The number of distinct real roots of `|(sinx, cosx, cosx),(cos x,sin x,cos x),(cos x,cos x,sin x)|=0` in the interval `-(pi)/4 le x le t (pi)/4` is

Options

A. 4
B. 1
C. 2
D. 3

Explanation

To find the number of distinct real roots of the determinant

\[ \begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0 \]

in the interval \(-\frac{\pi}{4} \leq x \leq t \frac{\pi}{4}\), we will follow these steps:

### Step 1: Calculate the Determinant
We start by calculating the determinant of the given matrix:

\[ D = \begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} \]

### Step 2: Apply Column Operations
We can simplify the determinant using column operations. We will perform the operation \(C_1 \leftarrow C_1 + C_2 + C_3\):

\[ C_1 = \sin x + \cos x + \cos x = \sin x + 2\cos x \]
\[ C_2 = \cos x + \sin x + \cos x = \sin x + 2\cos x \]
\[ C_3 = \cos x + \cos x + \sin x = 2\cos x + \sin x \]

The determinant now becomes:

\[ D = \begin{vmatrix} \sin x + 2\cos x & \sin x + 2\cos x & 2\cos x + \sin x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} \]

### Step 3: Further Simplification
Next, we can perform row operations. Let’s perform \(R_2 \leftarrow R_2 - R_1\) and \(R_3 \leftarrow R_3 - R_1\):

This gives us:

\[ D = \begin{vmatrix} \sin x + 2\cos x & \sin x + 2\cos x & 2\cos x + \sin x \\ \cos x - (\sin x + 2\cos x) & \sin x - (\sin x + 2\cos x) & \cos x - (2\cos x + \sin x) \\ \cos x - (\sin x + 2\cos x) & \cos x - (\sin x + 2\cos x) & \sin x - (2\cos x + \sin x) \end{vmatrix} \]

### Step 4: Set the Determinant to Zero
The determinant \(D\) must equal zero. This leads us to the equation:

\[ \sin x + 2\cos x = 0 \]

### Step 5: Solve the Equation
Rearranging gives:

\[ \sin x = -2\cos x \]

Dividing both sides by \(\cos x\) (where \(\cos x \neq 0\)) gives:

\[ \tan x = -2 \]

### Step 6: Find the Roots
The general solution for \(\tan x = -2\) is:

\[ x = \tan^{-1}(-2) + n\pi \]

### Step 7: Determine the Interval
We need to find the values of \(x\) in the interval \(-\frac{\pi}{4} \leq x \leq t \frac{\pi}{4}\).

Calculating \(\tan^{-1}(-2)\) gives us a specific angle, which we can denote as \(-\alpha\) where \(\alpha = \tan^{-1}(2)\).

Thus, the roots in the specified interval will be:

1. \(-\alpha\)
2. \(-\alpha + \pi\) (if it falls within the interval)

### Step 8: Count Distinct Roots
Now we need to check how many of these roots fall within the interval \(-\frac{\pi}{4} \leq x \leq t \frac{\pi}{4}\).

1. The root \(-\alpha\) is in the interval \(-\frac{\pi}{4} \leq -\alpha\) since \(-\alpha\) is negative.
2. The second root \(-\alpha + \pi\) needs to be checked against \(t \frac{\pi}{4}\).

### Conclusion
Thus, the number of distinct real roots of the determinant in the given interval is:

**1 distinct root if \(t \geq 1\) and 2 distinct roots if \(t > 1\).**