Mathematics-II Notes
Subject Overview
Mathematics-II extends the discrete-math foundation into applied mathematics for computer science — matrix algebra and solving simultaneous equations, statistics and logarithm applications, differential calculus, and integral calculus including numerical integration methods. A 6-credit core theory paper.
Subject at a glance
4
Units
24
Topics
6
Credits
4
Infographics
Unit-wise Syllabus
4 units — open any unit for detailed topic-wise notes, diagrams and important exam questions
- Types of matrices and operations
- determinants (without properties)
- minors
- cofactors
- adjoint and inverse of a matrix
- elementary transformations
- rank of a matrix
- solving simultaneous equations using Cramer's rule and matrix inversion
Inverse
A⁻¹ = adj(A) / det(A)
Exists only if det(A) ≠ 0
Cramer's rule
x = Dx / D, y = Dy / D
Matrix method
X = A⁻¹ B for AX = B
Rank
Order of the largest non-zero minor
Study guide
Matrices solve systems of equations. You study types of matrices and operations, determinants, minors, cofactors, adjoint and inverse, elementary transformations, rank and solving simultaneous equations by Cramer's rule and matrix inversion. Practise finding an inverse by the adjoint method.
- Measures of central tendency (mean, median, mode)
- measures of dispersion (mean deviation, standard deviation, coefficient of variation)
- applications of logarithms to compound interest
- depreciation and annuities
Mean
x̄ = Σx / n
Standard deviation
σ = √[Σ(x − x̄)² / n]
Coefficient of variation
(σ / x̄) × 100
Compound interest
A = P (1 + r/100)^n
Depreciation (reducing balance)
V = P (1 − r/100)^n
Study guide
You study measures of central tendency and dispersion (mean deviation, standard deviation, coefficient of variation) and applications of logarithms to compound interest, depreciation and annuities. Learn each formula with a small example.
- Introduction to differentiation
- derivative of a function of one variable
- power functions
- sum and product of two functions
- function of a function
- differentiation by substitution
- maxima and minima
Power rule
d/dx (xⁿ) = n xⁿ⁻¹
Product rule
(uv)' = u'v + uv'
Quotient rule
(u/v)' = (u'v − uv') / v²
Chain rule
dy/dx = dy/du × du/dx
Maxima and minima
f'(x) = 0; f'' < 0 max, f'' > 0 min
Study guide
Differential calculus studies rates of change. You study differentiation, derivatives of power functions, sums, products and functions of a function, differentiation by substitution and maxima and minima. Practise differentiation rules until they are automatic.
- Indefinite integral
- integration by substitution/parts/partial fractions
- definite integral
- numerical integration — trapezoidal rule
- Simpson's 1/3 and 3/8 rules
Power rule
∫xⁿ dx = xⁿ⁺¹ / (n + 1) + C
By parts
∫u dv = uv − ∫v du
Trapezoidal rule
(h/2) [y0 + 2(y1 + … + yn−1) + yn]
Simpson's 1/3 rule
(h/3) [y0 + 4(odd) + 2(even) + yn]
Study guide
Integral calculus is the reverse of differentiation. You study indefinite integrals, integration by substitution, parts and partial fractions, definite integrals and numerical integration with the trapezoidal rule and Simpson's 1/3 and 3/8 rules. Learn when to use each method.
Why Mathematics-II Matters
Matrices, statistics and calculus are the mathematical tools behind graphics, machine learning, simulations and numerical computing, and they are needed for competitive and entrance exams.
How to Prepare for Mathematics-II
- Solve problems daily and keep a formula sheet.
- Check inverse and equation solutions by substitution.
- Practise numerical integration with a table of values.
Mathematics-II: Frequently Asked Questions
Why is calculus needed in IT?
Calculus underpins optimisation, machine learning, graphics and scientific computing, where rates of change and areas under curves are used.
What is the rank of a matrix?
The rank is the number of linearly independent rows or columns in a matrix.