If `(cos e c^2theta-4)x^2+(cottheta+sqrt(3))x+cos^2(3pi)/2=0` holds true for all real `x ,` then the most general values of `theta` can be given by `n in Z)` `2npi+(11pi)/6` (b) `2npi+(5pi)/6` `2npi+-(7pi)/6` (d) `npi+-(11pi)/6`

Options

A. `2npi +(11 pi)/6`
B. `2n pi+(5 pi)/6`
C. `2n pi pm (7pi)/6`
D. `n pi pm (11 pi)/6`

Explanation

To solve the equation

\[ (\csc^2 \theta - 4)x^2 + (\cot \theta + \sqrt{3})x + \cot^2\left(\frac{3\pi}{2}\right) = 0 \]

for all real \( x \), we need to ensure that the coefficients of \( x^2 \), \( x \), and the constant term are all equal to zero.

### Step 1: Set the coefficient of \( x^2 \) to zero

The coefficient of \( x^2 \) is \( \csc^2 \theta - 4 \). For the equation to hold for all \( x \):

\[ \csc^2 \theta - 4 = 0 \]

### Step 2: Solve for \( \theta \)

Rearranging gives:

\[ \csc^2 \theta = 4 \]

Taking the reciprocal gives:

\[ \sin^2 \theta = \frac{1}{4} \]

Taking the square root:

\[ \sin \theta = \pm \frac{1}{2} \]

### Step 3: Find the angles corresponding to \( \sin \theta = \frac{1}{2} \)

The solutions for \( \sin \theta = \frac{1}{2} \) are:

\[ \theta = \frac{\pi}{6} + 2n\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2n\pi \]

### Step 4: Find the angles corresponding to \( \sin \theta = -\frac{1}{2} \)

The solutions for \( \sin \theta = -\frac{1}{2} \) are:

\[ \theta = \frac{7\pi}{6} + 2n\pi \quad \text{and} \quad \theta = \frac{11\pi}{6} + 2n\pi \]

### Step 5: Set the coefficient of \( x \) to zero

Next, we set the coefficient of \( x \) to zero:

\[ \cot \theta + \sqrt{3} = 0 \]

This gives:

\[ \cot \theta = -\sqrt{3} \]

### Step 6: Solve for \( \theta \)

The solutions for \( \cot \theta = -\sqrt{3} \) imply:

\[ \tan \theta = -\frac{1}{\sqrt{3}} \]

The angles corresponding to this are:

\[ \theta = \frac{5\pi}{6} + n\pi \quad \text{and} \quad \theta = \frac{11\pi}{6} + n\pi \]

### Step 7: Combine solutions

Now, we combine the solutions from both conditions. The general solutions for \( \theta \) that satisfy both conditions are:

1. \( \theta = \frac{5\pi}{6} + 2n\pi \)
2. \( \theta = \frac{11\pi}{6} + 2n\pi \)
3. \( \theta = \frac{7\pi}{6} + 2n\pi \)
4. \( \theta = \frac{11\pi}{6} + 2n\pi \)

### Final Answer

The most general values of \( \theta \) can be expressed as:

\[ \theta = n\pi \pm \frac{11\pi}{6} \quad (n \in \mathbb{Z}) \]