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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Differential Equations

NDA 2023 · Mathematics · Differential Equations

Paragraph: Let f(x)=P e^x+Q e^ 2 x +R e^ 3 x , where P, Q, R are...

Paragraph: Let f(x)=Pex+Qe2x+Re3xf(x)=P e^x+Q e^{2 x}+R e^{3 x}, where P,Q,RP, Q, R are real numbers. Further f(0)=6,f′(ln⁡3)=282f(0)=6, f^{\prime}(\ln 3)=282 and ∫0ln⁡2f(x)dx=11\int_0^{\ln 2} f(x) d x=11
Question: What is f′(0)f^{\prime}(0) equal to?

Options

  1. A1818
  2. B1616
  3. C1515
  4. D1414

Correct answer

D. 1414

Step-by-step solution

∵P+Q+R=6⇒P+2+3=6⇒P=1∴f(x)=ex+2e2x+3e3xf′(x)=ex+2⋅2e2x+3⋅3e3xf(0)=1+4+9=14\because \quad P+Q+R=6\Rightarrow P+2+3=6 \Rightarrow P=1\therefore \quad f(x)=e^x+2 e^{2 x}+3 e^{3 x}f^{\prime}(x)=e^x+2 \cdot 2 e^{2 x}+3 \cdot 3 e^{3 x}f(0)=1+4+9=14
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