Botswana BGCSE General Mathematics 1236 cheat sheet

Standard form writes a number as a × 10ⁿ where 1 ≤ a < 10; moving the decimal point k places changes the index by k.

Percentage change = (new value − original value) ÷ original value × 100%; a positive result is an increase.

Simple interest I = PRT ÷ 100; compound amount A = P(1 + r/100)ⁿ where n is the number of periods.

To divide a quantity in the ratio a : b : c, add the parts (a + b + c), divide the quantity by this total to get one part, then multiply.

Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

Order of operations: brackets, indices, multiplication and division (left to right), then addition and subtraction.

To convert a recurring decimal to a fraction, let x equal the decimal, multiply by 10ⁿ to shift the repeating block, then subtract to eliminate the recurring part.

HCF × LCM = product of the two numbers; find both from the prime factorisation written in index form.

To compare or order fractions, rewrite them with a common denominator or convert each to a decimal.

Notation: ξ is the universal set, ∅ or { } is the empty set, n(A) is the number of elements in A, ∈ means 'is an element of'.

A ∪ B contains every element in A or B or both; A ∩ B contains only elements common to both sets.

Two-set formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Complement A′ contains the elements of ξ not in A, so n(A′) = n(ξ) − n(A).

Venn regions: only A = n(A) − n(A ∩ B); only B = n(B) − n(A ∩ B); outside both = n(ξ) − n(A ∪ B).

Three-set formula: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C).

A set with n elements has 2ⁿ subsets, including the empty set and the set itself.

A ⊂ B means every element of A is also in B; sets are disjoint when A ∩ B = ∅.

When a word problem gives a total, fill the innermost Venn region first and work outwards.

A function assigns exactly one output to each input; the domain is the set of allowed inputs and the range the set of outputs produced.

f(3) means substitute x = 3 into the rule for f; the result is the value of the function at 3.

Composite functions: fg(x) = f(g(x)), so g is applied first; in general fg(x) ≠ gf(x).

To find an inverse, write y = f(x), swap x and y, then make y the subject; the result is f⁻¹(x) and f(f⁻¹(x)) = x.

A function is one-to-one when each output arises from exactly one input; it is many-to-one when different inputs share an output.

Vertical line test: a graph shows a function only if no vertical line meets it more than once.

The graph of y = mx + c is a straight line with gradient m and y-intercept c.

Restrictions on the domain: the denominator of a fraction cannot be zero, and the expression under a square root cannot be negative.

The graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x.

Expansions: a(b + c) = ab + ac; (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; (a + b)(a − b) = a² − b².

Factorising order: take out a common factor first, then test difference of two squares, then factorise a trinomial into two brackets.

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = [−b ± √(b² − 4ac)] ÷ 2a.

The discriminant b² − 4ac gives two distinct real roots when positive, one repeated root when zero, and no real roots when negative.

Simultaneous linear equations are solved by elimination (match coefficients, then add or subtract) or by substitution.

Inequalities are solved like equations, but the inequality sign reverses when both sides are multiplied or divided by a negative number.

Arithmetic sequence: nth term = a + (n − 1)d; sum of n terms = n/2 [2a + (n − 1)d].

Geometric sequence: nth term = arⁿ⁻¹; sum of n terms = a(rⁿ − 1) ÷ (r − 1) for r ≠ 1.

Changing the subject: perform the same operation on both sides and undo operations in reverse order; direct variation is y = kx, inverse variation is y = k/x.

Angles on a straight line sum to 180°, angles at a point sum to 360°, and vertically opposite angles are equal.

With parallel lines cut by a transversal: alternate angles are equal, corresponding angles are equal, and co-interior angles sum to 180°.

For a polygon with n sides the interior angles sum to (n − 2) × 180° and the exterior angles sum to 360°; in a regular polygon each exterior angle is 360°/n and each interior angle is (n − 2)180°/n.

Circle theorems: the angle at the centre is twice the angle at the circumference, the angle in a semicircle is 90°, and angles in the same segment are equal.

In a cyclic quadrilateral opposite angles sum to 180°, and an exterior angle equals the interior opposite angle.

A tangent is perpendicular to the radius at the point of contact, and two tangents drawn from the same external point are equal in length.

Congruence tests for triangles: SSS, SAS, ASA (or AAS) and RHS.

Transformations: reflection in a mirror line, rotation about a centre through a given angle and direction, translation by a column vector, enlargement by a scale factor from a centre.

Vectors: a column vector (x over y) has magnitude √(x² + y²); vectors add head-to-tail and a negative sign reverses direction.

Area formulas: rectangle l × w, triangle ½ × base × height, parallelogram b × h, trapezium ½(a + b)h.

Circle: circumference = 2πr = πd and area = πr².

Arc length = (θ/360) × 2πr and sector area = (θ/360) × πr², where θ is the angle at the centre in degrees.

Volume: cuboid lwh; any prism = area of cross-section × length; cylinder = πr²h.

Volume: cone = ⅓πr²h; sphere = 4/3 πr³; pyramid = ⅓ × base area × perpendicular height.

Surface area: cuboid 2(lw + lh + wh); cylinder 2πr² + 2πrh; sphere 4πr²; cone πr² + πrl where l is the slant height.

Total surface area of a prism = 2 × cross-sectional area + perimeter of cross-section × length.

If two similar solids have corresponding lengths in the ratio k : 1, their areas are in the ratio k² : 1 and their volumes in the ratio k³ : 1.

Area of a triangle from two sides and the included angle: ½ab sin C.

Mean = sum of values ÷ number of values; median = middle value of the ordered data; mode = most frequent value; range = highest − lowest.

For grouped data the mean is estimated by Σfx ÷ Σf, where x is the midpoint of each class and f is its frequency.

On a cumulative frequency curve the median is read at n/2, the lower quartile at n/4 and the upper quartile at 3n/4.

Pie chart sector angle for a category = (frequency ÷ total frequency) × 360°.

Probability P(E) = number of favourable outcomes ÷ number of equally likely outcomes, and 0 ≤ P(E) ≤ 1.

Complement rule: P(not E) = 1 − P(E).

Mutually exclusive events: P(A or B) = P(A) + P(B), because P(A and B) = 0.

Independent events: P(A and B) = P(A) × P(B).

Experimental probability = number of successful trials ÷ total number of trials.

SOHCAHTOA: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.

Pythagoras: in a right-angled triangle with hypotenuse c, a² + b² = c².

Exact values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3; sin 45° = cos 45° = 1/√2; sin 60° = √3/2, cos 60° = ½, tan 60° = √3.

Sine rule: a ÷ sin A = b ÷ sin B = c ÷ sin C; use it when a side and its opposite angle form a known pair.

Cosine rule: a² = b² + c² − 2bc cos A; rearrange to cos A = (b² + c² − a²) ÷ 2bc to find an angle from three sides.

Area of any triangle = ½ab sin C, using two sides and the angle between them.

The angle of elevation is measured upwards from the horizontal; the angle of depression is measured downwards from the horizontal.

A three-figure bearing is measured clockwise from north and written with three digits, for example 072° or 215°.

In a right-angled triangle tan θ equals the gradient of the slope, that is rise ÷ run.