Question

Let the mean and variance of 8 numbers \(x, y, 10, 12, 6, 12, 4, 8\) be \(9\) and \(9.25\) respectively. If \(x > y\), then \(3x - 2y\) is equal to:

• A. 25
• B. 1
• C. 24
• D. 13

Hint:

For problems involving mean and variance, set up equations based on the given definitions and solve systematically.

Answer 1

The Correct Option is A

Step 1: Use the mean to form an equation.
\[ \text{Mean} = \frac{x + y + 10 + 12 + 6 + 12 + 4 + 8}{8} = 9. \] \[ x + y + 52 = 72 \implies x + y = 20. \] Step 2: Use the variance to form another equation.
\[ \text{Variance} = \frac{\sum (a_i - \text{Mean})^2}{8}. \] \[ \frac{(x-9)^2 + (y-9)^2 + 3^2 + 3^2 + (-1)^2 + (-5)^2 + (-1)^2 + (-3)^2}{8} = 9.25. \] \[ (x-9)^2 + (y-9)^2 + 54 = 74 \implies (x-9)^2 + (y-9)^2 = 20. \] Step 3: Solve the equations.
\[ (x-9)^2 + (20 - x - 9)^2 = 20 \implies (x-9)^2 + (11-x)^2 = 20. \] Expanding: \[ (x-9)^2 + (11-x)^2 = (x^2 - 18x + 81) + (121 - 22x + x^2) = 20. \] \[ 2x^2 - 40x + 202 = 20 \implies x^2 - 20x + 91 = 0. \] Factoring: \[ (x-13)(x-7) = 0 \implies x = 13 \, \text{or} \, x = 7. \] Since \(x > y\), \(x = 13\) and \(y = 7\). 
Step 4: Calculate \(3x - 2y\).
\[ 3x - 2y = 3(13) - 2(7) = 39 - 14 = 25. \] Final Answer: \(3x - 2y = 25\).