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Given a(1)cos alpha(1)+alpha(2)cos alpha...

Given `a_(1)cos alpha_(1)+alpha_(2)cos alpha_(2)+…+a_(n)cos alpha_(n)=0` and `a_(1)cos(alpha_(1)+theta)+a_(2)cos(alpha_(2)+theta)+…+a_(n)cos(alpha_(n)+theta)=0 (theta ne k pi)`, then the value of `a_(1)cos(alpha_(1)+lambda)+a_(2)cos(alpha_(2)+lambda)+…+a_(n)cos(alpha_(n)+lambda)` is

A

`theta-lambda`

B

`theta+lambda`

C

`lambda`

D

0

Text Solution

Verified by Experts

The correct Answer is:
D
`a_(1)cos (alpha_(1)+theta)+a_(2)cos(alpha_(2)+theta)+…+a_(n)cos (alpha_(n)+theta)=0`
`rArr (a_(1)cos alpha_(1)+a_(2)cos alpha_(2)+…+alpha_(n))cos theta-(a_(1)sin alpha_(1)+a_(2)sin alpha_(2)+….+a_(n)sin alpha_(n)sin theta)=0`
`rArr a_(1) sin alpha_(1)+a_(2)sin alpha_(2)+....+a_(n)sin alpha_(n)=0`
(since `sin theta ne 0`)
and `a_(1) cos alpha_(1)+a_(2) cos alpha_(2)+...+ a_(n) cos alpha_(n)=0`
Now, `a_(1)cos (alpha_(1)+lambda)+a_(2)cos (alpha_(2)+lambda)+...+a_(n) cos (alpha_(n)+lambda)`
`=(a_(1)cos alpha_(1)+a_(2)cos alpha_(2)+...+a_(n) cos alpha_(n))cos lambda - (a_(1) sin alpha_(1)+a_(2) sin alpha_(2)+...+ a_(n) sin alpha_(n))sin lambda = 0`
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