Mathematics-III Exam Prep
Full explanation, worked examples & memory tricks in the app
Mathematics-III Exam Prep
English 21 leveled MCQs Free
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Sample questions with model answers
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10 marks Long answer
Find the Laplace transform of $f(t) = 3t^3 – 2e^{2t} + 4 \sin t$.
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$L\{f(t)\} = \frac{6}{s^4} - \frac{2}{s-2} + \frac{4}{s^2 + 1}$
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10 marks Long answer
Find the Laplace transform of $f''(t)$ if $L\{f(t)\} = F(s)$, $f(0) = 2$, $f'(0) = 3$.
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$s^2F(s) - 2s - 3$
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10 marks Long answer
Find $L \{\int_0^t (u^2 + 3u) \ du\}$.
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$\frac{2}{s^3} + \frac{3}{s^2}$
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10 marks Long answer
Find $L\{3\delta(t - 2) + 5\delta(t − 4)\}$.
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$3e^{-2s} + 5e^{-4s}$
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10 marks Long answer
Find the Laplace transform of the periodic function $f(t) = \begin{cases} 1, & 0 \le t < 1, \\ 0, & 1 \le t < 2, \end{cases}$ with period $T = 2$.
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$\frac{1}{s}(1 - e^{-2s})$
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10 marks Long answer
Find $L^{-1} (\frac{7s+11}{(s+1)(s+2)})$.
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$e^{-t} + 5e^{-2t}$
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10 marks Long answer
Solve $y'' + 4y = 0$ subject to $y(0) = 2$, $y'(0) = 0$.
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$y(t) = 2\cos(2t)$
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4 marks Medium answer
Find $L\{\sin 3 t\}$ and $L\{\cos 3 t\}$.
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$L\{\sin 3t\} = \frac{3}{s^2 + 9}, L\{\cos 3t\} = \frac{s}{s^2 + 9}$
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4 marks Medium answer
Find $L\{4 \cos 2 t - 3 \sin 5 t\}$.
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$\frac{4s}{s^2 + 4} - \frac{15}{s^2 + 25}$
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4 marks Medium answer
Find $L\{\int_0^t \sin u \ du\}$.
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$\frac{1 - e^{-st}}{s^2 + 1}$
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4 marks Medium answer
Find $L\{u(t - 2)(t − 2)\}$.
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$\frac{e^{-2s}}{s^2}$
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4 marks Medium answer
Find the Laplace transform of $f(t) = \begin{cases} 0, & 0 \le t < 2, \\ t - 2, & t \ge 2. \end{cases}$; using unit step function.
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$\frac{e^{-2s}}{s^2}$
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4 marks Medium answer
Find $L^{-1} (\frac{3s+5}{s^2+9})$.
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$e^{3t} \sin(3t) + \frac{5}{3} e^{3t} \cos(3t)$
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4 marks Medium answer
Using Laplace transform method, solve $\frac{dy}{dt} + 2y = 0$, $y(0) = 3$.
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$y(t) = 3e^{-2t}$
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2 marks Short answer
Find $L\{t^2\}$.
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$\frac{2}{s^3}$
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2 marks Short answer
Find $L\{u(t − 3)\}$, where $u(t – a)$ is the unit-step function.
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$\frac{e^{-3s}}{s}$
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2 marks Short answer
Find $L\{\delta(t − 2)\}$, where $\delta$ is the Dirac-delta function.
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$e^{-2s}$
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2 marks Short answer
Write first shifting property for Laplace transform.
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$L\{u(t-a)f(t-a)\} = e^{-as}F(s)$
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2 marks Short answer
Write second shifting property for Laplace transform.
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$L\{e^{at}f(t)\} = F(s-a)$
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2 marks Short answer
Write formula for $L\{f'(t)\}$.
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$L\{f'(t)\} = sF(s) - f(0)$
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2 marks Short answer
Find $L^{-1}\{\frac{5}{s^3}\}$.
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$\frac{5}{2}t^2$
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