Mathematics IGCSE Notes Index - WELCOME IGCSE

Mathematics IGCSE notesIndex1. Decimals and standard form2. Accuracy and Error3. Powers and roots4. Ratio & proportionclick on a topic tovisit the notes5. Fractions, ratios6. Percentages7. Rational and irrational numbers8. Algebra: simplifying and factorising9. Equations: linear, quadratic, simultaneous10. Rearranging formulae11. Inequalities12. Parallel lines, bearings, polygons13. Areas and volumes, similarity14. Trigonometry15. Circles16. Similar triangles, congruent triangles17. Transformations18. Loci and ruler and compass constructions19. Vectors20. Straight line graphs21. More graphs22. Distance, velocity graphs23. Sequences; trial and improvement24. Graphical transformations25. Probability26. Statistical calculations, diagrams, data collection27. Functions28. Calculus29. Sets{also use the intranet revision course of question papers and answers by topic }

1. Decimals and standard formtop(a) multiplying and dividing(i) 2.5 1.36 Move the decimal points to the right until each is a wholenumber, noting the total number of moves, perform the multiplication, thenmove the decimal point back by the previous total: 25 136 3400 , so the answer is 3.4{Note in the previous example, that transferring a factor of 2, or even better,4, from the 136 to the 25 makes it easier:25 136 25 (4 34) (25 4) 34 100 34 3400 }(ii) 0.00175 0.042 Move both decimal points together to the right until thedivisor is a whole number, perform the calculation, and that is the answer. 1.75 42 , but simplify the calculation by cancelling down any factors first.In this case, both numbers share a 7, so divide this out: 0.25 6 , and0.04166 0.25 , so the answer is 0.0416(iii) decimal placesTo round a number to n d.p., count n digits to the right of the decimal point. Ifthe digit following the nth is 5 , then the nth digit is raised by 1.e.g. round 3.012678 to 3 d.p. 3.012678 3.012 678 so 3.013 to 3 d.p.(iv) significant figuresTo round a number to n s.f., count digits from the left starting with the firstnon-zero digit, then proceed as for decimal places.e.g. round 3109.85 to 3 s.f., 3109.85 310 9.85 so 3110 to 3 s.f.e.g. round 0.0030162 to 3 s.f., 0.0030162 0.00301 62 , so 0.00302 to 3 s.f.(b) standard form(iii) Convert the following to standard form: (a) 25 000(b) 0.0000123Move the decimal point until you have a number x where 1 x 10 , and thenumber of places you moved the point will indicate the numerical value of thepower of 10. So 25000 2.5 104 , and 0.0000123 1.23 10 5(iv) multiplying in standard form: (4.4 105 ) ( 3.5 106 )elements are multiplied, rearrange them thus: (4.4 3.5) (105 106 ) 15.4 1011 1.54 10122As all the

(v) dividing in standard form:(3.2 2.5) (1012 103 ) 3.2 10122.5 1031.28 10 9Again, rearrange the calculation to(vi) adding/subtracting in standard form: (2.5 106 ) (3.75 107 ) Thehardest of the calculations. Convert both numbers into the same denomination,i.e. in this case 106 or 107, then add. (0.25 107 ) (3.75 107 ) 4 107Questions(a) 2.54 1.5(b) 2.55 0.015(c) Convert into standard form and multiply: 25 000 000 0.000 000 000 24(d) (2.6 103 ) (2 10 2 )(e) (1.55 10 3 ) ( 2.5 10 4 )Answers(a) 254 15 3810 , so 2.54 1.5 3.81(b) 2.55 0.015 2550 15 . Notice a factor of 5, so let’s cancel it first: 510 3 170(c) (2.5 107 ) (2.4 10 10 ) 6 10 3(d) (2.6 2) (103 10 2 ) 1.3 105(e) (1.55 10 3 ) (0.25 10 3 ) 1.3 10 33

2. Accuracy and ErrortopTo see how error can accumulate when using rounded values in a calculation,take the worst case each way: e.g. this rectangular space is3mmeasured as 5m by 3m, each measurement being to the nearestmetre. What is the area of the rectangle?5mTo find how small the area could be, consider the lower bounds of the twomeasurements: the length could be as low as 4.5m and the width as low as2.5m. So the smallest possible area is 4.5 2.5 11.25 m2. Now, the lengthcould be anything up to 5.5m but not including the value 5.5m itself (whichwould be rounded up to 6m) So the best way to deal with this is to use the(unattainable) upper bounds and get a ceiling for the area as5.5 3.5 19.25 m2, which the area could get infinitely close to, but not equalto. Then these two facts can be expressed as 11.25m2 area 19.25m2.Questions(a) A gold block in the shape of as cuboid measures 2.5cm by 5.0cm by20.0cm, each to the nearest 0.1cm. What is the volume of the block?(b) A runner runs 100m, measured to the nearest metre, in 12s, measured tothe nearest second. What is the speed of the runner?(c) a 3.0, b 2.5 , both measured to 2 s.f. What are the possible value ofa b ?Answers(a) lower bound volume 2.45 4.95 19.95 241.943625 cm3.upper bound volume 2.55 5.05 20.05 258.193875 cm3.So 241.943625cm3 volume 258.193875cm3distance, for the lower bound we need to take the smallesttimevalue of distance with the biggest value of time, and vice-versa for the upperbound.99.5100.5So, i.e.7.96ms-1 speed 8.739 ms-1 speed 12.511.5(b) Since speed (c) for the smallest value of a b , we need to take the smallest value of atogether with the biggest value of b, etc.0.4 a b 0.6So 2.95 2.55 a b 3.05 2.45 , i.e.4

3. Powers and rootstop1) x a x b x a b2) x a x b x a b3) ( x a )b x ab14) x a ax05) x 1(a) whole number powersNote that the base numbers (x’s) have to be the same;25 32 cannot be simplified any further.1) e.g. x 3 x 2 x 5 , 23 27 210If in doubt, write the powers out in full: a 3 a 2 means( a a a ) ( a a) which is a 52) x 6 x 2 x 4 , 58 52 56Again, if in doubt, spell it out:a a a a a awhich cancels down toa 6 a 2 meansa aa a a a a/ a/ a4a/ a/3) ( x 3 ) 2 x 6 , (32 ) 4 38To check this, ( x 3 ) 2 means ( x 3 ) ( x 3 ) which is x 64) x 3 1,x310 3 11 0.001310 10005) 100 1 .5

QuestionsSimplify the following as far as possible:(a) x 5 x 3(b) a 3 a 5 a 634 37(c) 53 3(d)25 41086 43Answers(a) x 5 x 3 x 5 3 x8(b) a 3 a 5 a 6 a 3 5 6 a 4(c)34 3734 7311 311 6 3555 163 333(d)25 41025 (22 )1025 220225 21386 43(23 )6 (22 )3218 26212(b) fractional powers16) x n n xpq7) x x p ( x ) p16) e.g. x 3 3 x ,qq192 9 3237) 27 ( 3 27) 2 (3)2 9qNote: if you can find the qth root of x easily then it’s better to use the ( x ) p version.6

Q. Simplify the following as far as possible:(a) 16121(b) 64 33(c) 4 23(d) 81 42(e) ( x 6 ) 3Answers.1(a) 16 2 16 41(b) 64 3 364 43(c) 4 2 ( 4)3 (2)3 83(d) 81 4 ( 4 81)3 (3)3 272(e) easier to use power law (3) above: ( x 6 ) 3 x76 23 x4

top4. Ratio and Proportion(a) Using ratiosThis is really a special case of proportion. If quantities are linearly related,either directly or inversely, (like number of workers and time taken to do a job),calculate by multiplying by a ratio:e.g.If 8 workers can together do a job in 6 days, how long would the same job takewith 12 workers?ans: it will take less time, so we multiply by the ratio 128 . So it takes6 128 4 days.e.g.If a workforce of 20 can produce 12 cars in 15 days, how many workersshould be used if 15 cars are needed in 10 days?1515ans:no. of workers 20 12 10 37 12 , ie 38.(b) ProportionWhere quantities are related not necessarily linearly.(i) Direct proportionThis is when an increase in one quantity causes an increase in the other.y x 2 , and you are given that y is 7.2 when x is 6.egRewrite asy kx 2 , and substitute the given values to find k:7.2 k 6 2 , sok 0.2 . The relationship can now be written asy 0.2 x 2 ,and any problems solved.(ii) Inverse proportionThis is when an increase in one quantity causes an decrease in the other.e.g. If y is inversely proportional to the cube of x,1theny 3xkRewrite asy 3 , and proceed as usual.x(iii) Multiplier methodFor the cunning, it is possible (but harder) to solve a problem without calculating k.e.g. Radiation varies inversely as the square of the distance away from the source. Insuitable units, the radiation at 10m away from the source is 75. What is the radiationat 50m away?ans: as distance increases by a factor of 5, radiation must decrease by a factor of 52 ,so the radiation is 75 25 3.8

Questions.(a) Water needs to be removed from an underground chamber before work cancommence. When the water was at a depth of 3m, five suction pipes were used andemptied the chamber in 4 hours. If the water is now at a depth of 5m (same crosssection), and you want to empty the chamber in 10 hours time, how many pipes needto be used?(b) y is proportional to x 2 and when x is 5 y is 6. Find(i) y when x is 25(ii) x when y is 8.64(c) The time t seconds taken for an object to travel a certain distance from rest isinversely proportional to the square root of the acceleration a. When a is 4m/s2, t is2s.What is the value of a if the time taken is 5 seconds?Answers5 4(a) No. of pipes 5 3 13 , so it would be necessary to use 4 pipes to be sure3 10of emptying within 10 hours.(b)y x2y kx 2and we know when x is 5, y is 6, so6 k 52 , so k y 625625, and we can write the relationship asx2 .(i) When x is 25, y 625 252 150 .(ii) When y is 8.64, 8.64 256 x 2 , so25 8.64no, don’t reach for the calculator yet!x2 6x 2 25 1.44 , so x 5 1.2 6 .1,akSo t . Substituting given values:ak2 , so k 4 , ie44t .a4416or 0.64 m/s2.When t 5, 5 , so a , and a 525a(c) t 9

5. Fractions and ratiostop(a) Fractions1219 5(i) Adding/subtracting: e.g. 3 1 . Convert to vulgar form first: ,6 36 3then find the lowest common denominator, in this case 6. Then19 5 19 2 591 1 .6 36621 716 7(ii) Multiplying/dividing: e.g. 5 . Convert to vulgar form: , and3 83 8then always cancel any factor in the numerator with a factor in thedenominator if possible, before multiplying together:2// 716 7162 714 1 .3 833 138/To divide, turn the into a and invert the second fraction.(iii) Converting to and from decimals: e.g. what is3as a decimal?400.0753is 0.075.40 3.000 so40But what is 0.075 as a fraction? 0.075 means753, then cancel down to.100040(b) Ratios(iv) To divide a quantity into 3 parts in the ratio 3: 4:5, call the divisions 3parts, 4 parts and 5 parts. There are 12 parts altogether, so find 1 part, andhence the 3 portions.(v) To find the ratio of several quantities, express in the same units then cancelor multiply up until in lowest terms e.g. what is the ratio of 3.0m to 2.25m to75cm?Perhaps metres is the best unit to use here, so the ratio is 3 :2.25: 0.75.Multiplying up by 4 (or 100 if you really insist) will render all numbersinteger. So the ratio is 12 : 9 : 3, and we can now cancel down to 4:3:110

Questions35(a) (2 ) 2 14111 31(b) (1 ) 23 55(c) What is 0.0875 as a fraction in lowest terms?(d) Split 5000 in the ratio of 1:2:5Answers1116121 16 (a) ( )2 41116 1111/ / / 116//121 1 111////16114 3 115 4 3 3 511 51(b) ( ) 3 55151115 113(c) 875357 10 00040080(d) 1:2:5 means 8 parts altogether. Each part is 5 5000000 8 625 , so the 5000 splits into 625, 1250, and 3125.11