Botswana BGCSE General Mathematics 1236 flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Standard form

A number written as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. For example, 4.2 × 10³ = 4200 and 3.1 × 10⁻² = 0.031.

Percentage change formula

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

Index law for division

When dividing powers with the same base, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Also a⁰ = 1 for any non-zero a, and a⁻ⁿ = 1/aⁿ.

Dividing in a given ratio

To share an amount in the ratio a : b, add the parts to get a + b, divide the amount by that total to find one part, then multiply by a and by b.

Union of two sets

The union A ∪ B is the set of all elements that belong to A, to B, or to both. On a Venn diagram it is every region inside the two circles.

Intersection of two sets

The intersection A ∩ B contains only the elements that are common to both A and B. On a Venn diagram it is the overlapping region shared by the two circles.

Complement of a set

The complement A′ contains every element of the universal set that is not in A. Its size is n(A′) = n(ξ) − n(A).

Number of subsets

A set with n distinct elements has 2ⁿ subsets in total, counting the empty set and the set itself. For example, a set of 3 elements has 8 subsets.

Composite function

The composite function fg(x) means f(g(x)): apply g first to x, then apply f to that result. Order matters, so fg(x) is usually not equal to gf(x).

Inverse function

The inverse function f⁻¹ reverses the effect of f, so f(f⁻¹(x)) = x. To find it, write y = f(x), swap x and y, then make y the subject.

Domain and range

The domain of a function is the set of all input values it is allowed to take; the range is the set of all output values it actually produces.

One-to-one function

A function is one-to-one when every output value comes from exactly one input value, so no two different inputs give the same answer.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = [−b ± √(b² − 4ac)] ÷ 2a. The discriminant b² − 4ac shows how many real roots exist.

Difference of two squares

The expression a² − b² factorises to (a + b)(a − b). This identity simplifies expressions and solves equations such as x² − 25 = 0.

nth term of arithmetic sequence

For an arithmetic sequence with first term a and common difference d, the nth term is a + (n − 1)d, and the sum of n terms is n/2[2a + (n − 1)d].

Inequality multiplied by negative

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign: if −2x > 6, then x < −3.

Angle sum of a polygon

The interior angles of a polygon with n sides add up to (n − 2) × 180°. A pentagon therefore has interior angles totalling 3 × 180° = 540°.

Angle in a semicircle

The angle in a semicircle is 90°. If AB is a diameter and C is any other point on the circle, then angle ACB is a right angle.

Cyclic quadrilateral opposite angles

Opposite angles of a cyclic quadrilateral add up to 180°. All four vertices lie on one circle, which makes each pair of opposite angles supplementary.

Exterior angles of a polygon

The exterior angles of any convex polygon add up to 360°, taking one at each vertex. In a regular polygon each exterior angle equals 360° ÷ n.

Volume of a cone

Volume of a cone = ⅓πr²h, where r is the radius of the circular base and h is the perpendicular height. A cone is one third of the matching cylinder.

Surface area of a sphere

Surface area of a sphere = 4πr², where r is its radius. The volume of the same sphere is given by 4/3 πr³.

Area of a sector

Area of a sector = (θ ÷ 360) × πr², where θ is the angle at the centre measured in degrees. Its arc length is (θ ÷ 360) × 2πr.

Volume scale factor for solids

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

Mean of grouped data

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

Mutually exclusive events

Events that cannot happen at the same time are mutually exclusive. For such events P(A or B) = P(A) + P(B), because P(A and B) equals zero.

Independent events

Two events are independent when the occurrence of one does not change the probability of the other. Then P(A and B) = P(A) × P(B).

Pie chart sector angle

The sector angle for a category equals (frequency ÷ total frequency) × 360°. All the sector angles in a complete pie chart add up to 360°.

Sine rule

In any triangle, a ÷ sin A = b ÷ sin B = c ÷ sin C, where side a is opposite angle A. Use it when a matching side and angle pair is known.

Cosine rule

a² = b² + c² − 2bc cos A. Use it to find a side when two sides and the included angle are known, or to find an angle from three sides.

Three-figure bearing

A bearing is measured clockwise from north and written with three digits, for example 045° for north-east or 210° for a direction in the south-west quadrant.

Exact trigonometric values

sin 30° = 0.5, cos 60° = 0.5 and tan 45° = 1. These exact values come from the 30-60-90 and 45-45-90 right-angled triangles.