How to use this sheet: take one topic, cover the formula, and say it before you look. Then answer a Core Mathematics past question that needs it, and write the working in full.
Print this page if you want a paper copy for revision at home. The navigation and the questions at the bottom stay off the paper. A printed copy is for home. GLearn does not set the rules of the examination hall.
The revision plan orders the weeks. The how to pass guide covers the timetable, past questions, written answers and exam day. These formulas are standard mathematics. GLearn does not publish a pass rate.
Indices
- Multiplying powers
am × an = am + n.
- Dividing powers
am / an = am − n, when a is not 0.
- A power of a power
(am)n = amn.
- A product or a quotient to a power
(a × b)n = an × bn. (a / b)n = an / bn, when b is not 0.
- Zero and negative indices
a0 = 1 when a is not 0. a−n = 1 / an when a is not 0.
- Fractional indices
a to the power 1/n is the nth root of a. a to the power m/n is the nth root of am, where that root is defined.
For a fractional index, take the base to be positive when you are not sure the root exists.
Logarithms
- What a logarithm says
If logb(x) = y, then by = x.
The base b is positive and not 1. The argument x is positive. The same rules apply to every log line below.
- Log of a product
logb(x × y) = logb(x) + logb(y).
- Log of a quotient
logb(x / y) = logb(x) − logb(y).
- Log of a power
logb(xn) = n × logb(x).
- Two values to remember
logb(b) = 1. logb(1) = 0.
- Change of base
logb(x) = logk(x) / logk(b).
k is any other valid base.
Quadratic equations
- Quadratic formula
For a × x2 + b × x + c = 0, with a not 0, x = [−b ± √(b2 − 4 × a × c)] / (2 × a).
- Discriminant
The discriminant is b2 − 4 × a × c.
If it is greater than 0, there are two different real roots. If it is 0, there is one repeated real root. If it is less than 0, there is no real root.
- Sum and product of the roots
The sum of the roots is −b / a. The product of the roots is c / a.
Sequences
- Arithmetic sequence
The difference d is constant. d = (second term) − (first term). Tn = a + (n − 1) × d.
a is the first term. n starts at 1 for that first term.
- Sum of an arithmetic sequence
Sn = (n / 2) × [2 × a + (n − 1) × d]. Also Sn = (n / 2) × (a + l).
l is the nth term, the last term in the sum. The two lines give the same total.
- Geometric sequence
The ratio r is constant. r = (second term) / (first term), when the first term is not 0. Tn = a × rn − 1.
- Sum of a geometric sequence
When r is not 1, Sn = a × (rn − 1) / (r − 1). The same sum is a × (1 − rn) / (1 − r). When r = 1, Sn = n × a.
- Sum to infinity
When r is greater than −1 and less than 1, the sum to infinity is a / (1 − r).
Mensuration
- Rectangle
Perimeter = 2 × (length + width). Area = length × width.
- Square
Perimeter = 4 × side. Area = side2.
- Triangle
Perimeter = a + b + c. Area = (1/2) × base × height.
The height is the perpendicular height that belongs to the base you used.
- Parallelogram
Area = base × height. Perimeter = 2 × (a + b).
The height is perpendicular to the base you used. a and b are a pair of adjacent sides.
- Trapezium
Area = (1/2) × (a + b) × height.
a and b are the two parallel sides. The height is perpendicular to those sides.
- Rhombus or kite
Area = (d1 × d2) / 2.
d₁ and d₂ are the diagonals.
- Circle
Circumference = 2 × π × r = π × d. Area = π × r2.
r is the radius and d is the diameter, so d = 2 × r. Use the value of π the question gives. 22/7 and 3.142 are approximations. Leave the answer in terms of π when the question allows it.
- Cube
Volume = side3. Total surface area = 6 × side2.
- Cuboid
Volume = length × width × height. Total surface area = 2 × (length × width + length × height + width × height).
- Prism
Volume = area of the cross section × length.
For a triangular prism, the cross section is the triangular end.
- Cylinder
Volume = π × r2 × h. Curved surface area = 2 × π × r × h. Total surface area = 2 × π × r × (r + h).
r is the radius of the circular end and h is the perpendicular height. The total surface area is for a closed cylinder, including both ends.
- Arc and sector, angle in degrees
Arc length = (θ / 360) × 2 × π × r. Sector area = (θ / 360) × π × r2.
θ is the angle at the centre, in degrees. The perimeter of the sector, along the arc and the two radii, is the arc length + 2 × r.
- Arc and sector, angle in radians
Arc length = r × θ. Sector area = (1/2) × r2 × θ.
Here θ is in radians, not degrees. To change degrees into radians, multiply by π / 180. To change radians into degrees, multiply by 180 / π.
- Smaller segment
When the angle at the centre is under 180°, the area of the smaller segment is the sector area minus the area of the triangle formed by the two radii and the chord.
- Right circular cone
Volume = (1/3) × π × r2 × h. Slant height l = √(r2 + h2). Curved surface area = π × r × l. Total surface area of a closed cone = π × r × (r + l).
h is the perpendicular height. l is the slant height along the side. The total includes the circular base.
- Sphere
Volume = (4/3) × π × r3. Surface area = 4 × π × r2.
- Hemisphere
Volume = (2/3) × π × r3. Curved surface area = 2 × π × r2. Total surface area including the flat face = 3 × π × r2.
The curved surface is the dome only. The total adds the circular face.
Trigonometry
- Right angled triangle ratios
sin θ = opposite / hypotenuse. cos θ = adjacent / hypotenuse. tan θ = opposite / adjacent.
Opposite and adjacent are named from the angle θ you are using. The hypotenuse is the side opposite the right angle.
- A link between the ratios
sin2θ + cos2θ = 1. tan θ = sin θ / cos θ, when cos θ is not 0.
- Complementary angles
sin(90° − θ) = cos θ. cos(90° − θ) = sin θ.
- Sine of 0°, 30°, 45°, 60° and 90°
sin 0° = 0. sin 30° = 1/2. sin 45° = √2/2. sin 60° = √3/2. sin 90° = 1.
√2/2 is the same value as 1/√2.
- Cosine of 0°, 30°, 45°, 60° and 90°
cos 0° = 1. cos 30° = √3/2. cos 45° = √2/2. cos 60° = 1/2. cos 90° = 0.
- Tangent of 0°, 30°, 45°, 60° and 90°
tan 0° = 0. tan 30° = 1/√3. tan 45° = 1. tan 60° = √3. tan 90° is undefined.
1/√3 is the same value as √3/3.
- Area from two sides and the included angle
Area = (1/2) × a × b × sin C.
C is the angle between sides a and b. This is for any triangle, not only a right angled one.
- Sine rule
a / sin A = b / sin B = c / sin C.
Side a is opposite angle A, and the same for b and B, and for c and C. Use it when you know a side and the angle opposite it, plus one more side or angle.
- Cosine rule
a2 = b2 + c2 − 2 × b × c × cos A. cos A = (b2 + c2 − a2) / (2 × b × c).
Side a is opposite angle A. Use the first line when you know two sides and the angle between them. Use the second when you know all three sides.
- Elevation and depression
An angle of elevation is measured upwards from the horizontal. An angle of depression is measured downwards from the horizontal.
Both sit in a right angled triangle, so the sine, cosine and tangent ratios apply.
Bearings
- How to write a bearing
A bearing is measured clockwise from north and written with three figures.
- The four main directions
North is 000°. East is 090°. South is 180°. West is 270°.
- Back bearing
If a bearing is under 180°, the back bearing is that bearing plus 180°. If a bearing is 180° or more, the back bearing is that bearing minus 180°.
Coordinate geometry
- Gradient
m = (y2 − y1) / (x2 − x1), when x2 is not equal to x1.
If x₂ = x₁, the line is vertical and it has no gradient.
- Distance
Distance = √[(x2 − x1)2 + (y2 − y1)2].
- Midpoint
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2).
- Equation of a line
y − y1 = m × (x − x1). Also y = m × x + c.
c is the value of y where the line crosses the vertical axis. The first line is for a line through the point (x₁, y₁) with gradient m.
- Parallel and perpendicular lines
Parallel lines that have gradients have the same gradient. If two lines are perpendicular and both have gradients, the product of the gradients is −1.
If one line is vertical, a line perpendicular to it is horizontal, and that horizontal line has gradient 0.
Statistics and probability
- Mean
Mean = (sum of the values) / (number of values).
- Mean from a frequency table
Mean = Σ(f × x) / Σf.
x is the value, or the midpoint of a class. f is the frequency of that value or class.
- Median
Sort the values. If the count is odd, the median is the middle value. If the count is even, the median is the mean of the two middle values.
- Mode
The mode is the value that occurs most often. A list can have more than one mode.
- Median of grouped data
Median = L + [((N / 2) − F) / f] × c.
L is the lower class boundary of the median class, not the lower limit written on the class. N is the total frequency. F is the cumulative frequency before the median class. f is the frequency of the median class. c is the class width.
- Mode of grouped data
Mode = L + [(f1 − f0) / (2 × f1 − f0 − f2)] × c.
L is the lower class boundary of the modal class. f₁ is its frequency. f₀ is the frequency of the class before it. f₂ is the frequency of the class after it. c is the class width. The formula needs a class on each side of the modal class.
- Standard deviation
Standard deviation = √[ Σ(x − mean)2 / n ]. With frequencies, √[ Σ(f × (x − mean)2) / Σf ].
School statistics divides by the number of values, n or Σf, not by one less than that number. A second form of the first line is √[ (Σx²) / n − (mean)² ]. The mean in that form is the same arithmetic mean.
- Mean deviation
Mean deviation = Σ|x − mean| / n. With frequencies, Σ(f × |x − mean|) / Σf.
- Probability of an event
P(A) = (number of favourable outcomes) / (number of equally likely outcomes).
- The range of a probability
0 ≤ P(A) ≤ 1.
- The complement
P(not A) = 1 − P(A).
- Events that cannot happen together
If A and B cannot happen together, P(A or B) = P(A) + P(B).
- Either event, including an overlap
P(A or B) = P(A) + P(B) − P(A and B).
Subtract the overlap so it is not counted twice. If A and B cannot happen together, that overlap is 0 and the line becomes the sum.
- Independent events
If A and B are independent, P(A and B) = P(A) × P(B).
Independent means neither event changes the chance of the other.
Variation
- Direct variation
If y varies directly as x, then y = k × x.
k is the constant of variation. It stays the same for that relationship.
- Inverse variation
If y varies inversely as x, then y = k / x, when x is not 0.
- Direct as a square
If y varies directly as x2, then y = k × x2.
- Joint variation
If y varies jointly as x and z, then y = k × x × z.
- Partial variation
If y varies directly as x and inversely as z, then y = (k × x) / z, when z is not 0.
Simple and compound interest
- Simple interest
I = (P × R × T) / 100
P is the principal. R is the rate in percent per year, so 5 percent means R = 5, not a decimal. T is the time in years. Four months is 4/12 of a year. The amount you have at the end is P + I.
- Amount with simple interest
A = P × (1 + (R × T) / 100)
This is the same result as adding the simple interest to the principal. Interest is not added back during the time.
- Compound interest, added once a year
A = P × (1 + R / 100)n
A is the amount after n years. R is the percent per year. The compound interest is A − P. Do not also apply the simple interest formula to the same sum.
- Compound interest, added m times a year
A = P × (1 + R / (100 × m))mn
R is still the percent per year. m is how many times the interest is added in one year. The power mn is the number of periods. The compound interest is A − P.
Modular arithmetic
- Congruence
a ≡ b (modulo m) when m divides a − b.
m is a whole number greater than 1. a and b leave the same remainder when they are divided by m.
- Remainders
The remainder when a whole number is divided by m is a whole number from 0 up to m − 1.
- Adding and multiplying
(a + b) modulo m = ((a modulo m) + (b modulo m)) modulo m. (a × b) modulo m = ((a modulo m) × (b modulo m)) modulo m.
You can reduce to a remainder at each step. The final remainder is the same as if you did the whole calculation first.
Sets
- Union of two sets
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
n(A) is how many elements are in A. The union is everything in A or B or both. The intersection is only what is in both. The subtraction stops the overlap being counted twice.
- Complement
n(A′) = n(U) − n(A).
U is the universal set. A′ is everything in U that is not in A. A and A′ have nothing in common, and together they make U.
- Number of subsets
A set with n elements has 2n subsets.
The count includes the empty set and the set itself.
- The empty set
The empty set has no elements, so n(∅) = 0. It is a subset of every set.
- Union of three sets
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C).
Each pairwise count includes elements that are also in the third set. The last term puts that triple overlap back in once.
Use the sheet with these pages
Kevin and Anna are the AI coaches on glearn.now. Kevin explains BECE working. Anna explains WASSCE working. The first questions need no account.
WASSCE practiceCore Mathematics past questionsWASSCE grade calculatorHow to pass the WASSCEWASSCE revision planWASSCE syllabusBECE Mathematics formula sheet
Practise with Kevin and Anna