Cuba 12th Grade Math Final Exam cheat sheet

Similarity scales lengths by k, areas by k², volumes by k³.

Inscribed angle equals half the intercepted central arc.

Sine law requires a known side–opposite-angle pair; watch the ambiguous case.

Cosine law handles SAS and SSS configurations.

Arc length and sector area formulas require radians.

Pythagorean identity sin²θ + cos²θ = 1 underpins most simplifications.

i² = −1; division uses multiplication by the conjugate.

|z| = √(a² + b²), and the argument must be placed in the correct quadrant.

Modulus–argument form is re^(iθ); multiplication adds arguments.

Real-coefficient polynomials have non-real roots in conjugate pairs.

|z − z₀| = r is a circle centred at z₀.

Median resists outliers; mean does not.

P(A ∪ B) subtracts the intersection to avoid double counting.

Conditional probability divides the joint probability by the condition's probability.

Combinations count selections; permutations count arrangements.

z = (x − μ)/σ places a value on the standard normal scale.

Random assignment, not random sampling alone, supports causal conclusions.

An implication and its contrapositive are equivalent; the converse is not.

Negating a quantified statement flips the quantifier and negates the predicate.

Induction needs a base case, a stated hypothesis, and the step.

Inclusion–exclusion corrects for double counting.

Pigeonhole principle forces a collision when items exceed categories.

Sum of vertex degrees equals twice the edge count.

AP sum: n/2 × (first + last) or n/2(2a + (n − 1)d).

GP sum to infinity exists only for |r| < 1 and equals a/(1 − r).

The general binomial term C(n, r)a^(n−r)b^r gives any coefficient directly.

Count the number of terms carefully when the index starts at zero.

A negative common ratio alternates signs, so check the parity of the power.

Consistency requires rank A = rank [A|b]; uniqueness also needs full column rank.

det A = 0 means no unique solution, not necessarily no solution.

Trace = sum of eigenvalues; determinant = product of eigenvalues.

a×b is perpendicular to both vectors; its magnitude is the parallelogram area.

Distance from point to plane: |n·P − d| / |n|.