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Your test showed weak areas — study them now:

➕ Study Materials

Mathematics
Study Notes

All notes follow the official WAEC and JAMB approved syllabus. Study a topic first, then take the practice quiz — after the test, come back here to see which topics you need to improve.

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Number Bases

Binary, octal, denary conversions

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Indices & Logarithms

Laws of indices, log rules, standard form

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Ratio & Percentages

Profit, loss, discount, VAT, ratios

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Simple & Compound Interest

SI and CI formulas with exam applications

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Equations & Factorisation

Quadratics, simultaneous, remainder theorem

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Variation & Functions

Direct, inverse, joint variation; function notation

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Sequences & Series

AP, GP — nth term and sum formulas

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Inequalities & Sets

Linear inequalities, Venn diagrams, set notation

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Plane Geometry

Angles, triangles, polygons, locus

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Circle Theorems

All 8 circle theorem rules tested by WAEC

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Mensuration

Areas, volumes of 2D and 3D shapes

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Coordinate Geometry

Gradient, midpoint, distance, equation of line

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Trig Ratios & Angles

SOHCAHTOA, special angles, trig identities

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Elevation & Bearings

Angles of elevation/depression, compass bearings

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Differentiation

dy/dx rules, gradient, maxima and minima

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Integration

Indefinite and definite integrals, area under curve

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Data & Averages

Mean, median, mode, range, variance, SD

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Probability & Permutation

Theoretical probability, permutations, combinations

Number Bases

✓ WAEC Paper 1 & 2✓ JAMB

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What is a Number Base?

A number base tells you how many digits are used. Base 10 (denary) uses 0–9. Base 2 (binary) uses 0 and 1 only. Base 8 (octal) uses 0–7. Base 16 (hexadecimal) uses 0–9 and A–F.

WAEC and JAMB test conversions between bases and arithmetic in different bases.

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Converting to Base 10 (Expansion Method)

Multiply each digit by its base raised to its position power, then add all results.

Example — Convert 11010₂ to base 10

Position:  4  3  2  1  0
Digit:     1  1  0  1  0
Value:  1×2⁴ + 1×2³ + 0×2² + 1×2¹ + 0×2⁰
      = 16 + 8 + 0 + 2 + 0 = 26

Example — Convert 101.11₂ to decimal

1×2² + 0×2¹ + 1×2⁰ + 1×2⁻¹ + 1×2⁻²
= 4 + 0 + 1 + 0.5 + 0.25 = 5.75

⬇️

Converting from Base 10 to Another Base (Division Method)

Divide the number repeatedly by the target base. Write the remainders from bottom to top.

Example — Convert 26 to binary (base 2)

26 ÷ 2 = 13 remainder 0
13 ÷ 2 = 6  remainder 1
6  ÷ 2 = 3  remainder 0
3  ÷ 2 = 1  remainder 1
1  ÷ 2 = 0  remainder 1
Reading remainders bottom to top: 11010₂ ✓

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Exam shortcut: For binary to octal or hex, group the binary digits in sets of 3 (octal) or 4 (hex) from the right, then convert each group directly.

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Arithmetic in Different Bases

Carry occurs when a digit reaches the base value. In base 2, carry when you reach 2. In base 8, carry at 8.

Binary Addition — 1101₂ + 1011₂

  1 1 0 1
+ 1 0 1 1
---------
1 1 0 0 0   (= 24 in base 10: 13 + 11 = 24 ✓)

⚠️

WAEC tests subtraction and multiplication in non-decimal bases. Always verify your answer by converting both the question and your answer to base 10 to check.

Indices & Logarithms

✓ WAEC✓ JAMB

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Laws of Indices

Law	Rule	Example
Multiplication	aᵐ × aⁿ = aᵐ⁺ⁿ	x³ × x⁴ = x⁷
Division	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	x⁶ ÷ x² = x⁴
Power of power	(aᵐ)ⁿ = aᵐⁿ	(x³)² = x⁶
Zero index	a⁰ = 1	7⁰ = 1, x⁰ = 1
Negative index	a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8
Fractional index	aᵐ/ⁿ = ⁿ√(aᵐ)	27⅓ = ∛27 = 3

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Most tested: Fractional and negative indices. Remember: 8⅔ = (∛8)² = 4. Evaluate the root first, then the power.

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Laws of Logarithms

Logarithm is the inverse of an index: if aˣ = N, then logₐN = x.

Law	Rule	Example (base 10)
Product	log(MN) = logM + logN	log(8 × 125) = log8 + log125
Quotient	log(M/N) = logM − logN	log(100/10) = log100 − log10 = 1
Power	log(Mⁿ) = n·logM	log(10³) = 3·log10 = 3
Base change	logₐN = logN / logа	log₂8 = log8 / log2 = 0.903/0.301 = 3
Special values	logₐ1 = 0; logₐa = 1	log 1 = 0; log 10 = 1

Worked Example

Simplify: log 8 + log 125 − log 10 (base 10)
= log(8 × 125) − log 10 = log 1000 − log 10
= log(1000/10) = log 100 = 2

⚠️

Standard form: Write numbers as A × 10ⁿ where 1 ≤ A < 10. E.g. 0.004562 = 4.562 × 10⁻³. To 3 significant figures: 4.56 × 10⁻³.

Fractions, Ratio & Percentages

✓ WAEC✓ JAMB

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Profit, Loss & Discount

Concept	Formula
Profit	Profit = Selling Price − Cost Price
% Profit	% Profit = (Profit / CP) × 100
Loss	Loss = Cost Price − Selling Price
% Loss	% Loss = (Loss / CP) × 100
Discount	Discount = Marked Price − Selling Price
% Discount	% Discount = (Discount / MP) × 100
VAT	VAT Amount = Rate% × Original Price

Example — % Profit

Bought for ₦15,000, sold for ₦18,000.
Profit = 18,000 − 15,000 = ₦3,000
% Profit = (3,000 / 15,000) × 100 = 20%

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Ratio, Proportion & Sharing

To share an amount in a given ratio, find the total parts, divide the amount by total parts, then multiply for each share.

Example — Sharing in ratio 3:5:2

Share ₦20,000 in ratio 3:5:2.
Total parts = 3+5+2 = 10
One part = ₦20,000 ÷ 10 = ₦2,000
Shares: ₦6,000 : ₦10,000 : ₦4,000

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Percentage increase/decrease trap: "Increased by 20% to give 60" → Original = 60 ÷ 1.2 = 50. Never divide by 0.2.

Simple & Compound Interest

✓ WAEC✓ JAMB

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Simple Interest (SI)

SI = (P × R × T) / 100
where P = Principal, R = Rate per annum, T = Time in years.
Amount = P + SI

Example

Find SI on ₦24,000 at 5% p.a. for 3 years.
SI = (24,000 × 5 × 3) / 100 = ₦3,600
Amount = 24,000 + 3,600 = ₦27,600

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Compound Interest (CI)

A = P(1 + R/100)ᵀ
CI = A − P

Example — Compound Interest

₦10,000 at 10% p.a. for 2 years, compounded annually.
A = 10,000 × (1 + 10/100)² = 10,000 × 1.1² = 10,000 × 1.21 = ₦12,100
CI = 12,100 − 10,000 = ₦2,100

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Key difference: Simple interest is calculated on the original principal only. Compound interest is calculated on the growing amount each period. CI always gives more than SI for the same rate and time.

Equations & Factorisation

✓ WAEC✓ JAMB

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Quadratic Equations

Three methods: Factorisation, Formula, and Completing the square.

Factorisation — 2x² + 5x − 3 = 0

Find two numbers that multiply to give (2 × −3 = −6)
and add to give +5: those are +6 and −1
2x² + 6x − x − 3 = 0
2x(x + 3) − 1(x + 3) = 0
(2x − 1)(x + 3) = 0
x = ½ or x = −3

Quadratic Formula — for ax² + bx + c = 0

x = [−b ± √(b² − 4ac)] / 2a
The discriminant b² − 4ac: if > 0 → 2 real roots; = 0 → 1 root; < 0 → no real roots.

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Simultaneous Equations

Two methods: Elimination and Substitution.

Elimination — 2x + y = 7 and x − y = 2

Add both equations: 3x = 9, so x = 3
Substitute: 2(3) + y = 7 → y = 1
Answer: x = 3, y = 1 ✓

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Remainder & Factor Theorem

Remainder Theorem: When f(x) is divided by (x − a), the remainder is f(a).
Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.

Example — Remainder when x³ − 5x + 3 is divided by (x − 2)

f(2) = 2³ − 5(2) + 3 = 8 − 10 + 3 = 1

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Factorisation of difference of two squares: a² − b² = (a + b)(a − b). E.g. x² − 9 = (x+3)(x−3). Dividing by (x+3) gives (x−3).

Variation & Functions

✓ WAEC✓ JAMB

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Types of Variation

Type	Statement	Equation	Example
Direct	y varies directly as x	y = kx	x=10,y=4 → k=0.4
Inverse	y varies inversely as x	y = k/x (xy = k)	x=4,y=5 → k=20
Joint	y varies jointly as x and z	y = kxz	Find k using given values
Partial	y = a + bx (partly constant)	y = a + bx	Form two equations, solve

Direct Variation Example

x varies directly as y. x = 10 when y = 4. Find x when y = 10.
y = kx → 10 = k × 4 → k = 2.5
When y = 10: x = 2.5 × 10 = 25

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Functions

A function f(x) maps each input x to exactly one output. To evaluate f(a), substitute x = a.

Example — f(x) = 3x² − 2x + 1, find f(−2)

f(−2) = 3(−2)² − 2(−2) + 1 = 3(4) + 4 + 1 = 12 + 4 + 1 = 17

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Inverse function: To find f⁻¹(x), replace f(x) with y, swap x and y, then solve for y.

Sequences & Series

✓ WAEC✓ JAMB

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Arithmetic Progression (AP)

Formula	Meaning
Tₙ = a + (n−1)d	nth term of AP. a = first term, d = common difference
Sₙ = n/2[2a + (n−1)d]	Sum of first n terms
Sₙ = n/2(a + l)	Sum using last term l

Example — AP: 3, 7, 11, 15... Find T₁₀ and S₁₀

a = 3, d = 4
T₁₀ = 3 + (10−1)×4 = 3 + 36 = 39
S₁₀ = 10/2 × [2(3) + 9(4)] = 5 × [6 + 36] = 5 × 42 = 210

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Finding the nth term from Sₙ: Tₙ = Sₙ − Sₙ₋₁. If Sₙ = n² + 2n, then T₅ = S₅ − S₄ = (25+10) − (16+8) = 35 − 24 = 11.

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Geometric Progression (GP)

Formula	Meaning
Tₙ = arⁿ⁻¹	nth term. a = first term, r = common ratio
Sₙ = a(rⁿ−1)/(r−1) if r > 1	Sum of first n terms
Sₙ = a(1−rⁿ)/(1−r) if r < 1	Sum of first n terms
S∞ = a/(1−r) if \|r\| < 1	Sum to infinity

Example — GP: 2, 6, 18... Find T₁₀

a = 2, r = 3
T₁₀ = 2 × 3⁹ = 2 × 19,683 = 39,366

Inequalities & Sets

✓ WAEC✓ JAMB

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Linear Inequalities

Solve like an equation, but flip the inequality sign when multiplying or dividing by a negative number.

Example — 3x − 7 > 2

3x > 2 + 7 → 3x > 9 → x > 3

Flip rule — −2x < 8

Divide by −2 and flip: x > −4

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Set Theory & Venn Diagrams

Notation	Meaning
A ∪ B	Union — all elements in A or B or both
A ∩ B	Intersection — elements in BOTH A and B
A'	Complement — elements NOT in A
n(A ∪ B)	n(A) + n(B) − n(A ∩ B)

Example — Venn Diagram Problem

40 students: 24 passed Maths, 18 passed English, 10 passed both.
n(M ∪ E) = 24 + 18 − 10 = 32 passed at least one.
Neither = 40 − 32 = 8 failed both.

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WAEC loves three-set Venn diagrams. Use: n(A∪B∪C) = n(A)+n(B)+n(C)−n(A∩B)−n(A∩C)−n(B∩C)+n(A∩B∩C).

Plane Geometry

✓ WAEC✓ JAMB

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Angles & Lines — Key Rules

Angles on a straight line add up to 180°
Angles at a point add up to 360°
Vertically opposite angles are equal
Alternate angles (Z-angles) are equal when lines are parallel
Co-interior angles (C-angles) add up to 180° when lines are parallel
Corresponding angles (F-angles) are equal when lines are parallel

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Triangles & Polygons

Shape	Interior Angles Sum	Each Interior (regular)	Exterior Angle
Triangle	180°	60° (equilateral)	= sum of two opposite interior angles
Quadrilateral	360°	90° (square)	360° ÷ n
Pentagon	540°	108°	72°
Hexagon	720°	120°	60°
n-sided polygon	(n−2) × 180°	(n−2)×180°/n	360°/n

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Quick formula: Sum of interior angles = (n − 2) × 180°. Each exterior angle of a regular polygon = 360° ÷ n. If exterior = 36°, then n = 360/36 = 10 sides.

Circle Theorems

✓ WAEC✓ JAMB

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All 8 Circle Theorems — Memorise These

#	Theorem
1	The angle at the centre is twice the angle at the circumference subtended by the same arc.
2	Angles in the same segment (subtended by the same chord) are equal.
3	The angle in a semicircle (angle subtended by a diameter) is 90°.
4	Opposite angles of a cyclic quadrilateral add up to 180°.
5	The exterior angle of a cyclic quadrilateral equals the interior opposite angle.
6	A radius drawn to the point of tangency is perpendicular to the tangent.
7	Two tangents from an external point are equal in length.
8	The perpendicular from the centre to a chord bisects the chord.

Example — Using Theorem 1

Angle subtended at centre = 140°. Angle at circumference = 140 ÷ 2 = 70°

Example — Using Theorem 4 (Cyclic Quadrilateral)

ABCD is cyclic. Angle A = 75°. Angle C = 180° − 75° = 105°

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Most tested: Theorem 1 (centre = 2 × circumference), Theorem 3 (angle in semicircle = 90°), Theorem 4 (cyclic quad opposite angles = 180°). These appear every year.

Mensuration

✓ WAEC✓ JAMB

📐

2D Shapes — Areas & Perimeters

Shape	Area	Perimeter / Circumference
Rectangle	l × w	2(l + w)
Triangle	½ × b × h	a + b + c
Trapezium	½(a + b) × h	Sum of all sides
Circle	πr²	2πr
Sector (angle θ)	(θ/360) × πr²	Arc = (θ/360) × 2πr

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Use π = 22/7 when the radius is a multiple of 7. Use π = 3.14 otherwise, unless the question specifies.

📦

3D Shapes — Surface Areas & Volumes

Shape	Volume	Total Surface Area
Cuboid	l × w × h	2(lw + lh + wh)
Cylinder	πr²h	2πr(r + h)
Cone	⅓πr²h	πr(r + l) where l = slant height
Sphere	4/3 πr³	4πr²
Pyramid	⅓ × base area × h	Base + ½ × perimeter × slant height

Example — Volume of cylinder (r = 7, h = 10, π = 22/7)

V = πr²h = 22/7 × 7² × 10 = 22/7 × 49 × 10 = 22 × 70 = 1,540 cm³

Coordinate Geometry

✓ WAEC✓ JAMB

📍

Core Formulas for Two Points (x₁,y₁) and (x₂,y₂)

Concept	Formula	Example: (1,3) and (4,9)
Gradient (slope)	m = (y₂−y₁)/(x₂−x₁)	m = (9−3)/(4−1) = 6/3 = 2
Midpoint	M = ((x₁+x₂)/2, (y₁+y₂)/2)	M = (5/2, 12/2) = (2.5, 6)
Distance	d = √[(x₂−x₁)² + (y₂−y₁)²]	d = √[9+36] = √45 = 3√5
Equation of line	y − y₁ = m(x − x₁)	y − 3 = 2(x − 1) → y = 2x + 1

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Parallel lines have equal gradients. Perpendicular lines: m₁ × m₂ = −1. If one gradient is 2, the perpendicular gradient is −½.

Trig Ratios & Angles

✓ WAEC✓ JAMB

📐

SOHCAHTOA

SOH: sin θ = Opposite / Hypotenuse
CAH: cos θ = Adjacent / Hypotenuse
TOA: tan θ = Opposite / Adjacent

Angle	sin	cos	tan
0°	0	1	0
30°	½	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	½	√3
90°	1	0	undefined

🎯

Key identity: sin²θ + cos²θ = 1. This is tested in simplification questions. Always remember it.

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Trigonometric Graphs & Quadrants

Use the CAST rule to determine which ratios are positive in each quadrant:

Q1 (0°–90°): All (sin, cos, tan) are positive
Q2 (90°–180°): Sine only is positive
Q3 (180°–270°): Tangent only is positive
Q4 (270°–360°): Cosine only is positive

Example — Find all values of x if sin x = 0.5, 0° ≤ x ≤ 360°

sin⁻¹(0.5) = 30°. Sine is positive in Q1 and Q2.
x = 30° and x = 180° − 30° = 150°

Angles of Elevation & Bearings

✓ WAEC✓ JAMB

🔭

Elevation & Depression

Angle of elevation: the angle measured upward from the horizontal to an object above.
Angle of depression: the angle measured downward from the horizontal to an object below.

Example — Tower Problem

A man stands 50m from the base of a tower. Angle of elevation = 30°.
tan 30° = height / 50
height = 50 × tan 30° = 50 × 1/√3 = 50/√3 = 50√3/3 ≈ 28.9m

⚠️

The angle of elevation from A to B equals the angle of depression from B to A (alternate angles). Draw a diagram for every elevation/depression question.

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Bearings

Bearings are measured clockwise from North and always written as 3 digits. N=000°, E=090°, S=180°, W=270°.

Bearing of N45°E = 045°
Bearing of S60°W = 180° + 60° = 240°
Back bearing (reverse direction): add or subtract 180°

Differentiation

✓ WAEC (Further Maths & Maths)✓ JAMB

📉

Basic Rules of Differentiation

Function	Derivative dy/dx
y = xⁿ	dy/dx = nxⁿ⁻¹
y = axⁿ	dy/dx = naxⁿ⁻¹
y = constant	dy/dx = 0
y = ax² + bx + c	dy/dx = 2ax + b

Example — Differentiate y = 4x³ − 3x² + 2x − 5

dy/dx = 12x² − 6x + 2

Gradient at a point — find gradient of y = 3x² − 5x + 1 at x = 2

dy/dx = 6x − 5. At x = 2: gradient = 6(2) − 5 = 7

🎯

Maximum & Minimum Points

At a maximum or minimum, dy/dx = 0. To distinguish:
• If d²y/dx² < 0 → Maximum
• If d²y/dx² > 0 → Minimum

Example — Find maximum of y = 4 − x²

dy/dx = −2x = 0 → x = 0
y_max = 4 − 0 = 4
d²y/dx² = −2 < 0 → confirmed maximum

🎯

JAMB and WAEC also ask for the equation of the tangent and normal to a curve at a point. Tangent gradient = dy/dx at the point. Normal gradient = −1 ÷ (dy/dx).

Integration

✓ WAEC✓ JAMB

📈

Basic Integration Rules

Integration is the reverse of differentiation.
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (indefinite integral, n ≠ −1)

Indefinite Integral — ∫ (2x + 3) dx

= 2x²/2 + 3x + C = x² + 3x + C

📊

Definite Integrals & Area Under a Curve

For ∫ₐᵇ f(x) dx: integrate, then evaluate at x = b and subtract evaluation at x = a.

Example — ∫₀² (x² + 1) dx

= [x³/3 + x]₀²
= (8/3 + 2) − (0 + 0)
= 8/3 + 6/3 = 14/3

⚠️

Area between a curve and the x-axis is the definite integral (take absolute value if the area is below the x-axis, i.e., the integral gives a negative result).

Data, Averages & Spread

✓ WAEC✓ JAMB

📊

Measures of Central Tendency

Measure	Definition	How to Find
Mean	Average value	Sum of all values ÷ number of values
Median	Middle value	Arrange in order; middle value (odd n) or average of two middle (even n)
Mode	Most frequent value	Value that appears most often
Range	Spread	Largest − Smallest

Example — Mean: 5 numbers with mean 12, four are 10,14,15,9

Sum = 5 × 12 = 60. Sum of known = 10+14+15+9 = 48. Fifth number = 60 − 48 = 12

📏

Variance & Standard Deviation

Variance (σ²) = mean of squared deviations from the mean = Σ(x − x̄)²/n
Standard Deviation (σ) = √(Variance)

Key Relationship

If Variance = 16, then Standard Deviation = √16 = 4

🎯

Frequency table mean: Use x̄ = Σ(fx) / Σf where f = frequency and x = midpoint of class interval for grouped data.

Probability & Permutation

✓ WAEC✓ JAMB

🎲

Probability

P(event) = Number of favourable outcomes / Total possible outcomes

Rule	Formula
Complement	P(A') = 1 − P(A)
Addition (mutually exclusive)	P(A or B) = P(A) + P(B)
Addition (not exclusive)	P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Multiplication (independent)	P(A and B) = P(A) × P(B)

Example — Ball Probability

Bag: 4 red, 6 blue balls. P(red) = 4/10 = 2/5

🔢

Permutations & Combinations

Concept	Formula	Example
Permutation (order matters)	ⁿPr = n! / (n−r)!	4 students in a row: ⁴P₄ = 4! = 24
Combination (order doesn't matter)	ⁿCr = n! / [r!(n−r)!]	Choose 3 from 5: ⁵C₃ = 10

Factorial examples

4! = 4 × 3 × 2 × 1 = 24 | 5! = 120 | 0! = 1 (by definition)

🎯

Key distinction: Arrangements (order matters) → Permutation. Selections/groups (order doesn't matter) → Combination. "How many ways can 4 sit in a row" → nPr. "Choose a committee of 3 from 8" → nCr.

⚡

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Your test showed weak areas — study them now:

MathematicsStudy Notes

Ready to test yourself? Jump straight into practice.

Number Bases

Indices & Logarithms

Ratio & Percentages

Simple & Compound Interest

Equations & Factorisation

Variation & Functions

Sequences & Series

Inequalities & Sets

Plane Geometry

Circle Theorems

Mensuration

Coordinate Geometry

Trig Ratios & Angles

Elevation & Bearings

Differentiation

Integration

Data & Averages

Probability & Permutation

Number Bases

Indices & Logarithms

Fractions, Ratio & Percentages

Simple & Compound Interest

Equations & Factorisation

Variation & Functions

Sequences & Series

Inequalities & Sets

Plane Geometry

Circle Theorems

Mensuration

Coordinate Geometry

Trig Ratios & Angles

Angles of Elevation & Bearings

Differentiation

Integration

Data, Averages & Spread

Probability & Permutation

Mathematics
Study Notes