If `|{:(x,sintheta,costheta),(-sintheta,-x,1),(costheta,1,x):}|=8`, then the value of x is :

Options

A. 2
B. 1
C. `-1`
D. `-2`

Answer

To solve the problem, we need to evaluate the determinant of the matrix given and set it equal to 8. The matrix is:

\[ \begin{vmatrix} x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x \end{vmatrix} \]

### Step 1: Calculate the determinant using the first row.

The determinant can be calculated using the formula for determinants of 3x3 matrices:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix, we have:

- \( a = x \)
- \( b = \sin \theta \)
- \( c = \cos \theta \)
- \( d = -\sin \theta \)
- \( e = -x \)
- \( f = 1 \)
- \( g = \cos \theta \)
- \( h = 1 \)
- \( i = x \)

Thus, the determinant becomes:

\[ D = x \left((-x)(x) - (1)(1)\right) - \sin \theta \left((-sin \theta)(x) - (1)(\cos \theta)\right) + \cos \theta \left((-sin \theta)(1) - (-x)(\cos \theta)\right) \]

### Step 2: Simplify each term in the determinant.

Calculating each term:

1. The first term:
\[ x \left(-x^2 - 1\right) = -x^3 - x \]

2. The second term:
\[ -\sin \theta \left(-\sin \theta x - \cos \theta\right) = \sin^2 \theta x + \sin \theta \cos \theta \]

3. The third term:
\[ \cos \theta \left(-\sin \theta + x \cos \theta\right) = -\sin \theta \cos \theta + x \cos^2 \theta \]

### Step 3: Combine all terms.

Now, combining all the terms we have:

\[ D = -x^3 - x + \sin^2 \theta x + \sin \theta \cos \theta - \sin \theta \cos \theta + x \cos^2 \theta \]

Notice that \(\sin \theta \cos \theta\) cancels out:

\[ D = -x^3 - x + \sin^2 \theta x + x \cos^2 \theta \]

### Step 4: Factor out \(x\).

We can factor out \(x\) from the remaining terms:

\[ D = -x^3 - x + x(\sin^2 \theta + \cos^2 \theta) \]

Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\[ D = -x^3 - x + x(1) = -x^3 - x + x = -x^3 \]

### Step 5: Set the determinant equal to 8.

Since we know that the determinant equals 8:

\[ -x^3 = 8 \]

### Step 6: Solve for \(x\).

Multiplying both sides by -1 gives us:

\[ x^3 = -8 \]

Taking the cube root of both sides:

\[ x = -2 \]

### Final Answer:

Thus, the value of \(x\) is \(-2\).

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