2017 A Level 9758 H2 Math Paper 1 Solution and Paper 2 Likely Questions



Use of elimination or substitution, especially when solving for 2 complex variables



Use of quadratic formula and (-1) = i when needed



Use of conjugate roots for polynomial with real coefficients (where the third root is a real root)



Factorise half of angle to simplify complex numbers



Illustrate accurately complex number(s) using an Argand diagram

2.2

Important Tips

Five Things to Do / Remember 1.

Write down all the formulas you may forget once the exam begins.

2.

Look through the paper to attempt the questions you are more confident first. You do not need to attempt the paper chronologically.

3.

For questions that requires a non-calculator approach, ALWAYS validate using a G.C. where possible. For questions that allows the use of G.C. but you’re not confident in, write down the intermediate steps.

4.

When you cannot attempt a question part, write a placeholder answer (e.g. answer for (a) is 10, or position vector is (1, 2, 5) and carry on with the placeholder answer for the subsequent parts). You get full E.C.F marks!

5.

When you cannot attempt questions that require you to show or proof, work from the reverse way and combine both workings together for a seemingly coherent proof. You may get full marks for that question part!

Three Things to Remember 1.

If you find the paper tough, many other candidates also feel so, make sure you’re not as discouraged as many of them.

2.

If you find the paper manageable, make sure your workings are free of careless mistakes.

3.

If you attempt a question correctly, it pays off (i.e. your effort – which may cumulatively be a month or more worth of time). If you attempt a question wrongly, pragmatically speaking, you wasted your effort studying for that chapter.

KEIRAN TAN ACADEMICS | [email protected] | 81867080

4

FINAL WORDS BEFORE GCE ‘A’ LEVELS 2017 H2 MATHEMATICS (9758) PAPER 2 2.1

Probable Topics for Paper 2 Section A (Pure Mathematics)

Review these topics / questions. Questions that have yet to appear in Paper 1 

Parametric Equations



Rate of Change



Arithmetic and Geometric Progression



Functions



Graphing Techniques: Special curve y = 1/f(x)



Applications of Integration: Area / Volume of Revolution

2.2

Checklist (Have you revised and remembered these?)

Review these before you turn in and make sure you can execute/recall these, at least for the duration when you’re in the examination hall.

2.2.1 Permutations and Combinations 

Remember general steps o

Consider overarching issues (e.g. circular permutations, grouping of objects, slotting, multiple cases, use of complement)

o



Consider between choosing and permuting 

Is ordering important? Arrangement versus Selection



If ordering is important, can the objects be repeated?



If the objects cannot be repeated, are the objects distinct?

Explain each case, workings, intermediate steps.


5

2.2.2 Probability 

Consider P&C approach or probability approach in solving



Remember the concept of conditional probability when question includes a phrase “given that …”



Conditions for Independency and Mutual Exclusivity, relation between Union and Intersection (which can be applied generically)



For Venn diagram questions, consider special scenarios such as mutual exclusivity and subset.

2.2.3 Distributions 

Properties of Binomial and Normal distribution



Combinations of distinct RV and continuous RV (the latter which applies beyond normal distribution) o

Concept of mean and variance involving addition and subtraction



Assumptions of Binomial and Normal distributions, with an emphasis of contextual explanation



Operations of modulus signs (e.g. P(|x|
For normal distribution 

Remember to input standard deviation (square root of variance) in G.C.



Use of Standard Normal distribution, especially when solving for unknowns



Concept of symmetry about the mean (i.e. If P(x b), μ = 0.5(a + b))

2.2.4 Sampling and Estimation Random and Non-random Sampling 

Concept of random and non-random sampling, with an emphasis of contextual explanation



Concept of population parameters and sample statistics



Manipulate unbiased estimates of mean and population variance o

Concept of mean and variance involving addition and subtraction

Central Limit Theorem 

Use of Central Limit Theorem (CLT) based on question phrasing of sample mean, sample sum



Differentiate between sample size n and fixed number of trials N for a binomial distribution



Use of CLT even when n < 50


6

2.2.5 Hypothesis Testing 

Check the sample statistics and null hypothesis to validate alternative hypothesis under a singletail test.



Definition for level of significance, p-value and critical value.



Steps in hypothesis testing o

State hypothesis

o

State assumptions (i.e. X is normally distributed) if needed

o

Express test-statistic

o

Level of significance

o

Finding the p-value to compare against level of significance (for standard questions), or compare the zcal expression with critical region (for questions involving unknowns).

2.2.6 Correlation and Linear Regression 

Concept of Product Moment Correlation Coefficient (r-value)



Concept of least square regression line



o

Passes through the mean of X and Y values

o

Minimises the sum of squared residuals

o

Interpret contextually the meaning of the intercept and gradient (regression coefficient)

Infer r-value and shape of graph (gradient, concavity) to choose most appropriate model or determine if a model is appropriate.



Infer outliers based on sketch of graph



Choice of regression line of y on x or x on y should first be based on question, and next be based on context.



Determine interpolation or extrapolation and r-value to account for reliability of prediction.

2.2.7 For the entire Section B 

Are intermediate steps in 5 significant figures?



Are final answers in 3 significant figures?



Are you familiar with all the steps to use G.C.?



Are there any concepts of probability, such as conditional probability, beyond the question on probability?


7

2.3

Suggested Solutions and Comments for Paper 1 (For your interest)

The following suggested solutions are provided by the tutor, who shall not be held liable for any loss or damage whatsoever caused by errors, omissions, misprints or misinterpretations of the contents hereof. Question 1:

e 2x ln 1  ax 

  a 2x 2 a 3x 3  1  2x  2x 2  ax       2 2  a 3  a2   ax  2a   x 2    2a  a 2  x 3    3 2   a2 a Since 2a   0, a 2    0  2 2   a  0 a  0 or a  4 Comments: Standard Maclaurin’s expansion using MF26. Candidates who were observant were able to relate this question to that of Question 9(c).


8

Question 2(i):

y

y  b x a

y 

y  0

1 x a

x

x a Question 2(ii):

1 x a 2 1 x  a   b 1 x  a b 1 x  a or b

b x  a  

x a

Comments:

Standard question on graphing techniques and operations of modulus sign.


9

Question 3(i):

y 2  2xy  5x 2  10  0

1

Differentiate 1 w.r.t. x  dy dy    10x  0 2y  2 y  x  dx dx 

2

dy  0, dx 2y  10x  0 Since

 y  5x

5x 

2

 2 5x   5x 2  10  0

30x 2  10x  10  0 3x 2  x  1  0

x 

1

2

Question 3(ii):

Since x  0, x 

1

2  1  5 y  5x  5     2  2 Differentiate 2 w.r.t. x

 dy   dy dy d2y d2y  2y    2y 2  2    x 2   10x  0  dx   dx dx dx dx  5 Let y  , 2 1 x , 2 dy 0 dx d2y 10  0 2 dx 4 2  maximum point 2

Comments:

The key to solving this question is to obtain the relationship between y and x.


10

Question 4(i):

4x  9 1  4 x 2 x 2 dy 1   0 for all x  , x  2 2 dx x  2

y

Question 4(ii):

a  4, b  1  Asymptotes are y  4 and x  2

Question 4(iii):

1. Translation of graph 4 units in the negative y-direction. 2. Translation of graph 2 units in the positive x-direction.

Comments: Standard question on graphing techniques.


11

Question 5(i):

Let f x   x 3  ax 2  bx  c f 1  8

f  2  3

 a b c  7

 4a  2b  c  4

1

f 3  25  9a  3b  c  2

2

 3 

Solving via Simultaneous Equation Solver the system of equations,

a  1.5, b  1.5, c  7

Question 5(ii):

f ' x   3x 2  3x  1.5  2 3x 2  3x  0.5  0

x  0.145 or x  1.15

Question 5(iii):

f x   x 3  1.5x 2  1.5x  7 f ' x   3x 2  3x  1.5

 3 x  0.5  0.75 2

Since f ' x   0, f x  is a strictly increasing function with no stationary points.  It has only one root, x  1.33.

Comments: Standard question on Equations, and involves concept of Remainder Theorem (that is an assumed knowledge of H2 Mathematics (9758)).


12

Question 6(i): It represents the set of points that lies on the line which passes through a and is parallel to b.

Question 6(ii): It represents the set of points on the plane which has normal vector, n. d is the displacement of the plane from origin, O. (OR d is the distance of the plane away from O.)

Question 6(iii):

r  a +t b

rn =d

 a+ t b   n = d a  n +t b  n = d

d  a n b n d  a n  b r  a    bn  Since bn  0, line is not parallel to plane, the solution represents the t

point of intersection between the line and the plane.

Comments:

Question on Vectors involving proof, which is similar to that of Specimen Paper 2017 (9758).


13

Question 7(i):

 sin 2mx  sin 2nx  dx    cos 2mx  2nx   cos 2mx  2nx  dx 1 2

 1 1 1   sin 2mx  2nx   sin 2mx  2nx   C  2  2m  2n 2m  2n  1 1  sin 2mx  2nx   sin 2mx  2nx   C 4m  4n 4m  4n

Question 7(ii): 

 f x  0





 0 

2

dx

sin 2mx  sin 2nx  dx   2

 sin 2mx   2 sin 2mx  sin 2nx   sin 2nx  dx 1  cos 4mx  1  cos 4nx    dx   2 sin 2mx  sin 2nx  dx 2 2   x sin 4mx  x sin 4nx  1 1       sin 2mx  2nx   sin 2mx  2nx  2 4m  4n 4m  4n 2  4m  2 4n  2        0   0 since sin k    0 for k   2 2 

2

2

0 



0

0



Comments: Question on Techniques of Integration using Factor Theorem, or Integration by Parts. Candidates have to be acquainted with trigonometric identities to approach this question.


14

Question 8(a):

z 2 1  i  2z  5  5i  0 z z z

 2 

2  36

2  4 1  i5  5i 2 1  i 2

2 1  i 2  6i

2 1  i 1  3i 1  3i 1  i z    1  2i 1i 1i 1 i or 1  3i 1  3i 1  i    2i 1i 1i 1 i z  1  2i or z  2  i z

Question 8(b)(i):

w 2  1  i  1  2i  1  2i 2

w 3  w 2  w  2i  1  i  2  2i 2

w 4  w 2   2i  4 2

2

w 4  pw 3  39w 2  qw  58  0

 4  p 2  2i  39 2i   q 1  i  58  0

Compairing real and imaginary parts, Re w  : 4  2 p  q  58  0 Im w  : 2p  78  q  0

 q  2p  54

 q  2p  78

Solving 1 and 2 , p  6, q  66

1

2


15

Question 8(b)(ii):

w 4  6w 3  39w 2  66w  58  0

Since all coefficients of P w  are real and w  1  i is a root, w  1  i is also a root. w  1  i w  1  i  w 2  2w  2     w 4  6w 3  39w 2  66w  58  w 2  2w  2w 2  aw  b  By inspection, 2b  58  b  29

 2b  2a  66  2 29  2a  66  a  4

 w 4  6w 3  39w 2  66w  58  w 2  2w  2w 2  4w  29

Comments:

Standard question on Complex Numbers.


16

Question 9(a)(i):

U n  Sn  Sn 1

2   U n  An 2  Bn  A n  1  B n  1   U n  2An  A  B

Question 9(a)(ii):

U 10  48

 19A  B  48

U 17  90

 33A  B  90

1

2

Solving 1 and 2 , A  3, B  9

Question 9(b):

L.H.S.

=r 2 r  1  r  1 r 2 2

2

 r 2 r 2  2r  1  r 2 r 2  2r  1

2

 4r 3

k 4



n

r 1

r3

2 2 1 n  2  r r  1  r  1 r 2   r 1   4  2 2 1  12 1  1  1  1 12 4  2 2  22 2  1  2  1 22



 32 3  1  3  1 32 2





2

 n  1 n   n  2 n  1 2 2   n 2 n  1  n  1 n 2   2 1  n  1 n 2 4 1 4  n  2n 3  n 2  4 2

2

2

2


17

Question 9(c):

xr r! n 1

Let ar  an 1 an



x

n 1 ! xn n!



x n 1

x  0 1 n  n  1  The sequence converges. lim

xr  r 0 r !  e x 

from M.F.26

Comments: Question on Sequences and Series. The sum to infinity can be simply obtained from MF26.


18

Question 10 2017 H2 Mathematics (9740) Paper 1 Q10 presents the same question, without the context.

 1  5         Points P and Q have position vectors  2  and 7 respectively, where a is a constant. The     1 a      3    straight line L passes through the origin O and has direction 1 .   2 (i)

Find the value of a for which L intersects the line through P and Q.

[4]

It is now given that a = –3. The variable point R lies on L. (ii)

Show that angle PRQ cannot be a right angle.

(iii)

Find the coordinates of R which make PR a minimum and find the exact value of PR

in this case.

[4]

[5]

Question 10 (i)

    4  PQ   5    a  1  1       4    lPQ : r =  2     5  ,        1 a  1   For PQ and lPQ to intersect,  4   1   4              5    2     5  ,          a  1 1 a  1 3  1  4 1   2  5

2

2  1  a  1 

3

Solving 1 , 2 and 3 ,     a  4.4

3 5 ,  11 11


19

Question 10 (ii)

 3      Let OR    1  ,      2  3   1   3   1           PR    1    2      2        2 1 2   1  3   5   3   5           QR    1    7      7        2 3 2   3        3   1   3   1  PR  QR     2      2   14  2  35   22     2   1 2   1

  Since discriminant  7  0, there are no solutions for PR  QR  0

 PRQ  90

Question 10 (iii)

 2 2 2 PR  3   1     2  2   1  2 PR  14  2  14   6  2 d PR  28  14 d  2 For PR to be minimum, PR has to also be minium,  2 d PR  0 d 1 28  14  0    2  2 d2 PR  28  0 d 2  2 1  PR is a minimum when   2  R 1.5, 0.5  1  2 2 2 1 PR  3  12   1   21    2  2  12   1  10 2



 

 




20

Comments: Question on Vectors. Candidates who were able to comprehend the context would find this question manageable, with many parts that were tested in G.C.E. ‘A’ Level questions from previous years.


21

Question 11(a):

dv c dt

Question 11(b)(i):

 1 dv   c dt

v  ct  D, where D is an arbitrary constant. When t  0, v  4, D  4 When t  2.5, v  29

29  2.5c  4  c  10  v  10t  4

Question 11(b)(ii):

dv  10  kv dt 1  10  kv dv   1 dt 1  ln 10  kv  t  E , where E is an arbitrary constant. k 10  kv  e kE e kt 10  kv  Ae kt A  e kE  When t  0, v  0, A  10 10  kv  10e kt v 

1 10  10e kt  k

Question 11(b)(iii):

10  40 k t  v  4 10  10e 4 0 As t  , v 





k 

1 4

When v  36, t  9.21 seconds


22

Comments: Standard question on Differential Equations. The concept of terminal velocity can be easily interpreted from the question phrase “after a long time”


23

2017 A Level 9758 H2 Math Paper 1 Solution and Paper 2 Likely Questions

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