CSEC Mathematics cheat sheet

Number bases: to convert base 10 to base b, divide repeatedly by b and read remainders upward; to convert base b to base 10, expand as a sum of digit × b^place value.

Laws of indices: a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1 (a ≠ 0); a^(−n) = 1/a^n; a^(1/n) = nth root of a.

Standard form: A × 10^n with 1 ≤ A < 10. To multiply, multiply the A values and add the indices; to divide, divide the A values and subtract the indices.

Percentage change = (change ÷ original) × 100. Profit or loss percent is always calculated on cost price; simple interest I = PRT/100 with T in years.

Compound interest and depreciation: A = P(1 ± r/100)^n, using plus for growth and minus for depreciation; n is the number of compounding periods.

Currency and rates: convert by multiplying or dividing using the given exchange rate; speed = distance ÷ time, average speed = total distance ÷ total time.

Direct and inverse variation: y ∝ x gives y = kx; y ∝ 1/x gives y = k/x; find k from a given pair of values, then substitute to find the unknown.

Set notation: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); the complement A′ contains elements of the universal set not in A; a Venn diagram region count is found by working from the innermost intersection outward.

Expanding and factorising: (a + b)^2 = a^2 + 2ab + b^2; (a − b)^2 = a^2 − 2ab + b^2; a^2 − b^2 = (a + b)(a − b). For ax^2 + bx + c, find two numbers whose product is ac and whose sum is b.

Quadratic formula: x = [−b ± √(b^2 − 4ac)] ÷ 2a. The discriminant b^2 − 4ac > 0 gives two real roots, = 0 gives equal roots, < 0 gives no real roots.

Simultaneous equations: eliminate one variable by adding or subtracting the equations after matching coefficients, or substitute from a linear equation into a quadratic one.

Straight line: gradient m = (y2 − y1)/(x2 − x1); equation y = mx + c; parallel lines have equal gradients; perpendicular gradients multiply to −1; distance = √[(x2 − x1)^2 + (y2 − y1)^2].

Circle theorems: angle at centre = 2 × angle at circumference on the same arc; angle in a semicircle = 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; tangent is perpendicular to radius at the point of contact.

Areas and volumes: circle A = πr^2, circumference = 2πr; sector arc = (θ/360) × 2πr and area = (θ/360) × πr^2; cylinder V = πr^2h; cone V = (1/3)πr^2h; sphere V = (4/3)πr^3.

Trigonometry: SOH CAH TOA for right-angled triangles; sine rule a/sin A = b/sin B = c/sin C; cosine rule a^2 = b^2 + c^2 − 2bc cos A; area of triangle = (1/2)ab sin C.

Statistics: mean = Σfx ÷ Σf for grouped data; median is the middle value or the (n+1)/2 th value; mode is most frequent; range = highest − lowest; probability of an event = favourable outcomes ÷ total equally likely outcomes, and P(not A) = 1 − P(A).