HPAS Maths Optional PYQs: Laplace Transforms

On that page, you will find year-wise and question-wise solutions.

Based on the sources provided (HPAS Maths Optional Question Papers 2014–2024), here is a compilation of all questions related to the Laplace Transform topic, including direct transformation, inverse transformation, solving differential equations, and application theorems.

HPAS 2024 Laplace Transform Questions

Question 1(c): Find the Laplace transform of: $\mathbf{\frac{1}{t} \int_{0}^{t} e^{u} \sin u \,du}$.

Question 7(a): If $ C(t) = \int_{t}^{\infty} \frac{\cos x}{x} dx $, show that the Laplace transform of $C(t)$ is $\mathbf{ \log\frac{\sqrt{s^2+1}}{s} }$.

HPAS 2023 Laplace Transform Questions

Question 1(d): Determine the inverse Laplace Transform of the function: $\mathbf{ \tan^{-1}\left(\frac{2}{s^2}\right) }$.

HPAS 2021 Laplace Transform Questions

Question 1(e): Determine the inverse Laplace transform of: $\mathbf{ \log\frac{s+c}{s+d} }$, where $c$ and $d$ are constants.

Question 7(a): Determine the inverse Laplace transform of: $\mathbf{ \frac{1}{s^2 – e^{-as}} }$.

Question 7(b): Using the concept of Laplace Transform, find the solution of the initial value problem: $\mathbf{ t\frac{d^2y}{dt^2} + 2t\frac{dy}{dt} + 2y = 2 }$ with $y(0) = 1$, and $y'(0)$ is arbitrary.

HPAS 2020 Laplace Transform Questions

Question 6(b): Solve the following initial value problem using the Laplace transform: $\mathbf{ \frac{d^2y}{dx^2} – 3\frac{dy}{dx} – 4y = x^2 }$ with $y(0)=2$ and $y'(0)=1$.

Question 6(c): Find the Laplace transform of the following periodic function $f(x)$ with period $2\pi$: $$ \mathbf{ f(x) = \begin{cases} x & , \ 0 \le x \le \pi \\ 2\pi – x & , \ \pi \le x \le 2\pi \end{cases} } $$

HPAS 2019 Laplace Transform Questions

Question 1(c): Obtain the Laplace transform of the function $\mathbf{ f(t) = e^{-3t}u(t-2) }$.

Question 6(a): Obtain the inverse Laplace Transform of $\mathbf{ f(s) = \log\frac{s+1}{s-1} }$.

Question 6(b): Find the Laplace transform of the following periodic function $f(t)$ with period $2c$: $$ \mathbf{ f(t) = \begin{cases} t & , \ 0 < t < c \\ 2c - t & , \ c < t < 2c \end{cases} } $$

Question 6(c): Solve the following Initial Value Problem using the Laplace transform: $\mathbf{ x” + 2x’ + 5x = e^{-t}\sin t }$ with initial conditions $x(0)=0$ and $x'(0)=1$.

HPAS 2018 Laplace Transform Questions

Question 3(b): Find the value of $\mathbf{L\{x^{7/2}\}}$.

HPAS 2017 Laplace Transform Questions

Question 1(c): Evaluate the Laplace transform of the function $\mathbf{f(x) = (x+2)^2 e^x}$.

Question 7(a): Show that: $\mathbf{ L\left\{ \int_{0}^{x} \frac{1-e^{-u}}{u} du \right\}(p) = \frac{1}{p}\log\left(1+\frac{1}{p}\right) }$.

Question 7(b): Apply the convolution theorem to find: $\mathbf{ L^{-1}\left\{ \frac{p^2}{(p^2+a^2)^2} \right\}(x) }$.

HPAS 2016 Laplace Transform Questions

Question 1(c): Determine the inverse Laplace transform of $\mathbf{\frac{e^{-1/s}}{s}}$.

HPAS 2015 Laplace Transform Questions

Question 7(a): Find the Laplace transform of $\mathbf{\sin\sqrt{t}}$. Also show that: $\mathbf{ L\left\{\frac{\cos\sqrt{t}}{\sqrt{t}}\right\}=\sqrt{\frac{\pi}{p}}\cdot e^{-\frac{1}{4p}} }$.

HPAS 2014 Laplace Transform Questions

Question 7(a): If $ L_{n}(x)=\frac{e^{x}}{n!}\cdot\frac{d^{n}}{dx^{n}}(e^{-x}\cdot x^{n}) $, then find Laplace transform $\mathbf{L\{L_{n}(x); p\}}$, with $p>1$.