Vector question-sum of the integers

Question

If a and b are two unit vectors, then the sum of the integers in the range of $\frac{5}{2}|a+b|+6|a-b|$ is

(1) 0

(2) 25

(3) 64

(4) 81

Answer 1

Disclaimer: This is an answer suitable for multiple choice questions only.

Let, $$f(a,b)=\frac{5}{2}|a+b|+6|a-b|$$

For $b=a$, $f(a,b)=5$, for $b=-a$, $f(a,b)=12$. So the range of $f$ is atleast $[5,12]$. Sum of integrs from 5 to 12 is $$\frac{12*13}{2}-\frac{4*5}{2}=68$$

Therefore the only remaining choice is 81.