If ` x = 2 cos t - cos 2t , y = 2 sin t - sin 2t`, then the value of
` |(d^(2) y)/(dx^(2))|_(t= pi//2 ) `is

Options

A. `3//2`
B. `5//2`
C. `5//2`
D. `-3//2`

Answer

D

Explanation

`(dx)/(dt)=2cost-2 cos 2t`
`therefore (dy)/(dx)=(2 cos t-2 cos 2t)/(-2 sin t+2sin 2t)`
`=(cos -cos 2t)/(-2sin 2t-sin t)`
`=(cos t-cos 2t)/(sin 2t-sin t)`
`implies(2 sin""(3t)/(2)*sin""(t)/(2))/(2cos ""(3t)/(2)*sin ""(t)/(2))`
`=tan ""(3t)/(2)`
`(d^(2)y)/(dx^(2))=sec ^(2)""(3t)/(2)*3/2*(dt)/(dx)`
`=3/2sec^(2)""(3t)/(2)(1)/((2 sin 2t-2sin t))`
`implies(d^(2)y)/(dx^(2))|_(t=pi//2)=-3//2`