2016 A-level H2 Mathematics (9740) Paper 2 Suggested Solutions

All solutions here are SUGGESTED. KS will hold no liability for any errors. Comments are entirely personal opinions.

Numerical Answers (click the questions for workings/explanation)

Question 1: 0.0251 \text{~m/min}
Question 2: \frac{x^2}{2} \text{sin}nx + \frac{2x}{n^2} \text{cos}nx - \frac{2}{n^3} \text{sin}nx + C;~ a = 2 \text{~or~} 6;~ \pi (\frac{2}{5} - \text{ln} \frac{3}{\sqrt{5}})
Question 3: a + \frac{3}{4} - \text{sin}a - \text{cos}a + \frac{1}{4} \text{cos}2a;~ \frac{1}{4}(\pi + 1)^2
Question 4: z = 2.63 + 1.93i, ~ 3.37 + 0.0715i ;~8^{\frac{1}{6}}e^{i(-\frac{\pi}{12})}, ~8^{\frac{1}{6}}e^{i(-\frac{3 \pi}{4})}, ~8^{\frac{1}{6}}e^{i(\frac{7\pi}{12})};~ n=7
Question 5: \frac{11}{42};~ \frac{4}{11};~ \frac{4}{1029}
Question 6: 60;~ 10;~ \{ \bar{x} \in \mathbb{R}, 0 \textless \bar{x} \le 34.8 \};~ \{ \alpha \in \mathbb{R}, 0 \textless \alpha \le 8.68 \}
Question 7: 24;~ 576;~ \frac{1}{12};~ \frac{5}{12}
Question 8: r = -0.980;~ c = -17.5;~ d = 91.8;~ y = 85.9
Question 9: a = 7.41;~ p = 0.599 \text{~or~} 0.166;~ 0.792
Question 10: 0.442;~ 0.151 ;~ 0.800;~ \lambda = 1.85


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KSΒ Comments

360 thoughts on “2016 A-level H2 Mathematics (9740) Paper 2 Suggested Solutions”

  1. Hi there, if I didn’t find the probability for 7(iii) and 7(iv) but just the number of arrangements how many marks will I get? Thank you:)

    1. Based on the comments of your peers I think the bell curve ought to be pretty similar to that of last year so I’m afraid you’re at a borderline B. Don’t be discouraged though, work hard for your other papers!

  2. For question 6, for both sets of values, I wrote that they are subsets of R+, and then carelessly wrote that they are less than or equal to 0 for the lower limits. Would I lose the answer marks for both of these parts?

  3. What if for qn 7 (iii) and (iv) you did not realise that they are asking for probability? Then you just put the number of ways to arrange?

          1. Hi but what if i scored A for everthing excpt math and i wanna just retake math? Must i do all my subjects again the next year or just for math alone

          2. sorry. But you can’t do that. For entry to local Uni, they look at the cert as a whole. You will need to retake at least 3H2 and GP. A cert with just h2 math as the single subject is useless, even with an A.

  4. Hi, thank you so much for uploading the solutions. May I ask if we chose the wrong outlier in Q8, around how many marks will be lost?

    1. The pure math was tedious. Finding z for the complex requires student to use GC instead of drawing triangles, something I guess they aren’t used to. Finding the equation of bisector need them to realise it cuts the centre. Last part for complex is tricky as some might forget to check for the positive part. Stats is pretty okay. Sampling was a bit dubious with 6599. PnC was easy but I reckon a handful of students did not see the word probability and paid 2 marks for that.

      How did you find it?

      1. i think many ways to solve the complex-tedious way can be you just find all the different equations and then calculate intersect

        but good use of gc’s equation solver helped greatly (even for last question)-eqn solver is magical

        overall slightly more novel (with the roulette style probability question) but manageable with all the hypo 0 tricks

        But i thought it was slightly harder than previous years (since previous years it was almost all straightforward questions)

  5. Hi, ‘KS Teng’, can u guess what is the min A grade? It’s okay to get this wrong πŸ™‚ Not graded. To satisfy our curiosity

  6. for qn8 if we circle the wrong point, wil the explanation for ii and iii be awarded? it seems like it is independent of the point circled

    1. the thing is that when you remove the wrong point, your answers for iii and v will be different :/ which point did you remove?

  7. Thank you so much for the answers and the efforts, really appreciate that. Btw if I circled the point wrongly for the regression line question, will the whole question be wrong?

  8. For qn 8, if you accidentally removed the first point instead, do you still get credit for the remaining parts? And how come the first point is not the outlier? The first points is way too off

  9. hi!! thank you so much for the answers and solutions! much appreciated πŸ™‚
    can i ask, if i forgot to exclude the outlier, does that mean there’ll be no ecf and thus 0 for the subsequent parts?TT_TT

      1. Hi i forgot to exclude the outlier too, but since c,d,r are all based on GC answer, does that mean all 3 are wrong? And v will be wrong too?

  10. Hi! For question 6(iii) and (iv), if the arrow is wrong, assume all else correct, how many marks would i be penalised, out of 7?

    1. This is because the first point still follows the general trend. The equation of the curve is best suited to a y=c/x +d. An outlier doesnt follow the general trend of the curve, in this case a converging curve. Hence x=3 is most probable, although x=4 may be arguable as well.

  11. For question 4, you assumed the equation of the line bisector will pass through the center of the circle. Wouldn’t it be better if you had use the point of origin instead?
    Thus y=-2.5x

        1. It cant pass through the origin because the circle is on the right of the y axis and above the x axis and the bisector is downward sloping. I drew mine accurately using 2cm for every unit on the graph paper so it passed through the centre of the circle perfectly. If you’ve done many complex number qns, this is a technique required or else u wont be able to solve the question.

          1. Yeah i misinterpreted the qn and used the equation of the circle and equation of line representing arg(z)=alpha πŸ™

  12. Hi, can i ask is the paper 2 tougher than paper 1? It seems that many did better for P1 than P2. Also, I got 70 for Paper 1 and 82 for Paper 2 so the average is 76. Is that still possible to achieve an A? Previously, you’ve mentioned that an A usually required 78 and above. Thanks! πŸ™‚ and thanks for uploading the answers so promptly too!

  13. Hi For the complex no qn last part n=7 what if i misread qn for 2 smallest values and i say 1st n=7 and 2nd n=15? Any marks deducted? Thks!

  14. What are your thoughts for this paper? What do you think the bellcurve will be like?
    I made too many careless mistakes :'(

  15. Hi for Qn4, I carelessly wrote the arg for w as it’s conjugate. But the calculations were all right. Do you think the entire qn will be marked as wrong, including ii? Thank you so much for the answers btw!!

  16. Hi, for questions 6, if my variance is wrong ( I went to find unbiased estimate) , how will it affect my marking for iii and iv?

  17. I believe the reason why the first point cannot be the outlier is because you will have problem explaining reliability later on (since you dont have a proper data range!)

    1. The test in general has marks. But in this case, it’s really up to your presentation as there is no need to use any test statistic (no fault in stating though).

  18. Hi KS may I ask why is the outlier not 1? It is really far off from the other points. Also will I be awarded any marks if all I did was draw a perpendicular bisector in 4ii?

    1. Bisector got no marks. They didn’t give marks for finding them.
      Outlier is at x=3, partly due to the last part of the question (this is some hint). The other reason is observed if you calculate the r value when you remove the point x=3 and x=1 respectively.

  19. Hi why can’t x=5 be the outlier? And my answers were only arnd 0.02 different from urs for c,d and r so how many marks will I lose? Thank you for the answers btw!

  20. Hi for your question 9c) working, you forgot to include 100 as the upper limit. (I.e. W is between 30 to 100 inclusive as there are only 100 questions). While the answer when rounded off will still be the same, since it is a 4 mark question, I think they might penalise for the missing working.

  21. For qn 4, if I carelessly wrote |w| to be 8 instead of sqrt(8), and my cube roots are 2e^(i…) (the arg of the three cube roots are correct), do I lose all the marks for that part?

  22. For 7i, if my ans is wrong. My ans is 4.then , for 7ii, if I write probability= No restriction -No men (same mistake)- 4. Do I get marks for 7ii?

  23. I got 75/100 for paper 2 and 85/100 for paper 1.

    Did negative marking tho… like if I got an ans wrong i’d deduct everything

    Will I still get an A?

    Seriously damn demoralised after today’s paper bc I heard A is quite high? πŸ™

    1. are you trying to rub it in or you simply cannot phantom that an average of 80 would give you an A that you need to confirm it?

      1. I think she means no malice. It’s fine to have such concerns as there as a handful of students that find this paper more than manageable. Her concern is understandable πŸ™‚

  24. For the P and C question if i didnt find the probability and only found the no. of ways how many marks will I be deducted? I feel like Im too strict with myself. I deducted 4m out of 6m in my own working.

  25. Hi I have about 85 for Paper 01 and 77 for Paper 02 (without method marks taken into consideration) based on your expected bell curve can I get my A?

    1. this really depends on how you do the question. i didnt introduce Z test statistic at all, so I was using my \bar{X} distribution, thus there is a need to tell them how i derive my distribution.

      1. Hi, thanks for the reply!
        Is there any marks given for defining RV and u(in hypo)?
        Also, I calculated mod r wronglying but my argument is correct for ques 4. Will I get 1 marry for it?

        1. i dont think there is marks for the definition or rv and \mu, since they didnt say define the variables you used. Its just presentation. you mean for writing w? its just writing, so i think thats 1 mark for getting it right. you probably have 0. but if your roots finding steps are correct, you can find 1 mark there.

  26. Hi just asking, what if for qns 1 my V forget to put pi but all working correct except final ans and the h (cause missing pi) how many marks lost

  27. OMG, I just realised I forgot to exclude the outlier, may I know how many marks will I lose? Will they award method marks still for the following parts if my method used is correct?

      1. In that case is it safe to say that only the answer mark will be deducted from each part of the qn that requires R assuming everything else is correct?

  28. Hey there Mr Teng, for 1 and 2(b) i didn’t get the final answers but my methodology was exactly the same as you, would I still be awarded credit? Also, for my Correlation and Regression, I didn’t exclude the outlier before I calculated R, c and d, would I get 0 that 3 marks and the subsequent last part? Thanks for your constant efforts to provide timely answers! Cheers!

    1. I think you will still get marks for methods, though it is hard to say how much as we do not know how much you’ve written or shown.
      You’re supposed to remove the outlier, as stated in the questions, you might lose credit for the values of c, d and r-value. The same for the estimate, since all these are purely answer marks. You should get marks for your written explanations.

  29. Hi, can I ask for q6 last part does it matter if we exclude 8.68? Cos if p-value=significance level means null hypothesis still rejected.
    Oso sry for bringing up a previous qn. For the complex no qn last part n=7 what if i misread qn for 2 smallest values and i say 1st n=7 and 2nd n=15? Any marks deducted?

  30. Hi is it true that the national distinction rate is about 50% ? Doesn’t that mean that to get A you don’t really need a very high score?

    1. Bell curve, I quote from Cambridge, is described as, “The grading for A/O levels is comparable to judging a student’s ability to swim 100 metres. If a student has evidently shown that he has been able to clear the 100 metres, that student will get the ‘A’ regardless of the number of people who are able to swim around the world.” So it is hard to say a very high score is need. H2 Physics is 40-45% distinction. O’levels E Math is also 50% distinction.

        1. technically, if you put school distinction rate against national distinction rate, you will get a very different graph which suggest a very mean score. bell curve ensures students dont fail and the A deserve their A and Bs deserve their Bs.

      1. I heard from my math teacher that the number of As awarded is predetermined by the examination board and that they are likely to award more As in mathematics and physics due to the lack of engineers in Singapore haha.

  31. hi! i am a student from hci. guys please relax and jiayou for the papers ahead !! a lot of my peers and i in fact, found the paper quite tough tbh!! more difficult than the past year papers and this year’s paper 2 pure math was definitely more challenging πŸ™‚ point is, there really isn’t any use in knowing where you stand/ whether you can get your a for math i guess. because no one knows how the entire cohort is going to fare right πŸ™‚ let’s just jiayou and focus on the remaining papers!!! and continue to have faith!!!

    1. But HCI even when they say it’s tedious a majority will score 80++ I’m pretty sure
      You guys are just plain smart I matter how tough or time consuming you guys will still score very well πŸ™

      1. no la, theres just a large divide perhaps some people will consistently hit, while others don’t. Nobody is plain smart you can still make many careless mistakes.

      2. Since top tier schools like hci, rI, vjc,njc… have majority (abt 80%) of their students get A for math, the rest of us need some assurance for grading lol.

  32. Hi if i made a careless mistake in 1 by writing r=2h , subsequently everything is right (according to r=2h) except for ans mark will i lose everything???)-:

  33. we can actually use complement for 7(iv)

    Probability = 1 – {C, S, T together} – {any two of C, S, T together}
    = 1 – \frac{1}{12} (from (iii)) – (3C2 x 2! x 7C2 x 2! x 6!)/9! = 1 – 1/12 – 1/2 = 5/12

  34. Hi! For Q6(iii) hypo testing, I carelessly thought that Ho is not rejected and I put >0.05, my final ans was >34.8, but my values and method are all the same as yours. How many marks will I lose? Thanks!

  35. For the complex qn (4aii) values of z I thought it was just the modulus so I gave the length only πŸ™ how many marks would be deducted? Thank you πŸ™‚

  36. Hi KS, may I ask if for question 2b, instead of du/dx, i went to find dx/du by making x the subject is that ok? However I made some careless mistake along the way I ended with [root(9-u)]/u^2) instead of your (9-u/u^2), how many marks will I receive for this question?

        1. because your method for the integration is gone. unless somehow you integrated the correct expression. :/ Math has no ECF, as precision is one of the mathematical skills we look to test.

  37. Hi Mr Teng, is it true that every year you are teaching Math, the national average distinction is roughly 50% without much fluctuations?

    1. yes, for 9740 h2 math. its been roughly 50% national average. but i believe in statistics, and what’s truly reflective is the school distinction rate. it tells more.

  38. hey i got 85 for paper 1 and about 81 for paper 2 ? are my chances of A gone since most people in those good schools will probably do very well.

  39. For the scatter diagram i did not see the part that it needed to be drawn on graph paper, and drew it normally on the foolscap, but to scale. Will i get any marks?

  40. Hi if of excluded the correct outlier but I forgot to circle it on the graph (lol) will just minus 1 mark right coz for the next few parts my values are correct, and if I didn’t leave to 3 sf but 5 sf instead will there be another mark lost ?

  41. HOW DO YOU GUYS EVEN DARE TO CHECK THE ANSWERS? Aren’t you guys scared that if you find qns that you made mistakes in, you won’t be able to concentrate on other papers? I don’t understand how you guys do it

    1. Because on the other hand when u realise ur answers are correct u will be more confident in aceing the others😊 And knowing that u are doomed will also motivate u to study harder for the others right and not becoming too complacent. Anyway all the best for the remaining papers!

  42. Hi Mr Teng, do you think Cambridge will accept x=5 outlier? Since the r value is actually 0.977, and the idea of correlation should allow for a notion of ‘best fit’, omitting x=5 actually still gives a 1/x trend. :/

    1. I am also convinced that x=3 is the best candidate for a outlier, just curious as to is statistics so steadfast in forcing all data plots to fit a desired mathematical model. πŸ™‚

  43. Omg i’m so so disappointed in myself because i can’t handle exam stress so my mind totally shut down during exam… my paper was pretty bad and i think i got around 75 for paper 1 and 60 for paper 2 bc i was so nervous argh. Will that give me a B thoooooo

    1. I think you should get a B. Or maybe you were too harsh on yourself and minus all the method marks when you counted. don’t worry
      All the best for your other papers!

    1. You lose the marks for c, d and r-value. You also lose the marks for the estimation of y. Also the circle part. So that should be 5. Your written explanation will be eligible for marks

    1. That should be fine too actually. Since both r and h are variables, some of my students did implicit differentiation too. The final answer should be the same though.

  44. For paper 2, q4(a)(ii), I got the equation of the circle and the equation of the normal, but then didn’t equate them… how many marks can I get?

    for q4(b)(i), I got the modulus and argument of w, and I remembered to divide the argument by 3, but forgot to cube the modulus, how many marks can I get?

    for q4(b)(ii) I managed to get to the step (n-1)arg(w) = 0.5pi. but i couldn’t solve for n, how many marks can i get?

    For q8, I was super careless. I circled the 4th point in my graph, but then omitted the 3rd point in my workings…however, the value for d is the same as the ans (though my values of c and r) are wrong. Can I get 1 mark for my d value? and if i use these c and d values to get my answer for (v), will i be awarded the method mark at least?

    for q10 (iv), I got 0.80045, then at first i rounded it off to 0.8005 (4sf), then i rounded it off to 0.801 (3sf) though the correct ans is 0.800… will I be penalised?

    Many thanks for your patience πŸ™‚ We all appreciate your input and help very very much

      1. Hi, I’m sorry bout that. I clearly overlooked it.
        P2Q4aii, with the equations, correctly found. You should get 2 marks
        P2Q4bi, should be 1 mark max
        P2Q4bii, should be 1 mark max
        P2Q8, you should get 1 marks for the d if it’s correct. As for the (v), it’s only 1 mark for the answer, so you won’t get any.
        P2Q10, yes the answer is wrong. You will lose the answer mark.

  45. similar to the comment above, i did the wrong substitution for question 1 and calculated dr/dt instead of dh/dt, the answer i got was 0.0125 (exactly half). i am sure that i did the differentiation and subsequently calculation correctly. will i be awarded any marks at all?

  46. Hi, if I didn’t times three in 7(ii), how many marks will I lost out of the total three marks? Thanks for answering my question.

  47. i did the wrong substitution for question 1 and calculated dr/dt instead of dh/dt, the answer i got was 0.0125 (exactly half). i am sure that i did the differentiation and subsequently calculation correctly. will i be awarded any marks at all? sorry for reposting! πŸ™

  48. Hi if my inequalities for hypo testing I got more and equal to zero will I get the whole thing wrong? So minus 1 mark for each inequality? Thank you

      1. Will I get any method marks if I wrote this ?
        Qn 7i. No of ways= 4C3=24

        Qn 7ii. No restriction= 10C3
        No men=24
        No women= 6C3
        Total ways= 10C3-24-6C3

  49. hi,
    For the explanation for the scatter plot, could I say c is negative to reflect the shape of the graph to fit the shape of thr scatter plot
    And for d ti be positive, to translate the graph in the positive y direction to fit the positivevalue of y given.

  50. Sorry did not mean to spam, hey for question 7a pnc, if I did not permutate my answers, will I get any method mark? Especially for the 3 mark one

      1. Inverse and reciprocal have very different meanings. Awarding you the 1 mark will be based on whether the marker is lenient and like to try and understand what you’re saying.

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