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TU BCA First Semester: Mathematics I (CAMT104) — Complete Syllabus Overview
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TU BCA First Semester: Mathematics I (CAMT104) — Complete Syllabus Overview

Mathematics I is one of the five core courses in the First Semester of the Bachelor of Computer Applications (BCA) programme under Tribhuvan University’s Faculty of Humanities and Social Sciences (TU FOHSS). The course carries the code CAMT104 (also referenced as CACS104 in some official documents), is worth 3 credit hours, and runs with a class load of 5 hours per week — 3 hours theory, 1 hour tutorial, and 1 hour practical. It is offered alongside Computer Fundamentals and Applications (CACS101), Digital Logic (CACS105), Society and Technology (CASO102), and English I (CAEN101) in the First Semester.

The course provides the essential mathematical foundation for the entire BCA programme. Topics span algebra and analytical geometry — set theory, complex numbers, relations and functions, sequences and series, matrices and determinants, conic sections, vectors in space, and permutation and combination. Mathematical software tools — Mathematica and MATLAB — are integrated into the practical component, giving students hands-on experience with computational tools used in industry and research.

Subject Information at a Glance

  • Course Title: Mathematics I
  • Course Code: CAMT104 (also referenced as CACS104)
  • Credit Hours: 3 credits
  • Year/Semester: I/I (First Year, First Semester)
  • Class Load: 5 hours per week — Theory: 3 hours + Tutorial: 1 hour + Practical: 1 hour
  • Full Marks: 60 (Theory) + 20 (Practical/Internal) = 80 total
  • Pass Marks: 24 (Theory)
  • Theory Exam Format: Group A (all short questions compulsory) + Group B (attempt any 6 from 8 short questions × 5 marks) + Group C (attempt any 2 from 3 long questions × 10 marks)
  • Examination: Semester-based (end-of-semester exam conducted by TU FOHSS)
  • Conducting Institution: Faculty of Humanities and Social Sciences (FOHSS), Tribhuvan University
  • Software Tools Used: Mathematica (for Unit 2 — functions and graphs; Unit 5 — conic sections) and MATLAB (for Unit 4 — matrices; Unit 5 — conic sections and vectors)

Why Mathematics I for BCA Students?

Mathematics is the language of computing. Every algorithm has a mathematical basis, every data structure encodes a mathematical relationship, and every computational problem is ultimately a mathematical one. The topics in CAMT104 are not chosen arbitrarily — each one has direct applications throughout the BCA curriculum:

  • Set theory and logic → discrete mathematics, database design (relational algebra), Boolean operations in programming
  • Functions and relations → data structures (mapping, hashing), database tables, programming abstractions
  • Matrices and determinants → linear transformations in computer graphics, machine learning weight matrices, network analysis, solving systems of equations in simulation
  • Sequences and series → algorithm analysis (summation formulas for loop analysis), numerical methods, financial computing
  • Vectors → 3D computer graphics (translation, rotation, scaling using vector arithmetic), game development, physics simulation, machine learning (embedding vectors)
  • Conic sections → computer graphics rendering, path planning in robotics, satellite communication (elliptical orbits)
  • Permutation and combination → algorithm complexity analysis (how many ways can a problem be configured?), cryptography (key space size), database query optimization, testing (generating test cases)

CAMT104 Theory Syllabus — Unit-wise Detail

Unit 1: Set Theory and Real & Complex Numbers (8 teaching hours)

1.1 Set Theory

  • Concept, Notation and Specification of Sets: a set as a well-defined collection of distinct objects; Roster/Tabular method ({1, 2, 3, 4, 5}); Set-builder/Rule method ({x : x is a prime number less than 10}); Venn diagram notation; membership (∈ and ∉); empty set (∅ or {}); universal set (U)
  • Types of Sets: finite sets (countable elements — e.g., {vowels in English alphabet}), infinite sets (e.g., natural numbers ℕ), singleton set (exactly one element), null/empty set (no elements), equal sets (same elements regardless of order), equivalent sets (same cardinality), subsets (A ⊆ B — every element of A is in B), proper subsets (A ⊂ B — A is subset but A ≠ B), power set P(A) (set of all subsets of A — if |A| = n then |P(A)| = 2ⁿ), universal set; disjoint sets (no common elements)
  • Operations on Sets and Venn Diagrams:
    • Union: A ∪ B = {x : x ∈ A or x ∈ B} — Venn diagram (entire shaded area of both circles)
    • Intersection: A ∩ B = {x : x ∈ A and x ∈ B} — Venn diagram (overlapping region only)
    • Difference: A − B = {x : x ∈ A and x ∉ B} — Venn diagram (A region excluding overlap)
    • Complement: A’ = U − A = {x : x ∈ U and x ∉ A} — Venn diagram (everything outside A)
    • Symmetric difference: A △ B = (A ∪ B) − (A ∩ B) = (A − B) ∪ (B − A) — elements in A or B but not both
  • Laws of Algebra of Sets (stated without proof — for memorization and application):
    • Identity laws: A ∪ ∅ = A; A ∩ U = A
    • Domination laws: A ∪ U = U; A ∩ ∅ = ∅
    • Idempotent laws: A ∪ A = A; A ∩ A = A
    • Complement laws: A ∪ A’ = U; A ∩ A’ = ∅; (A’)’ = A; U’ = ∅; ∅’ = U
    • Commutative laws: A ∪ B = B ∪ A; A ∩ B = B ∩ A
    • Associative laws: (A ∪ B) ∪ C = A ∪ (B ∪ C); (A ∩ B) ∩ C = A ∩ (B ∩ C)
    • Distributive laws: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C); A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
    • De Morgan’s laws: (A ∪ B)’ = A’ ∩ B’; (A ∩ B)’ = A’ ∪ B’
    • Absorption laws: A ∪ (A ∩ B) = A; A ∩ (A ∪ B) = A
  • Cardinal Number of Sets and Problems Related to Sets: |A| or n(A) = number of elements in set A; cardinal number of power set — |P(A)| = 2^|A|; inclusion-exclusion principle for two sets: |A ∪ B| = |A| + |B| − |A ∩ B|; for three sets: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |B ∩ C| − |A ∩ C| + |A ∩ B ∩ C|; practical word problems using Venn diagrams — “In a class of 60 students, 35 study Mathematics, 28 study Physics, and 15 study both — how many study neither?” — common exam question type; survey problems

1.2 Real Number System

  • Real Number System: natural numbers ℕ ⊂ whole numbers W ⊂ integers ℤ ⊂ rational numbers ℚ ⊂ real numbers ℝ; irrational numbers (non-terminating, non-repeating — √2, π, e); field axioms of real numbers (closure, associativity, commutativity, identity, inverse, distributivity); order axioms (trichotomy, transitivity, addition preserving, multiplication preserving)
  • Intervals: open interval (a, b) = {x : a < x < b}; closed interval [a, b] = {x : a ≤ x ≤ b}; half-open intervals (a, b] and [a, b); infinite intervals (a, ∞), (−∞, b], (−∞, ∞) = ℝ; representing intervals on number line; interval notation vs. set builder notation
  • Absolute Value of Real Number: |x| = x if x ≥ 0; |x| = −x if x < 0; geometrical interpretation as distance from origin; properties — |x| ≥ 0; |xy| = |x||y|; |x/y| = |x|/|y| (y ≠ 0); |x + y| ≤ |x| + |y| (triangle inequality); solving equations and inequalities involving absolute value — |x − 3| = 5 (two solutions: x = 8 or x = −2); |x − a| < r ↔ a − r < x < a + r

1.3 Complex Numbers

  • Introduction to Complex Numbers: motivation (solving x² + 1 = 0 — no real solution); imaginary unit i = √(−1), i² = −1; complex number z = a + bi where a = real part (Re(z)), b = imaginary part (Im(z)); ℂ as extension of ℝ
  • Geometrical Representation (Argand Diagram): complex number a + bi represented as point (a, b) in the complex plane (Argand plane); real axis (horizontal), imaginary axis (vertical); modulus |z| = √(a² + b²) = distance from origin; argument arg(z) = θ = arctan(b/a) — angle from positive real axis; polar form z = r(cos θ + i sin θ) = re^(iθ) (Euler’s formula — mentioned but derivation not required at this level)
  • Simple Algebraic Properties of Complex Numbers:
    • Addition: (a + bi) + (c + di) = (a + c) + (b + d)i — add real and imaginary parts separately
    • Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i — use FOIL and i² = −1
    • Conjugate: z̄ = a − bi; properties — z + z̄ = 2Re(z); z − z̄ = 2Im(z)i; z · z̄ = a² + b² = |z|²
    • Absolute value (modulus): |z| = √(a² + b²); |z₁z₂| = |z₁||z₂|; |z̄| = |z|
    • Inverse (reciprocal): z⁻¹ = z̄/|z|² = (a − bi)/(a² + b²); division — multiply numerator and denominator by conjugate of denominator
    • Square root of complex number: finding √(a + bi) by setting (x + yi)² = a + bi and solving

Unit 2: Relation, Functions and Graph (8 teaching hours)

2.1 Relations

  • Ordered Pairs: (a, b) ≠ (b, a) unless a = b; equality of ordered pairs — (a, b) = (c, d) iff a = c and b = d
  • Cartesian Product: A × B = {(a, b) : a ∈ A, b ∈ B}; |A × B| = |A| × |B|; A × B ≠ B × A in general; A × ∅ = ∅; A × (B ∪ C) = (A × B) ∪ (A × C)
  • Relation: a subset R of A × B; (a, b) ∈ R written as aRb; domain of R = set of all first elements; range of R = set of all second elements; representing relations — arrow diagrams, matrix (relation matrix), coordinate graph
  • Inverse of a Relation: if R ⊆ A × B, then R⁻¹ = {(b, a) : (a, b) ∈ R} ⊆ B × A; domain of R⁻¹ = range of R; range of R⁻¹ = domain of R
  • Types of Relations (on a set A — i.e., R ⊆ A × A):
    • Reflexive: aRa for all a ∈ A (every element is related to itself); e.g., “≤” on ℝ is reflexive (a ≤ a always)
    • Symmetric: if aRb then bRa for all a, b ∈ A; e.g., “is a sibling of” is symmetric; “≤” is NOT symmetric
    • Transitive: if aRb and bRc then aRc for all a, b, c ∈ A; e.g., “<” is transitive; “is adjacent to” is NOT
    • Equivalence relation: reflexive AND symmetric AND transitive; e.g., “≡ (mod n)” — congruence modulo n; equivalence relations partition the set into equivalence classes
    • Antisymmetric: if aRb and bRa then a = b; e.g., “≤” is antisymmetric; “partial order” requires reflexive + antisymmetric + transitive

2.2 Functions

  • Definition of Function: a function f: A → B is a special relation where every element of domain A is related to exactly one element of codomain B; f(a) = b means element a maps to b (b is the image of a, a is the pre-image of b); domain, codomain, range (image set — subset of codomain); every function is a relation but not every relation is a function
  • Domain and Range of a Function: finding domain (all x values for which f(x) is defined — watch for: denominator ≠ 0, even root of non-negative, log of positive); finding range (all possible output values — by solving for x or by analysis)
  • Inverse Functions: f⁻¹: B → A exists only when f is bijective (one-to-one and onto); finding f⁻¹ — swap x and y, solve for y; e.g., f(x) = 2x + 3 → f⁻¹(x) = (x − 3)/2; verifying — f(f⁻¹(x)) = x and f⁻¹(f(x)) = x; graph of f⁻¹ is reflection of graph of f across y = x line
  • Types of Functions:
    • One-to-one (Injective): f(a₁) = f(a₂) implies a₁ = a₂ (different inputs give different outputs); horizontal line test
    • Onto (Surjective): range = codomain (every element of B is the image of some element in A)
    • Bijective (One-to-one Correspondence): both injective and surjective; required for inverse to exist
    • Identity function: f(x) = x for all x; maps every element to itself
    • Constant function: f(x) = c for all x (some fixed constant c); horizontal line graph
  • Special Functions and Their Graphs:
    • Algebraic functions: linear f(x) = mx + b (straight line, slope m, y-intercept b), quadratic f(x) = ax² + bx + c (parabola — vertex, axis of symmetry, roots using quadratic formula), cubic f(x) = ax³ + bx² + cx + d
    • Trigonometric functions: sin x, cos x, tan x, cot x, sec x, csc x — graphs, period, amplitude, domain, range, transformations (phase shift, vertical shift)
    • Exponential function: f(x) = aˣ (a > 0, a ≠ 1) — exponential growth (a > 1) and decay (0 < a < 1); graph always passes through (0, 1); horizontal asymptote y = 0; natural exponential f(x) = eˣ (e ≈ 2.71828)
    • Logarithmic function: f(x) = logₐx (inverse of exponential); domain x > 0; range all reals; graph passes through (1, 0); vertical asymptote x = 0; natural log f(x) = ln x (base e)
  • Composite Function: (f ∘ g)(x) = f(g(x)) — applying g first then f; domain of f ∘ g = {x ∈ domain(g) : g(x) ∈ domain(f)}; note — f ∘ g ≠ g ∘ f in general; worked examples with algebraic, exponential, trigonometric functions; finding formula, domain, and range of composite
  • Practical work (Mathematica): plotting graphs of all function types; exploring transformations; finding domain and range graphically; verifying inverse functions by composition and by reflecting graph

Unit 3: Sequence and Series (8 teaching hours)

  • Sequence and Series: a sequence is an ordered list of numbers (terms) — a₁, a₂, a₃, …, aₙ; a series is the sum of terms of a sequence — Σaₙ; finite vs. infinite sequences; general term (nth term formula)
  • Arithmetic Sequence (AP): each term differs from the previous by a constant (common difference d); general term aₙ = a₁ + (n−1)d; sum of n terms Sₙ = n/2(2a₁ + (n−1)d) = n/2(a₁ + aₙ); example — 2, 5, 8, 11, … (d = 3)
  • Arithmetic Mean (AM): AM of two numbers a and b = (a + b)/2; inserting n arithmetic means between a and b — common difference d = (b − a)/(n + 1); sum of n terms property (sum = n × AM)
  • Properties of Arithmetic Sequences: if a, b, c are in AP then b = (a + c)/2 (b is AM of a and c); three numbers in AP: take (a − d), a, (a + d) as representatives; sum of first n natural numbers = n(n+1)/2; sum of squares of first n natural numbers = n(n+1)(2n+1)/6; sum of cubes = [n(n+1)/2]²
  • Geometric Sequence (GP): each term is a constant multiple (common ratio r) of the previous; general term aₙ = a₁ · r^(n−1); sum of n terms Sₙ = a₁(rⁿ − 1)/(r − 1) for r ≠ 1; Sₙ = na₁ for r = 1; example — 3, 6, 12, 24, … (r = 2)
  • Geometric Mean (GM): GM of two positive numbers a and b = √(ab); inserting n geometric means between a and b — common ratio r = (b/a)^(1/(n+1)); product of terms of a GP = (GM)ⁿ for n terms
  • Sum of Infinite Geometric Series: S∞ = a₁/(1 − r) for |r| < 1; diverges for |r| ≥ 1; applications — recurring decimals as fractions (0.333… = 1/3; 0.272727… = 3/11 — worked examples); compound interest, population models
  • Harmonic Sequence (HP): a sequence is HP if the reciprocals form an AP; general term of HP has no simple direct formula (work through reciprocals); harmonic mean HM of two numbers a and b = 2ab/(a + b)
  • Properties of AP, GP, HP sequences (with proofs where required): if A, G, H are the AM, GM, HM of two positive unequal numbers a and b respectively, then: A ≥ G ≥ H (AM ≥ GM ≥ HM — standard inequality); G² = A × H (GM² = AM × HM — proof: A = (a+b)/2, G = √(ab), H = 2ab/(a+b) → G² = ab = (a+b)/2 × 2ab/(a+b) = A × H ✓); A, G, H are themselves in GP (A/G = G/H); 2025 past exam question: “If arithmetic mean, geometric mean and harmonic mean between two unequal positive numbers are A, G, H respectively, then prove that (a) G² = A × H and (b) A ≥ G ≥ H” — classic CAMT104 Group C question
  • Taylor’s Theorem (without proof): if f(x) is infinitely differentiable at x = a, then f(x) = f(a) + f'(a)(x−a)/1! + f”(a)(x−a)²/2! + f”'(a)(x−a)³/3! + …; for a = 0 this is Maclaurin’s series; approximating functions as polynomials (truncating the series)
  • Taylor’s Series for Common Functions (from Taylor’s theorem): eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + … (for all x); sin x = x − x³/3! + x⁵/5! − x⁷/7! + … (for all x); cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + … (for all x); ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + … (for |x| ≤ 1, x ≠ −1); (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + … (binomial series — for |x| < 1, n real)
  • Exponential Series: eˣ Taylor series reviewed; computing e = 1 + 1 + 1/2! + 1/3! + … = Σ(1/n!); natural exponential as a series — convergence for all x; applications in computing (exponential function approximation in numerical libraries)

Unit 4: Matrices and Determinants (10 teaching hours — joint largest unit)

  • Introduction to Matrices: a matrix as a rectangular array of numbers in rows and columns; notation A = [aᵢⱼ]ₘₓₙ where aᵢⱼ is the element in row i, column j; order (m × n — m rows, n columns); matrix as a way to represent systems of equations, data tables, transformations
  • Types of Matrices: row matrix (1 × n), column matrix (m × 1), square matrix (m = n), null/zero matrix (all elements 0), identity matrix I (square — 1s on main diagonal, 0s elsewhere — Iₙ), diagonal matrix (square — non-zero only on main diagonal), scalar matrix (diagonal with all diagonal elements equal), upper triangular matrix (all elements below main diagonal = 0), lower triangular matrix, symmetric matrix (A = Aᵀ — aᵢⱼ = aⱼᵢ), skew-symmetric matrix (A = −Aᵀ — aᵢⱼ = −aⱼᵢ; diagonal elements = 0)
  • Equality of Matrices: A = B iff they have the same order AND aᵢⱼ = bᵢⱼ for all i, j
  • Algebra of Matrices:
    • Addition/Subtraction: only defined for matrices of same order; (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ; commutative (A + B = B + A), associative, identity (A + 0 = A), inverse (A + (−A) = 0)
    • Scalar multiplication: (kA)ᵢⱼ = k · aᵢⱼ; k(A + B) = kA + kB; (k + l)A = kA + lA
    • Matrix multiplication: A (m×n) × B (n×p) = C (m×p); (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ — row of A dot product with column of B; number of columns of A must equal number of rows of B; NOT commutative (AB ≠ BA in general); associative (A(BC) = (AB)C); distributive (A(B + C) = AB + AC); AI = IA = A; A · 0 = 0
  • Transpose of a Matrix: Aᵀ (or A’) — rows and columns interchanged; (Aᵀ)ᵢⱼ = aⱼᵢ; order of Aᵀ is n×m if A is m×n; properties — (Aᵀ)ᵀ = A; (A + B)ᵀ = Aᵀ + Bᵀ; (kA)ᵀ = kAᵀ; (AB)ᵀ = BᵀAᵀ (order reverses)
  • Determinant: a scalar value associated with every square matrix; |A| or det(A); 2×2: |A| = ad − bc for [[a,b],[c,d]]; 3×3: expansion along first row — |A| = a₁₁(M₁₁) − a₁₂(M₁₂) + a₁₃(M₁₃) where Mᵢⱼ is the (n−1)×(n−1) minor obtained by deleting row i and column j
  • Minors and Cofactors: minor Mᵢⱼ = determinant of submatrix obtained by deleting row i and column j; cofactor Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ (sign depends on position — checkerboard of +/−); determinant = sum of products of elements of any row (or column) with their cofactors — cofactor expansion (Laplace expansion); expanding along any row or column gives same result
  • Properties of Determinants (stated without proof, applied in problems):
    • Reflection: |Aᵀ| = |A|
    • Switching rows/columns: determinant changes sign
    • Scaling: multiplying one row/column by k multiplies determinant by k; |kA| = kⁿ|A| for n×n matrix
    • Adding multiple of one row to another: determinant unchanged
    • Repeated rows/columns: determinant = 0
    • Row/column of zeros: determinant = 0
    • Triangular matrix: determinant = product of diagonal elements
    • Product: |AB| = |A| · |B|
  • Singular and Non-Singular Matrix: singular if |A| = 0 (no inverse exists); non-singular if |A| ≠ 0 (inverse exists); significance — system Ax = b has unique solution iff A is non-singular
  • Adjoint and Inverse of Matrices: adjoint (adj A) = transpose of the cofactor matrix [Cᵢⱼ]ᵀ; inverse A⁻¹ = adj(A) / |A| (only when |A| ≠ 0); verification: A · A⁻¹ = A⁻¹ · A = I; for 2×2: A = [[a,b],[c,d]], A⁻¹ = (1/(ad−bc))[[d,−b],[−c,a]]; for 3×3: compute all 9 cofactors, form cofactor matrix, transpose to get adj(A), divide by det(A)
  • Linear Transformations: a function T: ℝⁿ → ℝᵐ is linear if T(u + v) = T(u) + T(v) and T(cu) = cT(u); every linear transformation can be represented by a matrix — T(x) = Ax; standard matrix of T; examples — rotation (by angle θ: [[cos θ, −sin θ],[sin θ, cos θ]]), reflection (across x-axis: [[1,0],[0,−1]]), scaling (by factor k: [[k,0],[0,k]]), shear; composition of transformations = product of matrices; applications in computer graphics — 2D transformations of images and game objects
  • Orthogonal Transformations: a transformation T is orthogonal if it preserves lengths and angles (dot products); orthogonal matrix Q satisfies QᵀQ = QQᵀ = I (i.e., Q⁻¹ = Qᵀ); |det(Q)| = 1; examples — rotation, reflection; orthogonal matrices preserve distance (||Qx|| = ||x||); used in computer graphics (rotations in 3D) and numerical methods (QR decomposition concept)
  • Rank of a Matrix: rank r = maximum number of linearly independent rows (= maximum number of linearly independent columns); equivalently, rank = order of the largest non-zero minor; finding rank using row echelon form (Gaussian elimination — systematic row operations); rank of A = rank of Aᵀ; rank(A) ≤ min(m, n) for m×n matrix; rank < n → columns are linearly dependent → |A| = 0 for square matrix
  • Solving System of Linear Equations using Matrices: Ax = b (matrix form); using inverse: x = A⁻¹b (only when A is non-singular — unique solution); Cramer’s rule: xᵢ = |Aᵢ|/|A| where Aᵢ is the matrix with column i replaced by b; Gaussian elimination (row reduction to row echelon form — applicable to any system including inconsistent and dependent); consistency analysis — unique solution (rank(A) = rank([A|b]) = n), infinitely many solutions (rank(A) = rank([A|b]) < n), no solution (rank(A) < rank([A|b]))
  • Practical work (MATLAB): matrix operations in MATLAB (defining matrices, matrix arithmetic, det(), inv(), rank(), linsolve()); solving systems of equations; computing eigenvalues (eig() command — concept of eigenvalues and eigenvectors introduced); plotting linear transformations geometrically

Unit 5: Analytical Geometry (10 teaching hours — joint largest unit)

5.1 Conic Sections

  • Definition of Conic Sections: a conic section is the intersection of a plane with a right circular double cone; four standard conics based on angle of the intersecting plane:
    • Circle: plane perpendicular to cone axis (eccentricity e = 0)
    • Ellipse: plane intersects one nappe at angle (0 < e < 1)
    • Parabola: plane parallel to one slant side of cone (e = 1)
    • Hyperbola: plane intersects both nappes (e > 1)
  • Related Terms: focus (F), directrix (d), vertex, eccentricity (e = distance from point to focus / distance from point to directrix), latus rectum (chord through focus perpendicular to axis), axis (of symmetry), centre (for ellipse and hyperbola)
  • Circle:
    • Definition: locus of all points equidistant (radius r) from a fixed point (centre (h, k))
    • Standard equation: (x − h)² + (y − k)² = r²; at origin: x² + y² = r²
    • General equation: x² + y² + Dx + Ey + F = 0; centre (−D/2, −E/2), radius √((D/2)² + (E/2)² − F)
    • 2025 exam question: “Find the equation of a circle passing through points (1,2), (3,1) and (0,1)” — solving the system of three equations in three unknowns D, E, F
    • Graphs and related problems
  • Parabola:
    • Definition: locus of points equidistant from focus and directrix (e = 1)
    • Standard forms: y² = 4ax (horizontal axis, focus (a,0), directrix x = −a); y² = −4ax (opening left); x² = 4ay (vertical axis, focus (0,a), directrix y = −a); x² = −4ay (opening downward)
    • Vertex at origin; vertex form with vertex at (h, k): (y − k)² = 4a(x − h) or (x − h)² = 4a(y − k)
    • Key features: vertex, focus, directrix, axis of symmetry, latus rectum (length 4a)
    • Equation and graph; finding equation given conditions; applications — parabolic reflectors (satellite dishes, headlights, telescope mirrors)
  • Ellipse:
    • Definition: locus of points where sum of distances from two fixed points (foci F₁, F₂) is constant (= 2a)
    • Standard equation (major axis along x-axis): x²/a² + y²/b² = 1 (a > b > 0); foci at (±c, 0) where c² = a² − b²; eccentricity e = c/a (0 < e < 1)
    • Standard equation (major axis along y-axis): x²/b² + y²/a² = 1 (a > b > 0)
    • Key features: semi-major axis a, semi-minor axis b, centre (0,0), vertices (±a, 0) or (0, ±a), co-vertices, foci, latus rectum (length 2b²/a); equation of ellipse with centre at (h, k)
    • Classifying by eccentricity: e close to 0 → nearly circular; e close to 1 → very elongated
    • Applications — planetary orbits (Kepler’s first law — planets orbit the sun in elliptical paths with the sun at one focus — relevant to satellite communication); elliptical room acoustics (Statuary Hall, US Capitol)
  • Hyperbola:
    • Definition: locus of points where |difference| of distances from two foci = constant (= 2a)
    • Standard equation (transverse axis along x-axis): x²/a² − y²/b² = 1; foci at (±c, 0) where c² = a² + b²; eccentricity e = c/a (e > 1)
    • Standard equation (transverse axis along y-axis): y²/a² − x²/b² = 1
    • Key features: vertices (±a, 0), transverse axis (2a), conjugate axis (2b), asymptotes y = ±(b/a)x; centre, foci, latus rectum
    • Applications — hyperbolic navigation (LORAN — Long Range Navigation); cooling tower design; Cassegrain telescope
  • Polar Equations of Conic Sections: with focus at the pole (origin) — r = ed/(1 − e cos θ) or r = ed/(1 + e cos θ); e < 1 → ellipse, e = 1 → parabola, e > 1 → hyperbola; polar equations of lines, circles; converting between polar and Cartesian
  • Practical work (Mathematica/MATLAB): plotting all conic sections; exploring effect of changing parameters; converting between standard and general forms; finding conic from general second-degree equation by completing the square

5.2 Vectors in Space

  • Vectors in Space: a vector as a quantity with both magnitude and direction; position vector of point P(x, y, z) from origin O = xi + yj + zk (in terms of unit vectors i, j, k along x, y, z axes); 2D vectors extended to 3D; column vector notation [x, y, z]ᵀ
  • Algebra of Vectors in Space: addition (u + v = (u₁+v₁)i + (u₂+v₂)j + (u₃+v₃)k), subtraction, scalar multiplication (ku = ku₁i + ku₂j + ku₃k); commutative, associative, distributive properties; zero vector (0, 0, 0); negative vector
  • Length (Magnitude) of a Vector: |u| = √(u₁² + u₂² + u₃²); Euclidean norm; length = 0 iff zero vector
  • Distance Between Two Points: d(P₁, P₂) = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²); Pythagorean theorem extended to 3D
  • Unit Vector: vector of magnitude 1; û = u/|u| (dividing by magnitude); standard unit vectors i = (1,0,0), j = (0,1,0), k = (0,0,1); expressing any vector as a linear combination of i, j, k; direction cosines (cosα = u₁/|u|, cosβ = u₂/|u|, cosγ = u₃/|u|) and direction ratios (u₁ : u₂ : u₃)
  • Null Vector: zero vector 0 = (0, 0, 0); zero magnitude, no defined direction; additive identity in vector algebra
  • Scalar (Dot) Product: u · v = u₁v₁ + u₂v₂ + u₃v₃ = |u||v|cos θ (θ = angle between u and v); commutative (u · v = v · u); distributive over addition; u · u = |u|²; perpendicularity condition: u · v = 0 iff u ⊥ v; angle between vectors: cos θ = u·v/(|u||v|); projection of u onto v: (u·v/|v|²)v; application — finding angle between two lines in 3D; work done by force (W = F · d); 2025 past exam: “Define scalar product. Using vector method, in triangle ABC prove that c = a cos B + b cos A” — direct CAMT104 Group C question
  • Cross (Vector) Product of Two Vectors: u × v = (u₂v₃−u₃v₂)i − (u₁v₃−u₃v₁)j + (u₁v₂−u₂v₁)k; computed as 3×3 determinant with first row i, j, k and subsequent rows u and v; magnitude: |u × v| = |u||v|sin θ; direction: perpendicular to both u and v (right-hand rule); NOT commutative (u × v = −v × u); distributive; u × u = 0; i × j = k, j × k = i, k × i = j; geometric interpretation — |u × v| = area of parallelogram with sides u and v; 2025 past exam: “Find the area of the parallelogram determined by vectors 2j + 3k and k” — applying cross product
  • Scalar Triple Product: u · (v × w) = determinant of 3×3 matrix with rows u, v, w; geometric interpretation — |u · (v × w)| = volume of parallelepiped with edges u, v, w; three vectors are coplanar iff their scalar triple product = 0; cyclic permutation property: u · (v × w) = v · (w × u) = w · (u × v)
  • Vector Triple Product: u × (v × w) = (u · w)v − (u · v)w (BAC − CAB rule); used in advanced geometry and physics
  • Geometrical Interpretation and Related Examples: using vectors to prove geometric theorems (midpoint theorem, diagonals of parallelogram, triangle inequality); finding equation of line in vector form r = a + tb; finding equation of plane in vector form n · (r − a) = 0; practical work (MATLAB): vector operations, dot product, cross product, plotting vectors in 3D

Unit 6: Permutation and Combination (9 teaching hours)

  • Basic Principle of Counting:
    • Multiplication principle (Fundamental counting principle): if task 1 can be done in m ways and task 2 can be done in n ways (independently), then both tasks can be done in m × n ways; generalization to k tasks: m₁ × m₂ × … × mₖ; examples — outfit combinations, license plates, multiple-choice tests
    • Addition principle: if task 1 can be done in m ways and task 2 in n ways and they are mutually exclusive (cannot both occur), then either task can be done in m + n ways
    • Combined use of both principles in complex counting problems
  • Factorial Notation: n! = n × (n−1) × (n−2) × … × 2 × 1; 0! = 1 (defined); 1! = 1; growing rapidly — 5! = 120, 10! = 3,628,800, 20! ≈ 2.4 × 10¹⁸; using factorials in permutation and combination formulas
  • Permutation — Set of Objects All Different: a permutation is an arrangement of objects in a definite order; the number of permutations of n distinct objects taken r at a time: P(n, r) = nPr = n!/(n−r)!; P(n, n) = n! (all n objects arranged); derivation (first position n choices, second n−1, …, rth position n−r+1 choices — multiply); worked examples — number of ways to arrange 4 books in 4 positions, number of 3-digit numbers from digits {1,2,3,4,5} without repetition
  • Permutation — Set of Objects Not All Different: n objects where n₁ are of type 1, n₂ of type 2, …, nₖ of type k (n₁ + n₂ + … + nₖ = n); number of distinct arrangements = n!/(n₁! × n₂! × … × nₖ!); 2025 past exam question: “The letters of the word are to be shuffled to form different words such that (a) the vowels always come together (b) the vowels may occupy only the odd positions” — standard CAMT104 permutation problem
  • Circular Arrangement: arranging n distinct objects in a circle; fixing one object as reference, remaining (n−1) arranged in (n−1)! ways; circular permutations of n = (n−1)!; distinguishing clockwise from counterclockwise; arrangements when objects can be flipped (necklace problems — divide by 2 for identical clockwise/counterclockwise); examples — seating people around a circular table, arranging colored beads on a ring
  • Permutation — Repeated Use of the Same Object: when repetition is allowed; number of r-permutations from n objects with repetition = nʳ (each position has n choices independently); examples — number of 4-character passwords from 26 letters with repetition, number of functions from a set of r elements to a set of n elements
  • Combination — Things All Different: a combination is a selection without regard to order; number of ways to choose r objects from n distinct objects (without repetition, without regard to order): C(n, r) = nCr = n!/[r!(n−r)!]; nCr = nPr/r! (divide permutations by r! to remove arrangements of chosen items); worked examples — choosing committee of 3 from 10 people, number of ways to deal a hand of 5 cards, choosing r problems to attempt from n given problems
  • Properties of Combinations (with proofs):
    • nC₀ = 1; nCₙ = 1 (choosing none or all)
    • nCr = nCₙ₋r (complementary — choosing r same as rejecting n−r)
    • nCr + nCr₋₁ = (n+1)Cr (Pascal’s identity — choosing r from n+1 = choose r from n + choose r−1 from n; foundation of Pascal’s triangle)
    • nC₁ = n; nC₂ = n(n−1)/2
    • Sum of all combinations: nC₀ + nC₁ + nC₂ + … + nCₙ = 2ⁿ (total subsets of n-element set)
    • r · nCr = n · (n-1)C(r-1) (used in summing combination series)
  • Combined Problems: problems requiring both permutation and combination reasoning; constrained arrangement problems — “arrange 3 men and 2 women in a row such that no two women are adjacent”; selection-and-arrangement problems — “choose 4 people from 10 and arrange them in a line”; problems involving restrictions (certain objects must be included/excluded); distributing objects among groups

Laboratory Work (Practical — 1 hour per week using Mathematica and/or MATLAB)

  • Unit 1 — Set Theory practical: using Mathematica to verify set operations; computing union, intersection, complement, and symmetric difference; generating power sets; solving Venn diagram problems numerically; visualizing sets using Venn diagrams
  • Unit 2 — Functions and Graphs (Mathematica): plotting algebraic (linear, quadratic, cubic), trigonometric, exponential, and logarithmic functions; exploring domain and range graphically; verifying inverse functions by plotting f and f⁻¹ (reflection across y=x); plotting composite functions; exploring function transformations (shifts, stretches, reflections)
  • Unit 3 — Sequences and Series: computing finite and infinite series sums; verifying AM ≥ GM ≥ HM numerically; plotting partial sums of Taylor series vs. exact function; exploring convergence of geometric series
  • Unit 4 — Matrices and Determinants (MATLAB): defining and operating on matrices; computing determinant (det()), inverse (inv()), rank (rank()); solving systems of linear equations (using backslash operator A\b or linsolve()); plotting 2D linear transformations (rotation, reflection, scaling); computing eigenvalues and eigenvectors (eig())
  • Unit 5 — Conic Sections and Vectors (Mathematica/MATLAB): plotting all four conic sections with varying parameters; finding focus, directrix, and eccentricity from equations; performing vector operations in 3D (dot product, cross product, magnitude); visualizing vectors in 3D; computing areas of parallelograms and volumes of parallelepipeds using cross and scalar triple products
  • Unit 6 — Permutation and Combination: computing factorials, permutations (nchoosek complement), and combinations; verifying Pascal’s identity and triangle; exploring counting problems computationally

Recommended Textbooks and References

Primary Textbook

  • Thomas, G.B. and Finney, R.S. Calculus with Analytic Geometry, 9th Edition. Addison Wesley — the officially prescribed primary textbook for CAMT104; covers calculus and analytic geometry content; widely available across Nepal’s BCA colleges; analytical geometry and vector sections directly relevant to Units 5

Reference Books (as listed in the official CAMT104 syllabus)

  • Bajracharya, D.R., Shrestha, R.M. et al. Basic Mathematics I and II. Sukunda Pustak Bhawan, Nepal — the most accessible Nepal-authored reference; directly aligned with TU BCA mathematics content; widely available across Kathmandu and other cities
  • Bydnick, F.S. Applied Mathematics for Business, Economics and the Social Sciences. McGraw-Hill Ryerson — useful for application-oriented mathematics problems
  • Monga, G.S. Mathematics for Management and Economics. Vikas Publishing House Pvt. Ltd., New Delhi
  • Paudel, K.C., GC, F.B. et al. Higher Secondary Mathematics. Asmita Publication and Distributors Pvt. Ltd., Nepal — familiar for NEB Grade 12 graduates; provides continuity from pre-university mathematics
  • Upadhayay, H.P., Paudel, K.C. et al. Elements of Business Mathematics. Pinnacle Publication
  • Yamane, T. Mathematics for Economists. Prentice Hall of India

Difference Between NEB Grade 12 Mathematics and TU BCA Mathematics I

Students entering TU BCA First Semester from NEB Grade 12 Mathematics will find both familiar territory and significant new content. NEB Grade 12 Basic Mathematics covers algebra (permutations, combinations, binomial theorem, group theory, complex numbers, quadratic equations, sequences, matrices), trigonometry, coordinate geometry (conic sections), vectors, calculus, computational methods, and mechanics — providing substantial overlap with CAMT104’s content.

However, CAMT104 differs from NEB in three important ways. First, the course is specifically oriented toward computing applications — set theory (directly relevant to database design and programming), functions (directly relevant to algorithm design), and matrix transformations (directly relevant to computer graphics) are taught with these applications explicitly in mind. Second, the practical component using Mathematica and MATLAB introduces computational mathematics tools used in the IT industry — these are entirely absent from the NEB curriculum. Third, some topics are more formally rigorous at university level — types of relations (reflexive, symmetric, transitive, equivalence — with formal definitions and proofs), Taylor’s theorem (stated formally), properties of determinants (applied to rank and linear systems), and linear/orthogonal transformations are treated with greater precision than NEB equivalents.

Tips to Score High in CAMT104

  • Master the cardinal number formula and inclusion-exclusion principle for Unit 1: Set theory problems appear in almost every past paper — “In a survey of 100 people, 60 read newspaper A, 45 read B, 20 read both — how many read neither?” These word problems using |A ∪ B| = |A| + |B| − |A ∩ B| are straightforward once the formula is internalized. Practice 10–15 varied word problems to build speed and accuracy.
  • The AM-GM-HM inequality is a guaranteed Group C question: Proving G² = A × H and A ≥ G ≥ H is a classic CAMT104 long-answer question — confirmed in the 2025 past paper. Know both the algebraic proof (substituting formulas for A, G, H in terms of a and b) and be able to illustrate with numerical examples.
  • For matrices, practice inverse and rank computation for 3×3: The most common matrix exam questions are: finding the inverse of a 3×3 matrix using the adjoint method (computing all 9 cofactors, transposing, dividing by det), computing the determinant by cofactor expansion, finding the rank by row reduction, and solving a 3×3 system of equations using Cramer’s rule or Gaussian elimination. Practise the inverse computation especially — it is lengthy but completely mechanical once the steps are memorized.
  • Conic sections require recognizing standard forms immediately: For all four conics, you must be able to: identify the type from an equation, find all standard features (vertex/centre, focus/foci, directrix, eccentricity, latus rectum), graph the conic, and find the equation given certain features. The circle equation from three points (2025 past paper) requires solving a 3×3 linear system — connect this to Unit 4 (matrices).
  • Vectors — know both the dot product and cross product applications: Past papers consistently test “find the angle between two vectors” (dot product), “prove a geometric identity using vectors” (dot product + component form), and “find area of parallelogram/triangle” (cross product magnitude). The 2025 exam specifically asked for the area of the parallelogram determined by vectors 2j + 3k and k — a direct cross product computation.
  • Permutation problems with constraints are the most exam-frequent: The 2025 paper asked about arranging letters of a word such that vowels come together and such that vowels occupy only odd positions. These constrained permutation problems follow a pattern: (a) treat the constrained group as a single object for the permutation count, then multiply by internal arrangements; (b) for position-restricted problems, fill restricted positions first then arrange the rest. Practice 15–20 such problems across various constraint types.
  • Use Bajracharya et al. Basic Mathematics as your primary reference: This Nepali-authored textbook (Sukunda Pustak Bhawan) is the most closely aligned with TU BCA Mathematics I content and is widely available across Nepal. It covers all six units in a style directly matched to how TU examination questions are worded. Thomas and Finney (the official prescribed textbook) is more advanced — use it for analytical geometry and vector sections where it is strongest.
  • Solve the 2025 past paper under timed conditions: The 2025 TU FOHSS CAMT104 exam is now available on Studocu.com. Group B included questions on permutation with constraints, complex number operations, and function composition. Group C included the AM-GM-HM proof, vector proof of cosine rule, and the circle through three points. Timing yourself on this exact paper is the most targeted exam preparation available.

Frequently Asked Questions (FAQs)

What is the course code and credit hours for Mathematics I in TU BCA?

The course code is CAMT104 (also referenced as CACS104 in some official documents), worth 3 credit hours. Class load is 5 hours per week (3 theory + 1 tutorial + 1 practical using Mathematica/MATLAB). Full marks: 60 (theory) + 20 (internal/practical) = 80. Pass mark for theory: 24 out of 60.

How many units are in CAMT104 and what do they cover?

CAMT104 has 6 units: Unit 1 — Set Theory and Real & Complex Numbers (8 hours), Unit 2 — Relation, Functions and Graph (8 hours), Unit 3 — Sequence and Series (8 hours), Unit 4 — Matrices and Determinants (10 hours), Unit 5 — Analytical Geometry — Conic Sections and Vectors in Space (10 hours), and Unit 6 — Permutation and Combination (9 hours). Total: approximately 53 teaching hours of theory.

What software is used in the CAMT104 practical component?

The official CAMT104 syllabus specifies Mathematica for Units 2 (functions and graphs) and 5 (conic sections), and MATLAB for Units 4 (matrices and determinants) and 5 (vectors). Both are standard computational mathematics tools used in engineering, science, and data analysis. Some colleges substitute or supplement with Python (NumPy, Matplotlib, SymPy) or GeoGebra depending on available licences.

What is the difference between permutation and combination?

A permutation is an arrangement of objects where the order matters — choosing 3 people for three different roles (president, secretary, treasurer) from 10 is a permutation problem because the same 3 people in different roles give different outcomes. A combination is a selection where order does not matter — choosing 3 people for a committee from 10 is a combination problem because the same 3 people in any order give the same committee. The number of permutations P(n,r) = n!/(n−r)! and the number of combinations C(n,r) = n!/[r!(n−r)!]. Note C(n,r) = P(n,r)/r! — each combination of r items can be arranged in r! ways.

How is CAMT104 related to the BCA programme’s later courses?

CAMT104 directly feeds into Mathematics II (CAMT155 in Second Semester — linear algebra, calculus, differential equations), Probability and Statistics (CACS202 in Third Semester — built on the set theory, combinatorics, and series from CAMT104), Data Structures and Algorithms (where complexity analysis uses summation formulas from sequences), Computer Graphics (where matrix transformations and vectors from Unit 4 and 5 are directly applied), and Discrete Mathematics (where logic, set theory, relations, and combinatorics extend the foundations laid in CAMT104).

Where can I find CAMT104 notes and past papers?

You can find Mathematics I study resources on nneducation.com. The official CAMT104 syllabus PDF is also embedded at the top of this page. The 2025 TU FOHSS board exam paper for CAMT104 is available on Studocu.com (search “BCA TU 1 Semester 2025 Mathematics I”). Notes are available on bcapoint.com, hamrocsit.com, and bcastudyguide.com. Bajracharya et al. Basic Mathematics (Sukunda Pustak Bhawan, Nepal) is the most recommended Nepal-specific textbook for this course.

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