Real numbers include rational and irrational numbers. Their operations follow closure where applicable, commutative and associative laws for addition and multiplication, and the distributive law.
Example: √2 + 3 is real, and 2(3 + √5) = 6 + 2√5.
Practice: Simplify 4√3 + 2√3 and identify the property used in 5(2 + √2).
Remember: Rational and irrational numbers together form the real-number system.
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Laws of Exponents for Real Numbers
Exponent laws can be extended to suitable real-number expressions. They help simplify products, quotients and powers efficiently.
Example: am × an = am+n; (am)n = amn.
Practice: Simplify 35 × 3−2 and (23)4.
Remember: Check restrictions such as a ≠ 0 when negative powers or division are involved.
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Rationalisation of Irrational Denominators
Rationalisation rewrites a fraction so that its denominator is rational. For binomial surds, multiplication by the conjugate is especially useful.
Example: 1/√5 = √5/5. For 1/(√3+1), multiply numerator and denominator by √3−1.
Practice: Rationalise 3/√7.
Remember: Multiply numerator and denominator by the same non-zero expression.
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Representation of Real Numbers on the Number Line
Every real number corresponds to a unique point on the number line. Irrational lengths such as √2 can be constructed geometrically using right triangles.
Example: A right triangle with legs 1 and 1 has hypotenuse √2, which can be transferred to the number line.
Practice: Describe a geometric construction for locating √5.
Remember: The real-number line has no gaps.
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Introduction to Polynomials
A polynomial is an algebraic expression made from variables with non-negative integer exponents and numerical coefficients. Polynomials may be classified by degree and number of terms.
Example: 4x³ − 2x + 7 is a cubic polynomial.
Practice: State the degree and coefficients of 5x⁴ − 3x² + 8.
Remember: The degree is the greatest exponent with a non-zero coefficient.
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Zeros of a Polynomial
A zero of a polynomial p(x) is a value of x for which p(x)=0. Graphically, real zeros correspond to x-coordinates where the graph meets the x-axis.
Example: For p(x)=x−4, p(4)=0, so 4 is a zero.
Practice: Find the zeros of x²−9.
Remember: Substitute a suspected value into the polynomial to test it.
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Remainder Theorem
When polynomial p(x) is divided by x−a, the remainder equals p(a). This avoids performing full polynomial division when only the remainder is required.
Example: For p(x)=x²+2x+3 divided by x−1, remainder = p(1)=6.
Practice: Find the remainder when x³−2x+5 is divided by x−2.
Remember: Divisor x−a means substitute x=a.
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Factor Theorem
The factor theorem states that x−a is a factor of p(x) exactly when p(a)=0. It connects polynomial zeros with factorisation.
Example: For p(x)=x²−5x+6, p(2)=0, so x−2 is a factor.
Practice: Test whether x−3 is a factor of x²−4x+3.
Remember: Zero remainder means the divisor is a factor.
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Algebraic Factorisation Using Identities
Algebraic identities can convert polynomial expressions into products. Recognising patterns makes factorisation faster and more reliable.
Example: x²−16=(x−4)(x+4).
Practice: Factorise 9a²−25b².
Remember: a²−b²=(a−b)(a+b).
📈 Linear Equations and Coordinate Geometry
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Linear Equations in Two Variables
An equation such as ax+by+c=0 represents a linear relationship between two variables. It generally has infinitely many ordered-pair solutions.
Example: x+y=5 has solutions (0,5), (2,3) and (5,0).
Practice: Find four solutions of 2x+y=6.
Remember: A solution must satisfy the equation after substitution.
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Graphs of Linear Equations in Two Variables
The solution set of a linear equation in two variables forms a straight line. Two or more valid solution points can be plotted and joined.
Example: For x+y=4, points (0,4), (2,2) and (4,0) lie on the graph.
Practice: Draw the graph of y=2x+1 using at least three points.
Remember: Every point on the line satisfies the equation.
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Cartesian Plane and Coordinate System
The Cartesian plane uses perpendicular x- and y-axes, an origin and four quadrants to locate points precisely.
Example: Point (−4,3) lies in Quadrant II.
Practice: Identify the quadrant of (5,−2), (−3,−6) and (2,7).
Remember: The x-coordinate is written first.
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Plotting Ordered Pairs on the Cartesian Plane
Ordered pairs can represent locations and relationships between quantities. Accurate plotting requires attention to scale, signs and axis labels.
Example: To plot (−2,4), move 2 units left and 4 units up from the origin.
Practice: Plot (3,5), (−4,2), (−2,−3) and (4,−1).
Remember: A point on an axis is not inside a quadrant.
📐 Euclidean Geometry and Triangles
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Euclid’s Geometry and Fundamental Definitions
Euclidean geometry develops mathematical conclusions from definitions, common notions and postulates. Fundamental ideas include points, lines, planes and geometric relations.
Example: A line can be extended indefinitely in both directions in ideal Euclidean geometry.
Practice: Explain the difference between a line segment, ray and line.
Remember: Geometry is built through logical deduction from accepted foundations.
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Axioms and Postulates in Geometry
Axioms are accepted general mathematical statements, while postulates are assumptions used specifically in geometry. They provide starting points for proofs.
Example: Things equal to the same thing are equal to one another.
Practice: Give one example each of an axiom and a geometric postulate.
Remember: Theorems are proved; axioms and postulates are accepted as foundations.
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Angle Relationships Formed by Parallel Lines
When a transversal crosses parallel lines, corresponding and alternate interior angles are equal, while co-interior angles are supplementary.
Example: If one corresponding angle is 65°, its matching corresponding angle is also 65°.
Practice: If a co-interior angle is 112°, find the other.
Remember: These relationships are powerful tools in geometric proofs.
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Congruence of Triangles
Congruent triangles have exactly the same shape and size. Their corresponding sides and angles are equal.
Example: If ΔABC ≅ ΔPQR, then AB=PQ, BC=QR and ∠A=∠P.
Practice: Write the corresponding parts when ΔXYZ ≅ ΔLMN.
Remember: The order of letters in a congruence statement shows correspondence.
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Triangle Congruence Criteria
Common criteria for proving triangle congruence include SSS, SAS, ASA/AAS and RHS for right triangles.
Example: Two triangles with three corresponding equal sides are congruent by SSS.
Practice: Identify the criterion when two sides and the included angle are equal.
Remember: AAA establishes similarity, not congruence.
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Properties of Isosceles Triangles
In an isosceles triangle, equal sides lie opposite equal angles. The converse also holds: equal angles imply equal opposite sides.
Example: If AB=AC, then ∠B=∠C.
Practice: If ∠B=55° and AB=AC, find ∠C and ∠A.
Remember: Match each side with its opposite angle.
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Inequalities in Triangles
Triangle inequalities compare side lengths and angles. The sum of any two sides must exceed the third side, and the larger side lies opposite the larger angle.
Example: Lengths 3, 4 and 8 cannot form a triangle because 3+4 is not greater than 8.
Practice: Decide whether 5 cm, 7 cm and 10 cm can form a triangle.
Remember: Check all relevant triangle inequalities.
🔷 Quadrilaterals, Areas and Circles
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Properties of Parallelograms
A parallelogram has opposite sides parallel and equal, opposite angles equal, adjacent angles supplementary and diagonals that bisect each other.
Example: If one angle of a parallelogram is 70°, the adjacent angle is 110°.
Practice: Find all angles if one angle is 125°.
Remember: Diagonals of a general parallelogram need not be equal.
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Midpoint Theorem and Its Applications
The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
Example: If BC=12 cm and D, E are midpoints of AB, AC, then DE=6 cm.
Practice: If the midpoint segment is 7.5 cm, find the third side.
Remember: Midpoint segment = 1/2 × third side.
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Areas of Parallelograms and Triangles
Figures on the same base and between the same parallels have useful area relationships. Parallelograms on the same base and parallels have equal areas.
Example: Two triangles on the same base and between the same parallels have equal areas.
Practice: Explain why triangles with the same base and equal height have equal area.
Remember: Area comparisons depend on both base and perpendicular height.
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Circles, Chords and Arcs
A chord joins two points on a circle. Equal chords subtend equal angles at the centre, and equal chords are equidistant from the centre.
Example: If chords AB and CD are equal, then ∠AOB=∠COD at centre O.
Practice: State two properties of equal chords.
Remember: The diameter is the longest chord of a circle.
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Angle Properties of Circles
Angles formed by the same chord or arc have important relationships. The angle subtended by an arc at the centre is twice the angle subtended at a point on the remaining circle.
Example: If a chord subtends 100° at the centre, it subtends 50° at a suitable point on the circle.
Practice: Find the central angle corresponding to an inscribed angle of 42°.
Remember: Always identify whether the angles stand on the same arc.
📏 Mensuration, Statistics and Probability
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Heron’s Formula
Heron’s formula finds the area of a triangle when all three side lengths are known. If s=(a+b+c)/2, then Area=√[s(s−a)(s−b)(s−c)].
Example: For sides 3,4,5, s=6 and area=√(6×3×2×1)=6 square units.
Practice: Use Heron’s formula for a triangle with sides 5,5 and 6.
Remember: Calculate the semi-perimeter before substituting.
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Surface Area of Composite Solids
Composite solids are built from two or more familiar solids. Their exposed surface area is found by adding visible surfaces and excluding joined internal faces.
Example: For a cylinder topped by a hemisphere, the joined circular face is internal and should not be counted twice.
Practice: List the exposed surfaces of a hemisphere mounted on a cylinder.
Remember: Count only surfaces exposed on the outside.
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Volume of Composite Solids
The volume of a composite solid can often be found by adding volumes of its component solids or subtracting removed portions.
Example: A solid made from a cylinder and hemisphere has volume V=πr²h+(2/3)πr³.
Practice: Write a volume expression for a cone placed on a cylinder with common radius r.
Remember: Keep all measurements in compatible units before calculating.
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Statistical Data Distribution and Frequency Tables
Frequency distributions organise data into values or class intervals with their frequencies. They make large datasets easier to interpret and represent graphically.
Practice: Group the data 4,7,9,12,15,18,21,24 into intervals 0–10, 10–20 and 20–30.
Remember: Class intervals should be clearly defined so observations are counted consistently.
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Experimental Probability and Relative Frequency
Experimental probability is based on observed outcomes from repeated trials. It is calculated as the number of times an event occurs divided by the total number of trials.
Example: If a coin shows heads 47 times in 100 tosses, experimental probability of heads = 47/100=0.47.
Practice: A die shows 6 on 18 of 120 rolls. Find the experimental probability.
Remember: With many trials, experimental probability often becomes more stable.