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Question

If x=2costcos2t, y=2sintsin2t

then d2ydx2 at t=π2 is?

A
1
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B
32
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C
32
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D
52
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Solution

The correct option is A 1
Now, dxdt=2sint+2sin2t

Also, dydt=2cost2cos2t

Thus, dydx=dydtdxdt

Thus, dydx=2cost2cos2t2sint+2sin2t

Thus, d2ydx2=ddtdydx.dtdx

Thus, d2ydx2=ddt×2cost2cos2t2sint+2sin2t×1(2sint+2sin2t)

Thus, d2ydx2=ddt(2cost2cos2t(2sint+2sin2t)2)

Or, d2ydx2=((2sint+2sin2t)2(2sint+4sin2t)(2cost2cos2t)×2(2sint+4cos2t)(2sint+2sin2t)4)

Thus, d2ydx2 at π2 is given by,
d2ydx2=(2)2(2)(2)(2)(24)(2)4
Or, d2ydx2=1

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