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UPSC NDA I 2023 Mathematics Solved Paper

Question Numbers: 71-73

Consider the following for the next three (03) items that follow :

Let

f(x) = Pe^{x} + Qe^{2x} + Re^{3x}

where

P, Q, R

are real numbers. Further

f(0) = 6

f'(\ln 3) = 282

and

\int\limits_{0}^{\ln 2}f(x)dx=11

Question : 73 of 120

Marks: +1, -0

What is f'(0) equal to ?

Solution:

Concept:

The derivative of

e^{mx}

m e^{mx}

Explanation:

Given

f(x) = P e^{x} + Q e^{2x} + R e^{3x}

with

P=1

Q=2

R=3

f(x) = e^{x} + 2 e^{2x} + 3 e^{3x}

Differentiate term by term:

f'(x) = 1 \cdot e^{x} + 2 \cdot 2 e^{2x} + 3 \cdot 3 e^{3x}

Simplify:

f'(x) = e^{x} + 4 e^{2x} + 9 e^{3x}

Evaluate at

x=0

f'(0) = e^{0} + 4 e^{0} + 9 e^{0} = 1 + 4 + 9 = 14

Answer:

14

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