# Hard Questions for the New AS Maths!

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**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**

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!

Spoiler:

_gcx

zq01

AryanGh

orderofthelotus

CoffeeGeek

etothepiiplusone

dcencima

effystonem

Rory_Evan

Moltenmo

Heptagon

pinkypaz123

yeezyb

bjt1882

LEuphoria

ecila21

t.64

brownanya122

usfbullz

Shadow_12

Black Water

StudyJosh

Pastelx

PixelatedUnicorn

Yasmine2012

Reece.W.J

TeacupAndTragedy

cleverclogs15

Alexia_17

Notnek

Lemur12

Sonechka

Medicine4ever

Show

_gcx

zq01

AryanGh

orderofthelotus

CoffeeGeek

etothepiiplusone

dcencima

effystonem

Rory_Evan

Moltenmo

Heptagon

pinkypaz123

yeezyb

bjt1882

LEuphoria

ecila21

t.64

brownanya122

usfbullz

Shadow_12

Black Water

StudyJosh

Pastelx

PixelatedUnicorn

Yasmine2012

Reece.W.J

TeacupAndTragedy

cleverclogs15

Alexia_17

Notnek

Lemur12

Sonechka

Medicine4ever

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#2

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 )

(No need to tag me since I'm in yr 13 doing the old spec :P )

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(Original post by

This will be interesting, but I’m on the old spec.

**HateOCR**)This will be interesting, but I’m on the old spec.

You're more than welcome to post questions purely for the guidance and help to others, and in fact i'd appreciate it.

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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:****Where k is a constant.****When f(x) is divided by****the remainder is r.****When f(x) is divided by****the remainder is 3r.****A. find the value of k****B. Find the value of r****C. Show that****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!**
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#6

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

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#7

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.

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#8

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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#10

A.

B.

C.

D.

Thanks for the tag. (These are the answers to Rick's question, viewer discretion is advised)

Spoiler:

k=-3. Just some substitution.

Show

k=-3. Just some substitution.

B.

Spoiler:

r=-12

Show

r=-12

C.

Spoiler:

5^3+(-3*5^2)+(-7*5)-15=0

Show

5^3+(-3*5^2)+(-7*5)-15=0

D.

Spoiler:

x=5 is only solution as the other two are complex numbers

Show

x=5 is only solution as the other two are complex numbers

Thanks for the tag. (These are the answers to Rick's question, viewer discretion is advised)

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#11

I just came up with this randomly:

A circle has area with its centre at the point .

It is given that the line is tangent to the circle at the point such that is an integer.

Furthermore, is tangent to a quadratic at .

(i) Determine the points of intersection between and

(ii) Show that and are diametrically opposite on

(iii) Hence, determine the ratio into which the triangle splits the area of

A circle has area with its centre at the point .

It is given that the line is tangent to the circle at the point such that is an integer.

Furthermore, is tangent to a quadratic at .

(i) Determine the points of intersection between and

(ii) Show that and are diametrically opposite on

(iii) Hence, determine the ratio into which the triangle splits the area of

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#13

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

Spoiler:

Let's consider our polynomial Here, we have 2 crucial bits of information to deduce k, we know that when also,

Probably fancier ways to do this, however we have an obvious 2 variable simultaneous equation we can easily solve for k here, equation one being (when substituting x = -1, reduces to):

For our second one, where x = 3, this reduces to

We can clearly multiply our first equation by 3, in order to eliminate r and isolate variable k, so

By subtracting equation one from equation two we get

thus .

We can plop this back in to check for our results for -1 and 3 in a bit, now with this value of k, substitute into either equation to obtain , which satisfies both equations in the system. We can further conclude this in the next question.

for part c) all you need to do is utilise the factor theorem (x-a) is a factor of the polynomial if and only if f(a) = 0. So let's sub 5 into this cubic we have

which is . Sub in 5 and you get 125 - 3(25) - 35 - 15 which is 125 - 75 - 35 - 15, or 125 - 125, which equals 0, which means (x-5) is a factor of this polynomial.

For the next part, we need to do long polynomial division.

by dividing by x-5, (the cubic), you obtain this such value as your quotient, by doing the quadratic formula on this final set of terms, you will get -2plusminus sqrt -8 all over 2, as we know, there is no such real value as there cannot be a real square root of a negative number (we're not delving into the complex world just yet bois), hence there is no real roots of that quadratic, thus (x-5) is the only real root of that cubic.

Show

Let's consider our polynomial Here, we have 2 crucial bits of information to deduce k, we know that when also,

Probably fancier ways to do this, however we have an obvious 2 variable simultaneous equation we can easily solve for k here, equation one being (when substituting x = -1, reduces to):

For our second one, where x = 3, this reduces to

We can clearly multiply our first equation by 3, in order to eliminate r and isolate variable k, so

By subtracting equation one from equation two we get

thus .

We can plop this back in to check for our results for -1 and 3 in a bit, now with this value of k, substitute into either equation to obtain , which satisfies both equations in the system. We can further conclude this in the next question.

for part c) all you need to do is utilise the factor theorem (x-a) is a factor of the polynomial if and only if f(a) = 0. So let's sub 5 into this cubic we have

which is . Sub in 5 and you get 125 - 3(25) - 35 - 15 which is 125 - 75 - 35 - 15, or 125 - 125, which equals 0, which means (x-5) is a factor of this polynomial.

For the next part, we need to do long polynomial division.

by dividing by x-5, (the cubic), you obtain this such value as your quotient, by doing the quadratic formula on this final set of terms, you will get -2plusminus sqrt -8 all over 2, as we know, there is no such real value as there cannot be a real square root of a negative number (we're not delving into the complex world just yet bois), hence there is no real roots of that quadratic, thus (x-5) is the only real root of that cubic.

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#14

Perhaps.

Using the binomial theorem, prove from first principles that for . Justify why this method cannot be used to prove the derivative for all .

Using the binomial theorem, prove from first principles that for . Justify why this method cannot be used to prove the derivative for all .

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#15

(Original post by

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!

**Pastelx**)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!

Spoiler:

I got 5 but my brain has the reputation for being notably stupid on simpler questions,

Show

I got 5 but my brain has the reputation for being notably stupid on simpler questions,

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

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#16

**usfbullz**)

@Pastelx is it

Spoiler:

D

Show

D

?

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

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#17

(Original post by

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

**AryanGh**)
Spoiler:

I got 5 but my brain has the reputation for being notably stupid on simpler questions,

Show

I got 5 but my brain has the reputation for being notably stupid on simpler questions,

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

Spoiler:

Yep C is correct! That was my thought too when I first saw the question–I said "WELL I KNOW IM APPLYING FOR OXFORD AND NOT CAMBRIDGE NOW"

Turns out you are only given a few minutes for each Oxford multiple choice question though so it is a lot harder until time constraints!

Show

Yep C is correct! That was my thought too when I first saw the question–I said "WELL I KNOW IM APPLYING FOR OXFORD AND NOT CAMBRIDGE NOW"

Turns out you are only given a few minutes for each Oxford multiple choice question though so it is a lot harder until time constraints!

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#20

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 have in the range ? Justify your answer.

How many solutions does have in the range ? Justify your answer.

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