Mathematics Revision

Perimeter

Perimeter is the total boundary distance enclosing a two-dimensional closed figure.

Square Root

Square root definition: If x² = y, then x is the square root of y (x = √y); positive square root is the principal root.

Fractions

Addition and subtraction of fractions require a common denominator determined by the Least Common Multiple (LCM) of denominators.

Binomial Expressions

Expansion of square of a sum: (a + b)² = a² + 2ab + b² (first term squared + twice product + second term squared).

Congruence of Triangles

Two geometric figures are congruent if they possess identical shape and equal size, coinciding exactly when superimposed.

Area

Area measures the extent of two-dimensional surface within a boundary: Rectangle Area = length × breadth (A = lb).

Factors of Quadratic Expressions

Quadratic trinomials take standard algebraic form: ax² + bx + c, where a, b, and c are real constants and a ≠ 0.

Triangles

Angle sum of a triangle theorem: The sum of the three interior angles of any planar triangle is always 180°.

Inverse Proportion

Inverse proportion defines a relationship where an increase in one variable produces a proportional decrease in the other: product remains…

Data Representation

Pie charts represent component data as sectors of a circular disk proportional to their frequencies: Sector angle = (Frequency / Total…

LCM of Algebraic Expressions

The Least Common Multiple (LCM) of algebraic expressions is the simplest polynomial divisible by each given expression without remainder.

Algebraic Fractions

An algebraic fraction is a rational expression P(x) / Q(x), where P and Q are polynomials and Q(x) ≠ 0.

Percentages

A percentage represents a fraction with a denominator of 100: x% = x / 100.

Equations

Linear equations with one unknown are solved by isolating the variable using balanced inverse operations on both sides.

Parallelograms I

A parallelogram is a quadrilateral with opposite sides parallel.

Parallelograms II

Special parallelograms inherit all standard parallelogram properties while possessing distinct diagnostic characteristics.

Sets

A set is a well-defined collection of distinct objects; universal set (ε or ξ) contains all elements under consideration.

Logarithms I

Logarithm definition: The logarithm of a positive number y to a positive base a (a ≠ 1) is the exponent x to which a must be raised to…

Logarithms II

Common logarithms use base 10 (denoted log₁₀ x or simply lg x).

Linear Relationships

Standard equation of a straight line in the Cartesian coordinate plane is y = mx + c.

Rate

A rate expresses the relationship between two quantities measured in different units: Rate = (First Quantity) / (Second Quantity).

Formulae

Mathematical Formula: An equation expressing a mathematical or physical relationship between two or more variables (e.g., v = u + at, A =…

Triangle Angle Theorems

Angles on a straight line add up to 180°; angles around a single point add up to 360°.

Linear Inequalities

Inequality relations: < (strictly less than), ≤ (less than or equal to), > (strictly greater than), ≥ (greater than or equal to).

Frequency Distributions

Grouped frequency distributions organize continuous numerical data into mutually exclusive class intervals with frequencies (f).

Chord-Centre Theorem

Theorem 1: The perpendicular drawn from the centre of a circle to a chord bisects the chord (if OM ⊥ AB, then AM = MB).

Constructions

Geometric constructions require strictly a pair of compasses and an unmarked straight edge; all construction arcs must remain clearly…

Surface Area and Volume

Right circular cylinder: Curved Surface Area = 2πrh; Total Surface Area = 2πrh + 2πr² = 2πr(h + r).

Probability

Probability of an event A with equally likely outcomes: P(A) = n(A) / n(S), where n(A) is favorable outcomes and n(S) is total sample space…

Real Numbers

Real numbers (ℝ) comprise all Rational numbers (ℚ) and Irrational numbers (ℚ').

Indices and Logarithms - I

Multiplication and division index laws: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ.

Indices and Logarithms - II

Common Logarithms (Base 10): Logarithm of number N to base 10: log10 N = Characteristic + Mantissa.

Surface Area of Solids

Right Circular Cylinder: Curved surface area = 2πrh; Total surface area of closed cylinder = 2πrh + 2πr² = 2πr(h + r).

Volume of Solids

Uniform Prism Volume: Volume = Cross-sectional Area × Length = A × L (e.g., Cuboid V = l × b × h; Triangular prism V = (1/2) b h × L).

Areas of Plane Figures between Parallel Lines

Parallelograms on Same Base and Between Same Parallels: Parallelograms standing on the same base (or equal bases) and lying between the…

Share Market

Shares & Nominal Value: Capital of a company divided into equal units called Shares; Face/Nominal Value is the original statutory price per…

The Midpoint Theorem

The Midpoint Theorem: The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half…

Graphs

Quadratic graph standard form: y = ax² + bx + c (parabolic curve).

Equiangular Triangles

Equiangular Triangles Definition: Two triangles are equiangular if the three interior angles of one triangle are respectively equal to the…

Data Representation and Interpretation

Cumulative frequency is the running total of frequencies up to the upper boundary of each class interval.

Geometric Progressions

Geometric Progression (GP): Sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed…

Pythagoras' Theorem

Pythagoras' Theorem: In any right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares…

Trigonometry

Trigonometric ratios in a right-angled triangle relative to acute angle θ: Sine (sin θ) = Opposite / Hypotenuse.

Matrices

A matrix is a rectangular array of numbers arranged in rows and columns; the order (dimension) is written m × n (m rows by n columns).

Inequalities

Inequality Symbols: Greater than (>), Less than (<), Greater than or equal to (≥), Less than or equal to (≤).

Cyclic Quadrilaterals

Cyclic Quadrilateral Definition: A quadrilateral whose all four vertices lie on the circumference of a single circle.

Tangents

Theorem 1: The tangent at any point on a circle is perpendicular to the radius drawn to the point of contact (OP ⊥ PT).