#1Repeated 2 Times[4 Marks]Fourier Series and Fourier Transform
Find Fourier cosine integral of $f(x) = e^{-kx}, k > 0, x > 0$ and use it to evaluate $\int_0^\infty \frac{\cos\omega x}{k^2 + \omega^2}\, d\omega$.
Appeared in:2082 Chaitra2081 Chaitra
ENSH 201Bachelor in Mechanical Engineering · Semester 32 Papers Available
Engineering Mathematics III
Past examination question papers and complete curriculum syllabus for Engineering Mathematics III (ENSH 201), Bachelor in Mechanical Engineering Semester 3 under Institute of Engineering (IOE), Tribhuvan University.
Past Question Papers (PDF)
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Most Frequently Asked Questions
Top recurring IOE board exam questions for Engineering Mathematics III with verified mark schemes, formula notation, and recurrence frequency.
Showing 30 of 30 top repeated questions
Fourier Series and Fourier Transform
7 Questions#2Repeated 2 Times[6 Marks]Fourier Series and Fourier Transform
Obtain the Fourier series representation for $f(x) = x + x^2$ in $-\pi < x < \pi$. Hence show that $\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots = \frac{\pi^2}{6}$.
Appeared in:2083 Baishakh2081 Bhadra
#3Repeated 2 Times[6 Marks]Fourier Series and Fourier Transform
Find the Fourier transform of the Gaussian pulse $f(t) = e^{-a t^2}$ ($a > 0$). Show that the Fourier transform of a Gaussian function is also Gaussian.
Appeared in:2082 Baishakh2080 Baishakh
#4Repeated 2 Times[6 Marks]Fourier Series and Fourier Transform
Find the Fourier sine and cosine transforms of $f(x) = e^{-a x}$ ($a > 0$). Hence evaluate $\int_0^\infty \frac{x \sin(m x)}{x^2 + a^2}\,dx$ using the Fourier inversion formula.
Appeared in:2082 Shrawan2079 Bhadra
#5Repeated 1 Times[4 Marks]Fourier Series and Fourier Transform
Find the Fourier series for $f(x) = x^2$ in the interval $-\pi \le x \le \pi$ and deduce the relation $\frac{\pi^2}{12} = \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \dots$ and $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$.
Appeared in:2081 Chaitra
#6Repeated 1 Times[4 Marks]Fourier Series and Fourier Transform
If $f(x) = x + x^2$ for $-\pi < x < \pi$, find the Fourier series of $f(x)$.
Appeared in:2082 Chaitra
#7Repeated 1 Times[4 Marks]Fourier Series and Fourier Transform
Find Fourier sine transform of $f(x) = e^{-x}$ and hence use the result in Parseval's identity to show $\int_0^\infty \frac{x^2}{(1+x^2)^2}\, dx = \frac{\pi}{4}$.
Appeared in:2082 Chaitra
Functions of Complex Variable
11 Questions#1Repeated 2 Times[4 Marks]Functions of Complex Variable
Evaluate $\int_0^{2\pi} \frac{d\theta}{5 + 4\sin\theta}$ using the method of contour integration in the complex plane.
Appeared in:2082 Chaitra2081 Chaitra
#2Repeated 2 Times[4 Marks]Functions of Complex Variable
Define harmonic function. Prove that $u = y^3 - 3x^2 y$ is a harmonic function. Find its harmonic conjugate and hence construct the corresponding analytic function $w = f(z)$.
Appeared in:2082 Chaitra2081 Chaitra
#3Repeated 2 Times[2 Marks]Functions of Complex Variable
What will be the image of circle $|z - i| = 1$ under the mapping $w = f(z) = \frac{1}{z}$ in the complex plane?
Appeared in:2082 Kartik2081 Chaitra
#4Repeated 2 Times[6 Marks]Functions of Complex Variable
State and prove the necessary and sufficient conditions for a complex function $f(z) = u(x, y) + i v(x, y)$ to be analytic (Cauchy-Riemann equations in Cartesian and polar forms).
Appeared in:2083 Baishakh2082 Shrawan
#5Repeated 2 Times[6 Marks]Functions of Complex Variable
Show that $u(x, y) = e^x (x \cos y - y \sin y)$ is harmonic. Find its harmonic conjugate $v(x, y)$ such that $f(z) = u + i v$ is analytic, and express $f(z)$ in terms of $z$.
Appeared in:2082 Baishakh2081 Bhadra
#6Repeated 2 Times[6 Marks]Functions of Complex Variable
State Cauchy's Integral Formula. Evaluate $\oint_C \frac{e^{2z}}{(z+1)^3 (z-2)}\,dz$, where $C$ is the circle $|z| = 3$ oriented counter-clockwise.
Appeared in:2082 Shrawan2080 Bhadra
#7Repeated 2 Times[6 Marks]Functions of Complex Variable
Find the Laurent series expansion of $f(z) = \frac{1}{(z-1)(z-2)}$ valid for: (i) $1 < |z| < 2$, (ii) $|z| > 2$, and (iii) $0 < |z-1| < 1$.
Appeared in:2081 Bhadra2079 Baishakh
#8Repeated 2 Times[8 Marks]Functions of Complex Variable
State Cauchy's Residue Theorem. Evaluate the real improper integral $\int_{-\infty}^\infty \frac{x^2}{(x^2 + 1)(x^2 + 4)}\,dx$ by contour integration in the upper half of the complex plane.
Appeared in:2082 Baishakh2081 Bhadra
#9Repeated 2 Times[5 Marks]Functions of Complex Variable
Find the bilinear (Möbius) transformation that maps the points $z_1 = -i, z_2 = 0, z_3 = i$ into $w_1 = -1, w_2 = i, w_3 = 1$ respectively.
Appeared in:2082 Shrawan2078 Bhadra
#10Repeated 1 Times[2 Marks]Functions of Complex Variable
Find the bilinear transformation which maps the points $z = 0, -1, 1$ into $w = i, 0, \infty$.
Appeared in:2082 Chaitra
#11Repeated 1 Times[4 Marks]Functions of Complex Variable
Express the function $f(z) = \frac{1}{(z-1)(z-2)}$ in Laurent series form in the region $1 < |z| < 2$.
Appeared in:2082 Chaitra
Partial Differential Equations
4 Questions#1Repeated 2 Times[4 Marks]Partial Differential Equations
Apply Charpit's method to solve the PDE: $2zx - px^2 - 2qxy + pq = 0$, where symbols have their usual meanings.
Appeared in:2082 Chaitra2082 Kartik
#2Repeated 2 Times[2 Marks]Partial Differential Equations
Write down the general second-order partial differential equation and rules for classification, and hence use it to classify the equation $x^2 u_{xx} + u_{yy} = 0$.
Appeared in:2082 Chaitra2082 Kartik
#3Repeated 2 Times[6 Marks]Partial Differential Equations
Solve the partial differential equation by separation of variables: $4 \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 3u$ given that $u(0, y) = 3 e^{-y} - e^{-5y}$.
Appeared in:2082 Baishakh2081 Bhadra
#4Repeated 1 Times[4 Marks]Partial Differential Equations
Solve the PDE: $z(p - q) = z^2 + (x + y)^2$, using Lagrange's method where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$.
Appeared in:2082 Chaitra
Modelling through Partial Differential Equation
5 Questions#1Repeated 3 Times[4 Marks]Modelling through Partial Differential Equation
Derive Navier-Stokes equation of motion for fluid flow.
Appeared in:2082 Chaitra2082 Kartik2081 Chaitra
#2Repeated 2 Times[4 Marks]Modelling through Partial Differential Equation
Solve one dimensional heat equation $\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}$ subject to boundary conditions $u(0, t) = u(L, t) = 0$ and initial condition $u(x, 0) = u_0 \sin\left(\frac{\pi x}{L}\right)$ using the method of separation of variables.
Appeared in:2082 Chaitra2081 Chaitra
#3Repeated 2 Times[4 Marks]Modelling through Partial Differential Equation
A tightly stretched string fixed at two ends $x = 0$ and $x = \pi$ is initially at rest in equilibrium position. If it is set in vibration by giving to each of its points a velocity $\left(\frac{\partial u}{\partial t}\right)_{t=0} = 0.03\sin x - 0.04\sin 3x$, find the deflection $u(x,t)$ at any point of the string at any time.
Appeared in:2082 Chaitra2082 Kartik
#4Repeated 2 Times[8 Marks]Modelling through Partial Differential Equation
Derive the two-dimensional Laplace equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ for steady-state heat conduction in a thin rectangular plate. Solve it for a plate of width $a$ and infinite height with boundary conditions $u(0, y) = 0, u(a, y) = 0, u(x, \infty) = 0, u(x, 0) = f(x)$.
Appeared in:2082 Baishakh2080 Bhadra
#5Repeated 2 Times[8 Marks]Modelling through Partial Differential Equation
Solve the one-dimensional wave equation $\frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2}$ for a string of length $L$ fixed at both ends with initial displacement $y(x, 0) = k(Lx - x^2)$ and zero initial velocity.
Appeared in:2081 Bhadra2078 Bhadra
Z- transform and its Applications
3 Questions#1Repeated 2 Times[4 Marks]Z- transform and its Applications
Using Z-transform technique, solve the difference equation $x(k+2) - 4x(k+1) + 4x(k) = 0$ under initial conditions $x(0) = 0, x(1) = 1$.
Appeared in:2082 Chaitra2081 Chaitra
#2Repeated 2 Times[6 Marks]Z- transform and its Applications
State and prove the Time Shifting (Translation) and Initial Value theorems of Z-transforms. Find the Z-transform of $x(n) = n^2 a^n u(n)$.
Appeared in:2082 Shrawan2081 Bhadra
#3Repeated 2 Times[6 Marks]Z- transform and its Applications
Find the inverse Z-transform of $X(z) = \frac{z(2z + 1)}{(z - 1)(z - 0.5)^2}$ for Region of Convergence (ROC) $|z| > 1$ using the partial fraction expansion method.
Appeared in:2082 Baishakh2079 Baishakh
Curriculum Syllabus & Course Topics
Sourced from TU curriculum portalChapter-wise Units & Micro-Syllabus Topics (5 Units)
1. Fourier Series and Fourier Transform
- 1.1Review of periodic, odd and even functions
- 1.2Fourier series of a function over an interval of length 2l and 2π; Euler’s formula, Dirichlet’s condition for uniform convergence of Fourier series, Fourier series of discontinuous functions
- 1.3Half range Fourier sine and cosine series
- 1.4Complex form of Fourier series; frequency and amplitude of a function
- 1.5Fourier integral theorem, Fourier sine and cosine integrals, complex form of Fourier integral
- 1.6Fourier transform, Fourier sine transform, Fourier cosine transform and their inversion formulas
- 1.7Fourier transform of the derivative of a function
- 1.8Relation between Fourier and Laplace transform
2. Functions of Complex Variable
- 2.1Intuitive idea of limit, continuity and differentiability of functions of complex variable
- 2.2Analytic functions, the Cauchy Reimann equations both in Cartesian and polar form, construction of analytic functions
- 2.3Harmonic functions, the orthogonal system
- 2.4Application of analytic functions in flow problems
- 2.5Transformation (Mapping), conformal mapping, translation, rotation and magnification; inversion, bilinear transformation
- 2.6Complex integration, simply and multiply connected regions, Cauchy’s integral theorem and formula
- 2.7Series of complex terms, power series, circle of convergence and radius of convergence, Taylor’s and Laurent’s series
- 2.8Zeros, singularities, poles; residue at poles, Cauchy’s residue theorem and evaluation real and improper integrals
3. Partial Differential Equations
- 3.1Definition and formation of partial differential equations
- 3.2Partial differential equations solvable by direct integration
- 3.3Linear partial differential equation of the first order, Lagrange’s linear equations and their solution
- 3.4Nonlinear partial differential equation of first order; equations of the form 𝑓(𝑝,𝑞)=0 ,𝑧=𝑝𝑥+𝑞𝑦+𝑓(𝑝,𝑞), 𝑓(𝑧,𝑝,𝑞)=0 ,𝑓(𝑥,𝑝)=𝑓 (𝑦,𝑞) (cid:2869) (cid:2870)
- 3.5Charpit’s method of solving nonlinear partial differential equations of first order
4. Modelling through Partial Differential Equation
- 4.1Second order partial differential equation and classification
- 4.2One-dimensional wave equation
- 4.3One-dimensional heat equation
- 4.4Two-dimensional heat equation, Laplace equation in Cartesian form
- 4.5Mass balance equation; equation of continuity in fluid dynamics, Navier- Stoke’s equation
- 4.6Momentum balance equation; Euler’s equation of motion for inviscid fluid flow
5. Z- transform and its Applications
- 5.1Representation of a sequence and basic operations
- 5.2Definition and existence of Z-transform, Z-transform of standard sequences
- 5.3Properties of Z-transform; linearity, change of scale, shifting properties, initial and final value theorems
- 5.4Differentiations of Z-transform
- 5.5Inverse Z-transform; partial fraction and residue methods
- 5.6Convolution of sequences, convolution of Z- transform
- 5.7Difference equations, application of Z-transform to solve difference equations and to find the sum of series
Examination Scheme & Marks Distribution
Evaluation Structure
- Final Board Theory Exam: 60 Marks (Pass mark: 24)
- Internal Assessment: 40 Marks (Pass mark: 16)
- Practical / Lab Exam: 25 or 50 Marks (Continuous lab evaluation + viva, where applicable)
* This is the general current IOE 60/40 scheme; verify course-specific details in the syllabus above.
Exam Preparation Guidelines
- Review the 2 available past examination papers to identify recurring patterns, core problem types, and chapter weightage.
- Practice numerical problems step-by-step with clean formula derivations, clear units, and standard assumptions.
- Structure answers with labeled diagrams, concise bullet points, and highlight final answers in numerical solutions.
Frequently Asked Questions (Engineering Mathematics III)
Q: How can I download Engineering Mathematics III past question papers?
You can preview or download the Engineering Mathematics III question papers (PDF) directly using the built-in viewer on this page with zero redirects or paywalls.
Q: What is the pass mark for Engineering Mathematics III?
The general current scheme is a 60-mark final theory exam and a 40-mark internal assessment, with pass marks of 24 and 16. Verify the course-specific syllabus above.
Q: Where can I find the complete syllabus for this subject?
The available chapter-wise syllabus and topic breakdown is indexed in the Syllabus section above, with links to the curriculum PDF source.
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Curriculum Syllabus & Marking Scheme