Secondary 4 Mathematics-Cultural, Social and Technical Option Ministerial Examination - Québec cheat sheet

Order of operations: brackets, exponents, multiplication/division left to right, then addition/subtraction left to right; a fraction bar acts as a bracket.

Laws of exponents: 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.

Radicals: √(ab) = √a·√b; √(a/b) = √a/√b; rationalize 1/√a by multiplying numerator and denominator by √a.

Direct proportionality y = kx gives a straight line through the origin; inverse proportionality y = k/x gives a hyperbola with xy constant.

Slope m = (y2 − y1)/(x2 − x1); line forms y = mx + b (slope-intercept) and y − y1 = m(x − x1) (point-slope).

Systems of two linear equations: substitution isolates one variable; elimination adds multiples of equations to cancel a variable; parallel distinct lines give no solution.

Quadratic: x = (−b ± √(b² − 4ac))/(2a); discriminant b² − 4ac > 0 gives two real roots, = 0 one double root, < 0 no real roots.

Function notation f(x) replaces y; the domain is the set of admissible x-values and the range the resulting f(x)-values; a relation is a function only if each x has one image.

Plane figures: triangle A = bh/2; rectangle A = bh; parallelogram A = bh; trapezoid A = (b1 + b2)h/2; circle A = πr² and C = 2πr.

Solids: prism/cylinder V = base area × height; pyramid/cone V = (base area × height)/3; sphere V = 4πr³/3 and A = 4πr².

Pythagorean relation a² + b² = c² holds only in right triangles; c is the hypotenuse, the side opposite the right angle.

Trigonometric ratios in a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Sine law a/sin A = b/sin B = c/sin C; cosine law a² = b² + c² − 2bc·cos A; use cosine law when given SAS or SSS.

Similar figures: corresponding angles equal and corresponding sides proportional; area ratio = k² and volume ratio = k³ where k is the side ratio.

Metric relations in a right triangle with altitude h to hypotenuse c, segments m and n: h² = mn, a² = mc, b² = nc.

Transformations: translation preserves orientation; rotation and reflection preserve shape and size; dilation by factor k multiplies lengths by k and areas by k².

Mean = sum of values / number of values; median = middle value of ordered data (average of two middle values if n is even); mode = most frequent value.

Range = maximum − minimum; it uses only two data values and ignores the spread of the rest.

Weighted mean = Σ(value × weight)/Σ(weights); use it when categories have different counts.

Quartiles: Q1 and Q3 are medians of the lower and upper halves; interquartile range IQR = Q3 − Q1 contains the middle 50% of data.

A box-and-whisker plot marks minimum, Q1, median, Q3, maximum; outliers are often flagged beyond 1.5 × IQR from a quartile.

Histograms show grouped quantitative data with bars touching; bar charts compare categories and leave gaps; circle graphs show parts of a whole.

Sampling: a random sample gives every member an equal chance of selection; a biased sample (convenience, voluntary response) distorts conclusions.

Correlation describes the strength and direction of a linear link between two variables; correlation does not establish causation.

Probability of event A: P(A) = number of favourable outcomes / number of equally likely outcomes; 0 ≤ P(A) ≤ 1.

Complement rule: P(not A) = 1 − P(A); useful when counting failures is easier than counting successes.

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

Multiplication rule: P(A and B) = P(A) × P(B|A); for independent events P(B|A) = P(B).

Odds in favour of A = P(A) : P(not A); odds against A = P(not A) : P(A); convert odds a:b to probability a/(a + b).

Expected value = Σ(outcome value × its probability); it is the long-run average payoff per trial, not a guaranteed result.

With replacement, probabilities stay constant between draws; without replacement, the denominator and numerator decrease after each draw.

To count ordered arrangements use permutations; for unordered selections use combinations, where order does not matter.

Modelling cycle: understand the situation, identify variables, build a model, apply it, validate against the context, then communicate the result.

Choose a model by the pattern in the data: constant difference suggests linear, constant ratio suggests exponential, constant second difference suggests quadratic.

Validation checks include units, order of magnitude, sign, comparison with known reference values, and testing the model on a data point not used to build it.

A valid solution must answer the question asked, respect the constraints given, and be stated with its unit and appropriate precision.

When comparing options, define a single comparison quantity (total cost, unit price, time) and evaluate every option on the same basis.

Optimization under constraints: express the quantity to optimize as a function of one variable, then test boundary values and critical values.

Rounding rules: keep full precision during intermediate steps and round only the final answer, unless the context fixes the precision.

A complete written solution states the given data, the chosen model and why, the computation, and a conclusion sentence interpreting the result.

Simple interest I = P·i·t, where i is the annual rate as a decimal and t is time in years; total amount = P + I.

Compound interest: amount C = P(1 + i)^n, where i is the rate per period and n the number of periods; interest earned = C − P.

Nominal annual rate compounded m times per year: periodic rate = annual rate/m and n = years × m; more frequent compounding raises the effective yield.

Sales tax is applied to the price after discounts; a discount of d% leaves a multiplier of (1 − d/100), and successive discounts multiply.

Unit price = total price / quantity; the lowest unit price identifies the best value only when quality and quantity needs are comparable.

A budget balances net income against fixed expenses, variable expenses and savings; net income = gross income − deductions.

Credit cost: total repaid = payment × number of payments; compare with the cash price to measure the financing cost.

Break-even point occurs where total revenue equals total cost; profit = revenue − total cost, and a negative result is a loss.