BSc CSIT 1st Semester Mathematics (MTH117) Important Questions, Past Papers & Exam Preparation Guide (TU)
Introduction
Mathematics (MTH117) is one of the most important and most feared subjects in the BSc CSIT first semester under Tribhuvan University. It covers calculus, vectors, differential equations, and series — topics that form the analytical foundation for later subjects including algorithms, computer graphics, artificial intelligence, numerical computing, and machine learning. Students who treat this course seriously in the first semester consistently find these later subjects easier.
TU Mathematics papers follow a predictable structure. Numerical methods (Newton-Raphson, Simpson's Rule, Trapezoidal Rule), integration techniques, differential equations, and vector calculus appear repeatedly across different exam years. The exact numerical values change, but the underlying method tested is the same. A student who understands the process — not just the answer — can handle any variation.
This guide is not a set of complete notes. For unit-wise explanations, worked derivations, and the full formula quick-reference, see the Complete BSc CSIT Mathematics (MTH117) Guide. What this guide provides is the exam map: which topics appear almost every year, which formulas you must know cold, which question types lose students the most marks, and how to allocate your final week for maximum efficiency.
Official Course Information
| Particular | Details |
|---|---|
| Course Title | Mathematics I |
| Course Code | MTH117 |
| Semester | First Semester |
| Nature of Course | Theory |
| Full Marks | 80 + 20 |
| Pass Marks | 32 + 8 |
| Credit Hours | 3 |
How TU Sets the MTH117 Paper
Understanding the exam structure before studying changes how you allocate time.
Short questions (2 marks each): Definition recall, formula application, or one-step computation. Units with more conceptual vocabulary (limits, series convergence criteria) tend to generate these.
Medium questions (4–5 marks): Apply a technique to a problem — evaluate a limit, differentiate a function, find a derivative application. These test whether you can execute a method correctly.
Long questions (8–10 marks): Multi-step problems requiring setup, execution, and interpretation. Differential equations, numerical integration, vector applications, and optimization problems are the most common long-question formats.
Numerical/derivation questions: Show every step. TU markers award step marks — a correct setup with an arithmetic error still earns most of the marks. A correct final answer with no working shown earns almost nothing.
High-Priority Topics (Study These First)
These topics appear most consistently across TU MTH117 past papers. If preparation time is limited, depth on these outweighs breadth on lower-probability topics.
| Topic | Unit | Priority |
|---|---|---|
| Limits (including sin x / x form) | 2 | ★★★★★ |
| Derivatives and rules (chain, product, quotient) | 3 | ★★★★★ |
| L'Hôpital's Rule | 3 | ★★★★★ |
| Maxima and minima (optimization problems) | 4 | ★★★★★ |
| Definite and indefinite integrals | 5 | ★★★★★ |
| Fundamental Theorem of Calculus | 5 | ★★★★★ |
| Area between curves | 6 | ★★★★★ |
| Ordinary differential equations (separable, linear 1st order) | 7 | ★★★★★ |
| 2nd order homogeneous + non-homogeneous ODEs | 7 | ★★★★★ |
| Taylor and Maclaurin series | 8 | ★★★★☆ |
| Convergence tests (ratio, comparison) | 8 | ★★★★☆ |
| Dot product, cross product | 9 | ★★★★★ |
| Equations of lines and planes in 3D | 9 | ★★★★☆ |
| Partial derivatives | 10 | ★★★★☆ |
| Double integrals | 10 | ★★★★☆ |
| Newton's Method (Newton-Raphson) | 4 | ★★★★★ |
Unit-Wise Important Questions
Unit 1: Function of One Variable
Frequently Asked:
- Find the domain and range of given functions (polynomial, trigonometric, logarithmic, exponential).
- Sketch the graph of a given function and identify its key features.
- Determine whether a function is odd, even, or periodic.
- Differentiate between one-to-one and onto functions.
Short Questions:
- Define domain and range with examples.
- What is a composite function? Give an example.
- State the difference between an even and odd function.
Exam probability: ★★★☆☆ — Usually 1 short question; understanding this unit is prerequisite for everything that follows.
Unit 2: Limits and Continuity
Frequently Asked Long Questions:
- Evaluate limits using algebraic simplification, including indeterminate forms.
- Find horizontal, vertical, and slant asymptotes of rational functions.
- Determine continuity of a function at a point and on an interval.
- Prove that a function has a horizontal asymptote at y = L.
Frequently Asked Short Questions:
- State the precise definition of a limit.
- Evaluate: lim(x→0) [sin x / x]. (Appears almost every year — answer is 1)
- What is a vertical asymptote? How do you find it?
- Define continuity at a point.
Must-Know Limits:
- lim(x→0) [sin x / x] = 1
- lim(x→0) [(1 + x)^(1/x)] = e
- lim(x→∞) [(1 + 1/x)^x] = e
Exam probability: ★★★★☆ — Limit evaluation appears in almost every paper.
Unit 3: Derivatives
Frequently Asked Long Questions:
- Find dy/dx using the chain rule for composite functions.
- Apply the Mean Value Theorem — verify MVT for a given function on [a, b].
- Evaluate limits using L'Hôpital's Rule (0/0 or ∞/∞ forms).
- Find the equation of the tangent line to a curve at a given point.
Frequently Asked Short Questions:
- State the Mean Value Theorem.
- State L'Hôpital's Rule. When is it applicable?
- Find dy/dx if y = x³sin(x) (product rule).
- Differentiate y = ln(cos x).
Exam probability: ★★★★★ — Differentiation appears in every paper, multiple question types.
Unit 4: Applications of Derivatives
Frequently Asked Long Questions:
- Find the maximum and minimum values of a function on a closed interval.
- Solve an optimization problem (maximize area given perimeter constraint, minimize cost, etc.).
- Apply Newton's Method to approximate a root — show 3 iterations.
- Sketch a curve using derivatives (increasing/decreasing, concavity, inflection points).
Frequently Asked Short Questions:
- What is Newton's Method? Write the iteration formula: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ).
- State the First Derivative Test.
- Define inflection point.
Newton's Method Exam Tip: Always show: identify f(x), compute f'(x), write the iteration formula, then compute at least 2–3 iterations. Step marks are awarded throughout.
Exam probability: ★★★★★ — Optimization and Newton's Method are both high-frequency long questions.
Unit 5: Antiderivatives
Frequently Asked Long Questions:
- Evaluate definite integrals using the Fundamental Theorem of Calculus.
- Apply integration by substitution to evaluate indefinite integrals.
- Apply integration by parts: ∫u dv = uv − ∫v du.
- Find the position/velocity of an object given acceleration (rectilinear motion).
Frequently Asked Short Questions:
- State the Fundamental Theorem of Calculus (both parts).
- Evaluate: ∫₀¹ x² dx.
- What is an improper integral? When does it converge?
- Evaluate: ∫ sin x dx.
Exam probability: ★★★★★ — Integration is the most mark-dense section of the paper.
Unit 6: Applications of Antiderivatives
Frequently Asked Long Questions:
- Find the area between two curves over a given interval.
- Find the volume of a solid of revolution using the disk or shell method.
- Apply numerical integration: Trapezoidal Rule and Simpson's Rule with a given dataset.
- Find arc length of a curve on [a, b].
Simpson's Rule — Almost Guaranteed: Approximate ∫ₐᵇ f(x) dx using Simpson's Rule with n subintervals (n must be even): ∫ ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)], where h = (b-a)/n
Trapezoidal Rule: ∫ ≈ (h/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + f(xₙ)]
Exam probability: ★★★★★ — Simpson's/Trapezoidal Rule numerical is one of the most reliably repeated question types.
Unit 7: Ordinary Differential Equations
Frequently Asked Long Questions:
- Solve a first-order separable ODE by separating variables and integrating.
- Solve a first-order linear ODE using the integrating factor method.
- Solve a second-order homogeneous ODE with constant coefficients (characteristic equation method).
- Solve a second-order non-homogeneous ODE using undetermined coefficients.
Key Methods:
| ODE Type | Method | Key Step |
|---|---|---|
| Separable: dy/dx = f(x)g(y) | Separate variables, integrate both sides | Get all y terms left, all x terms right |
| Linear 1st order: dy/dx + Py = Q | Integrating factor μ = e^∫P dx | Multiply through by μ, integrate |
| 2nd order homogeneous: ay″+by′+cy=0 | Characteristic equation: ar²+br+c=0 | Roots give form of solution |
| Non-homogeneous: ay″+by′+cy=g(x) | Particular + homogeneous solution | Guess form of yₚ based on g(x) |
Exam probability: ★★★★★ — This unit carries 6 teaching hours and appears heavily. At least one ODE long question is almost certain.
Unit 8: Infinite Sequence and Series
Frequently Asked Long Questions:
- Determine whether a given series converges or diverges using the ratio test.
- Find the Taylor series expansion of f(x) = eˣ, sin x, or cos x around x = 0.
- Find the Maclaurin series of a given function.
- Apply the comparison test or integral test to a series.
Frequently Asked Short Questions:
- State the ratio test. What does L < 1 mean?
- Find the first four terms of the Taylor series for sin x.
- Define convergence of an infinite series.
Essential Taylor/Maclaurin Series:
- eˣ = 1 + x + x²/2! + x³/3! + ...
- sin x = x − x³/3! + x⁵/5! − ...
- cos x = 1 − x²/2! + x⁴/4! − ...
- ln(1+x) = x − x²/2 + x³/3 − ... (|x| ≤ 1)
Exam probability: ★★★★☆ — Series convergence and Taylor expansion appear regularly.
Unit 9: Plane and Space Vectors
Frequently Asked Long Questions:
- Find the dot product and angle between two vectors. Find projection of one vector onto another.
- Find the cross product of two vectors. Show it is perpendicular to both.
- Find the equation of a line or plane in 3D space.
- Find the arc length and curvature of a vector curve.
Frequently Asked Short Questions:
- Define dot product. State its formula: a·b = |a||b|cosθ.
- What does a·b = 0 mean geometrically?
- Find the cross product of [1,2,3] and [4,5,6].
- What is the unit normal vector?
Must-Know Formulas:
| Operation | Formula |
|---|---|
| Dot product | a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ |
| Cross product magnitude | |a×b| = |a||b|sinθ |
| Projection of a onto b | proj_b(a) = (a·b/|b|²)b |
| Line in 3D | r(t) = r₀ + tv |
| Plane equation | a(x−x₀) + b(y−y₀) + c(z−z₀) = 0 |
Exam probability: ★★★★★ — Vector problems appear in almost every MTH117 paper.
Unit 10: Partial Derivatives and Multiple Integration
Frequently Asked Long Questions:
- Find all first and second partial derivatives of a given function of two variables.
- Find local maxima and minima using the second derivative test (D = fₓₓfᵧᵧ − fₓᵧ²).
- Evaluate a given double integral over a rectangular region.
- Find the tangent plane to a surface at a given point.
Frequently Asked Short Questions:
- What is a partial derivative? How is ∂f/∂x computed?
- State the condition for a saddle point.
- Evaluate: ∫₀¹ ∫₀¹ xy dx dy.
Exam probability: ★★★★☆ — Partial derivatives and double integrals are consistent but sometimes appear as medium-length questions rather than full long questions.
Most Repeated TU Questions (Past Paper Analysis)
Appears Almost Every Year
- Evaluate limits including lim(x→0) sin(x)/x
- L'Hôpital's Rule — evaluate a given 0/0 or ∞/∞ form
- Newton's Method — 2–3 iteration steps for a given f(x)
- Simpson's Rule or Trapezoidal Rule numerical
- First-order ODE (separable or linear) — solve completely
- Second-order homogeneous ODE — characteristic equation method
- Vector dot and cross product problems
- Area between curves
- Fundamental Theorem of Calculus application
Appears Frequently (2–3 Times in 5 Years)
- Mean Value Theorem — verify for given function
- Integration by parts
- Taylor or Maclaurin series expansion
- Curve sketching with first and second derivatives
- Volume of revolution (disk method)
- Equations of lines and planes in 3D
- Partial derivatives and saddle point analysis
- Double integrals over rectangular region
- Series convergence (ratio test)
Occasionally Appears
- Arc length calculation
- Curvature and normal/binormal vectors
- Comparison test or integral test for series
- Projection of vector onto another
- Surface area of revolution
Formula Quick Revision Sheet
Print or memorize this before the exam.
Differentiation Rules
| Rule | Formula |
|---|---|
| Power rule | d/dx(xⁿ) = nxⁿ⁻¹ |
| Product rule | d/dx(uv) = u'v + uv' |
| Quotient rule | d/dx(u/v) = (u'v − uv')/v² |
| Chain rule | d/dx[f(g(x))] = f'(g(x))·g'(x) |
| d/dx(sin x) | cos x |
| d/dx(cos x) | −sin x |
| d/dx(tan x) | sec²x |
| d/dx(eˣ) | eˣ |
| d/dx(ln x) | 1/x |
| d/dx(aˣ) | aˣ ln a |
Integration Formulas
| Integral | Result |
|---|---|
| ∫xⁿ dx | xⁿ⁺¹/(n+1) + C |
| ∫eˣ dx | eˣ + C |
| ∫sin x dx | −cos x + C |
| ∫cos x dx | sin x + C |
| ∫sec²x dx | tan x + C |
| ∫1/x dx | ln|x| + C |
| ∫u dv | uv − ∫v du (by parts) |
Key Theorems
| Theorem | Statement (brief) |
|---|---|
| Mean Value Theorem | f'(c) = [f(b)−f(a)]/(b−a) for some c ∈ (a,b) |
| Fundamental Theorem | ∫ₐᵇ f(x)dx = F(b) − F(a) |
| L'Hôpital's Rule | lim f/g = lim f'/g' for 0/0 or ∞/∞ forms |
| Newton's Method | xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ) |
ODE Key Formulas
| ODE | Solution Method |
|---|---|
| dy/dx = f(x)/g(y) | ∫g(y)dy = ∫f(x)dx |
| dy/dx + P(x)y = Q(x) | μ = e^∫P dx; y = (1/μ)∫μQ dx |
| ay″+by′+cy = 0 | Characteristic: ar²+br+c = 0 |
| D = fₓₓfᵧᵧ − (fₓᵧ)² | D > 0 and fₓₓ > 0: min; D > 0, fₓₓ < 0: max; D < 0: saddle |
Exam Preparation Tips
Show every step in numerical questions. TU markers award marks per step, not just for the correct final answer. A Simpson's Rule question where you correctly set up h, write the formula, substitute values, and make a small arithmetic error still earns most marks. Skipping steps earns almost nothing even with a correct answer.
Memorize the ODE solution methods as procedures, not as formulas. The integrating factor formula looks complex but is mechanical once you've practised it five times: find P(x), compute e^∫P dx, multiply through, integrate. Practice the procedure with four different equations and it becomes automatic.
Draw graphs for curve-sketching and area questions. A rough sketch that correctly shows intersection points, which curve is on top, and the region of integration earns partial marks and helps you set up the integral correctly. Students who skip the sketch frequently set up limits of integration backwards.
For series questions: State which test you're using before applying it. "Applying the Ratio Test:" followed by the calculation is more mark-friendly than launching straight into computation.
Prioritize Unit 7 (ODEs) over Unit 10 (Partial Derivatives). Both are important, but ODEs have 6 teaching hours and appear more heavily in long questions. If time is very limited, depth on ODEs earns more marks.
One-Week Study Plan
Day 1 — Limits, Continuity, and Derivatives (Units 2–3)
- Practice 5 limit evaluations including indeterminate forms
- Derive using product rule, quotient rule, and chain rule
- Verify MVT for 2 functions
- Evaluate limits using L'Hôpital's Rule
Day 2 — Applications of Derivatives (Unit 4)
- Solve 2 optimization word problems (max area, min cost)
- Do Newton's Method for 3 iterations on f(x) = x³ − 2
- Sketch a curve using first and second derivative analysis
Day 3 — Integration (Units 5–6)
- Evaluate 5 indefinite integrals (substitution + by parts)
- Evaluate 3 definite integrals using Fundamental Theorem
- Apply Simpson's Rule and Trapezoidal Rule to a dataset
- Find area between two curves
Day 4 — Differential Equations (Unit 7)
- Solve 2 separable ODEs
- Solve 2 first-order linear ODEs using integrating factor
- Solve 2 second-order homogeneous ODEs (distinct roots, repeated roots, complex roots)
- Attempt 1 non-homogeneous ODE using undetermined coefficients
Day 5 — Vectors (Unit 9)
- Compute dot products and find angles between vectors
- Compute cross products (use determinant method)
- Find equations of lines and planes in 3D
- Practice arc length and curvature formulas
Day 6 — Series and Partial Derivatives (Units 8 + 10)
- Apply ratio test to 3 series
- Write Taylor series for eˣ, sin x, cos x from memory
- Find partial derivatives of 3 functions
- Evaluate 2 double integrals
- Find and classify critical points using second derivative test
Day 7 — Past Questions + Formula Revision
- Attempt 2 full past papers under timed conditions
- Rewrite the formula quick-reference sheet from memory
- Focus on your weakest unit from the week
- Review step-by-step structure of Simpson's Rule and ODE methods
Common Mistakes That Cost Marks
Forgetting constant of integration C in indefinite integrals. Consistently loses 1 mark per question — a simple but expensive error.
Using L'Hôpital's Rule when the form isn't 0/0 or ∞/∞. L'Hôpital requires an indeterminate form first. Applying it to a determinate limit gives a wrong answer.
Newton's Method — using wrong iteration formula. The formula is xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Confusing f and f' gives completely wrong iterations.
ODE characteristic equation — sign errors. For ay″+by′+cy = 0, the characteristic equation is ar²+br+c = 0. Students frequently make sign errors substituting coefficients.
Simpson's Rule — odd number of subintervals. Simpson's Rule requires an even number of subintervals (n must be even). Using n = 3 or n = 5 is a setup error.
Cross product — forgetting the alternating sign pattern. The i component is +, j component is −, k component is +. Forgetting the minus on j is the single most common cross product error.
Partial derivatives — treating the wrong variable as constant. ∂f/∂x treats y as a constant. ∂f/∂y treats x as a constant. Mixing these up produces completely wrong derivatives.
Frequently Asked Questions
Is MTH117 difficult for BSc CSIT students? It's challenging because of the breadth of topics (10 units from functions to multiple integrals) and the need for both conceptual understanding and computational accuracy. Students who practice numerical problems daily, rather than only reading theory, consistently find it more manageable.
Which unit carries the most marks? Unit 7 (ODEs, 6 hours) and Unit 5 (Antiderivatives, 5 hours) tend to generate the most long-question marks. Vectors (Unit 9) and Applications of Derivatives (Unit 4) also regularly produce full long questions.
Is Newton's Method asked every year? It appears very consistently — either as a standalone question ("apply Newton's Method to find a root of f(x) = x³ − 2 to 3 iterations") or as part of an applications chapter question. Treat it as near-certain.
Is Simpson's Rule and Trapezoidal Rule important? Yes — these are among the most reliably repeated computational questions in TU MTH117 papers. The procedure is mechanical once learned and earns consistent marks.
How many ODE types should I prepare? At minimum: separable ODEs, first-order linear (integrating factor), and second-order homogeneous (characteristic equation). Non-homogeneous (undetermined coefficients) is also worth preparing given Unit 7's teaching hours.
Do I need to memorize Taylor series? Yes — eˣ, sin x, and cos x Maclaurin series appear as both standalone questions and as tools in limit evaluation. Write each from memory ten times and you'll remember them.
Is vector calculus (Unit 9) heavily tested? Yes. Dot product, cross product, and equations of lines and planes in 3D are almost certain to appear. Arc length and curvature are occasionally tested. Practice the cross product using the 3×3 determinant method.
Can I pass MTH117 by focusing only on high-priority topics? You can pass, but the high-priority list should be your starting point for depth, not your entire preparation. Lower-priority topics appear as short questions that can add up to 10–15 marks across the paper.
Conclusion
Mathematics is the subject that rewards consistent daily practice more than any other in the first semester. The students who score well aren't necessarily the most mathematically gifted — they're the ones who did five integration problems every day, practiced Newton's Method until it was automatic, and learned ODE solution methods as repeatable procedures rather than formulas to memorize.
Use this guide to focus your preparation on the highest-probability topics, use the formula sheet for final-day revision, and work through past papers under timed conditions in the final week. For the full unit-by-unit explanations and worked derivations behind every topic listed here, see the Complete BSc CSIT Mathematics (MTH117) Guide.
For related subjects: C Programming (CSC115) Complete Guide, Physics (PHY118) Complete Guide, Digital Logic (CSC116) Complete Guide, and Introduction to Information Technology (CSC114) Complete Guide.
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