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Hard Questions for the New AS Maths!




Mod edit:
Since this thread has to cater both to students who want a bit of a challenge and to those who are looking for STEP/BMO-style questions which are considerably more difficult than AS maths, we would appreciate it if you indicated when a question is much harder than anything in AS maths or in a different style. Please make sure that nothing on this thread requires knowledge of post-AS maths notation or concepts.
Thank you for your co-operation :smile:




Hi everyone!

As some of you can probably remember this, a while ago, during last year, a GCSE thread was made which consisted purely of extremely hard GCSE questions, and some even stretched beyond the current new GCSE maths Spec, and majority of them were grade 9 or even higher questions to test the limits of students.

I personally found that extremely useful, since the resources for the GCSE maths was extremely lacking and there was barely any past papers, since the first years did it previous summer.

Therefore, considering that we are now doing the new AS maths spec as well, I thought it would be extremely useful to make another thread which will be same as the GCSE one, but instead it has questions for the new AS maths.

Note; The previous thread made for GCSE maths had questions so hard that even students who got grade 9, could not fully complete, so do not be put off, if you cannot do some of the questions, and you can still achieve an A* at the end of year 13.

The main point of this thread is to get students helping students and in order to continue the thread like the previous one, it'll be extremely useful for students to post their own questions too.

It would also be advised that students stay inside the specification as much as possible, and not stretch too far apart, that even the high grade students cannot answer.

If you have answered a question, and know the answer and would like to post it, please put the answers in the spoilers, as it allows the other members to answer it too.

Here's a link of all the specifications for the exam boards. Students do different options, and I think it'll be useful to put the exam board's link, so they can navigate from there themselves:

AQA

Edexcel

OCR

WJEC

OCR MEI

Since students have not fully completed the specifications, unless you're like me, who has completed the spec, please refrain from posting questions too far ahead, and stay in the first half of the spec for now. If you want to post questions far ahead, do so, but please do not expect quick replies.

If you're not in year 12, and are in the year 13, you're more than ready to post questions just for guidance and help. It's appreciated and i'll see to it myself (most likely a rep! :P)

Tag list: (Tell me in the future if you do not wish to be tagged!

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(edited 6 years ago)

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This seems like a good helpful idea, good on you for taking initiative and making a thread for the new AS spec!

(No need to tag me since I'm in yr 13 doing the old spec :P :biggrin:)
Reply 2
This will be interesting, but I’m on the old spec.
Original post by HateOCR
This will be interesting, but I’m on the old spec.


I've mentioned regarding people doing the old specification, or in a year above.

You're more than welcome to post questions purely for the guidance and help to others, and in fact i'd appreciate it.
I'll do the pleasure of posting the first question, which is to do with Polynomial divisions and students should be able to answer with applying knowledge from their first few months:

The polynomial has the function:

f(x)=x3+kx27x15 f(x) = x^3 + kx^2 - 7x - 15

Where k is a constant.

When f(x) is divided by (x+1) (x+1) the remainder is r.

When f(x) is divided by (x3) (x-3) the remainder is 3r.

A. find the value of k

B. Find the value of r

C. Show that (x5) (x-5) is a factor of the function f(x).

D. Show that there is only one real solution to the equation f(x) = 0.

Please provide your answers in spoilers. If you have any questions, ask me!
whilst i can do many of these questions comfortably now, even after getting a grade 9, then and still even now, some of these questions look plain foreign to me :rofl:
Sounds great! The new edexcel books have some (by no means all) interesting challenge questions ending each exercise and I've had a few ideas from them for other questions - I'll try and post them soon.
Reply 7
9D719863-ABB9-4184-9907-FF1E94A3D32A.jpg.jpeg
This is from a past MAT paper but completely accessible to AS candidates–I have a few STEP/MAT questions that I’ve been given for specific AS topics which I will post later on perhaps!
I'll review what's in AS maths and see if I can put anything together.
A.

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B.

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C.

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D.

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Thanks for the tag. (These are the answers to Rick's question, viewer discretion is advised)
(edited 6 years ago)
I just came up with this randomly:

A circle C\mathcal{C} has area 17π17\pi with its centre at the point (6,5)(6,5).
It is given that the line :y=mx+12\ell : y=mx+12 is tangent to the circle at the point AA such that mm is an integer.
Furthermore, \ell is tangent to a quadratic Q:y=x2+bx+c\mathcal{Q} : y=x^2+bx+c at AA.

(i) Determine the points of intersection A,B,CA,B,C between C\mathcal{C} and Q\mathcal{Q}

(ii) Show that BB and CC are diametrically opposite on C\mathcal{C}

(iii) Hence, determine the ratio into which the triangle ABCABC splits the area of C\mathcal{C}
(edited 6 years ago)
@Pastelx is it

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?

I'm terrible at circles/loci in complex plane.
@RickHendricks here's my proposed solution to your problem, in spoilers, as you requested. idk how much of the solution i am allowed to give but i'll just do it all in spoilers, please point out if i make an error, i tend to do that with simpler questions

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Perhaps.

Using the binomial theorem, prove from first principles that d(xn)dx=nxn1\displaystyle \frac{\mathrm d(x^n)}{\mathrm dx} = nx^{n-1} for nZ0+n \in \mathbb Z^+_0. Justify why this method cannot be used to prove the derivative for all nRn \in \mathbb R.
Original post by Pastelx
9D719863-ABB9-4184-9907-FF1E94A3D32A.jpg.jpeg
This is from a past MAT paper but completely accessible to AS candidates–I have a few STEP/MAT questions that I’ve been given for specific AS topics which I will post later on perhaps!


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this is a MAT (ish) question?! wow, perhaps hope is not entirely lost for me :rofl:
Original post by usfbullz
@Pastelx is it

Spoiler


?

I'm terrible at circles/loci in complex plane.


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(edited 6 years ago)
Original post by AryanGh

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this is a MAT (ish) question?! wow, perhaps hope is not entirely lost for me :rofl:


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Original post by Pastelx

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(edited 6 years ago)
Original post by usfbullz

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Seems like a very nice thread idea. From the same MAT paper as Pastelx, I think this should be on the spec.

How many solutions does 2sin2x+2cos2x=22^{\sin^2{x}}+2^{\cos^2{x}}=2 have in the range 0x<2π0 \leqslant x < 2\pi? Justify your answer.

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