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

Question

The number of values of a ∈ R such that the variance of 3,7,12 a, 43-a is a natural number is: (A) 0 (B) 2 (C) 5 (D) infinite

Tardigrade Admin · Accepted Answer

Mean =13 Variance =(9+49+144+ a 2+(43- a )2/5)-132 ∈ N ⇒ (2 a2-a+1/5) ∈ N ⇒ 2 a 2- a +1-5 n =0 must have solution as natural numbers its D=40 n-7 always has 3 at unit place ⇒ D can't be perfect square So, a can't be integer.

Q. The number of values of a∈R such that the variance of 3,7,12a,43−a is a natural number is:

0

2

5

infinite

Mean =13 Variance =59+49+144+a2+(43−a)2​−132∈N ⇒52a2−a+1​∈N ⇒2a2−a+1−5n=0 must have solution as natural numbers its D=40n−7 always has 3 at unit place ⇒D can't be perfect square So, a can't be integer.

Q. The number of values of $a \in R$ such that the variance of $3, 7, 12 a, 43 - a$ is a natural number is: