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Tell scalars from vectors and write vectors with unit vectors, add and subtract vectors by the triangle, parallelogram and polygon laws, resolve vectors into components, and find the dot and cross products with their geometric meaning.
Read articleDerive the three equations of uniformly accelerated motion by graphs and by calculus, solve free-fall and stopping-distance problems, define relative velocity in one dimension, and solve overtaking and meeting problems in a moving frame.
Read articleTell average and instantaneous velocity and speed apart, find velocity and acceleration as derivatives, read position-time and velocity-time graphs including area under a v-t graph, and sketch graphs for uniform and non-uniform motion.
Read articleWrite dimensional formulas in terms of M, L and T, test equations for dimensional consistency, derive relations such as the pendulum's time period by dimensional analysis, know its limits, and convert quantities between unit systems.
Read articleLearn the seven SI base units and how derived units are built from them, convert between units, count significant figures, round the results of calculations correctly, and combine uncertainties in measured quantities.
Read articleLearn the axioms that every probability assignment must satisfy, find probabilities of events with equally likely outcomes, use the addition rule for A or B, and find the probability of not A with the complement rule.
Read articleDescribe events as subsets of a sample space, classify them as impossible, sure, simple or compound, combine them with not, or and and, and decide when events are mutually exclusive and exhaustive.
Read articleDefine variance and standard deviation, compute them for raw data by definition and by the shortcut formula, for discrete distributions and for continuous distributions by step-deviation, and compare variability with the coefficient of variation.
Read articleMeasure how spread out data is using the range and mean deviation about the mean and the median, for raw data, discrete frequency distributions and continuous distributions with step-deviation, and see where mean deviation falls short.
Read articleFind derivatives from first principles, read a derivative as a rate of change and as the slope of a tangent, apply the sum, product and quotient rules, and differentiate polynomials, sin x and cos x.
Read articleUse the Sandwich Theorem to prove sin x/x tends to 1 and (1 - cos x)/x tends to 0, evaluate composite trigonometric limits, and apply the standard limits of (e^x - 1)/x and log(1 + x)/x.
Read articleUnderstand a limit through left-hand and right-hand limits, use the algebra of limits for polynomial and rational functions, remove 0/0 by factorising or rationalising, and apply the standard result for (x^n - a^n)/(x - a).
Read articleLearn the three coordinate axes and planes in space and the eight octants, locate points from their coordinates and signs, use the distance formula in three dimensions, and test collinearity and classify triangles.
Read articleDerive the standard equation of an ellipse with c² = a² - b² and eccentricity c/a, read off its axes, vertices, foci and latus rectum, then do the same for the hyperbola with c² = a² + b² and eccentricity greater than 1.
Read articleDefine a parabola by its focus and directrix and derive y² = 4ax, learn the four standard forms with their focus, directrix and axis, find the latus rectum and sketch the curve, and write parabolas from given conditions.
Read articleSee how cutting a double cone with a plane produces a circle, ellipse, parabola or hyperbola, recognise the degenerate cases, derive the equation of a circle and read its centre and radius, and find circles from given conditions.
Read articleWrite equations of lines parallel to the axes and in point-slope, two-point, slope-intercept and intercept forms, convert between forms, find the perpendicular distance of a point from a line, and find the distance between parallel lines.
Read articleFind the slope of a line from two points or from its inclination, use slopes to test for parallel and perpendicular lines and collinear points, and calculate the acute angle between two lines.
Read articleTreat a sequence as a function on the natural numbers and generate terms from rules and recurrences, find terms of a G.P., add finite and infinite G.P.s, and use arithmetic and geometric means with the A.M.-G.M. inequality.
Read articleState the binomial theorem for positive integer powers and read its coefficients from Pascal's triangle, expand expressions such as (a + b)^n and (x + 1/x)^n, and use (1 + x)^n for approximations and divisibility proofs.
Read articleDerive nCr and its link with nPr, use the properties of nCr to simplify and solve equations, count committees and teams under conditions, and decide when a problem needs permutations, combinations or both.
Read articleUse the fundamental principle of counting, work with factorial notation, count arrangements of distinct objects with nPr with and without repetition, and count arrangements of letters that repeat.
Read articleTell strict from slack inequalities and spot linear ones, solve linear inequalities in one variable with correct sign changes, show solutions on a number line and in interval form, and turn word problems into inequalities.
Read articleWrite complex numbers as a + ib and see why i is needed, add, multiply and divide them and simplify powers of i, find the modulus and conjugate with their key properties, and plot complex numbers on the Argand plane.
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