For Q1 you can use the factor theorem. You know (x+3) is a factor, so subbing in x=-3, will mean the function is equal to zero and you can solve for a.
@@cpt.battlecock5264: Unfortunately learned that the hard way when I first taught a kid A Level maths after a long time of not needing to look at a paper. A reminder that being able to do something is completely different to being able to teach it...
A-Level was pretty good - I'd recommend doing an A-Level further maths paper if you were to do another one of these. I did OCR maths and further maths but much rather would've done Edexcel. Happy new year daddy
I can’t speak for further maths I don’t take it but for regular a level maths OCR isn’t that bad and there’s this myth that some exam boards are way harder than others and they aren’t at all
Papa is evolving, once he finishes doing a Further Math exam he should do a Checkpoint paper, they're the hardest edit: in all seriousness though it'd be fun to see an expert solve a Further Math paper
@@jellybabiesarecool4657 in my first year of exdecel further right now, on my mocks I barely scraped an A, not gonna lie it’s definitely a step up I’m used to cruising through maths
my god.. the 2019 Edexcel exam was the exact one I did for my A Levels, it's so weird seeing it again after nearly 3 years. I can still remember what it was like doing some of these questions. quite bittersweet since I'm now near the end of my undergrad degree in Maths!
@@oneinabillion654 well, covid made the whole experience a bit unconvential.. but i've genuinely enjoyed everything i've learned. after graduating, i'll be on a grad scheme at an investment bank.
@@nerevarchthn6860 well yesnt u can just take an abitur exam which is meant for a similar age group from what i understand A-levels is 16+ kinda and usually people finish abitur while 17+ish and its the last certificate before university for both systems if i understand correctly
@@nerevarchthn6860 well for German schools Abitur is the norm before u study, fachabitur and other stuff is only really used for applied stuff for general academic comparisons Abitur should be right small edit: its not like u need A levels to become a baker the same way u wouldn't need Abitur for the profession
I literally hated maths just a few months ago because I found it to be too boring. Now I realize, I just needed to watch Papa Flammy to know how cool maths is :) Thanks a lot for helping me see how cool maths is.
I took this exam. It was the first exam with the new syllabus and we knew nothing about what the paper would be like. Everyone did terribly I remember people were upset after the exam. Loads of people complained and the grade boundaries were super low. Luckily I scraped an A and managed to get onto my first choice course at uni. Defo was a hard set of exams though that year
Review : Question 1 : (Inefficiency) Use Factor theorem, input x=-3 and equate to 0. Question 2a : (Optimal) Question 2b : (Blunder?) My student senses would tingle that solving a question can't be as easy as a linear, making me go to a higher accuracy to a quadratic. But it was pretty unclear. Question 3a : (Inefficiency) Reduce rational to simplest terms, realize numerator is 5(x+1)^2 - 5 Question 4a : Formula remembrance ¯\_(ツ)_/¯ Question 4b : (Optimal) Question 5a : (Optimal) Question 5b : (Optimal) Question 5c : To avoid these mistakes, it's best to write the whole sets of ranges as you are composing functions ( g(x) ∈ [7,∞), 21/g(x) ∈ (0,3] ) Question 6a : (Optimal) I'll be honest I would have missed θ=0 Question 6b : 😐 Question 7a : (Inefficiency) 2^25/5^6 = 2^31/10^6, 2^30 is roughly 1 billion (remember Gigabyte vs. Gibibyte), so 2 billion / 1 million = 2 thousand, pretty close Question 7b : (Optimal) Question 8a : (Optimal) Question 8b : (skipped) Evaluate area function from 0 to b and set equal to -20/3, and obtain 3b^4 - 8 b^3 + 48 b^2 + 40 = 0, proceed from there. Question 9a : (Sub-Optimal) You missed b>0 btw. Also I think examiners would expect you to explain why b=1 doesn't work, and my explanation would be that because log(1) = 0 but 1≠0, this would imply that log(a) = log(a-1), but log is monotone, invertible etc. Question 10i : (Optimal) Question 10ii : (Inefficiency ) You can observe that at 28/3, first is 0, but second is 1/3, but at x=9, first is 1 but second is 0. This method can be realized by trying to imagine the graphs with their intercepts and shapes. Honestly, the blunder in second part (28+9 rather than 28-9) will lose you marks. Question 11a : (Optimal) Question 11b : (Optimal) Question 11c : (Error) You pointed out the a_0 term problem Question 12a : (Optimal) Question 12b : (Optimal) Question 12c : (Error) Question 12d : (Error) you pointed this out Question 13a : (Optimal) Question 13b : (Optimal) Question 13c : (Error, but it's okay) well if this was mcq then bprp's hiding trick would be faster (multiply by the factor whose value you want, sub in a value for x which makes that factor 0). but here showing your work is important, so long form was a good call. Question 14a : (Inefficiency) Bro, at the origin means that y=0 is given, just put that in cos(2y) Question 14bi : (Optimal) Question 14bii : (Error) Idk about the gradient thing, but I think they were asking why both forms give the same derivative near the origin, and the answer to that would be that the small angle approximation from the Taylor series preserves the first derivative near the center point by design. Or one can say that sin(x)/x → 0 in the limit near 0, so derivative is preserved. Also please realize that dy/dx~y implies exponential behaviour. Question 14bii : (Optimal)
A better strategy for the first problem might be that we substitute -3 instead of x as -3 should be a zer9 of the polynomial, it can yield the answer very easily
Great Job Papa! I'm doing my A-levels right now so it was really fun seeing you complete an A-level paper. Like everyone else, I'd suggest doing a Further Maths paper next. But perhaps after that you could try out a STEP paper - these are the exams you must do to study maths at some universities in the UK. They're vile and visicious.
That's not strictly true. The top 5 or so may ask for Step 1, 2 or 3. I studied for Step 1 (although I never sat it) and I am in my 2nd year of university doing BSc mathematics at a good UK university (top 10 for maths).
Maybe try a Singapore H3 Mathematics paper someday? Singapore's H2 A level papers are analogous to UK As (but typically considered harder), while H3 papers are typically reserved for those intending to major in a subject. Would be fun to see how you fare
Yooo I sat this exam back then and I'm now an electronic engineering student! I remember the outrage afterwards when my classmates and I found out it got leaked. Giving me a weird sense of nostalgia for A Levels! Happy new year! :)
Very impressive that you can improvise all of that Papa. Love your solutions and your methods, as I'm not good at maths but I still admire good solutions. In the meantime, happy new years eve Papa ❤ 👏
I like seeing you actually make small mistakes after watching you do insane junk in most videos. It reminds me that some of my mistakes are just human error, regardless of skill or study.
UK Edexcel A-Level Maths teacher here. Q4 - We teach the Maclaurin's series expansion for A-Level Maths. (a + bx)^n = a^n(1 + bx/a)^n = a^n[1 + n(bx/a) + n(n - 1)((bx/a)^2)/2! + ... You'd have to factor out 4^-0.5 first and expand (1/2)(1 - x/4)^(-1/2) using Maclaurin expansion.
I have a more effective method for like question number 1, we all know one of the factors of that polynom is (x+3), so x=(-3). Then subtitute (-3) to the function, and then you'll get the a = 3
Yeah that definitely makes a lot of sense. f(x) = (x+3)g(x) g being the remaining factor, f(-3) =(-3+3)g(-3) => f(-3) = 0 So 0 = 3(-3)^3 + 2a(-3)^2 - 4(-3) + 5a => a = 3
I know the feeling. I did my A-levels last year. I really don't think they will cancel the real A-levels though as it would mean some people would leave school without sitting any formal exam!
I remember the 2019 one was when the structures of the paper changed and no one knew so everyone did terribly. They had to make the grade boundaries so low. If I can remember correctly it was like 55 percent for an A*
You should try the IBDP Further Maths HL. IB Maths HL (not further maths) is comparable to the A Level Further Maths, but IB Further Maths is on a whole other level (which is why only 200 people take it each year), and has been discontinued. Graph theory, number theory, advanced stats and calculus , linear algebra, sets, relations etc It was specifically designed for students planning to major in pure maths.
Try the Edexcel further pure 2 exam. It's pretty good. Its got basic group theory, some number theory, some good old fashioned calc that u have done on this channel as well as some quite challenging discrete recurrence relations stuff that if u haven't done in I while might be a challenge since it requires a bit of memorising a technique. But overall my top pick for Ur next exam!
If you want to remember the quotient rule then use nancipis phrase. "LoDHi minus HiDLo over LoLo". Lo is the function on the bottom, Hi is the function on the top, and D means the derivative of. So it's just bottom function multiplies by derivative of the top function minus the top function multiplied by the derivative of the bottom function, all divided by the bottom function multiplied by the bottom function.
Try the IB Math Analysis and Approaches HL paper next! Would be fun to see how you handle it. The May 2021 one was the first exam and had some interesting inductive proof.
20:30 You would be correct, with Edexcel then Taylor series expansion is only covered in further maths (as part of further pure 1) so you wouldn't be expected to use it there.
For 2b, we learn that the small angle approximation to be used in exams is 1-((x^2)/2). Of course it can be simplified to one, or more accurate. Perhaps the question specifying 3 d.p would tell you that exactly 0.25 would not be a correct answer.
A word of warning if you try to do STEP papers, they are sadistically hard, and a lot of the questions can make you go 'WTF How do I even get started on this ****?!' You can do any 6 questions on a paper, with an average of 30 minutes each for a 3-hour exam. Each question is worth 20 marks and you can't get extra marks by doing more than 6 questions. The first two questions are usually designed to be more 'accessible', but are still just ridiculously hard.
The first question can be solved as x=-3 3(-3)³+2a(-3)²-4(-3)+5a = 0 a = 3 Edit:- Zero of (x+3) is -3. As (x+3) is a factor or f(x) which means that, f(-3) = 0
For 10 (ii) I would sketch the graphs of both functions, and look for intersections using the graph and by solving appropriate equations. Based on the results (what's on the graph including the marked co-ordinates of intersections), you can figure out whether the statement is always, sometimes, or never true.
Interesting how you expand all the numerator in the partial fractions - we were taught to use the “x” that would reduce it to a single coefficient - for example use “x=-3” would give 24 = B(-6-4) and repeat.
@@PapaFlammy69 no don’t take it in any way, I was more interested in how other countries taught maths and how you in particular teach different topics. As long as you get to the same correct answer it does not matter how we get there (as long as you show working ;) )
10 (ii) |3x-28|≥0 for any x; x-9 can be either greater or less than 0, and in the latter case less than |3x-28|. But at x=28/3: x-9 =1/3 > |3x-28|=0. Therefore, the statement taken in quotation marks is SOMETIMES TRUE. This is the answer. In my opinion, the problem condition does not require any additional calculations.
1) Overcomplicated problem 1. Just use remainder theorem (remainder = 0). f(-3)=0, solve for a 8) What happened to 8b?? Just set absolute value of integral from 0 to b equal to 20/3. 9) For log question, just using b/=1 and b/=a with a/=0 is more general. ("/=" means "not equal to")
For 8b), bounded by x=b means that’s your upper integration limit so you needed to evaluate int_0^b x(x+2)(x-4) dx which gives you the desired result :P
Give the first 3 terms of the binomial expansion of... "I'm going to do a Taylor Series expansion" lél 5 minutes later... "I'm probably doing it completely wrong..."
Ahhh, fond memories. This was the paper that I was doing for my A-levels. (The stats paper was a bitch) I'm now doing a placement year for my chemistry degree!
You could do : maths D Ens (most difficult math exam for future professors in universities and « CPGE » considered as the future elites of engineer and professors
For exercise 7 b) you might not have done so many calculations your model is V(t)=20000*(4/5)^t. V(t=10)=2000 replace (4/5)^t=x So V(t=10)=20000*x=2000 x=0.1 now check if x is equal to (4/5)^10=0.107 in order to see if the model is right.
@@josephbrennan370 Because it was difficult, although it didn’t help that I only took A levels maths as an extra (My country follows a different curriculum but my school happened to offer a levels for enrichment purposes) and thus only got taught it for 2 hours a week.
I would recommend the OCR A Level Further Maths - Pure Core Papers! (But the one that I think would be very fun would be the 'Additional Pure' papers on the Further Maths OCR as this paper really stretches the ceiling for secondary school maths)!
1. f(-3)=0 , would be faster 3. In fact adding and subtracting 5 make some nice cancellations and then differentiation will be a lot easier 4. Long division and taking square root could be applied (like for numbers) or taking square root and then derivative sqrt(4-x)=2 - x/4 - x^2/64 - x^3/512 -4 -x | (4 + (-x/4))(-x/4) -(-x+x^2/16) -x^2/16 | (4 - x/2 + (-x^2/64))(-x^2/64) - (-x^2/16+x^3/128+x^4/4096) -x^3/128-x^4/4096 | (4 - x/2 - x^2/32 - x^3/512)(-x^3/512) d/dx (sqrt(4-x)) = 1/(2*sqrt(4-x))*(-1) d/dx (sqrt(4-x)) = -1/2*1/sqrt(4-x) 1/sqrt(4-x) = -2 (d/dx (sqrt(4-x))) -2(-1/4-1/32x-3/512x^2) =1/2+1/16x+3/256x^2 You forgot about chain rule when iyou calculated derivatives for Taylor series expansion
For the one you couldnt do, I think it was 8b, can you not integrate between 0 and b as you know that the area will be equal to (20/3) as proven in A? And then from some manipulation Im sure you could somehow find that equation, although I haven't tried. I did my A levels a little while ago with what effectively was half of a further maths course as well, and I know how stumped you can get so quickly even when its Maths you would certainly know how to do. The wording is also weird in these papers for native English speakers, and while you are fluent I imagine its only going to be harder.
Area R2 = ∫(from 0 to b)[0-x(x+2)(x-4)]dx =-b^4/4 +2b^3/3 +4b^2. By condition, area R2 = area R1 =20/3. -b^4/4 +2b^3/3 +4b^2=20/3 => 3b^4-8b^3-48b^2+80=0 The very condition of the problem gives us a hint. Divide the polynomial, which is on the left side of the equation, by (b+2)^2 = b^2 +4b+4b and we get another multiplier 3b^2 -20b+20 (without remainder).
Q5 is "Completing the Square". Sub in x = -1 then y = 7 that's the TP. One of the benefits of completed square form. And when x = 0, y=7 that's the y-intercept. c) f(x-2) = 2(x-2)^2 + 4(x-2) + 9 and it turns out that g(x) = f(x-2)+2 so it's translation with vector 2 right, 2 up Not sure the numbers are correct but that's the method
For the binomial expansion we are given a formula, 1 + nx + [n(n-1)/2!]x^2 + [n(n-1)(n-2)/3!]x^3 + .... (not written exactly if you have a coefficient of x that also gets taken to the power I put on x)
Bro, you did so well considering you haven't studied most of this stuff in so many years (especially if you originally studied this stuff in German?). I remember when I started teaching A Level, I had just finished my MSC and couldn't remember anything from this.
awesome video! love to see you try out a vce math methods exam from down under! I know it made my life pain! (btw units 1,2 are for 17 y/o and 3,4 for 18)
Got an idea for maths gone wrong done right, how about something like all (x - 69)^3 which equal x^3 - 69^3 . Basically breaking exponential expansion laws
For Q1 you can use the factor theorem. You know (x+3) is a factor, so subbing in x=-3, will mean the function is equal to zero and you can solve for a.
oh, nice :)
What I was thinking
When youre so good at maths, You gotta do it the hard way. Goddamn.
@@cpt.battlecock5264: Unfortunately learned that the hard way when I first taught a kid A Level maths after a long time of not needing to look at a paper. A reminder that being able to do something is completely different to being able to teach it...
Can someone explain why Poly Diviviosion?
A-Level was pretty good - I'd recommend doing an A-Level further maths paper if you were to do another one of these. I did OCR maths and further maths but much rather would've done Edexcel. Happy new year daddy
What's the difference between Edexcel and ORC further Math?
@@loneranger4282 OCR is torture. OCR MEI further maths is greater torture. Edexcel is more modular and simple. Also, lined paper
My school takes MEI A level maths and further maths
@@yondabigman4668 Why is OCR MEI torture?
I can’t speak for further maths I don’t take it but for regular a level maths OCR isn’t that bad and there’s this myth that some exam boards are way harder than others and they aren’t at all
It's so cool to see someone from the other side of the world solving the same problems you do.
germany is lit 2 countries away lol
@sir pill I live in Cyprus and take a level exams. Still not the other side of the World but looks pretty far to me.
Right, all the way in germany, an hour away😅
I love how you said the Quotient rule is just a product of the product rule
:D
Papa is evolving, once he finishes doing a Further Math exam he should do a Checkpoint paper, they're the hardest
edit: in all seriousness though it'd be fun to see an expert solve a Further Math paper
Yh. He should either do further maths, STEP/MAT or UKMT papers
Yes I'm struggling with Edexcel further maths and I need Papa Flammy to show me the way.
@@jellybabiesarecool4657 in my first year of exdecel further right now, on my mocks I barely scraped an A, not gonna lie it’s definitely a step up I’m used to cruising through maths
I'd love to see you try a STEP III Exam. This is what students have to sit to qualify for studying mathematics at Cambridge.
I cried when doing that paper 💀🤣
Ooh I dread if I ever want to do one of those... A level is hard enough
my god.. the 2019 Edexcel exam was the exact one I did for my A Levels, it's so weird seeing it again after nearly 3 years. I can still remember what it was like doing some of these questions. quite bittersweet since I'm now near the end of my undergrad degree in Maths!
How has ur journey been in uni? What plans do u have after u graduate?
@@oneinabillion654: Seconded, especially with the whole lockdown shit!
@@oneinabillion654 well, covid made the whole experience a bit unconvential.. but i've genuinely enjoyed everything i've learned.
after graduating, i'll be on a grad scheme at an investment bank.
@@teacup2301 niceee
Hi if you don’t mind me asking 1) what uni are you at 2) what a level options did u pick?
do a german high school exam next that would be really cool, i really want to see the comparison between uk a levels and german tests.
germany doesnt have high school so its not easy to compare
@@nerevarchthn6860 well yesnt
u can just take an abitur exam which is meant for a similar age group
from what i understand A-levels is 16+ kinda
and usually people finish abitur while 17+ish
and its the last certificate before university for both systems if i understand correctly
@@officer_baitlyn it isnt it entirely depends on your school and whether to leave after tenth grade the system and exams are compeltely different
@@nerevarchthn6860 well for German schools Abitur is the norm before u study, fachabitur and other stuff is only really used for applied stuff
for general academic comparisons Abitur should be right
small edit:
its not like u need A levels to become a baker
the same way u wouldn't need Abitur for the profession
I literally hated maths just a few months ago because I found it to be too boring. Now I realize, I just needed to watch Papa Flammy to know how cool maths is :) Thanks a lot for helping me see how cool maths is.
I took this exam. It was the first exam with the new syllabus and we knew nothing about what the paper would be like. Everyone did terribly I remember people were upset after the exam. Loads of people complained and the grade boundaries were super low. Luckily I scraped an A and managed to get onto my first choice course at uni. Defo was a hard set of exams though that year
Review :
Question 1 : (Inefficiency) Use Factor theorem, input x=-3 and equate to 0.
Question 2a : (Optimal)
Question 2b : (Blunder?) My student senses would tingle that solving a question can't be as easy as a linear, making me go to a higher accuracy to a quadratic. But it was pretty unclear.
Question 3a : (Inefficiency) Reduce rational to simplest terms, realize numerator is 5(x+1)^2 - 5
Question 4a : Formula remembrance ¯\_(ツ)_/¯
Question 4b : (Optimal)
Question 5a : (Optimal)
Question 5b : (Optimal)
Question 5c : To avoid these mistakes, it's best to write the whole sets of ranges as you are composing functions ( g(x) ∈ [7,∞), 21/g(x) ∈ (0,3] )
Question 6a : (Optimal) I'll be honest I would have missed θ=0
Question 6b : 😐
Question 7a : (Inefficiency) 2^25/5^6 = 2^31/10^6, 2^30 is roughly 1 billion (remember Gigabyte vs. Gibibyte), so 2 billion / 1 million = 2 thousand, pretty close
Question 7b : (Optimal)
Question 8a : (Optimal)
Question 8b : (skipped) Evaluate area function from 0 to b and set equal to -20/3, and obtain 3b^4 - 8 b^3 + 48 b^2 + 40 = 0, proceed from there.
Question 9a : (Sub-Optimal) You missed b>0 btw. Also I think examiners would expect you to explain why b=1 doesn't work, and my explanation would be that because log(1) = 0 but 1≠0, this would imply that log(a) = log(a-1), but log is monotone, invertible etc.
Question 10i : (Optimal)
Question 10ii : (Inefficiency ) You can observe that at 28/3, first is 0, but second is 1/3, but at x=9, first is 1 but second is 0. This method can be realized by trying to imagine the graphs with their intercepts and shapes. Honestly, the blunder in second part (28+9 rather than 28-9) will lose you marks.
Question 11a : (Optimal)
Question 11b : (Optimal)
Question 11c : (Error) You pointed out the a_0 term problem
Question 12a : (Optimal)
Question 12b : (Optimal)
Question 12c : (Error)
Question 12d : (Error) you pointed this out
Question 13a : (Optimal)
Question 13b : (Optimal)
Question 13c : (Error, but it's okay) well if this was mcq then bprp's hiding trick would be faster (multiply by the factor whose value you want, sub in a value for x which makes that factor 0). but here showing your work is important, so long form was a good call.
Question 14a : (Inefficiency) Bro, at the origin means that y=0 is given, just put that in cos(2y)
Question 14bi : (Optimal)
Question 14bii : (Error) Idk about the gradient thing, but I think they were asking why both forms give the same derivative near the origin, and the answer to that would be that the small angle approximation from the Taylor series preserves the first derivative near the center point by design. Or one can say that sin(x)/x → 0 in the limit near 0, so derivative is preserved. Also please realize that dy/dx~y implies exponential behaviour.
Question 14bii : (Optimal)
Happy new year!
You too Varad, thank you for the breakdown
A better strategy for the first problem might be that we substitute -3 instead of x as -3 should be a zer9 of the polynomial, it can yield the answer very easily
Great Job Papa!
I'm doing my A-levels right now so it was really fun seeing you complete an A-level paper.
Like everyone else, I'd suggest doing a Further Maths paper next. But perhaps after that you could try out a STEP paper - these are the exams you must do to study maths at some universities in the UK. They're vile and visicious.
yup!!! :)
That's not strictly true. The top 5 or so may ask for Step 1, 2 or 3. I studied for Step 1 (although I never sat it) and I am in my 2nd year of university doing BSc mathematics at a good UK university (top 10 for maths).
@@guyguy1811 Yep that's why i said "some universities"
Maybe try a Singapore H3 Mathematics paper someday? Singapore's H2 A level papers are analogous to UK As (but typically considered harder), while H3 papers are typically reserved for those intending to major in a subject. Would be fun to see how you fare
how would you be able to find these papers online tho
Yooo I sat this exam back then and I'm now an electronic engineering student! I remember the outrage afterwards when my classmates and I found out it got leaked. Giving me a weird sense of nostalgia for A Levels! Happy new year! :)
Haha :D You too
Id recommend doing a step 2 or 3 paper if you want it to be more of a challenge
we'll see what the new year will bring us :)
@@PapaFlammy69 don't do step unless you enjoy pain
@@iain8938 step is the most fun exam
Very impressive that you can improvise all of that Papa. Love your solutions and your methods, as I'm not good at maths but I still admire good solutions. In the meantime, happy new years eve Papa ❤ 👏
you too Sergio
Extremely excited to watch this
:))
I like seeing you actually make small mistakes after watching you do insane junk in most videos. It reminds me that some of my mistakes are just human error, regardless of skill or study.
I am in no way qualified to understand any of this maths but i'm watching this anyway
UK Edexcel A-Level Maths teacher here.
Q4 - We teach the Maclaurin's series expansion for A-Level Maths. (a + bx)^n = a^n(1 + bx/a)^n = a^n[1 + n(bx/a) + n(n - 1)((bx/a)^2)/2! + ...
You'd have to factor out 4^-0.5 first and expand (1/2)(1 - x/4)^(-1/2) using Maclaurin expansion.
thx! :)
Love these videos
I have a more effective method for like question number 1, we all know one of the factors of that polynom is (x+3), so x=(-3). Then subtitute (-3) to the function, and then you'll get the a = 3
thank you
For question 1 since x+3 is a factor, x=-3 is a root so just plug that in and you can immediately solve for a :)
Yeah that definitely makes a lot of sense. f(x) = (x+3)g(x)
g being the remaining factor, f(-3) =(-3+3)g(-3) => f(-3) = 0
So 0 = 3(-3)^3 + 2a(-3)^2 - 4(-3) + 5a => a = 3
It would be quite based if you did the MAT and STEP 1,2,3 too.
we'll see what the new year will bring us :)
@@PapaFlammy69 try 2015 step 3 q1. A very cool integral
If you want another challenge, consider trying out the MAT - it's the mathematics entrance exam for Oxford University
Him calling the quotient rule stupid was the highlight of the video
:D
i don't understand shit that you nare doing, but it's cool to watch
Good timing, since UK A-Level students might have to do "mock" exams since our real A-Levels might be cancelled
damnnn
I know the feeling. I did my A-levels last year. I really don't think they will cancel the real A-levels though as it would mean some people would leave school without sitting any formal exam!
I remember the 2019 one was when the structures of the paper changed and no one knew so everyone did terribly. They had to make the grade boundaries so low. If I can remember correctly it was like 55 percent for an A*
wtf o.O
You should try the IBDP Further Maths HL. IB Maths HL (not further maths) is comparable to the A Level Further Maths, but IB Further Maths is on a whole other level (which is why only 200 people take it each year), and has been discontinued. Graph theory, number theory, advanced stats and calculus , linear algebra, sets, relations etc
It was specifically designed for students planning to major in pure maths.
Haha SL math gang
Try the Edexcel further pure 2 exam. It's pretty good. Its got basic group theory, some number theory, some good old fashioned calc that u have done on this channel as well as some quite challenging discrete recurrence relations stuff that if u haven't done in I while might be a challenge since it requires a bit of memorising a technique. But overall my top pick for Ur next exam!
14:40 i appreciate this bit so much, i hate the quotient rule lmao
:D
Do that further maths. And STEP. I would ask you to also do the Oxford PAT but I never want to see those 3 letters in a row again.
If you want to remember the quotient rule then use nancipis phrase. "LoDHi minus HiDLo over LoLo". Lo is the function on the bottom, Hi is the function on the top, and D means the derivative of. So it's just bottom function multiplies by derivative of the top function minus the top function multiplied by the derivative of the bottom function, all divided by the bottom function multiplied by the bottom function.
Nah, I don't want to remember it on purpose^^ for a lot of applications it's actually more useful to not bring it onto the common denominator :)
The beauty of maths is the fact you don't need to memorise a lot if you know how to derive it 😉
@@fetchstixRHD that's number one the best part about maths, that's what makes it so beautiful and perfect.
Try the IB Math Analysis and Approaches HL paper next! Would be fun to see how you handle it. The May 2021 one was the first exam and had some interesting inductive proof.
I’m taking this exam in June, same exam board too
We got a month or two bro 😭😭
20:30 You would be correct, with Edexcel then Taylor series expansion is only covered in further maths (as part of further pure 1) so you wouldn't be expected to use it there.
kk
Isn't it covered in core pure 2?
@@jellybabiesarecool4657 No, it's in further pure 1
For the first question, you could also have just plugged in x=-3 and set it equal to zero :)
In these paper small angle approximation are:
Sinx=x, cosx=1-((x^2)/2), Tanx=x
Good vid again man happy new year 💷👌🏼👍🏼
like wtf, how can, by this logic, tan(x) be approximated as x then?! So stupid :D
@@tddupaid ye, I could see that for the binomial expansion lol xD At least I was able to derive it by the Taylor series expansion :'D
@@PapaFlammy69 is there something wrong with using these approximations?
For 2b, we learn that the small angle approximation to be used in exams is 1-((x^2)/2). Of course it can be simplified to one, or more accurate. Perhaps the question specifying 3 d.p would tell you that exactly 0.25 would not be a correct answer.
for the first question since x+3 is a factor then f(-3) = 0 and then you can very easily find ‘a’ from there
I want to see flammy write a pure physics exam and dotson a pure math exam lol
A word of warning if you try to do STEP papers, they are sadistically hard, and a lot of the questions can make you go 'WTF How do I even get started on this ****?!' You can do any 6 questions on a paper, with an average of 30 minutes each for a 3-hour exam. Each question is worth 20 marks and you can't get extra marks by doing more than 6 questions. The first two questions are usually designed to be more 'accessible', but are still just ridiculously hard.
Can't wait to have to do these papers. Already want to die after the mat
@@user-ox7uw2lu8d lol no. Happy I didn't. Wouldn't have managed there. All the preparation for step was just depressing
The first question can be solved as
x=-3
3(-3)³+2a(-3)²-4(-3)+5a = 0
a = 3
Edit:- Zero of (x+3) is -3.
As (x+3) is a factor or f(x) which means that, f(-3) = 0
why choose x = -3 out of all real numbers, seems like you have learnt the question
@@harrypathak3935 ?
He didn't randomly choose -3. Its basic quotient rule, search it up
@@sandarbh2764 yeah
@@harrypathak3935 just take the zero of the polynomial (x+3).
For 10 (ii) I would sketch the graphs of both functions, and look for intersections using the graph and by solving appropriate equations. Based on the results (what's on the graph including the marked co-ordinates of intersections), you can figure out whether the statement is always, sometimes, or never true.
great idea! Honestly, didn't think about this option XD
Interesting how you expand all the numerator in the partial fractions - we were taught to use the “x” that would reduce it to a single coefficient - for example use “x=-3” would give 24 = B(-6-4) and repeat.
hm, okay lol No one ever taught me this stuff, it's all self taught, so my methods may not be the most efficient ones :)
@@PapaFlammy69 i don't understand aren't you a teacher?
@@PapaFlammy69 no don’t take it in any way, I was more interested in how other countries taught maths and how you in particular teach different topics. As long as you get to the same correct answer it does not matter how we get there (as long as you show working ;) )
@@malygos5616 What does this have to do with being a teacher? PFD is not being taught in Germany
1st question , u can use factor theorem as f(-3)=0 and solve the constant a.
“Ted talk, I thank you very much for listening to my sh*t” - nice sentence 👍
10 (ii)
|3x-28|≥0 for any x; x-9 can be either greater or less than 0, and in the latter case less than |3x-28|.
But at x=28/3: x-9 =1/3 > |3x-28|=0. Therefore, the statement taken in quotation marks is SOMETIMES TRUE. This is the answer. In my opinion, the problem condition does not require any additional calculations.
1) Overcomplicated problem 1. Just use remainder theorem (remainder = 0). f(-3)=0, solve for a
8) What happened to 8b?? Just set absolute value of integral from 0 to b equal to 20/3.
9) For log question, just using b/=1 and b/=a with a/=0 is more general. ("/=" means "not equal to")
YESS THIS IS THE ONE I'VE BEEN WAITING FOR
:)
there is one british exam called the UKMT BMO1, you should try it (: - it is for 15-18 year olds
how many exams do you guys have
@@epicmorphism2240 that's just a special exam to go to our national maths team, only 1000 people per year do it
@@domdj9476 i did bmo 2 :)
:(
@@youtubeaccount1718 bmo2 is next term?
@@youtubeaccount1718 I'm in year 10 and I just got to senior kangaroo
I like your funny words magic man
Yassss! \o/ Thoroughly enjoying this. These questions are so much fun to do, and watch. I feel like I'm watching a gameplay of my fav game.
:))
For 8b), bounded by x=b means that’s your upper integration limit so you needed to evaluate int_0^b x(x+2)(x-4) dx which gives you the desired result :P
ohhhh
Give the first 3 terms of the binomial expansion of...
"I'm going to do a Taylor Series expansion" lél
5 minutes later...
"I'm probably doing it completely wrong..."
I guess a Japanese high school graduation math exam is a fitting target for you)
Yeah, that's me, someone who doesn't understand nothing about what he's doing( Atleast I try) but has fun watching his videos.
Ahhh, fond memories. This was the paper that I was doing for my A-levels. (The stats paper was a bitch) I'm now doing a placement year for my chemistry degree!
Very cool, I did my A Levels in 2021 and I'm now in my first year of Aerospace Engineering.
A fellow chemist I see. I’m going to uni later this year and cannot wait 😎
You could do : maths D Ens (most difficult math exam for future professors in universities and « CPGE » considered as the future elites of engineer and professors
Does that knit cap help in maths? If so -> merch!
it does :p
For exercise 7 b) you might not have done so many calculations your model is V(t)=20000*(4/5)^t.
V(t=10)=2000 replace (4/5)^t=x
So V(t=10)=20000*x=2000
x=0.1 now check if x is equal to (4/5)^10=0.107 in order to see if the model is right.
Q1a f(-3) = 0 (Factor Theorem) and solve it that way.
Happy new year, I am always happy to see your new upload
you too
Try AP Calculus BC or AP Physics C (both parts: Mechanics and Electricity and Mangnetism).
Ah a levels. The bane of my existence until a year ago or so
:D
Why were they the bane of your existence?
@@josephbrennan370 Because it was difficult, although it didn’t help that I only took A levels maths as an extra (My country follows a different curriculum but my school happened to offer a levels for enrichment purposes) and thus only got taught it for 2 hours a week.
doing this in 1 year taking my A level this summer, then I move onto further maths alongside my 2 year courses in OCR Physics and AQA History
I would recommend the OCR A Level Further Maths - Pure Core Papers! (But the one that I think would be very fun would be the 'Additional Pure' papers on the Further Maths OCR as this paper really stretches the ceiling for secondary school maths)!
1. f(-3)=0 , would be faster
3. In fact adding and subtracting 5 make some nice cancellations and then differentiation will be a lot easier
4. Long division and taking square root could be applied (like for numbers)
or taking square root and then derivative
sqrt(4-x)=2 - x/4 - x^2/64 - x^3/512
-4
-x | (4 + (-x/4))(-x/4)
-(-x+x^2/16)
-x^2/16 | (4 - x/2 + (-x^2/64))(-x^2/64)
- (-x^2/16+x^3/128+x^4/4096)
-x^3/128-x^4/4096 | (4 - x/2 - x^2/32 - x^3/512)(-x^3/512)
d/dx (sqrt(4-x)) = 1/(2*sqrt(4-x))*(-1)
d/dx (sqrt(4-x)) = -1/2*1/sqrt(4-x)
1/sqrt(4-x) = -2 (d/dx (sqrt(4-x)))
-2(-1/4-1/32x-3/512x^2)
=1/2+1/16x+3/256x^2
You forgot about chain rule when iyou calculated derivatives for Taylor series expansion
For the one you couldnt do, I think it was 8b, can you not integrate between 0 and b as you know that the area will be equal to (20/3) as proven in A? And then from some manipulation Im sure you could somehow find that equation, although I haven't tried. I did my A levels a little while ago with what effectively was half of a further maths course as well, and I know how stumped you can get so quickly even when its Maths you would certainly know how to do. The wording is also weird in these papers for native English speakers, and while you are fluent I imagine its only going to be harder.
Area R2 = ∫(from 0 to b)[0-x(x+2)(x-4)]dx =-b^4/4 +2b^3/3 +4b^2.
By condition, area R2 = area R1 =20/3.
-b^4/4 +2b^3/3 +4b^2=20/3 => 3b^4-8b^3-48b^2+80=0
The very condition of the problem gives us a hint. Divide the polynomial, which is on the left side of the equation, by (b+2)^2 = b^2 +4b+4b and we get another multiplier 3b^2 -20b+20 (without remainder).
The transformation question wanted you to say the curve is reflected through the y axis
Hope you and your wife enjoy the new movie and tell us in your next video if you two enjoyed watching it
It was great, thank you :31
Great please more!!!🎉🎉🎉🎉🎉
10:30 ahh yes... pi=3 as it should be.
For task 3.
y(x)=(5x^2+10x)/(x+1)^2 = 5 -5/(x+1)^2.
dy(x)dx = 0+ (-2)(-5)/(x+1)^3= 10/(x+1)^3.
nice!
For Q1 if x+3 is a factor doesn't that mean that you can write f(x) as (x+3)P(x) where P(x) is some polynomial. Then you do f(-3)=0 and solve a.
further maths please
We do use Taylor Series in Uk but in A Level Further Mathematics; we are first taught Maclaurin series and then Taylor depending on your modules : D
the first question can be solved much easier: you just need to solve for f(-3) = 0
there are always ways to solve something differently, so yeah
Oh that's really cool!
Yeah I was thinking of this, its called remainder theorem.
Factor theorem is a special version of remainder theorem where the remainder is 0
Question 1: ffs just plug in -3 into f(x), since x=-3 is a zero of f.
Q5 is "Completing the Square".
Sub in x = -1 then y = 7 that's the TP. One of the benefits of completed square form.
And when x = 0, y=7 that's the y-intercept.
c) f(x-2) = 2(x-2)^2 + 4(x-2) + 9
and it turns out that g(x) = f(x-2)+2 so it's translation with vector 2 right, 2 up
Not sure the numbers are correct but that's the method
For the binomial expansion we are given a formula, 1 + nx + [n(n-1)/2!]x^2 + [n(n-1)(n-2)/3!]x^3 + .... (not written exactly if you have a coefficient of x that also gets taken to the power I put on x)
Hey papa flammy in the first question just use the fact that f(-3)=0 (since f(x)=(x+3).g(x)) and then u get the value of a very straightforwardly
You must do a a level further maths test !, I recommend doing any of the pure papers, Fp2 is pne of the hardest 😏 edexell ofc
Bruh, Exercise 9 = Exercise π² (For context, look at the timestamps)
Nice vid, my admission exam for engineering school that I did last summer was quite similar, good memories, happy new year from Belgium
You too Antoine
*But something something infinity*
My favourite line ever
:DDD
Q14b: Use the fundamental theorem of engineering
Do a STEP or a MAT exam next. They're even harder than this.
Bro, you did so well considering you haven't studied most of this stuff in so many years (especially if you originally studied this stuff in German?). I remember when I started teaching A Level, I had just finished my MSC and couldn't remember anything from this.
awesome video! love to see you try out a vce math methods exam from down under! I know it made my life pain! (btw units 1,2 are for 17 y/o and 3,4 for 18)
Do some STEP or MAT papers please! they are Cambridge and Oxford entrance exams!
You should do the AQA A-Level Further Maths exam! More fun!
Soon :p
Good luck with the rest of it. No calculator. No Formula Sheet. And a tendency to make the questions waaaay more difficult than they need to be :-)
Love you man so much
❤
Got an idea for maths gone wrong done right, how about something like all (x - 69)^3 which equal x^3 - 69^3 . Basically breaking exponential expansion laws
if x+3 is a factor then you can do f(-3) = 0 and find ‘a’ from there
this hurts my head
good job fam u should do a IB higher maths aa exam...