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
• E.

Hint:

To solve such problems, equate corresponding elements of the matrices and use algebraic identities to simplify expressions efficiently.

Answer 1

The Correct Option is D

The given matrices are: \[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}. \] 

1. Equating Corresponding Elements: - From the first row, first column: \[ x + y = 6. \] - From the second row, second column: \[ xy = 8. \] 

2. Expression to Evaluate: We need to find: \[ \frac{24}{x} + \frac{24}{y}. \] 

Simplify the expression using the identity \( \frac{a}{x} + \frac{a}{y} = a \cdot \frac{x + y}{xy} \): \[ \frac{24}{x} + \frac{24}{y} = 24 \cdot \frac{x + y}{xy}. \] 

3. Substitute Known Values: - \( x + y = 6 \), - \( xy = 8 \). Substitute these values: \[ \frac{24}{x} + \frac{24}{y} = 24 \cdot \frac{6}{8}. \] 

Simplify: \[ \frac{24}{x} + \frac{24}{y} = 24 \cdot \frac{3}{4} = 18. \] 

Hence, the value of \( \frac{24}{x} + \frac{24}{y} \) is (D) 18.