ENSH 151Bachelor in Chemical Engineering · Semester 21 Paper Available

Engineering Mathematics II

Past examination question papers and complete curriculum syllabus for Engineering Mathematics II (ENSH 151), Bachelor in Chemical Engineering Semester 2 under Institute of Engineering (IOE), Tribhuvan University.

Sem 2 Subjects (5)View Full Semester Overview →

ENSH 151Engineering Mathematics II ENCE 101Engineering Mechanics ENEE 101Basic Electrical EngineeringENCH 151Organic ChemistryENCH 152Physical Chemistry

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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.

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4 Chapters · Click any chapter to expand

Year-wise & Session-wise Exam Papers

Complete questions from each exam session (Regular & Back exams from 2082 to 2069) for Engineering Mathematics II. Click any session paper to expand.

Showing 411 questions across 22 exam session papers

Institute of Engineering • 2083 Baishakh Board Examination

Q1[8 Marks]Calculus of Two and More Variables

(a) If

u = sin^{- 1} (\frac{x ^{2} + y ^{2}}{x + y}), t h e n

show that

x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = tan u

. (b) If

u = x y z, v = x y + y z + z x, w = x + y + z, f in d

jacobian

\frac{\partial ( u , v , w )}{\partial ( x , y , z )}

Q2[8 Marks]Multiple Integrals

Evaluate

0 \int a y \int a \frac{x d x d y}{x ^{2} + y ^{2}}

by changing order of integration.

Q3[8 Marks]Vector Calculus

Find the directional derivative of

ϕ (x, y, z) = x y^{2} + y z^{3}

at the point (2, -1, 1) in the direction of vector

i + 2 j + 2 k

OR If

a

is a constant vector and

r

be the position vector then prove that

(a \times \nabla) \times r = - 2 a

Q4[8 Marks]Vector Calculus

State Green's theorem in plane. Apply it to find the area of the curve

x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}

Q5[8 Marks]Vector Calculus

Apply Gauss divergence theorem, evaluate

S \iint F \cdot \overset{n}{^} d s, w h er e

F = (2 x y + z) i + y^{2} j - (x + 3 y) k

and

S

is region bounded by the surface of the planes:

2 x + 2 y + z = 6, x = 0, y = 0, z = 0

. \n**[OR]**\n Apply Stoke's theorem to evaluate the line integral

C \oint (x y d x + x y^{2} d y)

where

C

is the boundary of the square in xy-plane with vertices (-1,0), (1,0), (0,1) and (0,-1).

Q6[8 Marks]Laplace Transform

Solve by power series method the differential equation

y^{''} - 4 x y^{'} + (4 x^{2} - 2) y = 0

Q7[8 Marks]Matrices

(a) Find the rank of the matrix

6410123369

(b) Find the eigen values of the matrix

A = 310130002

Q8[8 Marks]Matrices

Test the consistency of the system of linear equations and find their solution if found consistent.

2 x - y + 3 z = 8

- x + 2 y + z = 4

3 x + y - 4 z = 0

Q9[8 Marks]Matrices

Reduce the quadratic form

Q (X) = 5 x_{1}^{2} + 2 x_{2}^{2} + 2 x_{3}^{2} + 4 x_{1} x_{3} + 2 x_{2} x_{3} + 4 x_{1} x_{3}

into canonical form.

Q10[8 Marks]Solution of Differential Equation in Series and Special Functions

Express the function

f (x) = x^{3} - 5 x^{2} + 6 x + 1

in terms of Legendre's polynomials.

Q11[8 Marks]Multiple Integrals

(a) Evaluate

1 \int 2 0 \int x \frac{1}{x ^{2} + y ^{2}} d x d y

(b) Find the mass of solid bounded by

z = 1 - x^{2}

and planes

z = 0, y = 1, y = - 1

with

ρ (x, y, z) = z (y + 2)

Q12[8 Marks]Laplace Transform

(a) State existence criteria of the Laplace transform. Evaluate the Laplace transform of

\frac{1 - e ^{t}}{t}

(b) Find the inverse Laplace transform of

\frac{1}{s ^{2} - 3 s + 2}

Q13[8 Marks]Calculus of Two and More Variables

Find the extreme value of the function

u (x, y) = x^{2} + x y + y^{2} + 3 z^{2}

under the constraint

x + 2 y + 4 z = 60

Q14[8 Marks]Vector Calculus

Prove that necessary and sufficient condition for the vector function of a scalar variable

t

have a constant direction is

a \times \frac{d a}{d t} = 0

Q15[8 Marks]Calculus of Two and More Variables

Show that

J_{\frac{5}{2}} (x) = \frac{2}{π x} {\frac{3 - x ^{2}}{x ^{2}} sin x - \frac{3}{x} cos x}

where the symbols have their usual meanings.

Curriculum Syllabus & Course Topics

Sourced from TU curriculum portal

Full Syllabus (PDF)

Chapter-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.

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Curriculum Syllabus & Marking Scheme

Past Question Papers (PDF)

2nd-sem_Engineering Mathematics II.pdf

Most Frequently Asked Questions

Calculus of Two and More Variables

Multiple Integrals

Vector Calculus

Solution of Differential Equation in Series and Special Functions

Year-wise & Session-wise Exam Papers

Curriculum Syllabus & Course Topics

1. Calculus of Two and More Variables

2. Multiple Integrals

3. Vector Calculus

4. Laplace Transform

5. Matrices

6. Solution of Differential Equation in Series and Special Functions

Examination Scheme & Marks Distribution

Evaluation Structure

Exam Preparation Guidelines