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If (cosx+sin(ax) is a periodic function and a is a root of equation x 2 +px+q=0 where p,q belong to integers , then show that pq cannot be an odd integer.

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.

Grade:12

1 Answers

askIITiansexpert nagesh
16 Points
11 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.

 

Please feel free to post as many doubts on our discussion forum as you can. If you find any question

Difficult to understand - post it here and we will get you the answer and detailed solution very quickly. We

are all IITians and here to help you in your IIT JEE preparation. All the best harshit agarwal

 

Regards,

Askiitians Experts

nagesh

 

 

 

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