Question

If the mean and the standard deviation of the data \(3, 5, 7, a, b\) are \(5\) and \(2\) respectively, then \(a\) and \(b\) are the roots of the equation :

• A. \(x^2 - 20x + 18 = 0\)
• B. \(2x^2 - 20x + 19 = 0\)
• C. \(x^2 - 10x + 18 = 0\)
• D. \(x^2 - 10x + 19 = 0\)

Hint:

Mean \(=5\) gives \(a + b\); variance \(=4\) gives \(a^2 + b^2\); then find \(ab\).

Answer 1

The Correct Option is D

Mean: \(\dfrac{3+5+7+a+b}{5} = 5 \Rightarrow a + b = 10\).
Variance \(= 4\): \(\dfrac{3^2+5^2+7^2+a^2+b^2}{5} - 5^2 = 4 \Rightarrow 83 + a^2 + b^2 = 145 \Rightarrow a^2 + b^2 = 62\).
\(ab = \dfrac{(a+b)^2 - (a^2+b^2)}{2} = \dfrac{100 - 62}{2} = 19\).
The quadratic with roots \(a, b\): \(x^2 - (a+b)x + ab = x^2 - 10x + 19 = 0\).
Answer: (4) \(x^2 - 10x + 19 = 0\)