Euclid’s division lemma states that for positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Repeated division gives an efficient method for finding the HCF.
Example: 455 = 42×10 + 35, 42 = 35×1 + 7, and 35 = 7×5 + 0, so HCF(455,42)=7.
Practice: Use Euclid’s algorithm to find the HCF of 867 and 255.
Remember: The last non-zero remainder is the HCF.
2
Fundamental Theorem of Arithmetic
Every composite positive integer can be expressed as a product of primes, and this prime factorisation is unique apart from the order of factors. It provides a systematic basis for HCF and LCM reasoning.
Example: 360 = 2³×3²×5.
Practice: Prime-factorise 756 and 1080, then find their HCF and LCM.
Remember: Prime factorisation is unique except for the order of the prime factors.
3
Proofs of Irrationality
Irrationality proofs often use contradiction. Assume a number such as √2 is rational in lowest terms, then derive a result showing numerator and denominator must share a factor, contradicting the assumption.
Example: If √2=p/q in lowest terms, squaring implies p²=2q², leading to both p and q being even—a contradiction.
Practice: Study the contradiction method and outline a proof that √3 is irrational.
Remember: A contradiction shows the original assumption cannot be true.
4
Terminating and Non-Terminating Decimal Expansions
For a rational number p/q in lowest form, its decimal expansion terminates exactly when q has no prime factors other than 2 and 5. Otherwise the decimal expansion is non-terminating recurring.
Example: 7/40 terminates because 40=2³×5, while 7/12 recurs because 12 contains the factor 3.
Practice: Predict whether 13/160, 17/75 and 29/48 terminate without performing long division.
Remember: Reduce the fraction to lowest terms before examining its denominator.
5
Relationship between Zeros and Coefficients of a Polynomial
For a quadratic ax²+bx+c with zeros α and β, α+β=−b/a and αβ=c/a. These relations connect factorisation with the coefficients of the polynomial.
Example: For 2x²−7x+3, the sum of zeros is 7/2 and the product is 3/2.
Practice: Find the sum and product of zeros of 3x²+5x−2 without solving for the zeros.
Remember: Check the sign carefully in α+β=−b/a.
6
Constructing Quadratic Polynomials from Their Zeros
If α and β are known zeros, a quadratic polynomial can be formed as k[x²−(α+β)x+αβ], where k is any non-zero constant. Choosing k=1 gives the monic form.
Example: Zeros 2 and −5 give x²+3x−10.
Practice: Form a quadratic polynomial whose zeros are 3/2 and −4.
Remember: Multiplying the whole polynomial by any non-zero constant keeps the same zeros.
➗ Linear Equations, Quadratics and Progressions
7
Pair of Linear Equations in Two Variables
Two linear equations can represent two straight lines. A common solution is an ordered pair that satisfies both equations simultaneously and corresponds to their intersection when one exists.
Example: x+y=7 and x−y=1 have the common solution (4,3).
Practice: Verify whether (2,5) satisfies both x+y=7 and 2x−y=−1.
Remember: A solution of the pair must satisfy both equations.
8
Graphical Solution of a Pair of Linear Equations
Graphing both equations shows whether the corresponding lines intersect, are parallel or coincide. These cases correspond to one solution, no solution or infinitely many solutions.
Example: Two distinct lines meeting at (3,2) have the unique common solution x=3, y=2.
Practice: Graph x+y=5 and x−y=1 and read their intersection.
Remember: The geometric relationship between the lines reveals the number of solutions.
9
Algebraic Methods for Simultaneous Linear Equations
Substitution and elimination solve a pair without graphing. The most efficient method depends on the coefficients and form of the equations.
Example: For x+y=9 and x−y=3, adding eliminates y and gives 2x=12, so x=6 and y=3.
Practice: Solve 2x+3y=13 and 3x−2y=4 using elimination.
Remember: Substitute the solution back into both original equations to check it.
10
Consistency Conditions for Linear Equation Pairs
For a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0, coefficient ratios determine whether the lines intersect, are parallel or coincide. This allows the number of solutions to be predicted without fully solving.
Example: If a₁/a₂ ≠ b₁/b₂, the pair has exactly one solution.
Practice: Determine the number of solutions for 2x+3y−5=0 and 4x+6y−9=0.
Remember: Parallel distinct lines represent an inconsistent pair.
11
Quadratic Equations and Factorisation
A quadratic equation has standard form ax²+bx+c=0 with a≠0. When the expression factorises conveniently, the zero-product property can be used to obtain its roots.
Example: x²−7x+12=(x−3)(x−4)=0, so x=3 or x=4.
Practice: Solve 2x²−7x+3=0 by factorisation.
Remember: Write the equation equal to zero before applying factorisation.
12
Solving Quadratic Equations with the Quadratic Formula
For ax²+bx+c=0, the roots are x=[−b±√(b²−4ac)]/(2a). The formula works even when simple factorisation is difficult.
Example: For x²−4x−1=0, x=2±√5.
Practice: Solve 3x²+2x−7=0 using the quadratic formula.
Remember: Substitute a, b and c with their signs.
13
Discriminant and Nature of Quadratic Roots
The discriminant D=b²−4ac indicates the nature of real roots: D>0 gives two distinct real roots, D=0 gives equal real roots and D<0 gives no real roots.
Example: For x²−6x+9=0, D=36−36=0, so the roots are equal.
Practice: Determine the nature of roots of 2x²+x+5=0 without solving it.
Remember: The discriminant classifies roots before they are calculated.
14
Arithmetic Progressions and the nth Term
An arithmetic progression has a constant difference d between consecutive terms. Its nth term is aₙ=a+(n−1)d, where a is the first term.
Example: In 5,9,13,…, a=5 and d=4, so the 20th term is 81.
Practice: Find the 35th term of 12,17,22,… .
Remember: The common difference may be positive, negative or zero.
15
Sum of an Arithmetic Progression
The sum of the first n terms is Sₙ=n/2[2a+(n−1)d]. If the last term l is known, Sₙ=n(a+l)/2 can be more convenient.
Example: The sum of the first 10 terms of 3,7,11,… is 10/2[6+36]=210.
Practice: Find the sum of the first 25 terms of 8,11,14,… .
Remember: Do not confuse the nth term with the sum of the first n terms.
📐 Geometry and Coordinate Geometry
16
Similarity of Triangles and Proportional Sides
Similar triangles have the same shape: corresponding angles are equal and corresponding sides are proportional. Similarity allows unknown lengths to be found without requiring triangles to be the same size.
Example: If two similar triangles have scale factor 2, every corresponding side in one is twice the matching side in the other.
Practice: A triangle with sides 3,4,5 is similar to one whose shortest side is 9. Find its other sides.
Remember: Match corresponding vertices in the correct order.
17
Criteria for Triangle Similarity
Triangles can be proved similar using AA, SAS or SSS similarity criteria. These criteria establish proportional structure without checking every angle and side separately.
Example: If two corresponding angles are equal, the third angles are also equal, so AA similarity applies.
Practice: Decide which similarity criterion applies when two sides are proportional and the included angles are equal.
Remember: SAS similarity requires the equal angle to be included between the proportional sides.
18
Distance Formula in the Coordinate Plane
The distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²]. It follows directly from the Pythagorean theorem.
Example: The distance between (1,2) and (4,6) is √(3²+4²)=5.
Practice: Find the distance between (−3,4) and (5,−2).
Remember: Squaring coordinate differences removes their signs before addition.
19
Section Formula and Internal Division
If a point P divides the segment joining A(x₁,y₁) and B(x₂,y₂) internally in the ratio m:n, its coordinates are ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)).
Example: The midpoint is the special case m:n=1:1.
Practice: Find the point dividing (2,3) and (8,15) internally in the ratio 1:2.
Remember: Keep the ratio order consistent with the endpoints.
📏 Trigonometry and Circles
20
Trigonometric Ratios in a Right Triangle
For an acute angle θ in a right triangle, sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse and tanθ=opposite/adjacent. Reciprocal ratios are cosecθ, secθ and cotθ.
Example: If opposite=3, adjacent=4 and hypotenuse=5, then sinθ=3/5, cosθ=4/5 and tanθ=3/4.
Practice: Find all six trigonometric ratios for a right triangle with sides 5,12,13.
Remember: Opposite and adjacent depend on the chosen acute angle.
21
Trigonometric Identities and Standard Relationships
Important identities include sin²θ+cos²θ=1, 1+tan²θ=sec²θ and 1+cot²θ=cosec²θ. Identities are true for all angles where the expressions involved are defined.
Example: Dividing sin²θ+cos²θ=1 by cos²θ gives tan²θ+1=sec²θ.
Practice: If sinθ=3/5 for an acute angle, use an identity to find cosθ.
Remember: An identity is different from an equation true only for particular values.
22
Applications of Trigonometry: Heights and Distances
Angles of elevation and depression allow inaccessible heights and distances to be modelled using right triangles. A clear diagram is essential before selecting a trigonometric ratio.
Example: If a point is 20 m from a tower and the angle of elevation to its top is 45°, the height above eye level is 20 m.
Practice: Draw and solve a diagram for a pole viewed from 30 m away at an elevation angle of 30°.
Remember: Include observer height when the problem requires total height from the ground.
23
Tangents to a Circle
A tangent touches a circle at exactly one point. The radius drawn to the point of contact is perpendicular to the tangent.
Example: If OT is a radius to tangent point T, then OT is perpendicular to the tangent line at T.
Practice: Draw a circle, radius and tangent and mark the right angle at the point of contact.
Remember: The perpendicular relation occurs at the point of tangency.
24
Equal Tangents from an External Point
If two tangents are drawn from the same external point to a circle, their tangent lengths are equal. This result follows by comparing right triangles formed with the centre and points of contact.
Example: If PA and PB touch the same circle from P, then PA=PB.
Practice: If PA=2x+3 and PB=5x−12, find x.
Remember: The equality applies to tangent segments drawn from the same external point.
⭕ Mensuration, Statistics and Probability
25
Areas of Sectors and Segments of Circles
A sector with central angle θ has area (θ/360°)πr². A segment can often be found by subtracting the area of an appropriate triangle from the corresponding sector.
Example: A 90° sector is one quarter of the circle, so its area is πr²/4.
Practice: Find the area of a 60° sector of radius 12 cm.
Remember: Sector area is proportional to its central angle.
26
Surface Areas and Volumes of Combined Solids
Real objects can be modelled by combinations of cylinders, cones, spheres or hemispheres. Volume is usually added or subtracted by components, while surface area includes only surfaces actually exposed.
Example: A capsule-shaped object can be modelled as a cylinder with two hemispherical ends.
Practice: Write an expression for the volume of a cylinder of radius r and height h topped by a hemisphere of the same radius.
Remember: Joined internal faces are not part of the exposed surface area.
27
Mean of Grouped Data
For grouped frequency data, class marks can represent intervals. The mean is calculated using weighted values, commonly as Σfᵢxᵢ/Σfᵢ, where xᵢ is the class mark.
Example: For the interval 20–30, the class mark is 25.
Practice: Create class marks for 0–10, 10–20, 20–30 and 30–40, then explain how frequencies would weight them.
Remember: A class with larger frequency contributes more strongly to the grouped mean.
28
Median and Mode of Grouped Data
The median locates the middle position in an ordered grouped distribution, while the mode identifies the most concentrated class region. Grouped-data formulas estimate these measures within class intervals.
Example: The modal class is the class interval with the highest frequency before the grouped mode is estimated.
Practice: Given frequencies 4, 9, 15, 8 and 3 for consecutive intervals, identify the modal class and locate the median position.
Remember: Cumulative frequency is essential for locating the median class.
29
Classical Probability of Equally Likely Outcomes
When all outcomes are equally likely, P(E)=number of favourable outcomes/total number of outcomes. A probability lies from 0 to 1 inclusive.
Example: For a fair die, P(number greater than 4)=2/6=1/3.
Practice: A card numbered 1 to 20 is chosen at random. Find the probability that its number is divisible by 4.
Remember: Count outcomes carefully and confirm that the model assumes equal likelihood.
30
Complementary Events and Probability Reasoning
An event E and its complement not-E cover all possible outcomes without overlap, so P(not E)=1−P(E). This is often simpler than counting a complicated event directly.
Example: If the probability of rain is 0.3 in a simplified model, the probability of no rain is 0.7.
Practice: If P(E)=7/12, find P(not E) and verify their sum.
Remember: An event and its complement always have probabilities summing to 1.