Paragraph: Let f(x)=P e^x+Q e^ 2 x +R e^ 3 x , where P, Q, R are real numbers. Further f(0)=6, f^ ( 3)=282 and ∫_0^ 2 f(x) d x=11 Question: What is the value of Q ?

Correct answer: B. 2 f(x)=P e^x+Q e^ 2 x +R e^ J x f(0)=P+Q+R=6 ....(1) f^ (x)=P e^x+2 Q e^ 2 x +3 R e^ 3 x f^ ( 3)=P e^ 3 +2 Q e^ 2 3 +3 R e^ 3 3 3 P+18 Q+81 R=282 P+6 Q+27 R=94 ....(2) Since ∫ f(x) d x=P e^x+(Q)/(2) e^ 2 x +(R)/(3) e^ 3 x ∫_0^ f(x) d x=11(P e^ In 2 +(Q)/(2) e^2 In 2+(R)/(3) e^3 In 2-P-(Q)/(2)-(R)/(3))=11 2 P+2 Q+(8 R)/(3)-P-(Q)/(2)-(R)/(3)=11 P+(3 Q)/(2)+(7 R)/(3)=11 ....(3) (2) - (1) and (3) - (1), 5 Q+26 R=88 ....(4) and, (Q)/(2)+(4 R)/(3)=53 Q+8 R=30 ....(5) From (5), R=(30-3 Q)/(8) From (4), 5 Q+26((30-3 Q)/(8))=88 20 Q+390-39 Q=352 19 Q=38 Q=2

NDA 2023

Exam: NDA
Year: 2023
Subject: Mathematics
Chapter: Indefinite Integration

NDA 2023 · Mathematics · Indefinite Integration

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

Paragraph: Let

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

, where

P, Q, R

are real numbers. Further

f(0)=6, f^{\prime}(\ln 3)=282

and

\int_0^{\ln 2} f(x) d x=11

Question: What is the value of Q ?

Options

A $1$
B $2$
C $3$
D $4$

Correct answer

B. $2$

Step-by-step solution

f(x)=P e^x+Q e^{2 x}+R e^{J x}f(0)=P+Q+R=6

....(1)

f^{\prime}(x)=P e^x+2 Q e^{2 x}+3 R e^{3 x}f^{\prime}(\ln 3)=P e^{\ln 3}+2 Q e^{2 \ln 3}+3 R e^{3 \ln 3}\Rightarrow \quad 3 P+18 Q+81 R=282\Rightarrow P+6 Q+27 R=94

....(2) Since

\int f(x) d x=P e^x+\frac{Q}{2} e^{2 x}+\frac{R}{3} e^{3 x}\Rightarrow \int_0^{\ln } f(x) d x=11\left(P e^{\operatorname{In} 2}+\frac{Q}{2} e^2 \operatorname{In} 2+\frac{R}{3} e^3 \operatorname{In} 2-P-\frac{Q}{2}-\frac{R}{3}\right)=11\Rightarrow \quad 2 P+2 Q+\frac{8 R}{3}-P-\frac{Q}{2}-\frac{R}{3}=11\Rightarrow \quad P+\frac{3 Q}{2}+\frac{7 R}{3}=11

....(3) (2) - (1) and (3) - (1),

5 Q+26 R=88

....(4) and,

\frac{Q}{2}+\frac{4 R}{3}=53 Q+8 R=30

....(5) From (5),

R=\frac{30-3 Q}{8}

From (4),

5 Q+26\left(\frac{30-3 Q}{8}\right)=88\Rightarrow 20 Q+390-39 Q=352\Rightarrow \quad 19 Q=38\Rightarrow \quad Q=2

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