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:

When solving matrix equations, compare corresponding elements to derive equations for unknowns. Simplify expressions by substituting these values back into the problem, ensuring all calculations align with the matrix structure.

Answer 1

The Correct Option is D

From the equality of the matrices: \[ \begin{bmatrix} x + y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}, \] 

Step 1: Use the relationship between \( x \) and \( y \) 
The given equations are: \[ x + y = 6 \quad \text{and} \quad xy = 8. \] These are the sum and product of the roots of a quadratic equation. Let the quadratic equation be: \[ t^2 - (x + y)t + xy = 0. \] 

Substitute \( x + y = 6 \) and \( xy = 8 \): \[ t^2 - 6t + 8 = 0. \] 

Factorize the equation: \[ t^2 - 6t + 8 = (t - 2)(t - 4) = 0. \] Thus, \( x = 2 \) and \( y = 4 \) (or vice versa). 

Step 2: Compute \( \frac{24}{x} + \frac{24}{y} \) 
Substitute \( x = 2 \) and \( y = 4 \): \[ \frac{24}{x} + \frac{24}{y} = \frac{24}{2} + \frac{24}{4}. \] Simplify: \[ \frac{24}{2} + \frac{24}{4} = 12 + 6 = 18. \]