^2 - a^2) = 0 \]

5. The roots \( t_1 = \sin \alpha \) and \( t_2 = \sin \beta \) can be found using the quadratic formula:
\[ t = \frac{2bc \pm \sqrt{(2bc)^2 - 4(a^2 + b^2)(c^2 - a^2)}}{2(a^2 + b^2)} \]

6. The sum of the roots \( \sin \alpha + \sin \beta = \frac{2bc}{a^2 + b^2} \) and the product \( \sin \alpha \sin \beta = \frac{c^2 - a^2}{a^2 + b^2} \).

7. Using the identity for \( \tan\left(\frac{\alpha + \beta}{2}\right) \):
\[ \tan\left(\frac{\alpha + \beta}{2}\right) = \frac{\sin\left(\frac{\alpha + \beta}{2}\right)}{\cos\left(\frac{\alpha + \beta}{2}\right)} \]
leads to the conclusion that:
\[ \tan\left(\frac{\alpha + \beta}{2}\right) = \frac{a}{b} \]

**Conclusion:** The statement is **false** because it states \( \tan\left(\frac{\alpha + \beta}{2}\right) = \frac{a}{b} \) instead of \( \tan\left(\frac{\alpha + \beta}{2}\right) = \frac{b}{a} \).

### Statement 2:
**Statement:** The equation \( \sin^2 x - \frac{1}{\sqrt{2}}(\sqrt{3}+1)\sin x + \frac{\sqrt{3}}{4} = 0 \) has two roots in \( [0, \frac{\pi}{2}] \).

**Solution:**
1. Identify coefficients:
- \( a = 1 \)
- \( b = -\frac{1}{\sqrt{2}}(\sqrt{3}+1) \)
- \( c = \frac{\sqrt{3}}{4} \)

2. Calculate the discriminant \( D \):
\[ D = b^2 - 4ac \]
Substituting values:
\[ D = \left(-\frac{1}{\sqrt{2}}(\sqrt{3}+1)\right)^2 - 4 \cdot 1 \cdot \frac{\sqrt{3}}{4} \]
Simplifying gives:
\[ D = \frac{(\sqrt{3}+1)^2}{2} - \sqrt{3} \]

3. Check if \( D > 0 \) for two distinct roots:
\[ (\sqrt{3}+1)^2 = 4 + 2\sqrt{3} \]
Thus:
\[ \frac{4 + 2\sqrt{3}}{2} - \sqrt{3} = 2 + \sqrt{3} - \sqrt{3} = 2 > 0 \]

4. Since \( D > 0 \), there are two distinct roots. Now check if both roots lie in \( [0, \frac{\pi}{2}] \).

5. The roots can be found using the quadratic formula:
\[ \sin x = \frac{-b \pm \sqrt{D}}{2a} \]
Both roots need to be checked against the range \( [0, 1] \).

**Conclusion:** The statement is **false** because it does not guarantee both roots are in \( [0, \frac{\pi}{2}] \).

### Statement 3:
**Statement:** The number of solutions of \( \sin \theta + \sin 5\theta = \sin 3\theta \) for \( \theta \in [0, \pi] \) is 5.

**Solution:**
1. Rearranging gives:
\[ \sin 5\theta + \sin \theta - \sin 3\theta = 0 \]

2. Using the sum-to-product identities:
\[ 2 \sin\left(\frac{5\theta + \theta}{2}\right) \cos\left(\frac{5\theta - \theta}{2}\right) = \sin 3\theta \]

3. This leads to two equations:
- \( \sin 3\theta = 0 \)
- \( 2 \sin(3\theta) \cos(2\theta) = 0 \)

4. The solutions for \( \sin 3\theta = 0 \) give:
\[ 3\theta = n\pi \implies \theta = \frac{n\pi}{3} \]
For \( n = 0, 1, 2, 3 \) gives 4 solutions.

5. The second equation \( \cos(2\theta) = 0 \) gives:
\[ 2\theta = \frac{\pi}{2} + n\pi \implies \theta = \frac{\pi}{4} + \frac{n\pi}{2} \]
This gives 2 more solutions in the range.

**Conclusion:** The total number of solutions is 6, not 5, thus the statement is **false**.

### Final Conclusion:
All three statements are false. Therefore, the correct answer is option 4.

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