Hard Questions for the New AS Maths!

Watch
RickHendricks
Badges: 19
Rep:
?
#1
Report Thread starter 3 years ago
#1



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!
13
reply
S.H.Rahman
Badges: 15
Rep:
?
#2
Report 3 years ago
#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 )
1
reply
HateOCR
Badges: 22
Rep:
?
#3
Report 3 years ago
#3
This will be interesting, but I’m on the old spec.
1
reply
RickHendricks
Badges: 19
Rep:
?
#4
Report Thread starter 3 years ago
#4
(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.
0
reply
RickHendricks
Badges: 19
Rep:
?
#5
Report Thread starter 3 years ago
#5
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) = x^3 + kx^2 - 7x  - 15

Where k is a constant.

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

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

A. find the value of k

B. Find the value of r

C. Show that  (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!
5
reply
thotproduct
Badges: 19
Rep:
?
#6
Report 3 years ago
#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 :rofl:
1
reply
etothepiiplusone
Badges: 14
Rep:
?
#7
Report 3 years ago
#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.
1
reply
Pastelx
Badges: 18
Rep:
?
#8
Report 3 years ago
#8
Name:  9D719863-ABB9-4184-9907-FF1E94A3D32A.jpg.jpeg
Views: 1276
Size:  17.8 KB
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!
5
reply
_gcx
Badges: 21
Rep:
?
#9
Report 3 years ago
#9
I'll review what's in AS maths and see if I can put anything together.
1
reply
usfbullz
Badges: 14
Rep:
?
#10
Report 3 years ago
#10
A.
Spoiler:
Show


k=-3. Just some substitution.




B.
Spoiler:
Show


r=-12



C.
Spoiler:
Show


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



D.
Spoiler:
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)
1
reply
RDKGames
Badges: 20
Rep:
?
#11
Report 3 years ago
#11
I just came up with this randomly:

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

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

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

(iii) Hence, determine the ratio into which the triangle ABC splits the area of \mathcal{C}
5
reply
usfbullz
Badges: 14
Rep:
?
#12
Report 3 years ago
#12
@Pastelx is it
Spoiler:
Show

D

?

I'm terrible at circles/loci in complex plane.
0
reply
thotproduct
Badges: 19
Rep:
?
#13
Report 3 years ago
#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:
Show

Let's consider our polynomial  x^3 + kx^2 - 7x - 15. Here, we have 2 crucial bits of information to deduce k, we know that when  x = -1, f(-1) = r also,  for x = 3, f(3) = 3r

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):

 k - 9 = r

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

 9k - 9 = 3r

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

 3(k-9 = r) = 3k - 27 = 3r

By subtracting equation one from equation two we get

 -6k - 18 = 0,

thus  -6k = 18, thus k = -3 .

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  r = -12 , 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  x^3 - 3x^2 - 7x - 15 . 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  x^2 + 2x + 3 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.
4
reply
_gcx
Badges: 21
Rep:
?
#14
Report 3 years ago
#14
Perhaps.

Using the binomial theorem, prove from first principles that \displaystyle \frac{\mathrm d(x^n)}{\mathrm dx} = nx^{n-1} for n \in \mathbb Z^+_0. Justify why this method cannot be used to prove the derivative for all n \in \mathbb R.
1
reply
thotproduct
Badges: 19
Rep:
?
#15
Report 3 years ago
#15
(Original post by Pastelx)
Name:  9D719863-ABB9-4184-9907-FF1E94A3D32A.jpg.jpeg
Views: 1276
Size:  17.8 KB
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:
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 :rofl:
0
reply
Pastelx
Badges: 18
Rep:
?
#16
Report 3 years ago
#16
(Original post by usfbullz)
@Pastelx is it
Spoiler:
Show



D



?

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


Not quite, I'm afraid
It definitely helps if you draw a diagram
If you got D you have probably already put it into
 (x-a)^2 + (y-b)^2 = r^2
But now try considering the line from the centre to the circle to the origin

0
reply
Pastelx
Badges: 18
Rep:
?
#17
Report 3 years ago
#17
(Original post by AryanGh)
Spoiler:
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 :rofl:
Spoiler:
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!
0
reply
usfbullz
Badges: 14
Rep:
?
#18
Report 3 years ago
#18
(Original post by Pastelx)
Spoiler:
Show








Not quite, I'm afraid
It definitely helps if you draw a diagram
If you got D you have probably already put it into
 (x-a)^2 + (y-b)^2 = r^2
But now try considering the line from the centre to the circle to the origin







Spoiler:
Show





Oops, went too quickly. Correct answer is C. \sqrt {(-3-0)^2+(-4+0)^2} = 5




0
reply
Pastelx
Badges: 18
Rep:
?
#19
Report 3 years ago
#19
(Original post by usfbullz)
Spoiler:
Show






Oops, went too quickly. Correct answer is C. \sqrt {(-3-0)^2+(-4+0)^2} = 5





Spoiler:
Show

Yep correct!
1
reply
username3555092
Badges: 14
Rep:
?
#20
Report 3 years ago
#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 2^{\sin^2{x}}+2^{\cos^2{x}}=2 have in the range 0 \leqslant x < 2\pi? Justify your answer.
2
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Back
to top
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

Have you experienced financial difficulties as a student due to Covid-19?

Yes, I have really struggled financially (20)
13.16%
I have experienced some financial difficulties (42)
27.63%
I haven't experienced any financial difficulties and things have stayed the same (63)
41.45%
I have had better financial opportunities as a result of the pandemic (23)
15.13%
I've had another experience (let us know in the thread!) (4)
2.63%

Watched Threads

View All
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise