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The number of solutions of the equation ...

The number of solutions of the equation `16(sin^(5)x +cos^(5)x)=11(sin x + cos x)` in the interval `[0,2pi]` is

A

6

B

7

C

8

D

9

Text Solution

Verified by Experts

The correct Answer is:
A
`16 (sin^(5)x + cos^(5)x)-11(sin x + cos x) = 0`
`rArr (sin x + cos x) {16(sin^(4)x-sin^(3)x cos x + sin^(2)x cos^(2)x - sin x cos^(3) x + cos^(4) x)-11}=0`
`rArr (sin x + cos x){16(1-sin^()x cos^(2)x - sin x sin x cos x)-11}=0`
`rArr (sin x + cos x)(4 sin x cos x -1)(4 sin x cos x + 5) = 0`
As `4 sin x cos x + 5 ne 0` ,we have
`sin x + cos x = 0, 4 sin x cos x - 1 =0`
`rArr tan x = -1, sin 2x=(1)/(2)`
`rArr pi//12, 5pi//2, 9pi//12, 17pi//12, 21 pi//12`.
There are 6 solutions on `[0, 2pi]`
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