Question

lf the mean and standard deviation of 5.3.7. a.b are 5 and 2 respectively, then a and b are roots of the equation.

Answer 1

It has given that, the mean and standard deviation of 5, 3, 7, a , b are 5 and 2 respectively.

To find : find the quadratic equation of which a and b are roots.

solution : mean = sum of observations/total no of observations

⇒5 = (5 + 3 + 7 + a + b)/5

⇒25 = 15 + a + b

⇒a + b = 10 .......(1)

standard deviation = $\sqrt{\frac{\displaystyle\Sigma^n_{i=1}{(x_i-\bar{x})^2}}{n}}$

⇒2 = √[{(5 - 5)²+ (5 - 3)² + (5 - 7)² + (5 - a)² + (5 - b)²}/5]

⇒4 × 5 = (0) + 4 + 4 + (5 - a)² + (5 - b)²

⇒20 - 8 = 25 + a² - 10a + 25 - 10b + b²

⇒12 = 50 + (a² + b²) - 10(a + b)

⇒-38 = (a + b)² - 2ab - 10(a + b)

⇒-38 = 100 - 2ab - 100 [ from eq (1). ]

⇒ab = 19

so the equation equation is ..

x² - (a + b)x + ab = x² - 10x + 29

Therefore the correct option is (4) x² - 10x + 19