#1Repeated 2 Times[2 Marks]Calculus of Two and More Variables
If $u = \sin^{-1}\left(\frac{x^2 + y^2}{x + y}\right)$, show that $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = \tan u$.
Appeared in:2083 Baishakh2082 Bhadra
ENSH 151Bachelor in Electrical Engineering · Semester 21 Paper Available
Engineering Mathematics II
Past examination question papers and complete curriculum syllabus for Engineering Mathematics II (ENSH 151), Bachelor in Electrical Engineering Semester 2 under Institute of Engineering (IOE), Tribhuvan University.
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Most Frequently Asked Questions
Top recurring IOE board exam questions for Engineering Mathematics II with verified mark schemes, formula notation, and recurrence frequency.
Showing 30 of 30 top repeated questions
Calculus of Two and More Variables
8 Questions#2Repeated 2 Times[6 Marks]Calculus of Two and More Variables
Find the Fourier series representation of the function $f(x) = x^2$ in the interval $-\pi < x < \pi$. Hence deduce that $\frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \dots = \frac{\pi^2}{12}$.
Appeared in:2082 Bhadra2081 Chaitra
#3Repeated 2 Times[6 Marks]Calculus of Two and More Variables
Obtain the half-range Fourier sine series for $f(x) = x(\pi - x)$ in $0 < x < \pi$. Hence deduce that $\frac{1}{1^3} - \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + \dots = \frac{\pi^3}{32}$.
Appeared in:2081 Chaitra2078 Chaitra
#4Repeated 2 Times[6 Marks]Calculus of Two and More Variables
Solve Lagrange's linear partial differential equation: $x(y^2 - z^2)p + y(z^2 - x^2)q = z(x^2 - y^2)$, where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$.
Appeared in:2082 Bhadra2080 Chaitra
#5Repeated 2 Times[8 Marks]Calculus of Two and More Variables
Solve the one-dimensional wave equation $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ for a vibrating string of length $L$ fixed at both ends with boundary conditions $u(0, t) = 0, u(L, t) = 0$ and initial velocity zero using the method of separation of variables.
Appeared in:2081 Chaitra2079 Chaitra
#6Repeated 1 Times[2 Marks]Calculus of Two and More Variables
If $u = x+y+z, v = xy+yz+zx, w = x^2+y^2+z^2$, find the Jacobian $\frac{\partial(u,v,w)}{\partial(x,y,z)}$.
Appeared in:2083 Baishakh
#7Repeated 1 Times[4 Marks]Calculus of Two and More Variables
State existence criteria of the Laplace transform. Solve the differential equation $y"(t) + 4y'(t) + 4y(t) = e^{-t}; y(0) = 0, y'(0) = 0$, using Laplace transform method.
Appeared in:2083 Baishakh
#8Repeated 1 Times[4 Marks]Calculus of Two and More Variables
Solve $y" - 4xy' + (4x^2 - 2)y = 0$ by power series method.
Appeared in:2083 Baishakh
Multiple Integrals
7 Questions#1Repeated 2 Times[4 Marks]Multiple Integrals
Change the order of integration and evaluate $\int_0^1 \int_y^1 e^{x^2}\,dx\,dy$.
Appeared in:2083 Baishakh2082 Bhadra
#2Repeated 2 Times[5 Marks]Multiple Integrals
State Euler's theorem on homogeneous functions. If $u = \tan^{-1}\left(\frac{x^3 + 2y^3}{x + y}\right)$, show that $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = \sin 2u$.
Appeared in:2082 Bhadra2080 Chaitra
#3Repeated 2 Times[6 Marks]Multiple Integrals
Find the maximum and minimum values of $f(x, y, z) = x^2 + y^2 + z^2$ subject to the constraint $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ using Lagrange's method of undetermined multipliers.
Appeared in:2082 Bhadra2081 Chaitra
#4Repeated 2 Times[5 Marks]Multiple Integrals
Evaluate the integral by changing the order of integration: $\int_0^\infty \int_0^x x e^{-x^2/y}\,dy\,dx$.
Appeared in:2082 Bhadra2079 Chaitra
#5Repeated 2 Times[6 Marks]Multiple Integrals
Find the volume of the region bounded by the paraboloid $z = x^2 + y^2$ and the plane $z = 4$ using triple integration in cylindrical coordinates.
Appeared in:2081 Chaitra2078 Chaitra
#6Repeated 1 Times[4 Marks]Multiple Integrals
Find the extreme value of the function $u(x,y,z) = x^2 + xy + y^2 + 3z^2$ under the constraint $x + 2y + 4z = 60$ using Lagrange's multiplier method.
Appeared in:2083 Baishakh
#7Repeated 1 Times[2 Marks]Multiple Integrals
Find the mass of solid bounded by $z = 1 - x^2$ and planes $z = 0, y = 1, y = -1$ with density $\rho(x,y,z) = z(y + 2)$.
Appeared in:2083 Baishakh
Vector Calculus
7 Questions#1Repeated 2 Times[4 Marks]Vector Calculus
State Green's theorem in a plane. Apply it to find the area of the astroid (hypocycloid) $x^{2/3} + y^{2/3} = a^{2/3}$.
Appeared in:2083 Baishakh2082 Bhadra
#2Repeated 2 Times[4 Marks]Vector Calculus
Prove that the necessary and sufficient condition for the vector function $\vec{a}$ of scalar variable $t$ to have constant magnitude is $\vec{a} \cdot \frac{d\vec{a}}{dt} = 0$, and to have constant direction is $\vec{a} \times \frac{d\vec{a}}{dt} = 0$.
Appeared in:2083 Baishakh2082 Bhadra
#3Repeated 2 Times[5 Marks]Vector Calculus
Find the directional derivative of $\phi(x, y, z) = x^2 y z + 4x z^2$ at the point $(1, -2, -1)$ in the direction of the vector $2\hat{i} - \hat{j} - 2\hat{k}$. Find the direction in which it is maximum.
Appeared in:2082 Bhadra2081 Chaitra
#4Repeated 2 Times[6 Marks]Vector Calculus
Show that the vector field $\vec{F} = (y^2 \cos x + z^3)\hat{i} + (2y \sin x - 4)\hat{j} + (3x z^2 + 2)\hat{k}$ is conservative. Find its scalar potential function $\phi$ and evaluate $\int_C \vec{F} \cdot d\vec{r}$ from $(0, 1, -1)$ to $\left(\frac{\pi}{2}, -1, 2\right)$.
Appeared in:2081 Chaitra2079 Chaitra
#5Repeated 1 Times[4 Marks]Vector Calculus
Apply Gauss divergence theorem, evaluate $\iint_S \vec{F} \cdot \hat{n}\,dS$ where $\vec{F} = (2xy + z)\hat{i} + y^2\hat{j} - (x + 3y)\hat{k}$ and $S$ is the region bounded by the surface of the planes: $2x + 2y + z = 6, x = 0, y = 0, z = 0$.
Appeared in:2083 Baishakh
#6Repeated 1 Times[4 Marks]Vector Calculus
Apply Stoke's theorem to evaluate the line integral $\oint_C (xy\,dx + xy^2\,dy)$ where $C$ is the boundary of the square in $xy$-plane with vertices $(-1,0), (1,0), (0,1)$ and $(0,-1)$.
Appeared in:2083 Baishakh
#7Repeated 1 Times[2 Marks]Vector Calculus
Find the directional derivative of $\phi = 4x^2 + 3y + 4z$ at $(1, 2, 1)$ in the direction $2\hat{i} + 2\hat{j} + \hat{k}$.
Appeared in:2083 Baishakh
Matrices
7 Questions#1Repeated 2 Times[2 Marks]Matrices
Express the function $f(x) = 1 + x - x^2$ in terms of Legendre's polynomials.
Appeared in:2083 Baishakh2082 Bhadra
#2Repeated 2 Times[6 Marks]Matrices
Find the eigenvalues and corresponding eigenvectors of the matrix $A = \begin{pmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{pmatrix}$. Verify the Cayley-Hamilton theorem for $A$.
Appeared in:2082 Bhadra2081 Chaitra
#3Repeated 2 Times[6 Marks]Matrices
Test the consistency of the following system of linear equations and solve if consistent using Gaussian elimination: $x + 2y - z = 3$, $3x - y + 2z = 1$, $2x - 2y + 3z = 2$, $x - y + z = -1$.
Appeared in:2082 Bhadra2080 Chaitra
#4Repeated 2 Times[5 Marks]Matrices
Define rank of a matrix. Reduce the matrix $A = \begin{pmatrix} 1 & 2 & -1 & 4 \\ 2 & 4 & 3 & 5 \\ -1 & -2 & 6 & -7 \end{pmatrix}$ to its normal form and find its rank.
Appeared in:2081 Chaitra2079 Chaitra
#5Repeated 1 Times[4 Marks]Matrices
Reduce the quadratic form $Q(x) = 6x_1^2 + 3x_2^2 + 3x_3^2 - 4x_1 x_2 + 2x_2 x_3 - 4x_1 x_3$ into canonical form.
Appeared in:2083 Baishakh
#6Repeated 1 Times[2 Marks]Matrices
Find the rank of the matrix $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{pmatrix}$.
Appeared in:2083 Baishakh
#7Repeated 1 Times[4 Marks]Matrices
Apply Cayley-Hamilton theorem to find the inverse of matrix $\begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & -1 \\ 3 & -1 & 1 \end{pmatrix}$.
Appeared in:2082 Bhadra
Solution of Differential Equation in Series and Special Functions
1 Question#1Repeated 2 Times[5 Marks]Solution of Differential Equation in Series and Special Functions
State and prove the duplication formula for Gamma function: $\Gamma(m)\Gamma\left(m + \frac{1}{2}\right) = \frac{\sqrt{\pi}}{2^{2m-1}}\Gamma(2m)$.
Appeared in:2082 Bhadra2080 Chaitra
Curriculum Syllabus & Course Topics
Sourced from TU curriculum portalChapter-wise Units & Micro-Syllabus Topics (6 Units)
1. Calculus of Two and More Variables
- 1.1Partial differentiation
- 1.1.1Partial derivatives of first and higher order
- 1.1.2Homogeneous function: Euler’s theorem for two and three variables
- 1.1.3Total derivatives and differentials, differentiation of composite and implicit functions
- 1.1.4Jacobians and their properties
- 1.2Extreme values of two and three variables. Lagrange’s multiplier
- 1.3Application in optimization of function of two variables in one constraint
2. Multiple Integrals
- 2.1Double integrals in Cartesian and Polar form, change of order of integration
- 2.2Triple integrals in Cartesian, cylindrical and spherical coordinates
- 2.3Area, volume, moment of inertia, mass and centroid by double and triple integrals
3. Vector Calculus
- 3.1Review of scalar and vector products, scalar and vector triple product, scalar and vector product of four vectors
- 3.2Vector differentiation and integration, their geometrical meaning, velocity and acceleration
- 3.3Vector differential operators: Gradient, directional derivatives, divergence and curl
- 3.4Line integrals, independent of path, conservative and irrotational vector fields, scalar potential
- 3.5Introduction to Green’s theorem and its application
- 3.6Surface integrals, calculation of flux
- 3.7Volume integrals, Gauss divergence theorem (Without proof) and its application in evaluation of surface integrals
- 3.8Introduction to Stoke’s theorem and its application
4. Laplace Transform
- 4.1Definition of Laplace transform, condition for existence, Laplace transforms of some elementary functions, properties of Laplace transform, shifting and change of scale properties
- 4.2Inverse Laplace transform, uniqueness of inverse Laplace transform, properties of inverse Laplace transform
- 4.3Laplace transform of derivatives and integral, multiplication and division by tn the convolution theorem
- 4.4Laplace transform of Heaviside’s unit function, Dirac-delta function and periodic functions
- 4.5Application of Laplace transform to ordinary differential equations
5. Matrices
- 5.1Review of algebra of real and complex matrices
- 5.2Rank of matrices and its application in system of linear equations
- 5.3Vector space, linear dependence and independence
- 5.4Eigen values: Cayley Hamilton theorem and its applications
- 5.5Eigen vectors, diagonalization of matrices
- 5.6Reduction of quadratic forms into canonical forms (Three variables only)
6. Solution of Differential Equation in Series and Special Functions
- 6.1Power series method
- 6.2Bessel’s functions: Introduction, properties and application
- 6.3Legendre’s function: Introduction, properties and application
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 available past examination paper to understand question styling, typical derivation topics, and marks allocation.
- 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 II)
Q: How can I download Engineering Mathematics II past question papers?
You can preview or download the Engineering Mathematics II 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 II?
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