Kyoto Entrance Mathematics flashcards

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

Difference of cubes

a³ − b³ = (a − b)(a² + ab + b²). The quadratic factor has no real roots when a and b are real, so the factorization cannot be pushed further over the reals.

Rationalizing a denominator

Multiply numerator and denominator by the conjugate of the denominator. For example 1/(√5 − √3) becomes (√5 + √3)/2, since the denominator equals 5 − 3.

Vieta's formulas

For ax² + bx + c = 0 with roots α and β, the sum α + β equals −b/a and the product αβ equals c/a. Useful when roots are needed only through symmetric expressions.

Extraneous roots from squaring

Squaring both sides of an equation can introduce solutions that fail the original. After squaring, always substitute candidates back, or impose the sign condition required by the root.

De Morgan's laws

The complement of a union is the intersection of complements, and the complement of an intersection is the union of complements: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ.

Contrapositive equivalence

The statement P ⇒ Q is logically equivalent to its contrapositive ¬Q ⇒ ¬P. Proving the contrapositive therefore proves the original; the converse Q ⇒ P is a different statement.

Vertex of a parabola

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

Quadratic on a closed interval

To find the maximum or minimum of a quadratic on [α, β], compare the value at the vertex, if it lies inside the interval, with the values at both endpoints.

Cosine rule

In a triangle with sides a, b, c and angle A opposite side a, a² = b² + c² − 2bc cos A. It determines a side from two sides and the included angle, or an angle from three sides.

Synthesis of sine and cosine

a sinθ + b cosθ equals √(a² + b²) sin(θ + φ), where cos φ = a/√(a² + b²) and sin φ = b/√(a² + b²). This reduces the expression to a single wave.

Change of base formula

log_a b = log_c b / log_c a for any valid base c. A consequence is log_a b · log_b a = 1, which lets you swap the base and the argument.

Solving exponential equations by substitution

In equations such as 4ˣ − 3·2ˣ + 2 = 0, set t = 2ˣ with t > 0. Solve the resulting quadratic in t, then recover x by taking logarithms.

Chain rule

If y = f(u) and u = g(x), then dy/dx = f′(u)·g′(x). For example the derivative of sin(x²) is 2x cos(x²).

Second derivative test

At a stationary point where f′(a) = 0, f″(a) > 0 means a local minimum, f″(a) < 0 means a local maximum, and f″(a) = 0 is inconclusive.

Normal line to a curve

The normal at x = a is perpendicular to the tangent, so its slope is −1/f′(a) when f′(a) ≠ 0. It is vertical when the tangent is horizontal.

Area between two curves

If f(x) ≥ g(x) on [a, b], the enclosed area is ∫_a^b {f(x) − g(x)} dx. If the curves cross inside the interval, split at the crossing points.

Volume of revolution about x-axis

Rotating y = f(x) for a ≤ x ≤ b about the x-axis gives volume V = π∫_a^b {f(x)}² dx, provided f(x) ≥ 0 on the interval.

Integration by substitution

Set u = g(x), so du = g′(x) dx, and rewrite ∫f(g(x))g′(x) dx as ∫f(u) du. For definite integrals, convert the limits to u-values.

Infinite geometric series

The sum a + ar + ar² + … converges to a/(1 − r) exactly when |r| < 1. When |r| ≥ 1 the series diverges and has no finite sum.

Principle of mathematical induction

Show the statement holds for n = 1, assume it holds for n = k, and use that assumption to prove it for n = k + 1. This establishes it for all positive integers.

Telescoping sum

When each term is written as a difference, consecutive terms cancel. For example Σ 1/(k(k+1)) from 1 to n equals 1 − 1/(n+1).

Dot product perpendicularity test

For nonzero vectors a and b, a·b = 0 exactly when a and b are perpendicular. In coordinates, a₁b₁ + a₂b₂ + a₃b₃ = 0 in space.

Equation of a plane

A plane through point A with normal vector n consists of points P satisfying n·(OP→ − OA→) = 0. Expanded, it reads n₁x + n₂y + n₃z = k.

Modulus and conjugate

For z = a + bi, the conjugate is z̄ = a − bi and the modulus is |z| = √(a² + b²). Their product z z̄ equals a² + b², a nonnegative real number.

De Moivre's theorem

For z = r(cosθ + i sinθ) and a positive integer n, zⁿ = rⁿ(cos nθ + i sin nθ). It converts powers of complex numbers into scaling and rotation.

nth roots of a complex number

The equation zⁿ = w has exactly n distinct complex roots, equally spaced on a circle of radius |w|^{1/n}, with arguments (arg w + 2kπ)/n for k = 0 to n − 1.

Expected value of a random variable

E(X) = Σ xₖ pₖ over all values with their probabilities. It is linear, so E(aX + b) = aE(X) + b for constants a and b.

Variance shortcut formula

V(X) = E(X²) − {E(X)}². The variance is never negative, and V(aX + b) = a²V(X), so adding a constant does not change spread.

Binomial probability

In n independent trials each with success probability p, the probability of exactly r successes is C(n, r) pʳ (1 − p)ⁿ⁻ʳ.

Euclidean algorithm

Replace gcd(a, b) by gcd(b, r) where r is the remainder of a divided by b, and repeat until the remainder is zero. The last nonzero remainder is the gcd.

Solving linear congruences

The congruence ax ≡ b (mod m) has solutions exactly when gcd(a, m) divides b. It then has gcd(a, m) solutions modulo m, found by dividing through by the gcd.

Fermat's little theorem

If p is prime and a is not divisible by p, then a^{p−1} ≡ 1 (mod p). It reduces large powers modulo a prime and gives the inverse a^{p−2} mod p.