If (cosx+sin(ax) is a periodic function and a is a root of equation x2+px+q=0 where p,q belong to integers , then show that pq cannot be an odd integer.

16 Points
12 years ago

Dear harshit agarwal,

Given (cosx+sin(ax)) is a periodic function

This implies that 'a' should be rational.

'a' is root of x2+px+q=0, where p, q are integers.

As 'a' is rational number, let us consider a = m/n, where m,n are coprimes.

(m/n)2 + p(m/n) + q = 0.

m2 + pmn + qn2 = 0

m2 = -n(pm+qn)

hence m should be divisible by n, but m,n are coprimes, hence n=1, hence the equation will be

m2 +pm + q = 0, m is an integer.

q = -m(m+p)

It is clear that q is divisible by m.

If m is even then q is even and p can be even or odd.

If m is odd and p is odd then q is even

If m is odd and p is even then q is odd.

These are possible cases which doesn't include the case in which both q, p are odd.

if q is odd, then m should be odd and m+p should be odd

i.e., if q is odd, then m should be odd and p should be even.

Hence, pq cannot be odd integer.

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