MHT CET · Maths · Differentiation
If \(y=3 e^{5 x}+5 e^{3 x}, \quad\) then \(\frac{d^{2} y}{d x^{2}}-8 \frac{d y}{d x}=\)
- A \(-10 y\)
- B \(15 y\)
- C \(-15 y\)
- D \(10 y\)
Answer & Solution
Correct Answer
(C) \(-15 y\)
Step-by-step Solution
Detailed explanation
\(y=3 e^{5 x}+5 e^{3 x} \)
\( \therefore \frac{d y}{d x}=3 e^{5 x} \times 5+5 e^{3 x} \times 3=15 e^{5 x}+15 e^{3 x} \)
\( \frac{d^{2} y}{d x^{2}} =15 e^{5 x} \times 5+15 e^{3 x} \times 3 \)
\( =75 e^{5 x}+45 e^{3 x} \)
\( \therefore \frac{d^{2} y}{d x^{2}}-8 \frac{d y}{d x} =75 e^{5 x}+45 e^{3 x}-8\left(15 e^{5 x}+15 e^{3 x}\right) \)
\( =75 e^{5 x}+45 e^{3 x}-120 e^{5 x}-120 e^{3 x} \)
\( =-45 e^{5 x}-75 e^{3 x} \)
\( =-15\left(3 e^{5 x}+5 e^{3 x}\right) \)
\( =-15 y\)
\( \therefore \frac{d y}{d x}=3 e^{5 x} \times 5+5 e^{3 x} \times 3=15 e^{5 x}+15 e^{3 x} \)
\( \frac{d^{2} y}{d x^{2}} =15 e^{5 x} \times 5+15 e^{3 x} \times 3 \)
\( =75 e^{5 x}+45 e^{3 x} \)
\( \therefore \frac{d^{2} y}{d x^{2}}-8 \frac{d y}{d x} =75 e^{5 x}+45 e^{3 x}-8\left(15 e^{5 x}+15 e^{3 x}\right) \)
\( =75 e^{5 x}+45 e^{3 x}-120 e^{5 x}-120 e^{3 x} \)
\( =-45 e^{5 x}-75 e^{3 x} \)
\( =-15\left(3 e^{5 x}+5 e^{3 x}\right) \)
\( =-15 y\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(\mathrm{I}=\int \frac{\mathrm{e}^x}{\mathrm{e}^{4 x}+\mathrm{e}^{2 x}+1} \mathrm{~d} x\) and \(\mathrm{J}=\int \frac{\mathrm{e}^{-x}}{\mathrm{e}^{-4 x}+\mathrm{e}^{-2 x}+1} \mathrm{~d} x\) then for any arbitrary constant c, the value of \(\mathrm{J}-\mathrm{I}\) equalsMHT CET 2023 Medium
- The angle between the lines \(x=\mathrm{y}, \mathrm{z}=0\) and \(\mathrm{y}=0, \mathrm{z}=0\) isMHT CET 2025 Easy
- The ratio in which the plane \(\bar{r}\). \((\hat{i}-2 \hat{j}+3 \hat{k})=17\) divides the line joining the points \(-2 \hat{i}+4 \hat{j}+7 \hat{k}\) and \(3 \hat{i}-5 \hat{j}+8 \hat{k}\)MHT CET 2022 Medium
- Let \(I=\int \tan ^{-1}\left(\frac{2 x}{1-x^2}\right) d x\), then \(I-2 x \tan ^{-1} x=\)MHT CET 2025 Medium
- The area of the parallelogram whose diagonals are represented by the vectors \(\bar{a}=3 \hat{i}-\hat{j}-2 \hat{k}\) and \(\bar{b}=-\hat{i}+3 \hat{j}-3 \hat{k}\) isMHT CET 2021 Medium
- If \(f: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}\) are two functions defined by \(f(x)=2 x-3, \mathrm{~g}(x)=x^{3}+5\)
then \((\operatorname{fog})^{-1}(x)=\)MHT CET 2020 Hard
More PYQs from MHT CET
- If the lines \(\frac{2 x-4}{\lambda}=\frac{y-1}{2}=\frac{z-3}{1}\) and \(\frac{x-1}{1}=\frac{3 y-1}{\lambda}=\frac{z-2}{1}\) are perpendicular to each other, then \(\lambda=\)MHT CET 2021 Easy
- The following cross, YyRr x yyrr in garden pea plant, would result in the formation ofMHT CET 2022 Medium
- Which one of the following four graphs showing lines \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) and \(\mathrm{S}\) between maximum kinetic energy (E) and intensity of incident light (I) is correct?
MHT CET 2020 Easy - In Mirabilis jalapa, when red and white varieties are crossed, the hybrid obtained will be _________ pink.MHT CET 2018 Easy
- A bullet is fired on a target with velocity V. Its velocity decreases from \(v\) to \(v / 2\). When it penetrates 30 cm in a target. Through what thickness it will penetrate further in the target before coming to rest?MHT CET 2024 Medium
- Which among the following functional groups is reduced by diborane?MHT CET 2025 Medium