ASSEB HSLC General Mathematics flashcards

36 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 > 0, there exist unique integers q and r such that a = bq + r and 0 ≤ r < b. It is the basis of Euclid's algorithm for the HCF.

HCF–LCM product rule

For any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. Knowing either value lets you compute the other directly by division.

Terminating decimal condition

A rational number p/q in lowest terms has a terminating decimal expansion exactly when the prime factors of q are only 2 and 5, that is q = 2^m·5^n.

Fundamental Theorem of Arithmetic

Every integer greater than 1 can be expressed as a product of primes, and this prime factorisation is unique except for the order in which the primes are written.

Rationalising a binomial surd

To remove a surd from a denominator such as a + √b, multiply numerator and denominator by the conjugate a − √b. The denominator becomes a² − b, which is rational.

Conjugate surd pair

The expressions a + √b and a − √b are conjugates. Their product is a² − b, a rational number, which is why conjugates are used to rationalise denominators.

Comparing surds by common index

To compare surds with different orders, rewrite them with a common index, for example as sixth roots, then compare the radicands. Larger radicand means larger value for positive numbers.

Remainder Theorem

When a polynomial p(x) is divided by x − a, the remainder equals p(a). If p(a) = 0, then x − a is an exact factor of the polynomial.

Sum and product of roots

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

Quadratic from given roots

If α and β are the roots, the quadratic equation is x² − (α + β)x + αβ = 0. Any non-zero constant multiple of this polynomial has the same roots.

Discriminant test

For ax² + bx + c = 0, the discriminant D = b² − 4ac decides the roots: D > 0 gives two distinct real roots, D = 0 gives equal real roots, and D < 0 gives no real roots.

Consistency of linear pairs

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the pair is consistent with one solution if a₁/a₂ ≠ b₁/b₂, inconsistent if a₁/a₂ = b₁/b₂ ≠ c₁/c₂, and dependent if all three ratios are equal.

Quadratic formula

The roots of ax² + bx + c = 0 with a ≠ 0 are x = [−b ± √(b² − 4ac)]/(2a). The formula works for every quadratic, whether or not it factorises over the integers.

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. It grows linearly with n.

Sum of n AP terms

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

Arithmetic mean condition

Three numbers a, b, c are in arithmetic progression exactly when 2b = a + c. Then b is the arithmetic mean of a and c, lying midway between them.

Section formula

The point dividing the segment joining (x₁, y₁) and (x₂, y₂) internally in the ratio m:n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). Setting m = n gives the midpoint.

Perpendicular line slopes

Two non-vertical lines are perpendicular exactly when the product of their slopes is −1, so each slope is the negative reciprocal of the other.

Collinearity by area

Three points are collinear exactly when the area of the triangle they form is zero. This is tested by evaluating the coordinate area formula and setting it to zero.

Pythagorean identity

For every angle θ, sin²θ + cos²θ = 1. Dividing this identity by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.

Angle of elevation

The angle of elevation is the angle measured upward from the horizontal observer's line to the line of sight of an object above. Problems use tan θ = height/horizontal distance.

Angle of depression

The angle of depression is measured downward from the horizontal through the observer's eye to an object below. It equals the angle of elevation from the object to the observer.

Exterior angle of a triangle

An exterior angle of a triangle equals the sum of the two interior opposite angles. This follows from the angle sum of 180° and angles on a straight line.

Basic Proportionality Theorem

A line drawn parallel to one side of a triangle and intersecting the other two sides divides those two sides in the same ratio. Its converse also holds.

Similar triangle area ratio

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

Alternate Segment Theorem

The angle between a tangent and a chord drawn from the point of contact equals the angle subtended by that chord in the alternate segment of the circle.

Tangent–secant power relation

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

Cyclic quadrilateral property

A quadrilateral is cyclic exactly when its opposite angles are supplementary, summing to 180°. The exterior angle then equals the interior opposite angle.

Heron's formula

The area of a triangle with sides a, b, c is √[s(s − a)(s − b)(s − c)], where the semi-perimeter s = (a + b + c)/2. No height is needed.

Cone slant height

For a right circular cone of radius r and height h, the slant height is l = √(r² + h²). Curved surface area is πrl and total surface area is πr(l + r).

Grouped data median formula

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

Complementary probability rule

For any event E, the probability that E does not occur is P(not E) = 1 − P(E). This is useful when counting the complement is simpler than counting E itself.

Number of subsets

A set with n distinct elements has 2ⁿ subsets, including the empty set and the set itself. It has 2ⁿ − 1 proper subsets, which exclude the set itself.

De Morgan's laws

For sets A and B, the complement of A ∪ B equals A′ ∩ B′, and the complement of A ∩ B equals A′ ∪ B′. These laws mirror negation rules in logic.

Contrapositive equivalence

The contrapositive of p → q is ¬q → ¬p. A conditional statement and its contrapositive always have the same truth value, unlike the converse q → p.

Counterexample

A single case in which the hypothesis of a universal statement is true but the conclusion is false is enough to disprove that statement. One counterexample settles the matter.