Question

If \( \left| \begin{array}{cc} 2x & 5 \\ 4 & x \end{array} \right| = \left| \begin{array}{cc} 3 & 5 \\ 4 & 6 \end{array} \right| \), then the value of \( x \) is:

• A. $\frac{3}{2}$
• B. $6$
• C. $3$
• D. $\pm 3$

Hint:

Don’t forget to solve quadratic equations completely — always check for $\pm$ when square roots are involved.

Answer 1

The Correct Option is D

Use the formula for the determinant of a \( 2 \times 2 \) matrix: \( \left| \begin{array}{cc} a & b \\ c & d \end{array} \right| = ad - bc \).
Left side: \( \left| \begin{array}{cc} 2x & 5 \\ 4 & x \end{array} \right| = 2x \cdot x - 4 \cdot 5 = 2x^2 - 20 \).
Right side: \( \left| \begin{array}{cc} 3 & 5 \\ 4 & 6 \end{array} \right| = 3 \cdot 6 - 4 \cdot 5 = 18 - 20 = -2 \).
So, \( 2x^2 - 20 = -2 \Rightarrow 2x^2 = 18 \Rightarrow x^2 = 9 \Rightarrow x = \pm 3 \).