If $x = 2 cos t - cos 2 t, y = 2 sin t - sin 2 t$ , then the value of $y_{2}$ at $t = \frac{π}{2}$ is

1 / 2

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3 / 2

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- 1 / 2

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- 3 / 2

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Solution

The correct option is C $- 3 / 2$
$\frac{d y}{d x} = \frac{\frac{\frac{d y}{d t}}{d x}}{d t} = \frac{2 c o s t - 2 c o s 2 t}{- 2 s i n t + 2 s i n 2 t}$

$= \frac{c o s t - c o s^{2} t}{s i n 2 - s i n t}$

$\frac{d^{2} y}{d x^{2}} = \frac{d}{d t} (\frac{d y}{d x}) \frac{d t}{d x}$

$= \frac{(s i n t + 2 s i n^{2} t) (s i n 2 t - s i n t) (2 c o s 2 t - c o s t) (c o s t - c o s 2 t)}{(s i n 2 t - s i n t)^{2} (- 2 s i n t + 2 s i n 2 t)}$

$= \frac{(- 1) (- 1) - (- 2) (+ 1)}{- 2}$

$= \frac{- 3}{2}$

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x = 2 cos t - cos 2 t

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\frac{d^{2} y}{d x^{2}}

t = \frac{π}{2}

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\frac{d^{2} y}{d x^{2}} |_{t = π /_{2}}

x = 2 s i n t - sin 2 t, y = 2 c o s t - cos 2 t

\frac{d^{2} y}{d x^{2}}

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\frac{d^{2} y}{d x^{2}} at t = \frac{π}{2}

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