Question

If

\[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}, \]

then the value of

\[ \left(\frac{24}{x} + \frac{24}{y}\right) \]

is:

• A. \( 7 \)
• B. \( 6 \)
• C. \( 8 \)
• D. \( 18 \)

Hint:

To solve matrix element problems, equate corresponding elements and solve algebraically using known identities.

Answer 1

The Correct Option is D

Step 1: Equate corresponding elements of the matrices. \[ x + y = 6, \quad xy = 8 \] Step 2: Solve for the required expression. We need to evaluate: \[ \frac{24}{x} + \frac{24}{y} = 24 \left(\frac{1}{x} + \frac{1}{y}\right) \] Using the identity: \[ \frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy} \] Substituting values: \[ \frac{1}{x} + \frac{1}{y} = \frac{6}{8} = \frac{3}{4} \] Step 3: Final calculation. \[ \frac{24}{x} + \frac{24}{y} = 24 \times \frac{3}{4} = 18 \] Final Answer: \[ \boxed{18} \]