JCE Mathematics (Syllabus 13) cheat sheet

Order of operations: evaluate grouping symbols first, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right.

Fraction rules: a/b + c/d = (ad + bc)/(bd); a/b × c/d = ac/bd; a/b ÷ c/d = a/b × d/c (b, d ≠ 0).

Exponent laws: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); a^m/a^n = a^(m−n); a^0 = 1 for a ≠ 0; a^(−n) = 1/a^n.

Divisibility: by 3 if the digit sum is divisible by 3; by 4 if the last two digits form a multiple of 4; by 9 if the digit sum is divisible by 9; by 6 if divisible by both 2 and 3.

Percent relationships: part = percent × whole; percent change = (new − old)/old × 100%; increasing a quantity by p% multiplies it by (1 + p/100).

For positive integers a and b, gcd(a, b) × lcm(a, b) = a × b.

A proportion a/b = c/d holds exactly when ad = bc (b, d ≠ 0); in direct variation y/x is constant, while in inverse variation xy is constant.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiplying such numbers adds the exponents and dividing subtracts them.

Distributive property: a(b + c) = ab + ac; factoring reverses it, so ab + ac = a(b + c).

Solving ax + b = c with a ≠ 0 gives x = (c − b)/a.

Slope through two points: m = (y₂ − y₁)/(x₂ − x₁); point-slope form of a line: y − y₁ = m(x − x₁).

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a) for ax² + bx + c = 0 with a ≠ 0.

The discriminant b² − 4ac gives two distinct real roots when positive, one repeated real root when zero, and no real roots when negative.

Special products: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; (a − b)(a + b) = a² − b².

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol; adding or subtracting a number does not.

A linear system has one solution when the lines have different slopes, no solution when the slopes are equal but intercepts differ, and infinitely many solutions when both equations describe the same line.

Radicals: √(ab) = √a · √b for a, b ≥ 0, and √(a/b) = √a/√b for a ≥ 0 and b > 0.

Area: rectangle A = bh; triangle A = ½bh; trapezoid A = ½(b₁ + b₂)h; circle A = πr².

Circumference C = 2πr = πd; arc length = (central angle/360°) × 2πr.

Volume: prism or cylinder V = Bh; pyramid or cone V = ⅓Bh; sphere V = (4/3)πr³.

Surface area: cylinder = 2πr² + 2πrh; sphere = 4πr².

Pythagorean theorem: in a right triangle a² + b² = c², where c is the hypotenuse; the converse holds, so triples such as 3-4-5 and 5-12-13 form right triangles.

The interior angles of a triangle sum to 180°, and an exterior angle equals the sum of the two nonadjacent interior angles.

When parallel lines are cut by a transversal, alternate interior angles and corresponding angles are equal, and consecutive interior angles are supplementary.

Similar figures have proportional corresponding sides; for scale factor k, areas compare as k² and volumes as k³.

Coordinate geometry: distance = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Mean = (sum of values)/n; median = middle value of ordered data; mode = most frequent value; range = maximum − minimum.

Weighted mean = Σ(w · x)/Σw.

For equally likely outcomes, P(E) = favorable outcomes / total outcomes, and 0 ≤ P(E) ≤ 1.

Complement rule: P(not E) = 1 − P(E).

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the intersection term is zero.

Independent events satisfy P(A and B) = P(A) · P(B); conditional probability is P(A|B) = P(A and B)/P(B).

Counting: permutations nPr = n!/(n − r)!; combinations nCr = n!/[r!(n − r)!].

Expected value E(X) = Σ x · P(x); IQR = Q₃ − Q₁, and values lying more than 1.5 × IQR beyond a quartile are potential outliers.

Standard deviation measures the typical distance of values from the mean; a larger standard deviation indicates greater spread.

A relation is a function when every input has exactly one output; the vertical line test identifies functions on a graph.

Linear function f(x) = mx + b has slope m and y-intercept b; parallel lines have equal slopes, and perpendicular slopes multiply to −1.

Quadratic f(x) = ax² + bx + c has its vertex at x = −b/(2a); vertex form f(x) = a(x − h)² + k has vertex (h, k) and axis of symmetry x = h.

Transformations: f(x − h) + k shifts the graph h units right and k units up; −f(x) reflects it over the x-axis; a · f(x) stretches it vertically by |a|.

Exponential f(x) = a · b^x grows when b > 1 and decays when 0 < b < 1; its graph has horizontal asymptote y = 0.

Composition (f ∘ g)(x) = f(g(x)); inputs where g(x) lies outside the domain of f are excluded from the domain of the composition.

An inverse function f⁻¹ reverses input-output pairs and reflects the graph over y = x; it exists only when f is one-to-one, which the horizontal line test checks.

Even functions satisfy f(−x) = f(x) and are symmetric about the y-axis; odd functions satisfy f(−x) = −f(x) and are symmetric about the origin.

Graph features: x-intercepts solve f(x) = 0; the y-intercept is f(0); a function is increasing on intervals where its graph rises from left to right.