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Write a circle in standard, diameter, general and parametric form, read its centre and radius, find the circle through three points, and work out axis intercepts and the relative position of two circles.
Read articleDescribe random experiments and write their sample spaces in set notation, classify events as sure, impossible, mutually exclusive or exhaustive, and build sample spaces and tree diagrams when outcomes are not equally likely.
Read articleCalculate variance and standard deviation by the direct, short-cut and step-deviation methods, combine the means and standard deviations of two groups, and compare consistency with the coefficient of variation.
Read articleWrite lines in normal and general form, find the distance of a point from a line and between parallel lines, use the family of lines through an intersection and the angle bisectors, and derive the equation of a locus.
Read articleUse the section formula, the incentre formula and the area of a triangle to test collinearity, find slopes and angles between lines with the parallel and perpendicular conditions, and write lines in slope-intercept, two-point and intercept form.
Read articleUse the nth term, finite sum and infinite sum of a geometric progression, insert geometric means and compare them with arithmetic means, apply the sums of n, n squared and n cubed, and find nth terms by the method of differences.
Read articleDerive and use the nth term and sum formulae of an arithmetic progression, insert arithmetic means between two numbers, take three or four terms symmetrically, and model word problems as APs.
Read articleState and prove the binomial theorem for a positive integral index, generate coefficients with Pascal's triangle, find general and middle terms, and use the theorem for specific coefficients and numerical approximations.
Read articleDerive the nCr formula with its identities and Pascal's rule, count selections from different items and from identical items, and solve mixed problems that first select and then arrange.
Read articleApply the fundamental principle of counting and factorials, derive nPr and handle items that must or must not be together, form numbers and words with and without repetition, and count circular arrangements.
Read articleSolve linear inequalities and show them on the number line, solve quadratic inequalities by the method of intervals including perfect-square cases, and handle rational inequalities without the cross-multiplying mistake.
Read articleRead the nature of roots from the discriminant and link it to the sum and product of roots, find the sign of a quadratic expression, sketch parabolas with their maximum or minimum values, and solve applied problems.
Read articleUse the fundamental theorem of algebra and the quadratic formula, solve equations that reduce to quadratics by substitution, handle equations with a common root, and form new equations from given or transformed roots.
Read articlePlot complex numbers on the Argand plane and convert to polar form, interpret distance, circles and rotation geometrically, find the cube roots of unity and prove their properties, and use them to simplify algebraic expressions.
Read articleAdd, subtract, multiply and divide complex numbers in a + ib form, find the conjugate, modulus and argument with their properties, work out additive and multiplicative inverses, and find the square root of a complex number.
Read articleConvert sums and differences of sines and cosines into products, turn products such as 2 sin A cos B into sums, and use both sets of transformation formulae to simplify expressions and solve trigonometric equations.
Read articleDerive and apply the compound angle formulae for sin, cos and tan including tan(A + B + C), obtain double and triple angle formulae as special cases, use half and one-third angle formulae, and solve mixed identity problems.
Read articleProve trigonometric identities from the reciprocal, quotient and Pythagorean relations, find the periods of all six functions, sketch the graphs of sin, cos, tan, sec, cosec and cot, and use periodicity to handle angles of any size.
Read articleConvert between degrees and radians and use arc length, define all six trigonometric functions on the unit circle and prove the Pythagorean identity, find the area of a sector, and fix signs in each quadrant.
Read articleFind the sum, difference, product and quotient of two real functions with the correct domain, exclude the zeros of a denominator, evaluate combined functions at given points, and read domain, range and behaviour from a graph.
Read articleTreat a function as a rule y = f(x) with a domain and range, then sketch the standard graphs: constant, identity, polynomial and rational functions, the modulus, signum and greatest integer functions, and exponential and logarithmic curves.
Read articleForm Cartesian products and count their elements, define relations as subsets of A × B using arrow diagrams, roster and set-builder form, find domain, co-domain and range, and tell a relation from a function.
Read articleWrite sets in roster and set-builder form, classify them as empty, finite or infinite, use interval notation and power sets, shade unions and complements on Venn diagrams, and simplify problems with the complement laws.
Read articleUnderstand the utilitarian and ethical reasons to conserve biodiversity, the hotspot approach and India's three hotspots, protected areas and sacred groves, and ex situ methods from zoos to seed banks along with international commitments.
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