BSEH Secondary Mathematics - Basic flashcards

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

Euclid's division lemma

For integers a and b with b positive, there exist unique integers q and r such that a = bq + r and 0 ≤ r < b. It is the basis of the Euclidean algorithm for HCF.

Fundamental Theorem of Arithmetic

Every integer greater than 1 can be expressed as a product of primes, and this factorization is unique apart from the order of the factors.

Terminating decimal condition

A rational number p/q in lowest terms has a terminating decimal expansion exactly when the denominator q is of the form 2^m times 5^n for non-negative integers m and n.

HCF–LCM product rule

For any two positive integers a and b, the product of their highest common factor and lowest common multiple equals a times b: HCF(a, b) × LCM(a, b) = ab.

Remainder Theorem

If a polynomial p(x) is divided by the linear polynomial x − a, then the remainder is p(a), the value of the polynomial at x = a.

Factor Theorem

A polynomial p(x) has (x − a) as a factor if and only if p(a) = 0; equivalently, a is a root or zero of the polynomial p.

Sum and product of roots

For the quadratic ax² + bx + c = 0 with roots α and β, the sum α + β equals −b/a and the product αβ equals c/a.

Consistent pair of linear equations

Two linear equations are consistent when they have at least one common solution; their coefficient ratios satisfy a₁/a₂ ≠ b₁/b₂ (unique solution) or all three ratios are equal (infinitely many).

Quadratic discriminant

For ax² + bx + c = 0, D = b² − 4ac. If D > 0 there are two distinct real roots, if D = 0 one repeated root, and if D < 0 no real roots.

Quadratic formula

The roots of ax² + bx + c = 0 with a ≠ 0 are x = (−b ± √(b² − 4ac)) / (2a), obtained by completing the square on the general quadratic.

nth term of an AP

In an arithmetic progression with first term a and common difference d, the nth term is aₙ = a + (n − 1)d; the difference between consecutive terms is constant.

Sum of n terms of AP

The sum of the first n terms of an AP is Sₙ = (n/2)[2a + (n − 1)d], which also equals (n/2)(a + l) when l is the last term.

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) in the plane is √[(x₂ − x₁)² + (y₂ − y₁)²], derived from the Pythagoras theorem on the coordinate differences.

Section formula

The point dividing the segment joining (x₁, y₁) and (x₂, y₂) internally in the ratio m : n has coordinates ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)).

Area of a triangle by coordinates

The area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; a zero result means the points are collinear.

Perpendicular lines slopes

Two non-vertical lines are perpendicular exactly when the product of their slopes equals −1, so a line perpendicular to slope m has slope −1/m.

Basic Proportionality Theorem

A line drawn parallel to one side of a triangle intersects the other two sides in the same ratio; conversely, if it divides them proportionally, it is parallel to the third side.

Similarity area ratio

If two triangles are similar with corresponding sides in the ratio k, then their corresponding altitudes, medians and perimeters are in ratio k, and their areas are in ratio k².

Pythagoras theorem

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides; its converse identifies a triangle as right-angled.

Midpoint theorem

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is exactly half the length of that third side.

Tangent perpendicular to radius

A tangent to a circle is perpendicular to the radius drawn to the point of contact; from an external point, the two tangent segments to a circle are equal in length.

Angle in a semicircle

The angle subtended by a diameter at any point on the circle is a right angle, since the angle at the centre is 180 degrees and the inscribed angle is half of it.

Cyclic quadrilateral angles

In a cyclic quadrilateral the opposite angles are supplementary, summing to 180 degrees; an exterior angle equals the interior opposite angle of the quadrilateral.

Tangent–secant relation

From an external point P, if PT is a tangent and a secant cuts the circle at A and B, then PT² = PA × PB, the power of the point P.

Trigonometric identity

For any acute angle θ, sin²θ + cos²θ = 1; dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Angle of elevation

The angle of elevation of an object is the angle between the horizontal line through the observer's eye and the line of sight to the object, measured upward.

Angle of depression

The angle of depression is measured downward from the horizontal through the observer to the object; it equals the angle of elevation from the object to the observer.

Volume of a cone

A cone of base radius r and height h has volume (1/3)πr²h, one third of the volume of the cylinder with the same base and height.

Sphere surface and volume

A sphere of radius r has surface area 4πr² and volume (4/3)πr³; a hemisphere has curved surface 2πr² and total surface 3πr².

Heron's formula

The area of a triangle with sides a, b, c is √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter (a + b + c)/2.

Median of grouped data

Median = l + [(n/2 − cf)/f] × h, where l is the lower limit of the median class, cf the cumulative frequency before it, f its frequency and h the class size.

Empirical mean-median-mode relation

For a moderately skewed frequency distribution, Mode = 3 Median − 2 Mean, an approximate relation used when one of the three measures is unknown.

Classical probability

If a random experiment has n equally likely outcomes and m of them are favourable to event E, then P(E) = m/n, a number between 0 and 1 inclusive.

Complementary event rule

For any event E, the probability of E not occurring is P(not E) = 1 − P(E), because the probabilities of an event and its complement sum to 1.