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Write equations of lines with the point-slope and two-point forms, find lines parallel or perpendicular to a given line through a point, find the median, altitude and perpendicular bisector of a triangle, and work out intercepts and intersections.
Read articleFind the slope of a line from two points, from its inclination and from its equation, use slopes to test for parallel and perpendicular lines, read the gradient and intercept from y = mx + c, and check whether three points are collinear.
Read articleUse the section formula to find a point dividing a segment in a ratio, find the ratio in which a point or an axis divides a segment, use mid-points to find missing vertices, and locate the centroid and points of trisection.
Read articleFind the image of a point reflected in the x-axis, the y-axis, the origin and the lines x = a and y = a, spot invariant points, name the mirror line from a point and its image, and plot successive reflections to identify the figure formed.
Read articleTest whether a sequence is a geometric progression, find its nth term and which term equals a value, add its first n terms, and use the geometric mean in growth and depreciation problems.
Read articleFind the sum of an AP with both sum formulas, work out how many terms give a stated sum, recover any term from the sums, and solve savings, instalment and stacking problems.
Read articleTest whether a sequence is an arithmetic progression, use the nth term formula, count the terms of a finite AP and find which term equals a value, and work out the first term and common difference from two terms.
Read articleLearn when two matrices can be multiplied and what order the product has, multiply 2x2 and column matrices, verify that matrix multiplication is not commutative but is distributive, and solve AX = B and A² = kA + mI.
Read articleFind the order of a matrix and name row, column, square, zero and identity matrices, use equality of matrices to find unknowns, add and subtract 2x2 matrices, and multiply by a number to solve simple matrix equations.
Read articleUse the Remainder Theorem to find remainders in one step, test factors with the Factor Theorem, find unknown coefficients from factor and remainder conditions, and factorise cubic polynomials completely.
Read articleLearn invertendo, alternendo, componendo and dividendo, use componendo and dividendo to solve equations for x, handle equations with surds and cubes, and prove proportion identities with the k-method.
Read articleWork out compounded, duplicate, sub-duplicate, triplicate and sub-triplicate ratios, divide quantities in a ratio and handle ratios that change, find fourth, third and mean proportionals, and test continued proportion.
Read articleTurn word problems on numbers, ages, two-digit numbers, trains and boats, fields, paths and pipes into quadratic equations, solve them, and learn how to reject the root that cannot fit the situation.
Read articleUse the discriminant to decide the nature of roots without solving, find the value of k that gives equal roots, find an unknown coefficient from a known root, and solve quadratics with surd or letter coefficients.
Read articleRecognise a quadratic equation and write it in standard form, solve it by splitting the middle term, use the quadratic formula with correct rounding, and handle equations with fractions or reciprocals that reduce to quadratics.
Read articleSolve linear inequations and write the solution set for natural numbers, whole numbers, integers and real numbers, show it on a number line, handle combined inequations, and find greatest or smallest values in word problems.
Read articleTell face value from market value, count how many shares an investment buys at par, at a premium or at a discount, calculate dividend income and percentage return, compare investments, and solve inverse share problems.
Read articleSee where the recurring deposit interest formula comes from, calculate interest and maturity value step by step, and work backwards to find the monthly instalment, the rate of interest or the number of months.
Read articleCalculate input tax credit and net GST at each stage from manufacturer to retailer, find the total GST reaching the government, fill in a GST invoice with items at different rates, and solve problems that combine discount, profit and GST.
Read articleWork out CGST, SGST and IGST on any bill, find the price paid after a discount, calculate GST when a shopkeeper makes a profit or loss, and reverse the working to find a list price, cost price or GST rate from the tax paid.
Read articleUnderstand the greenhouse effect and global warming, see how acid rain forms and how the ozone layer is thinned, learn how pollution affects climate, health and wildlife, and review practical measures from CNG to sewage treatment.
Read articleLearn what pollution is and where air, water and soil pollutants come from, understand biomedical waste, radiation and noise pollution, tell biodegradable from non-biodegradable waste, and see why plastics, glass, thermocol and DDT cause lasting harm.
Read articleLearn what demography, population density, birth rate, death rate and growth rate mean with worked calculations, why human numbers rose so sharply, what problems a population explosion creates, and how population education and family planning methods help.
Read articleTell autosomes from sex chromosomes, see how an XX and XY cross decides whether a baby is a boy or a girl, follow the inheritance of haemophilia and colour blindness, and understand how mutation and variation differ.
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