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In a curve y = t2 + 2t and x = t3, find the slope \(\rm dy\over dx\) at t = 5
1. \(\rm \frac{4}{5}\)
2. \(\rm \frac{4}{3}\)
3. \(\rm \frac{12}{5}\)
4. \(\rm \frac{4}{25}\)
5. None of these

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Correct Answer - Option 4 : \(\rm \frac{4}{25}\)

Concept:

Parametric Form:

If f(x) and g(x) are the functions in x, then 

\(\rm df(x)\over dg(x)\) = \(\rm \frac{df(x)\over dx}{dg(x)\over dx}\) 

 

Calculation:

Given y = t2 + 2t

\(\rm {dy\over dt}\) = 2t + 2

Also x = t3

\(\rm dx\over dt\) = 3t2 

Now \(\rm dy\over dx\) = \(\rm \frac{dy\over dt}{dx\over dt}\) 

\(\rm dy\over dx\) = \(\rm \frac{2t +2}{3t^2}\)

At t = 5,

\(\rm {dy\over dx}\) = \(\rm \frac{2(5) +2}{3(5)^2}\) = \(\rm \frac{4}{25}\)

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