JEE Main 2020 · 5 September · evening · Single correct · +4 / −1

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 :

  1. (a) x² – 20x + 18 = 0
  2. (b) 2x² – 20x + 19 = 0
  3. (c) x² – 10x + 18 = 0
  4. (d) x² – 10x + 19 = 0
Answer and solution

Answer (d) x² – 10x + 19 = 0

Solution

Maths sir · 4 steps

The mean gives you a + b, the variance gives you a² + b², and together they give ab.

  1. Use the mean. There are 5 values with mean 5, so their sum is 25. That gives 15 + a + b = 25, so a + b = 10.
  2. Use the variance. The standard deviation is 2, so the variance is 4. Use variance = (mean of squares) - (mean)².

    (9 + 25 + 49 + a² + b²)/5 - 5² = 4 ⇒ 83 + a² + b² = 145 ⇒ a² + b² = 62

  3. Find the product. 2ab = (a+b)² - (a² + b²) = 100 - 62 = 38, so ab = 19.
  4. So. With sum 10 and product 19, a and b are the roots of x² - 10x + 19 = 0, which is option (d).

The idea to keep. Two unknowns in a data set? The mean fixes their sum and the variance fixes the sum of their squares. Then (a+b)² - (a²+b²) = 2ab gives you the quadratic.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Statistics

Asked about 100 times in JEE Main since 2020. This question is from 2020.