The least positive integral solution of `"sin" pi (x^(2) + x)-"sin" pi x^(2) = 0`, is

Options

A. a rational number
B. an irrational number of the form `sqrt(lambda)`
C. an irrational number of the form `(sqrt(lambda)-1)/(4)`, where `lambda` is an odd integer
D. an irrational number of the firm `(sqrt(lambda)+1)/(4)`, where `lambda` is an even integer.

Answer

C

Explanation

We have,
`"sin"pi(x^(2) + x) = "sin"pi x^(2)`
`rArr pi(x^(2) + x) = n pi + (-1)^(n) pix^(2), n in Z`
`rArr x^(2) + x = n + (-1)^(n)x^(2) " "… (i)`
Now, two cases arise.
CASE I When is an even integer, say `n = 2m, m in Z`
In this case, equation (i) reduces to
`x^(2) + x = 2m + x^(2) rArr x = 2m, m in Z.`
CASE II When n is an odd integer, say `n = 2m + 1, m in Z`
In this case, equation (i) reduces to
`x^(2) + x = (2m + 1)-x^(2)`
`rArr 2x^(2) + x - (2m+1)= 0`
`rArr x = (-1+-sqrt(1+8 (2m+1)))/(4)`
`rArr x = (-1 +- sqrt(lambda))/(4), " where " lambda = 8 (2m + 1) +1` is an odd integer