Question

Select the value(s) of \(x\) for which the determinants,

• A. \(\sqrt{3}\)
• B. \(3\)
• C. \(-1\)
• D. \(-\sqrt{3}\)

Hint:

For a \(2\times2\) matrix \[ \begin{vmatrix} a & b \\ c & d \end{vmatrix}, \] the determinant is calculated as \[ ad-bc. \]

Answer 1

The Correct Option is A, D

Step 1: Find the first determinant.
\[ \begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = (2)(1)-(4)(5) \] \[ =2-20 \] \[ =-18 \]

Step 2: Find the second determinant.
\[ \begin{vmatrix} 2x & 6 \\ 4 & x \end{vmatrix} = (2x)(x)-(6)(4) \] \[ =2x^2-24 \]

Step 3: Equate the determinants.
Since both determinants are equal,
\[ 2x^2-24=-18 \]

Step 4: Simplify the equation.
Add \(24\) on both sides.
\[ 2x^2=6 \]

Step 5: Solve for \(x^2\).
Divide both sides by \(2\).
\[ x^2=3 \]

Step 6: Find the values of \(x\).
Taking square roots,
\[ x=\pm \sqrt{3} \] Thus,
\[ x=\sqrt{3} \quad \text{or} \quad x=-\sqrt{3} \]

Step 7: Final conclusion.
Therefore, the correct values of \(x\) are
\[ \boxed{\sqrt{3},\,-\sqrt{3}} \]
Hence, the correct answers are options (A) and (D).