14.???
The set of values of 'a' for which the equations cos2x + a sin x = 2a-7 posses a solution is
1. (-∞,2)
2. [2.6]
3. (6, ∞)
4. (-∞, ∞)
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7 Answers
1This question was given in subjective section in this year's WBJEE paper ....
23convert cos2x to sinx , and write delta ≥ 0 for the obtained quadratic
62exactly what qwerty has said... but with slight more caution...
the root should lie between -1 and 1
cos2x + a sin x = 2a-7
1-2sin2x + a sin x = 2a - 7
so 2 sin2x - a sin x + 2a-8 =0
We want real roots between -1 and 1..
so minimum requirement is D>=0
a2-16(a-4)≥0
a2-16a+64≥0 which is always true....
we also want f(1)f(-1)≤0 where f(1) means sin x = 1 not x=1
so (2-a+2a-8)(2+a+2a-8)≤0
(a-6)(a-2)≤0
so a lies between 2, 6
There is one condition that i have left out...
(which one :P) (That my calculation shows has no solution!)
62i left the case is when both the roots lie between -1 and 1..
when -1<-b/2a<1
and b2-4ac≥0 and f(1)>0, f(-1)>0