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UPSC NDA I 2021 Mathematics Solved Paper

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Question : 104 of 120

Marks: +1, -0

If

x = e^{t} \cos t

y = e^{t} \sin t

\frac{dx}{dy}

t = 0

equal to?

0
1
2e
$-1$

Solution:

Concept:

Parametric differentiation: if

x = f(t)

and

y = g(t)

, then

\frac{dx}{dy} = \frac{f'(t)}{g'(t)}

or its reciprocal.

Explanation:

Given

x = e^{t} \cos t

and

y = e^{t} \sin t

.

Differentiate

x

with respect to

t

:

\frac{dx}{dt} = e^{t} \cos t - e^{t} \sin t

.

Differentiate

y

with respect to

t

:

\frac{dy}{dt} = e^{t} \sin t + e^{t} \cos t

.

Now compute

\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{e^{t} \sin t + e^{t} \cos t}{e^{t} \cos t - e^{t} \sin t}

.

At

t = 0

: numerator

= e^{0}(\sin 0 + \cos 0) = 1(0+1)=1

, denominator

= e^{0}(\cos 0 - \sin 0)=1(1-0)=1

. Hence

\frac{dy}{dx} = 1

.

Therefore

\frac{dx}{dy} = \frac{1}{dy/dx} = 1

.

Answer:

1

(Option B)

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