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Define the moment of a force and calculate it in newton metre and dyne centimetre, balance a metre rule with the principle of moments, tell translational from rotational equilibrium and static from dynamic, and locate the centre of gravity of regular and irregular bodies.
Read articleList the sample space of an experiment, compute the probability of an event by counting favourable outcomes, use the complement rule to avoid long lists, recognise sure and impossible events, and handle marbles, defective items, spinners and two dice.
Read articleBuild a cumulative frequency table, locate the median class and compute the median with its formula, identify the modal class and compute the mode, recover a missing frequency from a given median, and decide which average describes a real data set best.
Read articleFind the mean of grouped data by the direct method using class marks, shift the numbers with an assumed mean, divide them down with the step-deviation method, decide which method the data suits, and work backwards to a missing frequency.
Read articleRecall the volume formulas for all six basic solids, add volumes when two are joined, subtract when one is carved out, use conservation of volume for melting, recasting and rising water, and convert a capacity into litres and a cost.
Read articleRecall the surface area formulas for all six basic solids, add only the exposed curved surfaces when two are joined, handle a solid with a part scooped out of it, and cost a painting or polishing job from a rate per square metre.
Read articleWork out the length of an arc and the area of a sector from the central angle, subtract a triangle to get the area of a segment, and apply both to clock hands, a wiper blade and a tethered animal.
Read articleTell a tangent from a secant, count how many tangents a point admits, prove that a tangent is perpendicular to the radius at the point of contact, prove that two tangents from an outside point are equal, and use both results on circumscribed quadrilaterals and concentric circles.
Read articleDraw the right-triangle diagram for any height-and-distance problem, tell the angle of elevation from the angle of depression, find a height or a distance from an angle of thirty, forty-five or sixty degrees, and handle the two-triangle problems that carry the most marks.
Read articleProve that sin squared plus cos squared is always one, derive the two identities that follow from it, simplify expressions that collapse to a single number, prove identities by converting everything to sine and cosine, and get all six ratios from one.
Read articleWrite the six trigonometric ratios of an acute angle from the sides of a right triangle, recover the other five from any one of them using Pythagoras, learn the values for zero, thirty, forty-five, sixty and ninety degrees, and solve equations for an unknown angle.
Read articleUse the section formula to find the point dividing a segment in any ratio, take the mid-point as its simplest case, work backwards to the ratio in which a point or an axis divides a segment, find the points of trisection, and complete a parallelogram from three vertices.
Read articleFind the distance between any two points with the distance formula, test three points for collinearity, classify a triangle as isosceles, equilateral, scalene or right-angled, identify the quadrilateral four points make, and locate a point equidistant from two others.
Read articleUse the AA, SSS and SAS criteria to prove two triangles similar, compute unknown sides from the proportionality of corresponding sides, extend the same ratio to altitudes, medians and angle bisectors, and measure a tower from its shadow.
Read articleSeparate congruence from similarity, state the two conditions two polygons must satisfy to be similar, prove the Basic Proportionality Theorem, use it and its converse to find unknown lengths and to test whether a line is parallel, and apply it inside a trapezium.
Read articleAdd the first n terms of an arithmetic progression with either sum formula, work backwards from a given sum to find n, recover any term from the sums, and solve savings, salary, seat, log-stack and prize-money problems.
Read articleTest a list of numbers for a constant difference, write the nth term of an arithmetic progression, find the missing first term, common difference or position, decide whether a given number belongs to a sequence, and count a term from the end.
Read articleUse the quadratic formula on any equation, compute the discriminant to decide whether the roots are real and distinct, equal or not real, find the value of k that forces equal roots, and solve speed, work and dimension problems that turn out quadratic.
Read articleTest whether an equation is really quadratic, rewrite it in standard form and read off a, b and c, solve it by splitting the middle term, and learn how to reject the root that a real-world problem cannot accept.
Read articleSolve a pair of linear equations by substitution and by elimination, choose the faster method for the equations in front of you, and turn word problems on ages, digits, fractions, costs, boats and shared work into two equations you can actually solve.
Read articleCompare two lines using the ratios of their coefficients, decide whether a pair of equations has one solution, none, or infinitely many, solve graphically, find an unknown k from a consistency condition, and get the area of a triangle formed by two lines and an axis.
Read articleFind the zeroes of a quadratic by factorisation, verify them with the sum and product relations, build a polynomial from a given sum and product, and use the same relations to find an unknown coefficient without solving anything.
Read articleUse the fundamental theorem of arithmetic to find HCF and LCM from prime factorisation, check your answer with the product rule, prove that root two is irrational, and solve the bell and tanker problems that follow.
Read articleSee how ozone forms and how chlorofluorocarbons break it down, sort waste into biodegradable and non-biodegradable, compare landfilling, composting, recycling and incineration, and learn what actually reduces the load.
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