Question

The number of values of a ∈ N such that the variance of 3, 7, 12, a , 43 - a is a natural number is :

• A. 0
• B. 2
• C. 5
• D. Infinite

Answer 1

Answer: (A)

0

Answer 2

The correct answer is (A) : 0
Mean $= \frac{3+12+7+a+43-a}{5} =13$
Variance $= \frac{9+49+144+a²+(43-a)²}{ 5} - 13² ∈ N$
$\frac{2a²-a+1}{5} ∈ N$
$2a² - a + 1 = 5n [ n ∈ N ]$
$2a² - a + 1 = 5n = 0$
D = 1 – 4(1 – 5 n)2
= 40 n – 7
D cannot be a perfect square as all perfect squares will be of the form of 4λ or 4λ + 1
So, a cannot be natural number
Therefore , Number of values = 0