Top 10 Hardest Further Maths Questions 💀 • A-Level Further Maths, Core Pure Edexcel
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- Опубліковано 3 чер 2024
- Of course, this is just one method of ranking challenging questions, there are many different methodologies I could have used. But I think you'll agree these are very tough! Which did you find the hardest?
This video contains questions from A-Level Edexcel Further Maths Core Pure Papers from 2019-2022.
If you want the PDFs I used in the video, they're saved here: drive.google.com/drive/folder...
Check out everything else I have to offer on my channel! / bicenmaths
0:00 Intro
1:49 Number 10
4:46 Number 9
7:42 Number 8
10:36 Number 7
14:04 Number 6
18:35 Number 5
22:58 Number 4
26:40 Number 3
30:45 Number 2
34:30 Number 1
![Top 10 Hardest Pure Questions 💀 [UPDATED] • A-Level Maths, Edexcel](http://i.ytimg.com/vi/CtbremWlV8M/mqdefault.jpg)
Exactly what I've been waiting for. I must have searched for this about 5000000000 million times after seeing you make the "Hardest A level Maths Questions" video. Thanks for making this video, you legend!
And you've got the ones I've done for normal maths too! Thank you :)
The answer for number 7 is so much simpler and makes more sense for me as oppose to using roots of unity! Thank youuu
I discovered your channel not two hours ago and I was hoping you'd post this video, the timing could not be better thank you so much
Nice timing! I hope you enjoy it!
Mr Bicen Maths Sir you are an absolute gift to humanity. Thank you so much. This video could not have come at a better time.
You are very welcome!
He’s only gone and done it again!! 🙌🙌🙌 thank you for this video sir great video just a few days before the exam 🤞
I was rushing to get it done before that exam! I feel bad that it wasn't in time for AS exam today, but most of the stuff here is from A2 anyway!
this would be a whole different list if 2023 was included 😂
I think I'd agree here! I couldn't use any 2023 questions as they are all locked on the Edexcel website. But 2021 is really quite something, as well...!
@@BicenMathsHi Sir, if you remember could you please tell me which questions were the hardest on the 2023 papers?
@@Hakman1 I just did a 2023 paper 1 mock exam and question 7 was quite hard. It was a series question but it wasn't standard and required a decent bit of thinking. Question 8 was also fairly hard (matrices) and question 5 was hard (mainly just because I'm not great at vectors).
Apparently, the exam isn't meant to be widely available yet, but this came up when I searched for it:
www.exampaperspractice.co.uk/wp-content/uploads/2306-9FM0-01-A-level-Core-Pure-Mathematics-1-June-2023-pdf.pdf
Paper 1, Q7 and Q8 are really challenging. The matrices one is soooo fiddly.
@@BicenMaths I have the 2023 papers, if you want me I can send them to you
this is soo useful thank you so much
23:05 That question was absolutely brilliant. Not only was it relatively easy but I just enjoyed doing it in general. Though that might be because calculus (and integration in particular) is my favourite area of maths.
For question 7 part b, I modified A, B and C to make it easier to calculate the area
So A was previously 6 + 2i is now 2√10 + 0i, B is now -√10 + i√30 and C is now -√10 - i√30 (did this by calculating the magnitude of A and then set the imaginary part of A to 0 to make A only have a real part to it, then rotated by 2pi/3 radians to find B and C)
Now, to find base and height of ABC, I did abs(-√30) + √30 = 2√30 for the base and 2√10 + abs(-√10) = 3√10 for the height
Then found the area of ABC by using 1/2 * b * h
I've omitted some steps just because writing maths on youtube comments is annoying lol
I just did my further core AS paper this afternoon! So not sure why I’m watching this, but making me glad I dropped to only half of it 😂
Did you do OCR? that last roots of equations question cooked me...
by the time i was finished i didnt have any left over for the last question
@@IRedBerryI don’t worry, it should be fine! no I did edexcel :( but similar in that I was deffo rushing towards the end, usually I have spare time but the paper was harder than the practice ones I’d done 🥲
@@hannah2113 seems most of them were. I think I can pull through on stats and mechanics though. Best of luck to you
@@hannah2113 for reals I haven’t ever ran out of time that much in any other practise or mock paper
Sir, you should do some OCR MEI/OCR A questions. they are the hardest ones IMO
OCR MEI is way easier imo. OCR A is definitely hard though.
@@tombaker1643 MEI differential equations and Vectors outweigh edexcel. however edexcel have harder calculus. OCR A is the hardest though by far
Great video!, Sir will you update and add into your 'exam questions by topic' pdf on your google drive with the new exam paper questions ?
As soon as 2023 are unlocked on the Edexcel website I'll be updating all my PDFs very quickly with them included!
honestly relieved seeing how easy these are
At 3:37, instead of doing algebraic division, I went for a substitution of x = tanθ / 4 and then went about 1/32 ∫ tan^2 θ dθ after reducing the fraction and it lead me to the same answer.
Oh nice, that works really well!
Or v = (1 + 16x²)/32
at 32:00 if you couldn’t get the part A would choosing any number for alpha allow you to pick up marks on the other parts ? thanks
Yes! Not all the marks, but the method ones for sure. Good point!
I do OCR MEI further maths and these edexcel questions seem so much harder! Our core is just one 2 hour 40 min paper but the questions seem a lot more straightforward than any of the ones here ahah.
that's what I'm saying
At 16:06, for the graphical calculators, does exam mode disable the function to sketch polar graphs?
I have the cg50, and no, you can still plot graphs (including polar) in exam mode
@@anon1534 Thank you
Also mr bicen at 6:52 do you normally have to show ur working out for when solving a quadratic when you can just simply put it into calculator and give ur solutions for x
You can do it on the calc unless it says something like 'show using algebra' or 'show that' - then I'd do the extra line of working!
for number 1 you can just do the matrix multiplied by xy and equal it to xy then show that when you form simultaneous equations x and y = 0 so there's no point that is invariant
That only shows there are no invariant points - there are some cases of invariant lines where the points themselves are not invariant, but the lines actually are! So that method wouldn't be awarded marks.
For the first question what do you think about taking the antiderivative of v' to be v=(x^2)/2+c (since this will always be a valid antiderivative since d/dx[c]=0) where you then find that to cancel the top with the bottom of the fraction in the resultant integral you set c=1/16, it skips the alg. division and finding the integral of 1/(a^2+x^2) to be (arctan(x/a))/a?
Wow, I have never seen this approach before! Very cool!
hi sir rlly rlly nice vid, just had a question about 36:55 how can we assume this line goes through the origin as this would surely only work if the +c is 0?
I'm pretty sure that it's the exam board spec that says you only need to consider when c=0. However, it isn't much harder. When you factorise out the x, you are left with some terms with no x, from which you can factorise out a c. This will leave you with the quadratic in m, and also a linear expression for m (eg. c(3m-2)). The solution to my example would be 2/3. Therefore, if a solution to the quadratic in m is the same as the solution to this (2/3) then the line is invarient for any value of c. If not, then clearly c has to = 0 for the whole expression to = 0, therefore the invarient line is only for y=mx. I know this sounds wordy, but it's actually no more difficult just a little more algebra.
For this question though, since you are only trying to prove there aren't any invarient lines, if there are no solutions to the quadratic then there will never be any invarient lines, regardless of whether we account for c. Essentially, c is a special case which may or may not work when we have an invarient line, but accounting for differing values of c will never introduce more invarient lines.
Agreed - thanks for the replies.
11:05 it can also be done simpler by using (6+2i)exp(2i(pi)/3) because multiplication of complex numbers represents a rotation in the complex plane. This is far easier and can just be plugged into the calculator
Nice - thanks for sharing this alternative approach!
gives you horrible numbers unless you have the graphing calculator which gives it in exact form
@@grintzz6728 it's A level so you have a graphical calculator so does just give it exactly.
@@olivia242 not everyone has a graphing calculator though? its optional to have
Hi Sir, great video- thank you! Why can't we use integration by parts twice for question 1, once we reach the 2x^2/1+16x^2 stage? I tried and it doesn't get the answer, but I don't know why it's not a valid method.
Because your v' = (1+16x^2)^-1 which isn't simple to integrate - so perhaps you made the mistake there?
Hello sir thank you.
Can you make one for further stats and further mechanics?
I am not sure I'll have the time this year but I'll do my best!
Sir, for number 8 part a, could you use the cross product to prove it instead by maxing it a matrix?
Yep, that works too!
Thanks so much for the video! for question 7 is it also possible to approach it as a roots of unity question? if you find the roots of unity of z^3=1 then multiply by 6+2i to find the coordinates if B and C?
that does work yeah
Yep!
Thank you so much! I have one question though: how did you know 2x^2 / (1 + 16x^2) is a top heavy fraction that needed division ? 3:29
Because the order of the numerator is 2, and the order of the denominator is 2. If the order of the numerator is greater than or equal to the denominator, then it's top heavy!
The denominator differentiates to a linear term so there is no way that it can "match" the numerator hence needing division.
Could you please make a video on the proof by induction for a differentiation equation question that came up in the 2023 paper?
Can't do videos on 2023 until they're unlocked on the Edexcel website unfortunately!
Hi sir, for the last Q why do you need to use a different symbol for x if it is invariant?
Because it isn't an invariant point - we are saying that the line is invariant, which means all the points on the line stay on the same line, NOT that they necessarily map to exactly the same point.
Hi mr bicen, for the first question when asking for the mean value, how do i know whether my limits are meant to give answers in degrees or radians as i gave my answer in degrees and got it wrong
if it is integration and their are trig values in it it should be on radians
Yep, must be in radians for any calculus!
@@BicenMaths Not strictly correct but as far as A-Level is concerned! The small angle approximations need "ajusting" to work with degrees.
i know youver already done one with alevel pure maths but if you did that for mechanics and stats THAT WOULD BE AMAZING !!
I am hoping to do one... soon!
@@BicenMaths Thank you !!!!!!!
I just took the 2021 CP1 paper today, 1 day before my CP1 exam and I have to say it was a nightmare. I scrapped the timer and just ended up working without a time limit.
Yeah it's a hell-scape of a paper. If something like this happens tomorrow, know that everyone is finding it challenging, and move forward onto the next questions quickly to make sure you get the easier marks that are available. Here's hoping for a relatively nice paper!
Yes. Let’s hope they nice us this year.
Can you make a video like this for AS core pure? That would be amazing!
Something I can add to my list!
Do you think there any particularly challenging AS Further Maths questions? I think the June 2022 AS Paper 1 is still probably the most trickest AS paper I've ever seen. I'm probably not going to spend much more time revising the AS content for next Wednesday, but I just want your take on the AS papers.
There's definitely some challenging questions in AS papers, for sure - and it is so worth revising AS content, as you can see in this video, there's quite a few AS topics (vectors, linear transformations, series - and more series and matrices would be here if 2023 were included too!).
For the last question, could you find the characteristic equation of the matrix and use discriminant to prove no real eigenvectors, so no invariant lines?
Yes I believe that would be accepted!
Thank you so much - this is really fantastic! Quick question: for the invariant lines question, why does it not matter if you don't do the +c? I had thought that it would be incorrect to assume from the start that c=0...
I know! I agree with you! But it's what the mark scheme said, so they obviously (for once) thought they'd be nice!
@@BicenMaths Thanks for confirming! 👍
its because regardless of what c is, if that quadratic in m has no solutions then there will be no invariant lines either way
question, for the last question when u equated the coefficients of x, assuming the question stated to find the invarient lines, u would need to find the value of m by equating the x coefficients in the final equation. i found out that you can also equate the c terms as well to find the value when c = 0 when u factorise it and another value of m. my question is, what would i do for this value of m because i keep doing questions like this but the answer isnt the value of m i got when equating the c coefficients which is confusing
You should look at the x part first to see what values of m give that part as 0 - and then for those values of m, look at the other part of the equation to see what corresponding values of c will make that 0. Then you have them as pairs!
Do u have any tips for how we could prepare for these questions since they are nothing like what we have seen, 2021 2023 and specimen papers are really difficult i cant imagine sitting them the first time round especially with time pressure. Do you think 2024 will be as difficult ?
I do think it will be as hard (at least it easily could be!) - don't over allocate too much time to hard questions - make sure you spend the time being accurate and successful elsewhere. It'd be terrible to lose 3 marks in an easy question and spend 30 minutes trying to get 3 marks on a hard question. It's all a strategy about how to get the most marks possible.
The 2019 question with the complex numbers I would've put in polar form and just shifted the argument round by π/3, but then again maybe that's just because I've forgotten matrices
The trouble is that this doesn't give you exact values (at least I don't think that it does...). Would be interested to see how it works out!
Thank you so much legend
For Number 2 at 30:59 if we couldn't do part a) and just quoted a random value of alpha to use for part b) would we still get marks if we did b) correctly?
Yes, that's a great strategy. You will get (at least) all the method marks that way!
thanks boss
number 4 is beautiful
_I agree. Really hope a ‘tricky’ hyperbolic substitution comes up cause I really enjoy them._
For question 6, why wouldn’t you just multiply by e^(2π/3)i to get the other two complex numbers (roots of unity)?
That works too!
6:52 exactly what I do lol. Make them think I factorised it
You should do some aqa questions they are quite difficult compared to the edexcel questions imo
AQA give out copyright strikes for their exams, which is why you won't see any of theirs on UA-cam!
@@BicenMaths ohh right i see thats so annoying i literally do aqa n theres basically no walkthroughs for papers or anything😭
@@kris2804z litterally me, i wish i did edexcel😭
hi, in your opinion how hard did you think the 2023 core pure papers were? I have recently just done them in mocks and found them very difficult
Paper 1 was crazy crazy. Paper 2 was hard but more expected. 2021 is the hardest I’ve seen, I think.
@@BicenMaths yeah in my opinion the matrix question really stumped me. also do you have any predictions on what type of questions might come up this year on the core pure, eg there wasn’t a really tricky polar coordinates area question last year, do you think it’s likely to come up this year
@@user-ji4op8xn8i I sadly don't think the examiners behave in that way - I wish I knew how they planned things! I am always wary about predictions before any paper has been sat, as it's impossible to know - once Paper 1 has been sat I'll be able to offer some better guidance for what's next. I hope that makes sense! :)
@@BicenMaths yeah that makes sense, edexcel have been very unpredictable recently with some of these questions from more recent papers
Would you please be able to do this but for normal A-Level maths ?
I already have done! ua-cam.com/play/PL0SSkmc4r_BaZX7c_oXZ8LpQo5U4FCRu3.html&si=28zu_cUi4mLCbliS all in this playlist
For the last question, I don’t understand why only the x part has to be proven not equal to zero? When I was doing it I tried to say that one component would have to be the negative of the other for the whole thing to equal zero but there wasn’t any way to solve that. Not sure if that makes sense haha but that’s the one question I can’t really figure out conceptually!
Think of it like a comparing coefficients type problem, like ax + b = cx + d
Clearly here, a=c and b=d
We just have ax + b = 0 in this case, which means that a=0 and b=0 for it to be true (as really it is ax+b = 0x + 0)
In this case, as we have shown that a cannot be equal to 0, it doesn't matter what b is as the whole thing can't equal 0.
The reason we aren't saying 'what values of x could make this true' is because we need it to be true for ALL values of x, as an identity, rather than just a single value that would work. Hope that helps!
at 28:28 you get the formula for the sum with a 1 at the start, whereas the actual sum just starts at 0.5e^itheta. How come this diesn't affect your answer?
Oh yeah I think its an error on your part but because you're only considering the imaginary part you get the same answer.
Does anyone know how to get access to the 2023 papers ?
For Number 7 isn’t it easier to use roots of unity since the question is almost identical to one of the example questions in the textbook ( Example 18 EX1G)
I don't think it is as simple as that, as we want them all in exact form.
@@BicenMaths you can convert the exponential form to complex form for each e^i2kpi/3 as an example e^i2pi/3 is -1/2 + root3/2 i and you can multiply it with 6+2i to get an exact value
Nice - thanks for sharing, of course that makes sense too!
@@BicenMaths Thank you Sir it’s an honor, and I just wanted say your Google drive has been a massive help and I’m sure everyone would agree you’re a life saver!
22:30 wouldnt it be easier to just create a table? it is pretty clear 99800 is pretty small in the scheme of things, (as evident by n=6), so at that rate it is faster than expanding those brackets.
would this get you marks? you are still using the closed form, and you kind of cheat by using the calculator to solve the cubic for you anyway
I don't think that that is allowed in the mark scheme - but would be worth trying if you got stuck with the algebra!
Do you think you'll get full marks on any of the further maths papers
The arrogant version of me says 'yes'! But the realist version of me knows how easy it is to make mistakes, so I think it is highly unlikely! I will be trying my best to score full marks somewhere, I hope!
For #8, I used a different method and i'm confused why it didnt work:
Firstly, we have a point on the plane (3,3,2). I labelled this as B. Since A lies on the line it can be written as (4+t, -5+6t, 2-3t).
To get the shortest distance between the point A and the plane, since B lies on the plane I figured i could simply find the vector AB, which is the line joining point A on the line and point B on the plane. Then, I found the dot product between AB and:
1) The 2 direction vectors of the plane
2) The direction vector of the line
... and for each of these i equated the dot product to 0 (since AB is perpendicular to the line and plane) and then solved for t each time.
However, i ended up getting 3 different values of t - none of which were correct. I don't understand why this didn't work?
This is because you've simplified the problem down to the shortest distance between a point (A) and another point (B), which is not the same as the shortest distance between a point and a plane. We cannot say that the shortest distance would be with coordinate B, it might be to a different point on that plane!
I'm curious as to whether statistically any of the 2023 questions landed in the top 10 & superseded even the 2021 ones?
I did this statistical analysis a while ago before the 2023 data was out, so I can't comment on this I'm afraid!
🔥🔥🔥
@@neofx645 lock in buddy
FM1 edition next? 😃
I might have to do some option modules too!
FM1 is SO hard
None are as bad as the notorious ant question.
For those wondering what this means… ua-cam.com/video/0k6wqPAyZ2E/v-deo.htmlsi=1oARAPk_7Brcqj3r 😂 🐜
na the ant question has no reason being in exercise 1e bro 🤣🤣🤣🤣
anyone know when the 2023 papers will be unlocked?
Your teachers should have it. My teachers sent me copies because they aren’t available for students online unless you have a special code :)
On question #3 b)i) surely a=0.5e^iθ as there is no 1 at the start of the series?
Oh wait just realised it doesn’t matter because 1 isn’t imaginary
12:28 first thing i thought too haha
Can't believe it's already been a year since TotK!
I’m so confused. How was number 1 the worst possible answered question? Where did students go wrong on this one?
They've not asked anything quite like this before, so I think the unusual nature of it meant that many weren't properly prepared!
@@BicenMaths its come up a few times since though, and its on the practice sets etc. however cuz it was on 2023 doubt its on todaays paper
cool chain
1st
every one of these is ez lol