ENSH 101Bachelor in Industrial Engineering · Semester 12 Papers Available

Engineering Mathematics I

Past examination question papers and complete curriculum syllabus for Engineering Mathematics I (ENSH 101), Bachelor in Industrial Engineering Semester 1 under Institute of Engineering (IOE), Tribhuvan University.

Sem 1 Subjects (5)View Full Semester Overview →

ENSH 101Engineering Mathematics I ENSH 102Engineering Physics ENCT 101Computer Programming ENME 101Engineering Drawing ENEE 101Basic Electrical Engineering

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1st-sem_Engineering Mathematics I.pdf

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Most Frequently Asked Questions

Top recurring IOE board exam questions for Engineering Mathematics I with verified mark schemes, formula notation, and recurrence frequency.

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5 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 I. Click any session paper to expand.

Showing 471 questions across 27 exam session papers

Institute of Engineering • 2083 Baishakh Board Examination

Q1[8 Marks]Derivatives and its Applications

State Leibnitz's theorem. If

y = (x^{2} - 1)^{n}

, then use the Leibnitz's theorem to prove

(x^{2} - 1) y_{n + 2} + 2 x y_{n + 1} - n (n + 1) y_{n} = 0

. State Rolle's theorem and verify the theorem for the function

f (x) = e^{- x} sin x, x \in [0, π]

Q2[5 Marks]Derivatives and its Applications

State L'Hospital's Rule and evaluate

lim_{x \to 0} (\frac{s i n x}{x})^{1/ x^{2}}

Q3[5 Marks]Derivatives and its Applications

Define asymptotes and explain their types. Find the asymptotes of the curve

x^{2} y^{2} - x^{2} y - x y^{2} + x + y + 1 = 0

Q4[5 Marks]Derivatives and its Applications

Find the pedal equation of the curve

y^{2} = 4 a (x + a)

Q5[5 Marks]Antiderivatives and its Applications

Show that

\int_{0}^{π} \frac{x s i n x}{1 + c o s ^{2} x} d x = \frac{π ^{2}}{4}

Q6[8 Marks]Antiderivatives and its Applications

Apply differentiation under the integral sign to evaluate

\int_{0}^{\infty} e^{- a x} \frac{s i n x}{x} d x

and then show that

\int_{0}^{\infty} \frac{s i n x}{x} d x = \frac{π}{2}

Q7[8 Marks]Antiderivatives and its Applications

Define Gamma function and use it to show that

\int_{0}^{π /2} cos^{4} θ sin^{2} θ d θ = \frac{π}{32}

Q8[5 Marks]Antiderivatives and its Applications

Find the surface area of solid generated by revolving the cycloid

x = a (θ + sin θ), y = a (1 + cos θ)

about its base. **[OR]** Find the area bounded by the cardioid

r = a (1 - cos θ)

Q9[5 Marks]Ordinary Differential Equations and its Applications

Solve the differential equation:

\frac{d y}{d x} + x sin 2 y = x^{3} cos^{2} y

Q10[5 Marks]Ordinary Differential Equations and its Applications

Find the general solution of

y = 2 p x + y^{2} p^{3}

, where

p = \frac{d y}{d x}

Q11[5 Marks]Ordinary Differential Equations and its Applications

Solve the differential equation:

(D^{2} + 2 D + 5) y = e^{- x} sin x

Q12[5 Marks]Ordinary Differential Equations and its Applications

Solve the differential equation:

x^{2} \frac{d ^{2} y}{d x ^{2}} + x \frac{d y}{d x} + y = lo g x

Q13[5 Marks]Plane Analytic Geometry

Find the angle through which the axes must be turned so that the equation

x + 2 y + 5 = 0

may be reduced to the form

x = c

and determine the value of

c

Q14[5 Marks]Plane Analytic Geometry

Find the condition that the line

x cos α + y sin α = p

touches the hyperbola

\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1

, and also find the point of contact.

Q15[5 Marks]Plane Analytic Geometry

Identify the conic

x^{2} + 6 x y + y^{2} - 4 x + 4 y - 12 = 0

, and hence find its center and the length of axes. **[OR]** Describe and sketch the graph of the polar equation:

r = \frac{1}{1 + 3 c o s θ}

Curriculum Syllabus & Course Topics

Sourced from TU curriculum portal

Full Syllabus (PDF)

Chapter-wise Units & Micro-Syllabus Topics (5 Units)

1. Derivatives and its Applications
- 1.1Review of derivative and differentiability, mean value theorems with interpretations
- 1.2Indeterminate forms, types and their real life examples, L-Hospital's rule
- 1.3Power series of single valued functions
- 1.3.1Taylor's series
- 1.3.2Maclaurin's series
- 1.4Asymptotes to C artesian and Polar curves
- 1.5Pedal equation to Cartesian and Polar curves
- 1.6Curvature and radius of curvature for Cartesian curves
2. Antiderivatives and its Applications
- 2.1Review of definite and indefinite integrals
- 2.2Differentiation under integral sign
- 2.3Improper integrals
- 2.4Application of Beta and Gamma functions
- 2.5Area, arc length, volume and surface of revolution in plane for Cartesian curves
- 2.6Centroid and moment of inertia under area of curve
3. Ordinary Differential Equations and its Applications
- 3.1Review of order, degree, solution of first order first degree differential equations by variable separation method and solution of homogeneous equations
- 3.2Linear differential equation and equations reducible to linear differential equation of first order Bernoulli’s equation, modeling electric circuit
- 3.3First order and higher degree differential equations; Clairaut’s form
- 3.4Linear second order differential equations with constant coefficient and variable coefficients reducible to constant coefficients, Cauchy’s equations and modeling mass spring system
- 3.5Application in physical sciences and engineering
4. Plane Analytic Geometry
- 4.1Transformation of coordinates: Translation and rotation
- 4.2Equation of conic in Cartesian and Polar form, identification of conics
5. Three dimensional geometry
- 5.1The straight line: Symmetrical and general form
- 5.2Coplanar lines
- 5.3Shortest distance
- 5.4Sphere: General equation, plane section by planes, tangent planes
- 5.5Introduction to right circular cone and right circular cylinder

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.

Authentic IOE Past Papers

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Curriculum Syllabus & Marking Scheme

Past Question Papers (PDF)

1st-sem_Engineering Mathematics I.pdf

Most Frequently Asked Questions

Derivatives and its Applications

Antiderivatives and its Applications

Ordinary Differential Equations and its Applications

Plane Analytic Geometry

Three dimensional geometry

Year-wise & Session-wise Exam Papers

Curriculum Syllabus & Course Topics

1. Derivatives and its Applications

2. Antiderivatives and its Applications

3. Ordinary Differential Equations and its Applications

4. Plane Analytic Geometry

5. Three dimensional geometry

Examination Scheme & Marks Distribution

Evaluation Structure

Exam Preparation Guidelines