Conditional permutation deals with arrangements limited by specific rules. You will learn to count outcomes when items must stay together or stay apart. This logic is essential for advanced scheduling and security where specific conditions apply.

You will master the tie-together method for items that stay together; apply the gap method for items that must stay apart; solve problems with fixed positions; and calculate arrangements with restricted letter and digit placements.

Conditional Permutation

Permutation with identical objects covers arrangements where some items are not distinguishable. Swapping identical items creates no new pattern, so you must divide to find unique results. This is essential for calculating arrangements in words, passwords, and digital data accurately.

You will master the formula for n items with repetitions; calculate arrangements for words with repeated letters; solve grid-based path problems; and use factorial division to remove duplicate counts.

Permutation with Identical Objects

This chapter introduces the logic of counting required for all arrangement and selection problems. You will master the foundational rules used to calculate total outcomes without manual listing, forming the essential groundwork for the entire course.

You will understand and apply the product rule for sequential events; the sum rule for separate choices; the difference rule for total minus unwanted cases; and the division rule to remove duplicate counts.

Introduction

Permutation focuses on the arrangement of objects where the specific order is essential. You will learn to calculate total outcomes when selecting and ordering items from a set, a skill required for creating secure PINs and scheduling tasks. This chapter provides the mathematical formulas needed to handle arrangements of all available items or just a specific portion of them.

You will master the use of factorials to count arrangements of all n objects; apply the nPr formula to arrange r items from a larger set; and solve practical problems involving distinct objects in fixed positions.

Permutation

Cyclic permutation covers arrangements in a circle where the starting position is not fixed. You will learn to remove repeated patterns to ensure accurate counting. This section explains the difference between arrangements viewed from one side and those that can be turned over.

You will master the formula for circular arrangements where clockwise and anticlockwise orders differ; calculate unique patterns for items that can be flipped like beads; and solve problems involving symmetry.

Cyclic Permutation

This chapter applies combinatorial logic to solve geometry problems involving points, lines, and shapes. You will learn to calculate the number of possible geometric figures that can be formed from a given set of points, which is essential for spatial reasoning and advanced coordinate geometry.

By the end of this chapter, you will master calculating the number of straight lines from points; determining the number of triangles and quadrilaterals; and finding the total diagonals in a polygon.

Geometric Applications

Derangements cover scenarios where no item in a set returns to its original position. This chapter explains how to count failed matches, such as putting letters into wrong envelopes or seating people in different chairs. Mastering this prevents overcounting and solves problems where specific items must not occupy their natural spots.

You will learn the derangement formula, calculate subfactorials for small sets, solve problems with restricted positions, and apply these rules to real-world logic puzzles.

Derangements

Combination focuses on selection where order does not matter. It is essential for picking committees or groups where picking A then B is the same as B then A. You will learn to identify selection tasks to avoid overcounting and ensure accurate results.

You will master the nCr formula; solve committee problems with group restrictions; calculate lines and triangles in geometry; and handle selections from multiple sets.

Combination

Stop counting one by one. This course provides the mathematical tools to calculate arrangements and selections with speed. You will master the fundamental counting rules including product, addition, subtraction, and division. The curriculum covers linear and cyclic permutations, identical objects, and restricted placements where items must be together or apart. We conclude with combinations, derangements, and the calculation of shapes within geometric frameworks.

Counting is the engine of probability and modern computing. These skills are vital for cryptography, software development, and statistical analysis. You will apply these methods to calculate network paths, secure passwords, and optimise logistics. In any technical career, the ability to quantify possibilities is the difference between guessing and knowing.

You will solve complex arrangement problems using standard formulas and logical shortcuts. You will handle specific constraints like fixed positions or items that cannot be next to each other. You will learn to distinguish between permutations and combinations in practical scenarios. You will also calculate the number of lines, triangles, and diagonals in geometric figures using combinatorial principles.

This foundation is for undergraduate students and secondary school leavers aiming for high scores in competitive exams. It is also essential for professionals in engineering, finance, and data science who require a refresher on discrete mathematics. Anyone seeking to improve their logical deduction and analytical speed will find value here.

Our mission is to simplify complex concepts by teaching from first principles. We go beyond simple problem-solving to build a deep, foundational understanding of the "why" behind the "how."

Our content is meticulously designed for undergraduates navigating the rigorous curricula of top engineering and physical science programmes in Nigeria and globally.

As a core academic partner on UniDrills, we are committed to upholding the highest standards of clarity, precision, and professional excellence in education.

Max Engineering Tutorials

An academic and graduate engineer dedicated to explaining complex concepts in mathematics, computer science and engineering. I founded UniDrills to bring this vision to life, creating a world-class e-learning marketplace with a global outlook. I apply a first-principles approach to software architecture to build a professional and intuitive learning experience.

NUC CCMAS

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science.

This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions.

Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Elementary Mathematics I

Algebra and Trigonometry

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Mathematics

This chapter covers basic binomial expansion and Pascal's triangle. Identifying these patterns early allows you to skip long multiplication and build algebraic speed. Accuracy in these fundamentals is required for all advanced scientific computations.

You will learn to find the total number of terms; use Pascal???s triangle for coefficients; observe the symmetry of powers; and expand simple binomials to small positive indices.

This chapter covers the formal binomial formula, combination notation, and rigorous proof by mathematical induction. These tools allow you to find specific terms and coefficients for large powers without full expansion. Mastering these techniques is essential for accurate decimal approximations and university-level algebraic proofs.

You will master the nCr formula for coefficients; calculate the general term of an expansion; identify terms independent of x; prove the theorem using mathematical induction; and solve small decimal approximations.

The Binomial Formula

This chapter extends binomial expansion to expressions with negative and fractional indices. These expansions result in infinite series rather than finite terms, providing a critical tool for approximating roots and reciprocal functions. Engineers and scientists use these series to simplify complex equations where absolute precision is required for very small values.

You will master the general expansion formula for any rational index; determine the range of values for which an expansion is valid; calculate approximations for square and cube roots; and find the first few terms of infinite binomial series.

Negative and Fractional Powers

This chapter focuses on isolating specific parts of a binomial expansion without writing out the full polynomial. It is a critical skill for solving complex engineering and probability problems where only a single value or coefficient is required.

By the end, you will master calculating the general term formula; finding the coefficient of a specific power of x; and identifying the term independent of x. You will also learn to determine middle terms and specific term positions.

Individual Terms

Stop manually multiplying brackets. This course teaches the systematic expansion of binomial expressions using Pascal's triangle and the formal Binomial Theorem. We cover the fundamental proof for positive integers before moving to individual term identification and the general expansion for negative or fractional powers. You will master the mechanics of series expansion from simple squares to complex infinite series.

The Binomial Theorem is a core tool in probability, statistics, and financial engineering. It allows for the approximation of complex functions and the calculation of compounding interest or risk factors. Engineers and data scientists use these principles to manage error margins and optimise algorithms. Understanding these patterns is essential for any career involving quantitative analysis or technical forecasting.

You will expand binomials to any power and identify specific terms within a series without full expansion. You will demonstrate the theorem's proof and apply Pascal's triangle for rapid computation. You will also calculate expansions for non-integer indices and determine the range of validity for these series. These skills enable the simplification of advanced algebraic expressions.

This curriculum is for undergraduate students and secondary school leavers entering science or technology tracks. It provides the necessary foundation for engineering, physics, and economics degrees. Professionals needing a mathematical refresher for data analysis will find the direct approach efficient. Any learner requiring precise algebraic tools for higher education will gain immediate value.

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This chapter defines the principle of mathematical induction and its role in formal logic. You will understand the domino effect analogy and why this method is necessary for proving statements across infinite sets of integers. It provides the essential foundation for all subsequent proof techniques in this course.

You will master defining the principle of induction, verifying the base case for n equals one, formulating the inductive hypothesis for n equals k, and executing the inductive step to prove the statement for n equals k plus one.

This chapter covers proofs for the sums of series with terms written as fractions. It is a vital section because it forces you to use advanced algebraic manipulation and common denominators to link the inductive hypothesis to the final goal.

By the end, you will master handling terms with product denominators, using common factors to simplify fractional expressions, and proving telescoping series identities through the inductive step.

Fractional Series

This chapter teaches how to use induction to prove that algebraic expressions are exactly divisible by specific integers. It is a critical skill for number theory and cryptography, moving beyond simple addition to master the manipulation of factors and remainders in a logical sequence.

You will master expressing divisibility as a linear equation, manipulating exponents to isolate common factors, and applying the inductive hypothesis to prove that expressions like n cubed plus 2n are multiples of three.

Divisibility

This chapter focuses on using induction to prove formulas for the sum of arithmetic and geometric sequences. It is the most common application of the method and essential for verifying that a general summation formula holds true for any number of terms.

By the end, you will master proving linear and quadratic series identities, applying the inductive hypothesis to algebraic summations, and simplifying complex fractional series expressions to reach the required right-hand side.

Series

This chapter covers using induction to prove mathematical inequalities where one expression is always greater than another. It is a vital section because it requires logical estimation and transitivity rather than simple equality. This skill is essential for analyzing algorithm efficiency and bounding errors in numerical analysis.

You will master handling base cases where n is greater than one, applying the inductive hypothesis to inequalities, and using algebraic estimation to prove that exponential growth eventually exceeds polynomial growth.

Inequalities

This chapter applies induction to sequences where each term depends on those before it. It is critical for computer science and engineering because it provides the mathematical proof for how recursive algorithms and loops function. You will learn to verify that a general formula correctly predicts the value of any term in a repeating sequence.

You will master defining terms using recurrence relations, proving explicit formulas for recursive sequences, and applying the inductive step to link successive terms.

Recursive Sequences

This chapter covers miscellaneous induction proofs that do not fit into standard categories. It tests your ability to apply logical steps to unfamiliar mathematical structures, ensuring you can use the base case and inductive step on any technical problem.

By the end, you will master proving formulas for the nth power of a matrix, calculating total subsets for a set with n elements, and applying inductive logic to non-standard algebraic identities.

Subsets

This chapter covers the algebraic manipulation of sets to form new collections through rigorous logical rules. Mastering these operations is foundational for database querying, probability theory, and complex data stratification in engineering and analytical systems.

You will master four objectives: performing unions, intersections, and set differences; calculating absolute and relative complements; applying De Morgan's laws to simplify set expressions; and identifying disjoint sets.

Operations on Sets

This chapter defines the rigorous classification of numerical collections and their hierarchical inclusion. Mastery of these standard sets is mandatory for defining domains in calculus, establishing computational constraints, and ensuring logical consistency in mathematical proofs.

You will master four objectives: categorising natural, integer, rational, and irrational numbers; identifying the real and complex number systems; applying interval notation for continuous subsets; and mapping the nested relationship between numerical sets.

Sets of Numbers

This chapter covers the fundamental laws governing set operations and the formal proof of set identities. Mastering this symbolic logic is essential for simplifying complex boolean expressions and ensuring the validity of analytical arguments in computer science and mathematical engineering.

You will master four objectives: applying idempotent, associative, and commutative laws; implementing distributive and absorption laws; proving identities using De Morgan's laws; and simplifying expressions using identity and complement laws.

Algebra of Sets

This chapter introduces the visual representation of set relationships and the spatial logic of inclusions. Mastery of Venn diagrams is essential for mapping data overlaps, solving combinatorial problems, and verifying set identities through geometric interpretation.

You will master four objectives: constructing diagrams for two and three sets; shaded representation of unions and intersections; interpreting regions of relative and absolute complements; and solving practical survey problems using the principle of inclusion-exclusion.

Venn Diagrams

This chapter defines the construction of ordered pairs and the product of distinct sets. Understanding these mappings is mandatory for defining relations, functions, and the multidimensional coordinate systems used in database schema design and linear algebra.

By the end of this chapter, you will master: constructing ordered pairs and n-tuples; calculating the cardinality of product sets; applying the non-commutative property; and executing distributive operations over unions and intersections.

Cartesian Products

This chapter explains how elements from different sets interact through specific rules. These principles are the building blocks for computer programming and advanced calculus. You will learn to link data points accurately and transform inputs into valid outputs.

You will master identifying relations and functions; performing algebraic and composite operations; classifying injective, surjective, and bijective mappings; and calculating inverse functions.

Relations, Mappings and Functions

Mathematics begins with sets. This course covers everything from basic definitions and membership notations to complex set algebra and De Morgan's laws. You will master cardinality, power sets, and the classification of number systems including rational, irrational, and complex numbers. The curriculum moves from simple operations like union and intersection into element-wise proofs, Cartesian products, and the mechanics of relations and functions.

Set theory is the language of modern data and logic. These concepts are essential for computer programming, database management, and statistical analysis. Understanding functions and mappings allows you to model real-world dependencies in engineering, economics, and the sciences. Mastering these foundations provides the exact logical framework needed to solve complex problems in technology and research.

Upon completion, you will be able to simplify set expressions and solve grouping problems using Venn diagrams and the inclusion-exclusion principle. You will know how to perform element-wise proofs and calculate set cardinalities. You will also gain the ability to evaluate composite functions and prove whether a mapping is one-to-one, onto, or bijective.

This course is designed for undergraduate students and secondary school leavers entering STEM disciplines. It provides a necessary logical foundation for anyone moving into calculus, data science, or advanced mathematics. The clear, direct instruction ensures that any student can develop the systematic thinking required for professional technical roles.

set-theory-mathematics-undergraduate-foundation

This chapter covers arithmetic sequences where terms increase or decrease by a constant common difference. Mastering these linear patterns is essential for understanding uniform growth, simple interest, and steady accumulation in science and finance.

You will master calculating general terms and sequence lengths; evaluating finite series sums; inserting arithmetic means; and using symmetric notation to solve algebraic progression problems involving ratios and unknown variables.

Arithmetic Sequences and Series

This chapter covers geometric sequences where terms change by a constant ratio. Mastering these exponential patterns is essential for modelling compound interest, asset depreciation, and physical rebound heights in science and finance.

You will master calculating general terms and finite sums; inserting geometric means; and applying symmetric notation or logarithms to solve for unknown variables in real-world growth and decay scenarios.

Finite Geometric Sequences and Series

This chapter establishes the fundamental definitions and notation for ordered numerical lists and their summations. Technical precision here is vital for distinguishing between sequences and series before advancing to complex progressions.

You will master defining formal sequence notation; distinguishing between finite and infinite structures; identifying individual terms via subscripts; and utilising sigma notation for compact series representation.

This chapter covers the algebraic mechanics of ordered numerical lists defined by functional rules. Mastery of these structures is essential for translating discrete patterns into precise mathematical models using both direct and step-by-step methods.

You will master generating terms from explicit formulae; defining sequences through recurrence relations; deriving general rules from numerical patterns; and verifying term accuracy through algebraic substitution.

Sequences

This chapter covers the algebraic mechanics of discrete summation and the rigorous application of sigma notation. Technical proficiency here is vital for condensing and expanding series, providing the foundational logic for calculating total accumulations in financial and physical models.

You will master executing summations using sigma notation; applying linear properties to simplify complex sums; extracting individual terms from partial sums; and classifying series as convergent, divergent, or oscillatory.

This chapter covers infinite geometric series that sum to a finite limit. Mastering convergence is vital for limits, calculus, and professional financial modelling.

You will define conditions for series convergence; calculate the sum to infinity; expand recurring decimals into geometric series; and convert these decimals into exact fractions.

Infinite Geometric Series

This chapter covers complex sequences and special series that fall outside standard linear or exponential models. You will learn to synthesise arithmetic and geometric properties to resolve hybrid patterns and use standard identities for power summations.

You will master identifying harmonic progressions through reciprocals; calculating nth terms for arithmetic-geometric sequences; and applying formulae for the sums of natural numbers and their squares.

Other Sequences and Series

Master the algebraic rules governing ordered lists of numbers and their total sums. This course establishes a solid foundation in the mechanics of sequences and series. We investigate precise definitions for arithmetic and geometric progressions; use formal notation for general terms; and handle finite and infinite series using sigma notation. The curriculum focuses on the structural properties required for university-level mathematical analysis.

Knowing how to handle numerical patterns is vital for many modern careers. Financial analysts use geometric series to calculate compound interest, loan repayments, and annuities. Computer scientists use sequence structures to organise data sets and analyse computer loops. This knowledge is necessary for measuring growth, decay, and accumulation in business and science with absolute accuracy.

You will learn to identify and define sequences using general terms and recurrence formulas. You will calculate the nth term and the sum of terms for arithmetic, geometric, and harmonic progressions. You will master sigma notation; determine when infinite series converge; convert recurring decimals into fractions; and use formulas to sum natural numbers, squares, and cubes.

This course is for students entering university foundation programmes in mathematics, finance, or computer science. It serves as an intensive refresher for professionals returning to academia or shifting into roles that demand precise numerical work. Even those outside these fields will benefit from the sharp logical thinking and problem-solving speed developed through pattern analysis.

sequences-and-series-mathematics-undergraduate-foundation-2

This chapter covers determinants, the specific values used to identify singular matrices and solve linear systems. Mastering these is vital for calculating inverses and analysing matrix properties.

You will master evaluating determinants of any order; applying minors and cofactors; using algebraic properties to simplify calculations; and solving linear systems using Cramer's rule.

Determinants

This chapter covers matrix transposition — the swapping of rows and columns. Mastery of these patterns is essential for simplifying calculations in engineering and science.

You will define transpose notation; identify symmetric and orthogonal types; compute conjugate transposes; and classify Hermitian and unitary matrices.

Transposition of Matrices

This chapter covers elementary row and column operations to simplify complex matrices. Mastering these transformations is vital for finding the rank, nullity, and reduced row echelon form of any matrix.

You will master executing elementary row and column transformations; identifying equivalent and elementary matrices; reducing matrices to row echelon and reduced row echelon forms; and calculating the rank and nullity of a matrix.

Elementary Transformations

This chapter moves matrix theory from paper to software. Mastering spreadsheet tools is essential for automating calculations and handling large data sets in professional environments.

You will perform matrix arithmetic and transposes; calculate determinants and inverses; and solve square systems of linear equations using built-in spreadsheet functions.

Computer-Aided Handling

This chapter establishes the basic building blocks of matrix theory. Recognising these structures is essential for solving complex linear systems and performing matrix transformations in advanced mathematics.

You will define matrix order and notation; identify row, column, and null matrices; distinguish between square, scalar, and identity types; and classify triangular, banded, and sparse matrices.

This chapter covers solving multiple equations using matrix algebra. Mastering these methods is essential for identifying solution types and modelling complex systems in physics and engineering.

You will master converting equations to matrix forms; testing consistency with ranks; solving homogeneous and nonhomogeneous systems; and using pivots to handle infinite solutions.

Systems of Linear Equations

This chapter covers matrix inversion and its core uses. Mastering the inverse is vital for solving square systems of equations and reversing matrix operations.

You will compute matrix adjoints; determine invertibility; apply inverse properties to products and transposes; solve square linear systems; and define similar matrices.

Matrix Inverses

This chapter establishes the core arithmetic of matrices. These operations provide the necessary tools for solving linear equations and handling large data sets in professional fields.

You will define matrix equality; perform addition and subtraction; apply scalar multiplication; execute matrix multiplication; and identify key properties like non-commutativity and associativity.

Algebra of Matrices

Master advanced matrix algebra and linear systems through rigorous technical training. This course covers matrix definitions; algebraic operations; transpositions; and elementary row transformations. You will investigate determinants; matrix inverses using the adjoint method; and the structural properties of similar matrices. Computational methods using MS-Excel or Google Sheets are included to handle large-scale data sets efficiently.

These mathematical structures are the engines behind modern engineering and data science. Structural engineers use matrices to calculate stress in bridges; computer scientists use them to render 3D graphics; and economists use linear systems to model market equilibrium. Mastering these tools allows you to solve multi-variable problems that are impossible to handle manually.

By the end, you will possess the skills to execute matrix additions; multiplications; and scalar operations with absolute precision. You will demonstrate competence in reducing matrices to row echelon form; calculating rank and nullity; and evaluating determinants of any order. Furthermore, you will command the ability to solve complex systems of linear equations using Cramer's rule or matrix inversion; identify special matrix types like orthogonal or symmetric; and automate these calculations using software.

This curriculum is built for undergraduate students in engineering; mathematics; and physics requiring a high-level command of linear algebra. Professionals in data science or finance will find the content vital for understanding the algorithms governing their software. Even those in general sciences will benefit from the structured logical thinking and numerical methods required to manage complex variables in any technical field.

OAU, Ife

Advanced engineering mathematics covering solid analytical geometry, integrals, scalar and vector fields, matrices and determinants and complex variables.

Curated for third-year students of engineering at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Engineering Analysis I

Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors.

Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Mathematical Methods II

Master the mathematical language of engineering. This programme delivers the complete analytical toolkit required for a successful engineering career, covering single-variable calculus, multivariable calculus, linear algebra, and vector analysis. It provides the essential foundation for all subsequent engineering courses.

This programme is for second-year undergraduate students across all engineering disciplines. It delivers the official NUC CCMAS curriculum for Engineering Mathematics, providing the core training required for advanced modules in mechanics, thermodynamics, and circuit theory.

Model and analyse complex physical systems using calculus, linear algebra, and vector analysis. You will be equipped to solve problems in dynamics, statics, and field theory, providing the quantitative proficiency required for advanced engineering study and professional practice.

Engineering Mathematics I

Master the algebraic structures that underpin modern science and computation. This academic track delivers the complete NUC CCMAS MTH 204 curriculum, moving rigorously from abstract vector spaces to practical matrix theory. It provides the essential mathematical toolkit required for advanced problem-solving in high-demand STEM fields.

This programme is targeted at undergraduates in mathematics, engineering, and computer science requiring a firm grounding in linear structures. It also serves professionals in data science, cryptography, and machine learning needing a rigorous theoretical refresher on foundational concepts.

You will achieve competence in manipulating abstract vector spaces, determining basis and dimension, and analyzing linear transformations through their kernels and images. You will master matrix arithmetic, compute determinants, solve systems of linear equations using advanced methods, and apply techniques of eigenvalues and diagonalization. Completion establishes the critical foundation demanded for advanced studies in multivariate calculus, differential equations, and complex computational algorithms.

Linear Algebra I

Advanced linear algebra is the mathematical backbone of modern data science, engineering, and physics. This learning track delivers the rigorous MTH 205 curriculum based on NUC CCMAS standards, focusing on sophisticated matrix analysis and practical computational methods critical for solving complex technical problems.

This programme is targeted at undergraduates in mathematics, engineering, and computer science requiring a deep command of advanced matrix theory. It is equally essential for data scientists and engineers seeking a rigorous theoretical foundation for machine learning algorithms, cryptography, and complex system modelling.

You will master matrix manipulations to solve linear systems and compute determinants and inverses efficiently using various methods including software like Python and MATLAB. You will gain competence in determining eigenvalues and eigenvectors, applying diagonalization to analyze the stability of dynamical systems, and working with quadratic and canonical forms. Completion establishes the critical mathematical expertise required for advanced studies in multivariate statistics, differential equations, and algorithmic development.

Linear Algebra II

matrices-determinants-and-systems-of-linear-equations

Linear Algebra

This chapter provides a foundational classification of chemical transformations. It introduces the main types of chemical reactions, providing a framework for predicting the products of simple reactions.

Key learning objectives include: identifying and providing examples of the common reaction types, such as synthesis, decomposition, and displacement.

Chemical Reactions

This chapter provides the roadmap for the course. It outlines the foundational concepts of chemistry to be covered, from the scientific method and measurement to the classification of matter and reactions.

Key learning objectives include understanding the overall course structure and appreciating how these distinct topics interconnect to form the basis of chemical science.

This chapter covers the fundamental classification of matter as required by the syllabus. It provides the formal definitions for atoms, molecules, elements, compounds, and mixtures, and explores the changes matter undergoes.

Key learning objectives include: defining and differentiating between the classifications of matter; and distinguishing between physical and chemical changes.

Matter

This chapter consolidates your mastery of the non-negotiable concepts required for advanced study in the chemical sciences, acting as a final knowledge checkpoint before you proceed.

Key objectives are to review all core concepts and outline the immediate next steps required in your university-level chemistry curriculum.

Conclusion

Chemistry is the science of matter. This course provides a rigorous introduction to its fundamental principles, moving from the definition of matter and its states to the atomic and molecular structures that govern its behaviour. We will classify matter into elements, compounds, and mixtures, and establish the critical distinction between physical and chemical changes.

A command of chemistry is essential for any quantitative science. The principles covered here are foundational to medicine, engineering, environmental science, and materials development. Understanding how matter is structured and how it transforms is the basis for creating new materials, developing pharmaceuticals, and managing industrial chemical processes safely and efficiently.

Upon completion, you will possess a clear conceptual framework for chemistry. You will be able to classify different types of matter, distinguish between physical and chemical changes, and define elements, compounds, and mixtures. You will also be able to identify the fundamental roles of atoms, molecules, and ions in chemical reactions.

This course is designed for first-year university students beginning their studies in any scientific or engineering discipline. It serves as a mandatory prerequisite for further study in chemistry, chemical engineering, biology, and medicine. It is also suitable for anyone requiring a concise and formal review of foundational chemical concepts.

A passionate teacher of Chemistry who is not just interested in teaching but has a strong desire in seeing students enjoying the beauty of the subject 

This learning track delivers the complete NUC CCMAS curriculum for General Chemistry I. It is a comprehensive programme designed to build a robust, university-level foundation in modern chemistry. The track systematically covers all essential topics, from atomic theory, chemical bonding, and the states of matter, to the quantitative principles of stoichiometry, equilibrium, thermodynamics, and kinetics.

This programme is for first-year undergraduates in science, technology, engineering, and mathematics (STEM) faculties who are required to take CHM 101. It is also essential for any student or professional globally who needs a rigorous and complete foundation in first-year university chemistry for further study or career development.

This track delivers a full skill set in chemical theory and quantitative problem-solving. Graduates will be able to determine molecular structures, calculate reaction quantities, analyse the energetics and rates of reactions, and solve complex equilibrium problems. This programme provides the non-negotiable prerequisite knowledge for all subsequent chemistry courses and for any degree in the physical sciences, engineering, or medicine.

General Chemistry I

introduction-to-chemistry-undergraduate-foundation

Chemistry

This chapter provides the roadmap for the course. It outlines the progression from the fundamental nature of chemical bonds to the theories that predict the three-dimensional shapes of molecules.

Key learning objectives include: understanding the overall course structure and appreciating why the shape of a molecule is critical to its function.

This chapter focuses on mastering Valence Shell Electron Pair Repulsion (VSEPR) theory to accurately predict the three-dimensional geometry of molecules. Understanding molecular shape is non-negotiable; it governs all chemical properties, reactivity, and function, providing the framework for molecular design.

Upon completion, you will be able to: predict electron and molecular geometry for molecules with two to six electron domains, identify geometries in examples with multiple lone pairs, and master the step-by-step procedure for determining complex molecular shapes.

Shapes of Molecules

This chapter introduces Valence Bond Theory and the crucial concept of orbital hybridization. Hybridization is a necessary model for explaining molecular shapes and bond angles that cannot be accounted for by simple VSEPR theory alone, bridging the gap between bonding theory and observed molecular geometry.

Upon completion, you will be able to: define bonding as the overlap of atomic orbitals, explain the formation of sigma and pi bonds, and accurately determine the hybridization (from sp to sp3d2) for the central atom in any molecule.

Hybridization Theory

This chapter details intermolecular forces (IMFs) - the attractive forces between molecules. These non-covalent interactions dictate all physical properties, including melting points, boiling points, and solubility, which is critical for predicting substance behaviour.

Upon completion, you will be able to: define and identify London dispersion and dipole-dipole forces, explain the unique strength of hydrogen bonding, and use dipole moment and molecular symmetry to determine a molecule's overall polarity.

Intermolecular Forces

This chapter establishes the foundation by classifying and detailing the strong intramolecular forces that hold atoms together. Mastery of ionic and covalent bonding is the prerequisite for subsequent analysis of molecular structure. We also introduce the critical hydrogen bond.

Key objectives: Define ionic, covalent, and coordinate covalent bonds; contrast the physical properties of ionic and covalent substances; and understand hydrogen bonding.

Types of Bonds

This final chapter is dedicated to comprehensive application and synthesis of all course material. It reinforces the critical connections between Lewis structures, molecular geometry, and physical properties, ensuring foundational knowledge is secure for advanced topics.

Upon completion, you will be able to demonstrate mastery across: drawing Lewis structures, predicting VSEPR geometry, determining orbital hybridization and bond types, and identifying intermolecular forces in any given molecule.

This chapter establishes the Lewis structure framework, the fundamental method for mapping valence electron arrangement in molecules and ions. Mastering this structured approach is the prerequisite skill for predicting molecular geometry and understanding bond characteristics later in the course. We cover systematic rules for neutral species and polyatomic ions, including the essential concept of resonance.

By the end of this chapter, you will master: drawing valid Lewis structures for molecules and ions; identifying and drawing resonance structures; applying the three main octet rule exceptions; and calculating and using formal charges to select the most plausible structure.

Lewis Structures

The shape of a molecule dictates its function. This course provides a practical, foundational treatment of chemical bonding and molecular geometry - the principles governing how atoms connect to form substances. It covers ionic and covalent bonding, the drawing of Lewis structures, and methods for determining molecular shape using VSEPR theory and orbital hybridisation. Finally, we examine intermolecular forces that control a substance's physical properties.

A command of this material is essential for molecular design. Knowledge of bonding and geometry dictates the properties of everything from pharmaceuticals, where shape governs drug-receptor specificity, to new materials like polymers and catalysts. Mastery of this foundational knowledge is the starting point for developing or synthesising any new chemical compound. Our system uses comprehensive videos and allows you to ask questions directly, making it fully online and easy to master at your pace.

By the end of this course, you will be able to draw correct Lewis structures for any molecule, predict its three-dimensional geometry and polarity using VSEPR theory, determine the central atom's hybridisation and bond types, and identify the various intermolecular forces present in a substance.

This course is mandatory for all undergraduate students of chemistry, biochemistry, and materials science, and is a direct prerequisite for studying organic chemistry. It benefits anyone needing a rigorous, structured, student-centred refresher or initial exposure to this critical subject, assuming a complete understanding of atomic theory and electronic configuration.

chemical-bonding-and-shapes-of-molecules-chemistry

This chapter provides the roadmap for the course. It outlines the progression from early atomic theories to the modern understanding of atomic structure and its direct relationship to the periodic table.

Key learning objectives include: understanding the overall course structure and appreciating the historical and experimental development of atomic theory.

This chapter traces the origin of atomic theory, starting with the empirical laws that led to Dalton's foundational model. It then covers the pivotal discovery of the electron, the first experimental proof that the atom is divisible, which forced the initial theory to evolve.

Key topics include the laws of chemical combination, Dalton's atomic postulates and their critical shortcomings, and the experiments by Thomson and Millikan that discovered and characterised the electron, leading to the 'plum pudding' model.

Early Atomic Models

This chapter covers the development of the nuclear atom, from Rutherford's definitive gold foil experiment to Chadwick's discovery of the neutron. It establishes the modern planetary model of the atom and immediately addresses its critical failure, demonstrating why classical physics is insufficient.

Key topics include the experimental evidence for the nucleus, the discovery of the neutron, the shortcomings of Rutherford's classical model, and the fundamental properties of protons, neutrons, and electrons.

The Nuclear Model

This concluding chapter summarises the key concepts of atomic theory. It reinforces the understanding of atomic structure and its connection to the periodic properties of the elements.

This summary prepares the student for the next course, 'Chemical Bonding and Molecular Geometry', where electronic structure is used to predict how atoms form molecules.

This chapter introduces the modern quantum mechanical model, replacing Bohr's flawed orbits with probabilistic orbitals. Mastery of this model is non-negotiable, as it provides the definitive framework for describing electron behaviour and predicting all chemical properties.

Key topics include: the four quantum numbers, defining orbital shapes (s, p, d, f) and their spatial orientation, and applying the Aufbau principle, Pauli exclusion, and Hund's rule to write correct electronic configurations.

The Quantum Model

This chapter details Niels Bohr's model, the first attempt to solve the failures of the classical nuclear atom using quantum theory. Understanding this transitional model is critical to grasping the full conceptual leap to modern quantum mechanics.

Topics include: Bohr's postulates, explaining hydrogen's emission spectrum, energy level calculations, and the model's ultimate shortcomings.

Bohr's Model

This chapter details wave-particle duality, the principle that all matter has wave properties. This concept explains the failure of Bohr's fixed orbits and establishes the necessary theoretical foundation for the modern quantum mechanical model of the atom.

Key topics include: De Broglie???s wave hypothesis, applying the de Broglie relation (?? = h/mv), and calculating the wavelength of moving particles based on their mass and velocity.

Wave-Particle Duality

This chapter connects electronic configuration to the predictable, repeating properties of the elements. Mastery of these trends is the practical application of atomic theory and is essential for predicting chemical behaviour, bonding, and reactivity.

Key topics: trends in atomic radii, ionisation energy, electron affinity, and electronegativity; and the classification of elements as metals, non-metals, and metalloids.

Periodicity

This course provides a complete guide to the modern theory of the atom. It traces the historical development of atomic models, from Dalton's foundational theory to the modern understanding of electronic structure. The material covers subatomic particles, the contributions of Thomson and Rutherford, the Bohr model, wave-particle duality, and culminates in a full treatment of electronic configuration and its direct relationship to the periodic trends of the elements.

A command of atomic theory is the absolute foundation of modern chemistry. The electronic structure of the atom dictates all chemical properties and bonding behaviour. This knowledge is essential for understanding the periodic table, predicting chemical reactions, and is the prerequisite for the study of spectroscopy, materials science, and quantum mechanics.

By the end of this course, you will be able to describe the historical evolution of atomic theory. You will also be able to explain the atom's structure, including the roles of protons, neutrons, and electrons. You will determine the electronic configuration of any element and use periodic trends to predict and justify atomic properties like size, ionisation potential, and electronegativity.

This course is for students who have completed an introductory chemistry course. It is a mandatory prerequisite for any student pursuing a degree in chemistry, chemical engineering, materials science, or physics.

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This chapter establishes the rigorous classification and fundamental operations of the real number system. You will master the structural hierarchy of numbers and basic algebraic manipulations, providing the essential precision required for solving complex equation systems and inequalities in subsequent chapters.

You will master: classifying real numbers into rational and irrational subsets; applying order of operations to complex expressions; identifying properties of real number subsets; and executing fundamental algebraic simplifications.

This chapter establishes the rigorous mechanical framework for resolving equations of varying degrees and structures. Mastering these transformation techniques is essential for isolating variables in engineering models and scientific formulas with absolute logical consistency.

You will master the systematic isolation of unknowns in linear and quadratic forms; the resolution of rational and radical equations; and the reduction of cubic, biquadratic, and symmetric expressions into solvable factors.

Equations in One Unknown

This chapter governs the manipulation and factorisation of higher-degree algebraic expressions. You will master the structural logic of polynomial functions, establishing the essential skills required for advanced partial fraction decomposition and the analysis of non-linear systems in engineering and physics.

You will master: executing polynomial long division and synthetic division; applying the Remainder and Factor Theorems; factorising cubic and higher-order expressions; and identifying roots of polynomial equations.

Polynomials

This chapter governs the resolution of mixed systems where linear and non-linear dependencies intersect. You will master the algebraic techniques required to solve for variables across multiple constraints, providing the essential logic for intersecting geometric paths and complex equilibrium states in engineering and economics.

You will master: executing substitution and elimination for dual linear systems; identifying points of intersection between straight lines and curves; resolving homogeneous quadratic systems through variable substitution; and simplifying complex rational and product constraints to determine precise coordinate solution sets.

Simultaneous Equations

This chapter governs the mechanics of exponential notation and the foundational laws for manipulating powers. You will master these operations to establish the technical proficiency required for solving complex exponential equations and performing the logarithmic transformations essential to scientific and financial data analysis.

You will master: applying the product, quotient, and power laws; evaluating zero, negative, and fractional indices; and simplifying complex exponential expressions through systematic reduction.

Indices

This chapter covers the inverse operation of fraction addition by breaking down complex rational expressions into simpler parts. Mastery of this decomposition is a prerequisite for advanced calculus and engineering system analysis.

You will master identifying proper and improper functions; decomposing expressions with distinct or repeated linear factors; resolving irreducible quadratic denominators; and applying polynomial long division to handle high-degree numerators.

Partial Fractions

This chapter governs the algebra of non-equality and region-based constraints. You will master the rigorous handling of directionality in mathematical relationships, establishing the essential logic required for economic optimisation, engineering tolerance analysis, and defining valid data ranges in computational algorithms.

You will master: executing sign-reversal rules during manipulation; solving linear, quadratic, and rational inequalities; using interval notation; and shading valid regions.

This chapter establishes the inverse relationship between powers and exponents to handle nonlinear data. Mastery of logarithmic transformations is essential for solving complex exponential equations and modelling growth patterns in engineering and financial sectors.

You will master: converting between index and logarithmic forms; applying the product, quotient, and power laws; executing the change of base formula; and solving algebraic equations involving unknown exponents or bases.

Logarithms

This chapter covers surds, which are roots that do not have whole number answers. You must learn to use them to keep your calculations exact, as using decimals in engineering and science leads to errors.

You will master simplifying and performing arithmetic on surds; rationalising denominators using conjugates; finding square roots of compound surds; and solving algebraic equations involving radical terms.

Surds

This chapter governs the synthesis of indices and logarithms to solve equations where variables appear as exponents. You will master the algebraic techniques required to isolate and resolve unknown indices, providing essential analytical tools for modelling exponential growth, decay constants, and complex scaling in engineering and financial data.

You will master: equating bases for direct solution; applying logarithmic transformations; reducing equations to quadratic forms; and solving non-linear simultaneous systems involving mixed bases.

Equations with Unknown Index

Mathematics requires absolute precision. This course provides a rigorous grounding in real number operations, from integers and rational numbers to complex systems and algebraic laws. You will learn to handle polynomials, solve equations ranging from linear to biquadratic, and navigate simultaneous systems. The syllabus covers inequalities, sign tables, partial fractions, indices, logarithms, and the systematic simplification of surds.

These tools are essential for success in engineering, accounting, and the sciences. Clear mathematical thinking allows you to model financial risks, calculate structural loads, and write efficient computer code. Proficiency in these operations ensures accuracy in any career that relies on quantitative data and logical deduction.

You will gain the ability to classify number systems, apply algebraic theorems, and solve complex equations in one or more unknowns. You will acquire the skills to manipulate inequalities, resolve algebraic fractions, and simplify expressions using the laws of indices and logarithms. The course provides the technical competence required to handle surds and find roots of compound expressions effectively.

This training is built for undergraduate foundation students and secondary school leavers entering university. It provides a necessary bridge for anyone needing to strengthen their mathematical base before advanced study. Even those in non-technical roles will find value in the disciplined analytical approach required to master these foundational concepts.

operations-with-real-numbers-mathematics-undergraduate

This chapter shows how equation coefficients relate to their roots. Learning these shortcuts helps you study and build equations quickly without solving them first.

You will master calculating the sum and product of roots; evaluating expressions like the sum of squares or cubes; constructing new equations from given roots; and identifying conditions for shared roots between equations.

Properties of Roots

This chapter provides the procedural core of the course by teaching the three primary methods for solving quadratic equations. Mastering these techniques is mandatory for any further study in calculus, physics, or engineering.

You will master solving equations by factorisation; using the method of completing the square; applying the universal quadratic formula; and reducing complex fractional equations into standard quadratic form.

Solving Quadratic Equations

This chapter consolidates all theoretical and practical aspects of quadratic equations covered in the course. It ensures you have a firm command of every analytical tool and solution method before advancing to higher mathematics.

You will master factorisation, completing the square, and the quadratic formula; interpreting the discriminant for the nature of roots; applying sum and product properties; and identifying turning points for optimisation.

This chapter focuses on the discriminant to predict the properties of roots without solving the equation. It is a critical analytical tool for determining the existence and type of solutions in engineering and physics models.

You will master calculating the discriminant; classifying roots as real, identical, or imaginary; identifying rational and irrational solutions; and applying tangency conditions to curves and lines.

Nature of Roots

This chapter establishes the foundation for quadratic analysis by defining the standard form and its real-world relevance. Mastering these basics is essential before attempting complex solving techniques or applications.

You will master identifying quadratic components; rearranging equations into standard form; understanding the concept of roots; and applying the non-zero coefficient constraint.

This chapter focuses on finding the highest or lowest points of quadratic graphs. It is vital for solving real-world optimisation problems in physics, engineering, and finance.

You will master identifying maximum and minimum turning points; locating the vertex of a parabola; using the completing the square method for optimisation; and proving numerical boundary limits for quadratic expressions.

Maximum and Minimum Values

This chapter covers solving quadratic inequalities and finding the ranges of rational functions. These skills are vital for defining valid domains and identifying excluded values in advanced physics and engineering models.

You will master solving quadratic inequalities using graph shapes and critical values; applying the discriminant to find unknown coefficient ranges; rearranging rational expressions into quadratic form; and identifying the exact sets of values a function can or cannot take.

Inequalities and Rational Expressions

Quadratic equations are the bedrock of advanced algebra and the starting point for mastering non-linear mathematical models. This course provides a rigorous analysis of these equations, covering standard forms, solving methods like factorisation and completing the square, and the critical use of the discriminant to determine root nature. You will study turning points for optimisation, solve complex quadratic inequalities, and master the relationship between roots and coefficients to build equations from scratch.

Mastering these models is essential for careers in engineering, physics, and finance where trajectories and profit levels must be calculated precisely. You will use these principles to describe projectile motion, design parabolic structures, and identify boundary limits in economic models. This is the mathematics of optimisation, providing the necessary tools to find the highest or lowest possible values in any system governed by second-degree polynomials.

By the end of this course, you will solve any quadratic equation using multiple methods; use the discriminant to identify real, identical, or imaginary roots; and determine the location and value of turning points. You will master solving quadratic inequalities; identifying valid ranges for rational expressions; and using symmetric identities to evaluate root properties without solving the original equation. These skills form the essential foundation for calculus and mechanics.

This course is built for first-year university students in mathematics, physics, engineering, and economics who need a solid algebraic foundation. It also serves secondary school leavers preparing for advanced entrance exams or technical professionals looking to refresh their analytical skills. Even those in data science or basic programming will benefit from the logical structuring and problem-solving techniques required to handle non-linear variables.

theory-of-quadratic-equations-mathematics-undergraduate

This chapter provides the foundational roadmap for the study of electrochemistry. It defines the relationship between chemical reactions and electrical energy, establishing the framework for all subsequent course modules.

By the end of this chapter, you will be able to distinguish between galvanic and electrolytic cells, appreciate the industrial importance of electrochemical processes, and navigate the course structure effectively.

This chapter covers galvanic cells, which convert spontaneous redox reactions into electrical energy. Understanding these cells is essential for grasping the operation of batteries and portable power sources.

By the end of this chapter, you will be able to identify key cell components, represent cells using standard notation, explain the function of inert electrodes, and derive balanced chemical equations from cell notation.

Galvanic Cells

This concluding chapter consolidates essential redox concepts and energy transfer. It ensures readiness for industrial applications and further study of general chemistry.

You will master the distinction between galvanic and electrolytic cells; energy conversion principles; and the practical application of Faraday's laws.

This chapter covers electrolytic cells driven by external electrical energy. These systems are essential for electroplating and industrial metal extraction.

You will learn to apply Faraday's first and second laws; calculate electrochemical equivalent weights; and determine the quantitative relationship between current, time, and mass in single and series cells.

Electrolytic Cells

This chapter links electrochemistry to core thermodynamic principles. You will master the quantitative relationships that govern electrochemical spontaneity and cell efficiency, which are critical for predicting how batteries and fuel cells behave under real-world conditions.

By the end of this chapter, you will be able to calculate cell potentials, determine reaction spontaneity using Gibbs free energy, apply the Nernst equation for non-standard states, and derive equilibrium constants.

Thermodynamics and Cell Potential

Electrochemistry bridges chemical reactions and electrical energy. This course examines the mechanics of electron transfer. You will study galvanic cells that generate power and electrolytic cells that consume it. The curriculum covers cell structure, standard notation, and the role of inert electrodes.

This science drives portable energy and industrial manufacturing. You apply these principles when working with batteries, fuel cells, and corrosion prevention. Industrial applications rely heavily on this knowledge for metal extraction, electroplating, and purifying elements like aluminium and copper. Mastering this connects theoretical chemistry to practical engineering.

You will learn to identify cell components and write precise cell notations. You will calculate standard cell potentials, determine reaction spontaneity using Gibbs free energy, and apply the Nernst equation to non-standard conditions. You will also use Faraday's laws to compute mass deposited, time taken, and current passed during electrolysis.

This course is built for undergraduate foundation students in chemistry and chemical engineering. It is also directly useful for recent secondary school leavers preparing for university admissions. Engineering and materials science students will find this material essential for understanding thermodynamics and electrical systems.

electrochemistry-undergraduate-foundation

Courses

Permutation and Combination - Mathematics (Undergraduate Foundation)

Binomial Theorem - Mathematics (Undergraduate Foundation)

Mathematical Induction - Mathematics (Undergraduate Foundation)

Set Theory - Mathematics (Undergraduate Foundation)

Sequences and Series - Mathematics (Undergraduate Foundation)

Matrices, Determinants, and Systems of Linear Equations - Linear Algebra (Undergraduate Advanced)

Introduction to Chemistry (Undergraduate Foundation)

Chemical Bonding and Shapes of Molecules - Chemistry (Undergraduate Foundation)

Atomic Structure and Periodicity of Elements - Chemistry (Undergraduate Foundation)

Operations with Real Numbers - Mathematics (Undergraduate Foundation)

Theory of Quadratic Equations - Mathematics (Undergraduate Foundation)

Electrochemistry (Undergraduate Foundation)