If `x^(2) = |(sin theta,cos theta,0),(-cos theta,sin theta ,1),(sin theta,cos theta,2)|`, then the value of `4x^(2) + x "sin" (3pi)/(2) +5` is

Options

A. `13- sqrt2`
B. `13 + sqrt2`
C. `sqrt2-13`
D. Both a and b

Answer

To solve the problem, we need to evaluate the determinant and then find the value of the expression \(4x^2 + x \sin\left(\frac{3\pi}{2}\right) + 5\).

### Step 1: Calculate the Determinant

We are given:
\[ x^2 = \begin{vmatrix} \sin \theta & \cos \theta & 0 \\ -\cos \theta & \sin \theta & 1 \\ \sin \theta & \cos \theta & 2 \end{vmatrix} \]

To calculate this determinant, we can use the formula for the determinant of a \(3 \times 3\) matrix:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix:
- \(a = \sin \theta\), \(b = \cos \theta\), \(c = 0\)
- \(d = -\cos \theta\), \(e = \sin \theta\), \(f = 1\)
- \(g = \sin \theta\), \(h = \cos \theta\), \(i = 2\)

Calculating the determinant:
\[ x^2 = \sin \theta \left(\sin \theta \cdot 2 - 1 \cdot \cos \theta\right) - \cos \theta \left(-\cos \theta \cdot 2 - 1 \cdot \sin \theta\right) + 0 \]

This simplifies to:
\[ x^2 = \sin \theta (2 \sin \theta - \cos \theta) + \cos \theta (2 \cos \theta + \sin \theta) \]

### Step 2: Simplify the Expression

Now, simplifying further:
\[ x^2 = 2 \sin^2 \theta - \sin \theta \cos \theta + 2 \cos^2 \theta + \sin \theta \cos \theta \]
\[ x^2 = 2 \sin^2 \theta + 2 \cos^2 \theta = 2(\sin^2 \theta + \cos^2 \theta) = 2 \]

Thus, we have:
\[ x^2 = 2 \implies x = \sqrt{2} \text{ or } x = -\sqrt{2} \]

### Step 3: Evaluate the Expression

Now we need to evaluate:
\[ 4x^2 + x \sin\left(\frac{3\pi}{2}\right) + 5 \]

First, calculate \(4x^2\):
\[ 4x^2 = 4 \times 2 = 8 \]

Next, calculate \(\sin\left(\frac{3\pi}{2}\right)\):
\[ \sin\left(\frac{3\pi}{2}\right) = -1 \]

Now substituting \(x = \sqrt{2}\):
\[ 4x^2 + x \sin\left(\frac{3\pi}{2}\right) + 5 = 8 + \sqrt{2}(-1) + 5 = 8 - \sqrt{2} + 5 = 13 - \sqrt{2} \]

Now substituting \(x = -\sqrt{2}\):
\[ 4x^2 + x \sin\left(\frac{3\pi}{2}\right) + 5 = 8 - \sqrt{2}(-1) + 5 = 8 + \sqrt{2} + 5 = 13 + \sqrt{2} \]

### Final Result

Thus, the values of the expression are:
- For \(x = \sqrt{2}\): \(13 - \sqrt{2}\)
- For \(x = -\sqrt{2}\): \(13 + \sqrt{2}\)

Both values are possible, so the final answer is:
\[ \text{Both } 13 - \sqrt{2} \text{ and } 13 + \sqrt{2} \]