EJU Mathematics Course 2 flashcards

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

Absolute value inequality |x| < k

For k > 0, |x| < k is equivalent to −k < x < k. The solution is a single open interval centered at zero with radius k.

Discriminant of a quadratic

For ax² + bx + c = 0 with a ≠ 0, D = b² − 4ac. D > 0 gives two distinct real roots, D = 0 one repeated root, D < 0 no real roots.

Rationalizing a square-root denominator

Multiply numerator and denominator by the conjugate: 1/(√a + √b) becomes (√a − √b)/(a − b). This removes radicals from the denominator when a ≠ b.

Vertex of a parabola

For y = ax² + bx + c, the vertex is at x = −b/(2a) with y-value c − b²/(4a). It is the minimum when a > 0 and the maximum when a < 0.

Sign of a quadratic

With a > 0 and real roots α < β, ax² + bx + c is positive outside [α, β] and negative between the roots. If D < 0, the sign equals the sign of a everywhere.

Vieta's formulas

If α and β are the roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. These give symmetric expressions without solving the equation.

Inclusion–exclusion for two sets

n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The intersection is subtracted because its elements were counted twice when the two set sizes were added.

Combination formula

C(n, r) = n!/(r!(n − r)!) counts r-element subsets of an n-element set. It satisfies C(n, r) = C(n, n − r) and C(n, 0) = 1.

General term of binomial expansion

In (a + b)ⁿ, the term containing bᵏ is C(n, k) aⁿ⁻ᵏ bᵏ. Setting k appropriately finds a specific coefficient or power.

Conditional probability

P(B | A) = P(A ∩ B)/P(A), defined when P(A) > 0. It rescales probability to the reduced sample space where A has occurred.

Independent events

Events A and B are independent exactly when P(A ∩ B) = P(A)P(B), equivalently P(B | A) = P(B) when P(A) > 0.

Expected value of a random variable

E(X) = Σ xᵢ pᵢ, the probability-weighted average of the values. It is linear: E(aX + b) = aE(X) + b for constants a, b.

Euclidean algorithm

Repeatedly replace (a, b) with (b, r), where r is the remainder of a divided by b, until the remainder is zero. The last nonzero remainder is gcd(a, b).

Linear Diophantine solvability

The equation ax + by = c has integer solutions exactly when gcd(a, b) divides c. One solution plus multiples of (b/d, −a/d) gives all solutions, d = gcd(a, b).

Number of divisors

If N = p₁^e₁ p₂^e₂ … then N has (e₁ + 1)(e₂ + 1)… positive divisors, since each exponent can be chosen independently from 0 up to its maximum.

Inscribed angle theorem

An angle inscribed in a circle equals half the central angle subtending the same arc. Angles subtending the same arc from the same side are equal.

Power of a point

For a point P and a circle, the product PA · PB is constant for any secant through P. For a tangent, PT² equals that product.

Angle bisector theorem

In triangle ABC, the bisector of angle A meets side BC at D with BD/DC = AB/AC. The ratio of the divided segments equals the ratio of adjacent sides.

Pythagorean identity

sin²θ + cos²θ = 1 for every angle θ. Dividing by cos²θ gives 1 + tan²θ = 1/cos²θ, valid where cos θ ≠ 0.

Law of cosines

In triangle ABC, a² = b² + c² − 2bc cos A, where a is opposite angle A. It determines a side from two sides and the included angle.

Sinusoidal combination formula

a sin θ + b cos θ = √(a² + b²) sin(θ + φ), where cos φ = a/√(a² + b²) and sin φ = b/√(a² + b²). This reveals amplitude and phase.

Definition of logarithm

log_a b = c means aᶜ = b, with base a > 0, a ≠ 1, and argument b > 0. The logarithm is the exponent needed to produce b from a.

Change of base formula

log_a b = log_c b / log_c c for positive bases not equal to 1. A consequence is log_a b · log_b a = 1 for valid bases.

Number of digits of an integer

A positive integer N has ⌊log₁₀ N⌋ + 1 digits. The fractional part of log₁₀ N determines the leading digits of N.

Tangent line to a curve

At x = a, the tangent to y = f(x) is y − f(a) = f′(a)(x − a). The normal there has slope −1/f′(a) when f′(a) ≠ 0.

Second derivative test

If f′(a) = 0 and f″(a) > 0, then f has a local minimum at a. If f″(a) < 0, it has a local maximum there.

Definite integral and area

∫ₐᵇ f(x) dx = F(b) − F(a) for any antiderivative F. The area between two curves is the integral of upper minus lower over the interval.

Sum of a geometric series

For ratio r ≠ 1, the first n terms sum to a₁(rⁿ − 1)/(r − 1). As n grows without bound the sum tends to a₁/(1 − r) when |r| < 1.

Arithmetic sequence sum

The first n terms of an arithmetic sequence with first term a₁ and common difference d sum to n(2a₁ + (n − 1)d)/2, equivalently n(a₁ + aₙ)/2.

Linear recurrence fixed point

For aₙ₊₁ = paₙ + q with p ≠ 1, subtracting α = q/(1 − p) gives aₙ₊₁ − α = p(aₙ − α), a geometric sequence in aₙ − α.

Inner product and angle

a · b = a₁b₁ + a₂b₂ = |a||b| cos θ, so cos θ = (a · b)/(|a||b|). Nonzero vectors are perpendicular exactly when a · b = 0.

Internal division of a segment

The point dividing segment AB internally in ratio m : n has position vector (n a + m b)/(m + n), where a and b are the position vectors of A and B.