MHT CET · Maths · Differentiation
If \(y=\mathrm{A} \cos \mathrm{n} x+\mathrm{B} \sin \mathrm{n} x\), then \(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}=\)
- A \(-\mathrm{n}^2 y\)
- B \(\mathrm{n}^2 y\)
- C \(\mathrm{n}^2 x\)
- D \(\mathrm{n}^2 x^2\)
Answer & Solution
Correct Answer
(A) \(-\mathrm{n}^2 y\)
Step-by-step Solution
Detailed explanation
\(y=A \cos n x+B \sin n x ...(i)\)
Differentiating w.r.to \(x\), we get
\(\begin{aligned}
& \frac{d y}{d x}=-A n(\sin n x)+B n \cos n x \\
& \frac{d y}{d x}=-n A \sin n x+n \cdot B \cos n x
\end{aligned}\)
Again differentiating w.r.to \(x\), we get
\(\begin{aligned}
\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2} & =-\mathrm{n}^2 A \cos n x-n^2 B \sin n x \\
& =-n^2[A \cos n x+B \sin n x] \\
\frac{\mathrm{d}^2 y}{d x^2} & =-n^2 y
\end{aligned}\)
...[from (i)]
Differentiating w.r.to \(x\), we get
\(\begin{aligned}
& \frac{d y}{d x}=-A n(\sin n x)+B n \cos n x \\
& \frac{d y}{d x}=-n A \sin n x+n \cdot B \cos n x
\end{aligned}\)
Again differentiating w.r.to \(x\), we get
\(\begin{aligned}
\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2} & =-\mathrm{n}^2 A \cos n x-n^2 B \sin n x \\
& =-n^2[A \cos n x+B \sin n x] \\
\frac{\mathrm{d}^2 y}{d x^2} & =-n^2 y
\end{aligned}\)
...[from (i)]
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 \(\mathrm{f}(x)=(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\), then \(f^{\prime}(1)=\)MHT CET 2024 Medium
- \(\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2}\) is equal toMHT CET 2022 Easy
- The probability that a certain kind of component will survive a given test is \(\frac{2}{3}\). The probability that at most 2 components out of 4 tested, will survive isMHT CET 2025 Medium
- The domain of the function \(\mathrm{f}(x)={ }^{7-x} \mathrm{P}_{x-1}\)MHT CET 2025 Medium
- The volume of a parallelopiped whose coterminous edges are \(2 \overrightarrow{\mathbf{a}}, 2 \overrightarrow{\mathbf{b}}, 2 \overrightarrow{\mathbf{c}}\), isMHT CET 2010 Easy
- If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are non-negative distinct numbers and \(\mathrm{a} \hat{\imath}+\mathrm{a} \hat{\jmath}+\mathrm{c} \hat{k}, \hat{\imath}+\hat{k}\) and \(\mathrm{c} \hat{\imath}+\mathrm{c} \hat{\jmath}+\mathrm{b} \hat{k}\)
are coplanar vectors, thenMHT CET 2020 Easy
More PYQs from MHT CET
- A compound of Xe and F is found to have atomic ratio \(\mathrm{Xe}: \mathrm{F}\) as \(0 \cdot 4: 2 \cdot 4\), Find the oxidation number of Xe ?MHT CET 2025 Medium
- Calculate the temperature of 0.05 M sucrose solution in Kelvin if the osmotic pressure of the solution is 1.5 atm.
\(\left[\mathrm{R}=0.0821 \mathrm{dm}^3 \mathrm{~atm} \mathrm{~K} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right]\)MHT CET 2025 Easy - In Young's double slit experiment, the fringe width is \(2 \mathrm{~mm}\). The separation between the \(13^{\text {th }}\) bright fringe and the \(4^{\text {th }}\) dark fringe from the centre of the screen on same side will beMHT CET 2023 Hard
- Which from following properties is NOT exhibited by LDP?MHT CET 2023 Easy
- \(\tan \left(\cos ^{-1} \frac{1}{\sqrt{2}}+\tan ^{-1} \frac{1}{2}\right)=\)MHT CET 2024 Easy
- How many molecules of methyl iodide are required to obtain tetramethyl ammonium iodide from dimethyl amine?MHT CET 2021 Medium