JJC 2018 H2 Math revision package

l o &I/ JURONG JUNIOR COLLEGE Preparation for 2018 GCE A Level Examination H2 Mathematics 9758 Revision Package (Topical) Specially Prepared For: MF26 Contents Pure Mathematics Topic 1. Topic 2: Topic 3: Topic 4: Topic 5: Topic 6: Topic 7: Topic 8: Topic 9: Topic 10: Equations & Inequalities Graphing Techniques Functions Sequences and Series Techniques and Applications of Differentiation Techniques and Applications of Integration Maclaurin's Series & Binomial Expansion Differential Equations Vectors Complex Numbers Probability and Statistics Topic 11: Answers Topic 12: Topic 13: Topic 14: Topic 15: Topic 16: Permutations & Combinations Probability Discrete Random Variables Normal Distribution & Sampling Hypothesis Testing Correlation Coefficient and Linear Regression 101 109 117

SCHEME OF EXAMINATION PAPERS For the examination in H2 Mathematics, there will be two 3-hour papers, each carrying 50% of the total mark, and each marked out of 100, as follows: PAPER 1 (3 hours) ALevels: Friday 9 November 2018 08:00-11:00 A paper consisting of about 10 to 12 questions of different lengths and marks based on the Pure Mathematics section of the syllabus. There will be at least two questions on application of Mathematics in real-world contexts, including those from sciences and engineering. Each question will carry at least 12 marks and may require concepts and skills from more than one topic. Candidates will be expected to answer all questions. PAPER 2 (3 hours) ALevels: Wednesday 14 November 2018 08:00-11:00 A paper consisting of 2 sections, Sections A and B. Section A (Pure Mathematics — 40 marks) will consist of about 4 to 5 questions of different lengths and marks based on the Pure Mathematics section of the syllabus. Section B (Probability and Statistics — 60 marks) will consist of about 6 to 8 questions of different lengths and marks based on the Probability and Statistics section of the syllabus. There will be at least two questions on application of Mathematics in real-world contexts, including those from sciences and engineering. Each question will carry at least 12 marks and may require concepts and skills from more than one topic. Candidates will be expected to answer all questions. USE OF GRAPHIC CALCULATOR (GC) The use of GC without computer algebra system will be expected. The examination papers will be set with the assumption that candidates will have access to GC. As a general rule, unsupported answers obtained from GC are allowed unless the question states otherwise. Where unsupported answers from GC are not allowed, candidates are required to present the mathematical steps using mathematical notations and not calculator commands. For questions where graphs are used to find a solution, candidates should sketch these graphs as part of their answers. Incorrect answers without working will receive no marks. However, if there is written evidence of using GC correctly, method marks may be awarded. Students should be aware that there are limitations inherent in GC. For example, answers obtained by tracing along a graph to find roots of an equation may not produce the required accuracy.

PURE MATHEMATICS Algebraic series Binomial expansion: (a+b) =a" + ('12] a" b+ (Zj a" b + [’31) a" b +...+b", where n is a positive integer and ny_ n! e Maclaurin expansion: ) = F(0) + xF/(0) + e F7(0) 4 ..+ X F D (0) 4. 2 nl (1+x)"=1+nx+Mx2+...+n(n~1)”"(n~r+1)x"+... (]x]<1) ¥l X2 3 r ef =l x . . (all x) 203 7l 3 5 )7 2rt sinx=x—£—+£——...+u)—x——+... (all x) 3 8l (2r + 1! 2 4 AN L2 005x=1—f—+£——...+(——1—)—x—+... (all x) o204 (2r)! 2 3 4yt !n(1+x)=x—f——+f——...+(—l)———+... (-1<x<1) 2 3 r Partial fractions decomposition Non-repeated linear factors: px+gq A4 B = + (ax+b)(cx+d) (ax+b) (cx+d) Repeated linear factors: pxl vgx+r A4 N B + C (ax+b)ex+d)? (ax+b) (cx+d) (cx+d)? Non-repeated quadratic factor: pxl +gx+r A Bx+C (ax+b)(x? +c?) - (ax+b) (x?+c?)

Trigonometry sin(A4 £ B)=sin Acos B £ cos Asin B cos(4 + B)=cos Acos B¥sin 4sin B fan4+tan B 1F¥tan 4tan B sin2A4=2sin Acos 4 tan(4+ B)= cos2A4=cos? A—sin? A=2cos? A-1=1-2sin? 4 tan ZAE——Z—Ea—n—?—— 1-tan” 4 sin P+sinQ=2sin1(P+Q)cos}(P- Q) sin P—sin Q=2cos (P +Q)sin(P-Q) cos P+ cos Q=2cos % (P+ 0)cos (P - Q) cos P —cos Q=-2sin1(P+Q)sinL(P-0) Principal values: 0<cos™x <7 (o <1 1 -1 1 —2fl<tan xX<37 Derivatives f(x) f'(x) sin”' x ! 1—x2 4 1 cos™ x - 1—x2 tan™' x ! - 1+ x cosecx cosecx cotx Secx secxtanx

Integrals (Arbitrary constants are omitted; a denotes a positive constant.) f(x) fan x cot x cosecx seCx Vectors jflfldx In{sec x) In(sin x) —In(cosec x + cot x) In(sec x + tan x) The point dividing 4B in the ratio 1: 4 has position vector pat b A+u Vector product: a by a,bs - azh, axb=la, |xi b, |=| asb - a;b, as b3 a1b2 - a2b1 ([xl<a) (x>a) ([x[<a) (K <z7) (0<x<m) (O<x<r) (W <37)

PROBABILITY AND STATISTICS Standard discrete distributions Distribution of X' P(X =x) Mean Variance Binomial B(n,p) [Z)px (1-p) np np(1- p) Poisson Po(4) g™t }“—: A A xX: Geometric Geo(p) (1 py'p ;1; 1;2p Standard continuous distribution Distribution of X p.d.f. Mean Variance Exponential Ae ™ 1 1 A 22 Sampling and testing Unbiased estimate of population variance: 2_n (mx—f)"’]: 1 (sz_(Zx)ZJ n-1 n n—1 n Unbiased estimate of common population variance from two samples: s = I(x = %)% +2(x; — %) ny+n, —2 Regression and correlation Estimated product moment correlation coefficient: TxXy e o e J{Z(x~f) }{E(y——i) } \/(sz_(zx)Z]LZyz_(ZY)z) n n Estimated regression line of y on x: (x-%)y—-») y-y=b(x—Xx), where bzz st

THE NORMAL DISTRIBUTION FUNCTION If Z has a normal distribution with mean 0 and variance 1 then, for each value of z, the table gives the value of @(z), where D(z)=P(Z <L 2). For negative values of z use ®(-z) =1-d(z). 123 456 7 89 z 0 1 2 3 4 5 6 7 8 9 ADD 0.0 | 0.5000 | 0.5040 0.5080 0.5120|0.5160 0.5199 0.5239|0.5279 0.5319 0.5359|4 8 12|16 20 24|28 32 36 0.1 | 0.5398 | 0.5438 0.5478 0.5517|0.5557 0.5596 0.5636|0.5675 0.5714 0.5753|4 8 12|16 20 24|28 32 36 0.2 |0.5793 | 0.5832 0.5871 0.5910 |0.5948 0.5987 0.6026|0.6064 0.6103 0.6141|4 8 12[15 19 23|27 31 35 0.3 10.6179|0.6217 0.6255 0.6293 | 0.6331 0.6368 0.6406 |0.6443 0.6480 0.6517|4 7 11|15 19 22|26 30 34 0.4 | 0.6554 | 0.6591 0.6628 0.6664 |0.6700 0.6736 0.6772|0.6808 0.6844 0.6879|4 7 11|14 18 22|25 29 32 0.5 10.6915|0.6950 0.6985 0.7019 |0.7054 0.7088 0.7123|0.7157 0.7190 0.7224|3 7 10|14 17 20|24 27 31 0.6 | 0.7257 | 0.7291 0.7324 0.7357 | 0.7389 0.7422 0.7454|0.7486 0.7517 0.7549|3 7 10|13 16 19|23 26 29 0.7 | 0.7580 | 0.7611 0.7642 0.7673 |0.7704 0.7734 0.7764|0.7794 0.7823 0.7852|3 6 9[12 15 18|21 24 27 0.8 10.7881|0.7910 0.7939 0.7967 | 0.7995 0.8023 0.8051|0.8078 0.8106 0.8133|3 5 8|11 14 16[19 22 25 0.9 10.8159|0.8186 0.8212 0.3238 [ 0.8264 0.8289 0.8315|0.8340 0.8365 0.8389|3 5 8|10 13 15[18 20 23 1.0 | 0.8413|0.8438 0.8461 0.8485|0.8508 0.8531 0.8554 | 0.8577 0.8599 0.8621|2 5 7| 9 12 14|16 19 21 1.1 10.86430.8665 0.8686 0.8708|0.8729 0.8749 0.8770|0.8790 0.8810 0.8830|2 4 6| 8 10 12|14 16 18 1.2 10.8849|0.8869 0.8888 0.8907|0.8925 0.8944 0.8962|0.8980 0.8997 0.9015|2 4 6| 7 9 11|13 15 17 1.3 10.9032|0.9049 0.9066 0.9082|0.9099 0.9115 0.9131|0.9147 0.9162 0.9177|2 3 5| 6 8 10|11 13 14 1.4 10.9192|0.9207 0.9222 0.9236|0.9251 0.9265 0.9279|0.9292 0.9306 0.9319|1 3 4| 6 7 8|10 11 13 1.50.9332| 0.9345 0.9357 0.93700.9382 0.9394 0.9406|0.9418 0.9429 0.9441|1 2 4|5 6 7| 8 10 11 1.6 1 0.9452)0.9463 0.9474 0.9484|0.9495 0.9505 0.9515|0.9525 0.9535 0.9545|1 2 3| 4 5 6/ 7 8 9 1.7 | 0.9554 1 0.9564 0.9573 0.9582|0.9591 0.9599 0.9608 | 0.9616 0.9625 0.9633|1 2 3| 4 4 5/ 6 7 8 1.8 [0.9641|0.9649 0.9656 0.9664 |0.9671 0.9678 0.9686