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

Thoughts about Paper 2.

Numerical Answers (click the questions for workings/explanation)

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:

Question 6:

Question 7:

Question 8:

Question 9:

Question 10:

Question 11:

Graph for Q1

Comments/Explanations: Very easy question so long as you do not make any careless mistake. Make sure you write “or”. Students can also check their answers with the GC, with the graphical method.

(i)

When

When

(ii)

When

Tangent:

When

Tangent:

When they meet,

Coordinates

Comments/Explanations: Firstly, it is fine to give all answers in 3SF since the question did not mention anything about exact. I just happen to recognise such functions. Besides that, it can be easily done with the Graphic Calculator.

(i)

—(1)

—(2)

—(3)

(2) – (1) :

(3) – (2) :

Using GC,

(ii)

Required Sum

Comments/ Explanations: Proving the first part is not that easy as students just got to just try and figure something out, rather than staring for link. Use the GC to solve the next part, but be careful to copy the correct . The last part tests the students’ understanding on summation of GP and using of .

(i)

(ii)

For to be a unit vector,

(iii)

(rejected since )

Comments/ Explanations: Students could have evaluated the vector before doing the cross or dot product. Knowing that its a unit vector is a cue to students that the modulus is 1. This is also a fairly easy question

(i)

Let be the statement

When

Since is true.

Assume is true for some

Want to prove is true.

is true.

Since is true, and is true is true, by mathematical induction, is true for all positive integers n.

(ii)

(iii)

…

From (ii),

From (i),

Comments/Explanations: This is a standard tutorial question. Students should know that they are not required to memorise summation of or , though they are handy here. Thus, being a bit more observant, they should realise they can easily use (i) to resolve the sum. There should be no need to simplify the answer in (iii) since it is only 3 marks.

(a)

Let

Thus, is a root.

[Long Division] or [Quadratic Formula]

Long division

Required root

(b)

Since coefficients of given equation are all real, is also a root, given that is a root.

By comparing coefficients,

(rejected since )

Comments/ Explanation: First part of (a) is manageable, while the long division can be tedious. Students can also use the quadratic formula (b) is quite easy so long as students know their basics of roots well.

(i)

(ii)

When

(iii)

Let .

When

Comments/Explanations: Tricky a bit, as students are expected to leave (i) in terms of and . (iii) has several different methods, one can only use series expansion of

(i)

(a)

(b)

, where C is an arbitrary constant.

, where is an arbitrary constant.

When

, where D is an arbitrary constant

When

(ii)

, where B is an arbitrary constant

When

When

(iii)

Using GC, when , Time taken in (i)

Using GC, when , Time taken in (ii)

Comments/Explanations: This question is simply tedious. But definitely not hard.

(a)

(i)

Let

(ii)

Using GC,

Reject since

Since

This value is in both the domains of and so it should satisfy.

(b)

(i)

(ii)

Consider

We have that is not . Thus, the inverse does not exist.

Comments/ Explanation: The first part is your typical tutorial question. Picking the correct is a bit hard though, so students need to give it a good thought. The second part, as point out by Mr. Wee WS, is similar to an Olympiad Question on functional equations. To prove that inverse does not exists, there are other ways too. So feel free to try.

(i)

(a)

Since

, and

.

is parallel to both directions of ,

Since they intersect,

—(1)

—(2)

—(3)

Using GC,

(b)

Required planes are or

Simplifying, we have or

Required equations are: or

(ii)

For to not cut at a unique point,

Comments/Explanation: First part of the question is rather straightforward and students just need to be really meticulous. The next part tests students on their understand of a unit vector, or distance between planes. Students MUST leave the final answers in cartesian form. The last part is simply testing their knowledge of lines and planes. I will say that system of linear equations is tested in part 1 already.

### Relevant materials

MF15

### KS Comments

I think the paper is tedious, but definitely manageable. Hard ones, could have been q3, q7a, 10, 11ib. So if you did your tutorials and past year papers well, with proper time management and no careless, 70 is manageable. To get an A, you need to fight for that 30 marks which really test you on your comprehension skills and precisions. And these questions should distinct the students who deserve an A.

What are the least number of marks one must obtain to get a distinction?

Its ranges from year to year

Q6 44 is wrong since U3 is U2+3^2+3 = 14+9+3=26 Thanks

sorry it should be 44 because n^3

LOL

Omg I left 6 in terms of un… I think I lost 11 marks can I still achieve a distinction grade?

~~Wthell~~are you trying to rub salt on other’s wound? Only lost 11 marks and you’re asking if you can still get A.~~S.O.B~~i think there is no malice. so be nice. π

he maybe dk whats 100-11

U THINK EH

Yup definitely.:)

Hi!how was the H2 math paper overall?

Hi, do you think this paper is generally difficult? What would the bell curve be like?

Ur qn 4 (ii) got mistake in the working

yea, i made a typed and made careless. sorry

hi, can i know what’s the mistake in 4ii? has it been corrected?

yea, been corrected.

Hello just wondering but how did you find the paper? Like the standard of the paper. Thanks!

for qn4ii ) i let the first term be b(0.74)^n and used a/(1-r)….final ans is (50b/13)(0.74^n). Can someone tell me whats wrong with it?

yea, i made a typed and made careless. Trying to rush all out at the same time. sorry

Oh ty! π

Seems to be an error in 4ii

yea, i made a typed and made careless. sorry

Is this paper generally a difficult paper?

no kick

Does anyone know if private candidates have a different bell curve? I only scored 75…

For 8i, is it possible if I leave f”(x) and f”'(x) in terms of a and y instead of dy/dx

sure!

Why is it that for question one when I use graph to check it seems to suggest that the solutions is X less than -2 and X more than 1/3

you overlook the asymptote , try setting the GC windows and have ymax=100.

Hi for 8ii i left my f(x) in terms of y instead of x. All coefficients etc were correct. How many marks do i lose?

Hi, for 7b can I just sub z=1+ai and compare real and imaginary parts?

And for 7a instead of long division can also use comparing coefficients right? Still got the same answer

And for 11ia) isit ok if I went to find the normal vector and show that it is parallel to (-2,1,2)? Then I found the vector equation of the plane, and substitute to find the coordinates. Is that acceptable?

yup

Qn 2 if leave in decimal places instead of ln2 can? Qn say use calculator never say exact π

Yup yup. I wrote in the comments.

Is this paper difficult ? how many marks to get A?

85

how u know

Hi, for Q9, is it okay to leave answers to the first model and second model with arbitrary constants?

No, you should solve for them at the end since you’re given the initial conditions. And if you don’t solve for them, how do you do (iii)?

Hi, I solved for the arbitrary constants when doing (iii), but left the previous answers with the constants. Would that be accepted?

i see. but you still need to have it in the previous parts. they wanted x in terms of t, so that should refer to particular solutions instead of general solutions.

For 7 part 1, can I use quadratic formula to solve? I showed the working except for the step of removal of the squareroot term (but technically it can done without gc)

yea. but you need to show the removal of square root term though. cos you can’t use a GC for it. and they will take it as you did

Hi for 8ii i left f(x) in terms of y instead of x but everything else was correct. How many marks will i lose?

so you use x, instead of y like ? If so, its just the answer mark.

Ok thank you. For 6iii i didnt use the results from part (i). I split the summation r^3 + r into 2 different summations and solved it separately. Is this acceptable?

yea, if you memorise the formulas correctly.

Thanks! For 11ia the part where they asked to find ΞΌ and t my method and expressions were correct but i subbed 1 value into the gc wrongly and got a different answer. How many marks will i lose?

just the answer mark i’ll say, if equations are all good.

Ok thanks. Sigh lost around 26 marks even though i typically get 90+ for the past year papers. Is there still a chance for A?

Hi for the vector question 11(i)(a), can I ask to show that l is perpendicular to p, can I calculate the normal vector for the plane. And since the cross product of the normal vector and vector of the line =0, l and p are thus perpendicular? Thanks!

Sure. But i think you meant

DOTproduct of the normal vector and vector of the line =0But for a plane to be perpendicular to a line, since the normal vector is perpendicular to the plane, shouldn’t it to be the cross product=0?

oh right. i mistook it as the last part to show . sorry

Okay thank you! I almost freaked out cus I barely reached a low A without ecf marks.

math has no ecf actually. its only method marks and answer marks

Cus dot product=0 is for perpendicular vectors but cross product=0 is for parallel vectors

does it matter if i get a 75/100 for both papers or a 60/100 & 90/100 for each paper?

your score is the average of both papers. one can score 100/100 and 50/100 too.

How many marks do I need to get to get an A??

Hello may I know the marks for a safe A for paper1 cos it’s quite a tough paper.

hi guys i think just stop focusing on this since its alr over. and stop asking impossible qns like “what do u think the bellcurve will b like?” or “what is the standard to get A?” this varies on a national level, and im sure only cambridge will know lol π rest assured whats done is done, just hope yall did ur best! and focus on what really matters: the remaining papers!!! JIAYOU

yes, and all the best to your other papers! π

Hi, do you think this paper is tougher than the past year A levels paper? I hope I am not the only one who finds it tougher. thanks.

To be honest, every year the students find their own year hard… I feel that its a tedious paper, but still manageable. Quite a lot can be found in typical prelims/ tutorials.

Thanks. I’ll try to work harder for Paper 2…

You raised a very interesting point. Why do you students find the paper for their year uniquely difficult compared to past papers each year – even though an objective observer can see that they are more or less similar?

Edit: Why do you *think students

i think it’s psychological. But I’m not qualified to comment. Though every year, the students will lament that they got the hardest paper. This paper was tedious but definitely manageable. Some questions is definitely harder but that’s for every year; there is always a couple of tough questions every year to sieve the 90 from 80.

Hi if i scored 44 do you think i still can get B? Really praying hard now cause i made 20marks worth of careless like not reading qn properly etc …

You should work really hard for paper 2.

Hi, considering paper 2 will be of similar difficulty, what will be the minimum for an A?

Thanks!

I’ll say 80 will give an A definitely. But if you ask me the minimum, it is really hard to judge. If paper 2 is going to be the same standards, that is, tedious but manageable, you probably need 78/79. I know there are students who do quite well for this paper.

What mark do you thin is needed to get B overall? So that i can strive for that mark.. I know that getting 100 for p2 is close to impossible and furthermore math is not one of my strengths, compred to chem and physics.

65-68 that range I’ll say.

Hi, for question 7b, is it acceptable to just sub z=1+ai and compare real and imaginary, hence I did not state 1-ai is also a root blah….

And question 7a, I did not do long division, but compared coefficients instead.

And question 11ib) I let the normal vector be b multiplied by (-2,1,2), where b is a constant, and let the point (0,c/b,0) lie on the unknown plane p, with equation -2bx+by+2z=c. And solve to get the 2 equations. Is this method correct? Cause I assumed the normal vector dosent necessarily is (-2,1,2), it can also be like (-4,2,4) , as long as it is parallel to the plane.

yup, your method works. actually several methods will work, you can use quadratic formula for 7a too. I’ll say all the methods requires some precision.

for 7b, its okay the if you just sub.

11iib, yes that definitely works. and like 7a, its just tedious. What i did was just whichever is convenient.

Hi, I just want to enquire if it is possible for 75 to be an A? cause I find it quite unbelievable that 50% of the cohort can get 80 and above..

must see paper 2. and actually, i think 78/79 is good

Hi for 11(ii) i did it the same way but i made a careless mistake when i added the values together after using dot product and got a different answer. How many marks will i lose?

Hi for qn 8i) the differentiated answer should not contain dy/dx because the question said leave in terms of a and y?

Hi Paul, thanks for that! I didn’t see that too. I guess was rushing it too much. π

Is it possible to secure B around 140/200 this year? Thanks

yup

Sorry I know this is a very unrelated question, but I would like to ask about majoring in mathematics. Is it very difficult compared to h2 mathematics? And what are the job prospects a available?

actually there are a lot of prospects, i have several ex students doing quant analyst, actuaries, data mining, and even computing, aside from teaching.

H2 Mathematics is not reflective of uni Mathematics at all. This is a talk by one of the top hedge fund owners.

Difficulty varies based on one’s interests and abilities, but i suggest you studying something you truly like and enjoy. Mathematics needs interest, else you might struggle. An easy example will be Mathematical Induction in H2 Math, in uni, first year will consider doing Strong Mathematical Induction, a more rigorous proof. Much of Uni Math revolves around proofs. And Mathematics have many different fields too.

Here is a link to some year 1 probability that I did a few months ago for my students.

Interesting! For me, I’m thinking of doing a minor in Statistics.

Go ahead! I think that’s very useful knowledge for anybody.

Indeed π I really think Stats has much more to offer beyond what we see in H2 Math. I enjoy the concepts, but I think the way it’s assessed is a little too mechanistic, formulaic and well, easy i.e. standard steps, just press GC.

From your experience, how does Uni Stats differ from H2 Math Stats?

(For the record my major will probably be History or Political Science.)

A lot different. You get to learn more in university definitely. Application too. For example, when I was studying financial mathematics. One of the first project was to track and hedge a stock. And we were doing it real time for 13 weeks to decide. You learn more abstract things in Uni that test on applications.

Don’t mind me asking, but what college are you from? And you clearly are an arts student π

i kind of “geh kiang” and inverted the negative signs for the cartesian equations and wrote -2x+y+2z=-35 and -2x+y+2z=37 for q11ia) would i get some method marks for this? thanks in advance and really appreciate you for your work!

Hi for question 5οΌiiοΌi read the question wrongly and used a=b instead of 2a=-2b will all 4 marks be deducted??

You might get a method mark for the unit vector

okay thx!!! for question 3 i drew the graph on the negative side of the x-axis but all the other points are correct do iget any marks for that??

should have a mark for the shape.

May i know what is the graph paper for?

You can use it to draw your argand diagrams

Hi if I get around 50+ for this paper, do you think I can at least get a B or C? May I know how many marks do I have to get for paper 2 too? Thank u so much!!

I think aiming for 70 will be healthy. All the best for the other papers:)

Thank you!!

okay thx!!! for question 3 i drew the graph on the negative side of the x-axis but all the other points are correct do iget any marks for that??

Is it safe to get an A with around 144/200 in total?

For question 11, if I get one value wrong (t), and used it to solve for the other two unknowns, will I get method marks?

For 7b) I didn’t manage to find the roots but I stated there’s a conjugate and I multiplied to get a quadratic factor. How many marks is this working?

6iii), I didn’t use MOD so I lost the marks but I did stated that the equation is equal to n3+ n is there method marks for this?

And for question 4, very sadly I got the wrong GP from reading the question wrongly. Do I get marks for at least quoting what does Sn = to?

no, you could’ve used a GC to do it. There isn’t ECF for math. Solving by hand isn’t a method, but you should get marks if the equations are all correct.

thats a mark since its a root and you did state it.

After introducing , did you put in the (i) answers. It really depends on your presentation.

No marks for the formulas.

i did question 5 correctly but realised i didnt label my coordinates in terms of a, b and c π is that gonna make a huge difference?

I assume you refer to q3, they did ask you to state the coordinates. you might lose that 1 mark.

Hello I got around 76 for this paper. How much do you think I should aim for Paper 2 to get an overall A?

Thank you!

And thank you for the solutions too! Really efficient!

Hi, to be safe, getting an average of 80 is good. But i doubt its the minimum. But never hurts to be safe. π

Pleasure π All the best for your other papers.

Hi for 11(ii) i did it the same way but i made a careless mistake when i added the values together after using dot product and got a different answer. How many marks will i lose?

For 4ii i used Sum to infinity – sum to N but for the r value i used -0.74 instead of 0.74. Will i get zero for this?

I got 30 marks for paper 1. I’m doomed right?

Did you count method marks too or only answer makrs?

I dont think 50% of the level got 70+ marks as many people had trouble with time management and careless mistakes, so you won’t be doomed. Work harder for paper 2 so that you will lie in a safer region of the bell curve! π

I dont think 50% of the level got 70+ marks as many people had trouble with time management and careless mistakes, so you wonβt be doomed. Work harder for paper 2 so that you will lie in a safer region of the bell curve! π

How do you normally do for exams in sch?

hi, i think for question 11ii, the value of a should be 3. and for qn 8iii, is it okay if i use sin2x/cos2x then use maclaurin’s series expansion for sinx and cosx and divide to get the first two terms?

omgg okay its 4.5 ps

that will work too. since they didn’t say hence or anything

hi for the sum to infinity qn, i calculated the r value wrongly but i get the sum to inifinity minus sum to n part correctly just that the value of r i subbed is wrong, will i be able to get ecf for this? what mark will i get?

Math has no ECF. You only can get method mark and answer mark. So you probably will lose your answer mark.

thanks for ur reply!!!! is 85/100 for p2 possible and 75/100 overall able to secure an A? cuz i rly feel v sad that i lost 35m in p1 and mainly due to careless mistakes πππ thanks for the solutions anyw!!!

yes.. thats for sure. wait you lost 35m or 25m in p1? :/

All the best for your other papers!

35m… I’m aiming a 85 but i heard in order to get A in math, it’s a at least 80m ππππ thanks omg i feel that i neglected math so much so regretted

i see. Yes, its rather competitive. Sometimes go to 79. But better be safe than sorry. This year paper, if you do enough tutorial and past year papers and be careful, its a rather manageable paper. But you should look forward. π All the best!

thank u!!!! i should spam myself w math p2 this weekend and 90/100 :’)

Can u please spot which chapters might come out in pure maths for paper 2 thanks !

Hi what do you think is mark range for B?

You may read it here.

Hi there, just wondering which pure math topics do you think will come out in paper 2?

You may read it here.

For qns 6iii if I didn’t do MOD how ah? Cause my answer don’t have the +2 lol how many marks gone?

should simply lose 2 marks

Thank you Mr Teng!!!!

For the last question, for the line to NOT cut the plane at a unique point, the whole line can line in the plane itself too.

like you mentioned, the question says unique point. if the whole line is in the plane, thats no longer a unique point.

Hi for question 8i, can you leave your answers in factorised form?

they asked for in terms of a and y.

Yup so I wrote 2a^2y (1+y^2) and 2a^3 (1+y^2)(1+3y^2)

Oh that should be a problem. They won’t nitpick

Should or should not??

Should not

i got around 78 marks for paper 1. how much should i score for paper 2 to get an a? what do u think are the range for a and b grades, thank you?

To be honest, I think A should be around 78/79. This paper frankly isn’t difficult. It is tedious. Scoring an 80 for overall will be safe.

For qn 10bii, can I write “No, g(n) does not have an inverse since it is not one one function”?

Sure

But that’s just 1mark.

Thanks! What do u think is the minimum mark/ cut off to get an A? (Just a guess, does not need to be accurate)

78/79. Judging from what I see some students are scoring.

Just a guess, what do u think is an A for General Paper?

Can i ask for qn 8(i), i expressed the f”'(x) wrongly which caused my part (ii) and (iii) answer for the f”'(x) part wrong but the method is correct, how many marks will I lose? I had another similar mistake in another qn but the mistake was only carried over another part of the question. Thanks.

actually, when the answer is wrong, your answer mark is gone. assuming all else is fine, you might get 1 for (ii) and 2 for (iii)

Hi what if for qns 9, I made a terrible mistake and intergrate wrongly and get two totally different equations ? will i still get method marks or I will get zero from itοΌ

depends on whats your mistake is

Hi what is for qn 5(i) i had a careless mistake and I had b=2a, will i get method mark for 5(i) and 5(ii)? thank you so much π

yes, you can get method marks, probably just 1 mark for your evaluation of the unit vector. Because for (ii) 2 marks go to the answers.

Hi for the vector question, if my answer is in dot product form instead of cartesian form how many marks will be deducted? And if i never put f'(x) and f”(x) in terms of a and y how many marks will be deducted also? :((

Thanks KS for the solutions!

U could get into trouble for publishing the answers even before the paper is over. There is a chance some students may still see this before paper is up, esp if there is train delay. Hope you can publish responsibly.

Hi TanMa, thanks for your concern. Let’s first be clear that these are not the answers, just suggested solutions. And these suggested solutions were NOT up before the paper is over. It was up by 11:45am, when the paper has ended.

Yes, you may argue that there was a train delay so the paper ended for some. I can even tell you that some students with special reasons have extra time for the paper. These students might finish after 12pm. How can the students access it while doing the paper? Unless you know something, we don’t.

Moreover, this is a link. The title and header is stated explicitly clear. Clicking this link is an act of free will and one’s own responsibility. In addition, we have zero profits from doing this as there are no advertisements too.

Thank you for your responsible comment. Cheers π

This question is not on math paper 1. Can you help to explain how to define p-value in paper 2? π

its the smallest (extremum) probability of rejecting given that is true.

sorry again. one last question, how to define unbiased estimates in exam?

thank you for your p-value explanation and paper 1 ans π

Expected value of the estimate is its true population parameter

e.g. Expected value of is

hello π do you think a 75 is enough to get an a for this year?

for paper 2, the pure math part was more difficult than previous papers i thinkkk especially the complex numbers one! and a lot of people made careless mistakes at the back!! and if i circled wrongly for the scatter diagram, will i get 0 for the subsequent parts π

I think that you might have a high B. You still get marks for all the explanations. Pure math was more tedious and the complex was indeed slightly different from previous years.

i don’t even dare to check p2 ans now so scared if i got careless mistakes, just let it be :’) i hope the bellcurve can be down a bit by paper 2 cuz i rly screwed for my paper 1 πππ

Hi, this might be a little late, but for the very last vectors qn on the cartesian of the two planes at distance 12 away from p, will my answer be accepted if i write it as 2x the value of ur goven ans i.e. -4x+2y+4z = 35 ?

sorry i meant -4x+2y+4z = 70

Thats alright. I doubt they will be that strict, given this question is tough and many students can’t do it.

do you think this year’s paper or last year’s paper is harder?

they both have different expectations of students. To do well for this year, I think tenacity is rather important, and students have to be extremely surgical this time.

Hi thanks for the solutions! May I know how I fare if i got about 76/100 for paper 1 and 81/100 for paper 2? Should be studying econs now but am really bothered by this >< seeing your previous comments and I know i am very border line now. But like idk if my counting is correct later got mistakes i did not see or I am just worrying for nth maybe will have more method marks than I thought. How to know if u counted correctly ah? like how many method marks is reasonable?

Thanks!

I think you will get the A. No worries π and method marks entirely based on your method of solving, even if the values are wrong.

All the best for the rest of your papers!

Hi!

For part 2 of the vectors qn, I got the correct a.n (35 and 37), but then i put in the wrong normal,(careless), how many marks will i get?

For the functions qn, i forgot to say the domain of f(inv) must be a real number. Is that minus 1 mark?

For the reason why f(x)=f(inv)(x), I didnt say that the value will fall into both domains… Is that also minus 1 mark?

For the APGP qn, the last part, I wrote S(infinity) – Sn, but then i subsequently flipped the two around in the rest of my workings. Can i stikl get one mark for the first step?

Thanks so much!!

Vectors, you might lose 2marks since the equations will both be wrong

Functions, 1 mark if they are strict.

As for , should be necessary since you should find 3 answers for x, unless you justify otherwise.

APGP, should just 1 mark if lenient, since you probably did fill in the formulas

All the best for the other papers!

Hi what if i get 67 and 62 can i still get B? I heard the B range is rather wide like 60 to 75

I think that’s fine for a B

Is 75 an A or B?

Most probably a high B

How u know?

Will there be method marks for the complex number question part (II)? I think I subbed in 1 + ai and expanded the equation wrongly and got a weird value of a and k subsequently

Should get 1 mark.

Hi, for qn 3 will marks be deducted if I only labelled the coordinates on the graph but did not “state” as such? Also, is it true that Cambridge doesn’t give 0.5 marks? Thanks

yes, there is no 0.5m. there is only M1 (method) and A1 (answer). As for stating, we should have them explicitly said out