Maths: The Final Perceivable Lap
NO MORE
Always phrase your findings to properly answer the question
Later, use highlighters in statistics questions
PMCC (Product Moment Correlation Coefficient)
Faulty heating elements are independent of one another
Specimen Paper 1
I lost the transcript for what I did, but I have the marks lost written down and can work backwards from
there
1. Substitution of the wrong piece of information. All was well, other than you subbing in the radius
of the circle instead of D with respect to time. I don’t want to lose this mark, please make sure
you didn’t just pluck numbers out of nowhere, ask yourself where it came from, if it’s the
correct value to substitute
2. At least 3-4 marks lost to not writing exact coordinates when you were told to. Make it a habit
to highlight prerequisites, as well as the final requirements of the question such as a ratio of
r:h, or a final volume, etcetera.
3. Idk be very afraid of
, blind…ass..
4. I tend to forget this one, the transformation of two equations into the equation of a line
derivation of a set of vectors…
5. For imaginary numbers, the first order of business is we want to use the GC to determine the final
answer, by subbing all the values and working directly in. If we can get that first, and it appears to
make sense, we are more likely to get the other root without Borne Back Beat Ceaselessly Into
Past
Additionally, unless we are facing a last-resort situation, we don’t want to use w = a + ib, because
this is an inefficient method
6.
is not = arg(2iusinθ), so if you ever find yourself needing to find one or the other, use
this method
7.
If h is small compared to R, it doesn’t mean h+R roughly = R, it means h/R is small enough that
small angle approximation could come into play if you derive (1 + h/R)^n
8. In a differential equation, if there is more than one x^t term present, we should probably simplify
the equation so we can more clearly view its behaviour and eventual asymptote. Also! Often
times in DE or graph sketch, we’re going to see an asymptote somewhere, it probably makes
sense to search it out now than later
9. Cartesian equation for a line involves making the random variable the subject and equating x to y
and to z. Always check for if the vector question is asking for a cartesian equation.
Specimen Paper 2
1. Good for exposure: Quadratic expression means difference must be linear
But if difference is linear, it either has to increase consistently or decrease consistently, it can’t
increase and then decrease, which disproves the question’s suggestion
2.
They were searching for a coefficient of x squared, but while you first expanded e before
expanding that wrt to the power of a, it did not yield the correct answer. Instead, you should have
treated (ax) as (x) and expanded from there
3. How do we prove independence in context?
4. Additionally: Expression for the “mutually exclusive” concept
5. It is possible to form exact inequalities for adjacent binomial expressions. If the question asks you
for an “exact” inequality, this is your mode of action. (the principle is basically 6Cn / 6Cn-1
usually yields an integer, which simplifies the expression)
6. Phrasing for why 2-tail method is preferred:
7. Observe the usage of the keyword “sample mean (variable)”
8. Hypothesis testing: How does the value of population variance affect the final result?
Since we know z, we can also determine a critical value, independent of the features of the
normal distribution. Hence, this will also allow us to form an inequality for sigma.
Learning point is that so long as z is involved, we can find real values and solve for a single
unknown in the sample mean variable
9. What…the fuck?
Prelim A - P1
1. y = -¼ x^2 did not have its entire domain drawn out, because the other curve stopped at x=0 so
you yourself didn’t capture the other curve. As a result, the curve sketched was unfinished.
Before being done with the curve sketch, ask yourself if the GC has deceived you? If the domain
of one of the piecewise curves has led you to exclude the other portion of the curve
2. Rough sketching to take note of intercepts when the curves given have unknowns (like ln(4/x-a))
is a valid strategy. Just a rough sketch to take note of intercepts in terms of unknowns, when the
intersections between curves probably do occur.
3. Firstly, this (below) is good because the concept proven from the previous part (what is (x-a)(x+a)
where a is complex) was used. If you can recognise the previous section’s usefulness apply it.
Secondly, this is also good because of the time management strategy, shifting the position of each
of the terms so that I can be less confused. This will help me not get 45 Ranking Points.
4. Just to make sure I don’t lose this mark, when proving a curve is one-one:
“Every line y=a, a ∈ ℝ”, i should specify the subset
5. “Next time, subtract off”, repeatedly the strategy is just to subtract an expression for common
difference from an expression for common difference
6. There was a graph that hid its positive root extremely close to r=1. Earlier, i found the root of 1,
so when I saw the graph I didn’t realise there was a root hiding there and hence dismissed it as the
only one.
How do we know a third real root exists? By conjugate root theorem, since I have found 2 real
roots and the polynomial is to the third degree, the third root must be a real root, there can’t be a
single complex root and two real roots. Where do I search for this next time? In the places that
look like intersections, I need to get suspicious of those.
7. So, firstly, even-numbered n terms refers to second term, fourth term, sixth term..etc…rather than
the value itself being even. Secondly don’t assume the first term of the AP is the first term of the
GP as per questions you’ve done in the past. Read the question pls, the first term of the GP is
in this case, really, the second term of the AP. To make this clear, you may want to draw a
tiny table
8. Expressing 2 as 2e^i0 is the way to go
9. -1 on its own is really e^ipi according to the argand diagram, and so for any expression involving
re^iθ should be default have r be positive, as you can fix the factor of negative into the
theta
10. If you get stuck trying to solve a complicated algebraic expression for a GP, ask yourself if you
got the number of terms right, if you’re oversimplifying. You got the hard part of this question
right (which involves applying the clue that w^n is somehow = 2^n, good), but it makes no sense
if your number of terms was n, when it should have been n+2
11. I’m not subtracting marks because it’s technically right, but you do want to make sure that you
give x, so if you get an expression for x^2, and you know x>0, then do the assumption for them
that x must be square rooted.
12. At the same time, ask yourself if the DE question actually asked you to get rid of the value of k. If
not, then you can prevent wasting time by leaving your final solution in terms of k
13. I don’t get what’s happening
14. Most of the time, the trig function with a power sign attached to it is a death omen, unless this is
f’(x)[f(x)], this is bad and we can quickly get rid of it by using the double angle formula, to get
cos2t which is actually something we can actually integrate
15. Glad to see that you managed to figure it out, though the time you took was questionable…
for this question, while i dont have anything all that
interesting to say, do note that if you have to attach a pi sign anyways, you’re better off
multiplying by the pi sign both in your scientific calculator check and in your gc check
(1) (38/3)(π)=(π/4)(∫y dx)
But
(2) (38/3) ≠ (∫y dx)
And if you end up believing (2) should be true, you’re never going to be able to solve the
question
Prelim A - P2
1. A couple of pretty nifty tricks here for vector questions you should keep in mind. Also, I’m proud of
myself. Firstly, seeing that we’re looking for an area of trapezium, the cross product perpendicular can
come in handy to help us find the height of the parallelogram. Secondly, neat vector concepts are applied.
2. Tower floor rule, (the second value was erroneously determined to have upper limit=5, otherwise fine
3. “For 100 such hospitalisation claims”: Yes, these are independent observations. The only time Var(X) is
multiplied by a^2 over a is when there’s a multiplication factor of some sort, like a discount or a length
reduction, or a “twice or thrice than”. Otherwise it’s always independent observations.
4. If a “team leader” is now involved in the permutation, take them out of the universal set
5. Now, i dont know what i was trying to do but
I think I was trying to divide my permutation by a probability?? Anyway, the answer key’s version is
more elegant, and I would rather follow this
6. Is the distribution normal? Always ask yourself that question first, and check if it is circumvented with
common limit theorem. You didn’t do this preliminary step, and asserted the assumption that population
sd is same as sample sd. This assumption is more downstream, because to begin with I’m not even sure
the thing can be modelled by a damn normal distribution model
7. If product moment correlation is -1, then we say “that there is a strong negative linear correlation”,
but also that “It does not mean that red wine intake decreases the risk of heart disease.”
8. Did i accidentally swap sufficient and insufficient evidence
9. There was a question on the combined sample, and while comparison is possible based on general ±
changes to the value to the unknown (critical value in an inequality with expression), if the question tells
you the exact changes, it will save time to just repeat the hypothesis testing process.
10. If there’s an unknown in the normal distribution, standardise, standardise, standardise
11. If there is a selection process and then there is a P(A<VALUE), don’t combine the normal distribution
with the selection process.
Instead, you should be doing a step 1, where the selection process picks either A or B, followed by
multiplying by B. If you force them into each other, you’re going to distort the normal distribution
12. For 11(iii), you substituted in mean instead of standard deviation. Anyway, that’s not really
important, instead, what is is the possibility of
Being done in a single step, without showing much else of any other working
13. I call upon my angels
14. Ashes to ashes
15. Dust to dust
MISSING: 4, 5, 9
2019 P1
1. Shameful
2. There are many careless mistakes that are probably the outcome of me not doing this on the same copy
as the questions, so I'm turning a blind eye to all of them. They involve very simple things, like
transferring numbers onto the worksheet. We can’t have these kinds of errors for the actual A Levels, but I
think they are much less likely to happen at A Levels.
-did not multiply entire expression by ‘p’
-translation in the wrong order which would yield an entirely different expression
-sub in the wrong value of r into a sigma notation
-∈ℝ is not =ℝ
-placed in the wrong range
-must substitute in 2 different values, one is 32, the other is 30. You wrote both as 32
I think these are all the result of a lack of focus, having the questions printed out onto a different
copy, and fatigue…I hate A Levels, but it is almost over soon
3. If you want to find the angle between two intersecting lines, you may need to account for the height of
the y-intercept and subtract it off. This is because the y-coordinate itself wouldn’t actually be the height of
the triangle which you are trying to exploit….or it’s just the inverse tangent of the gradient…
4. Specifying the possible values of f, depending on what the n value is
6. Proud of you: You specified cases to make sure the value of u obtained satisfied the condition that it
must be negative. However, you used a more long-winded way, and maybe it would have been better if
you simply used the AP expression
7. Remember to include the i coefficient in a sine ratio for complex number = re^iθ
8. Hard question that involves using complex properties, keep this trick under your belt
9. Idk why you put in the version shown in your graph of the y-intercept when you had to substitute in
a= to a certain value, don’t do that it’s really fuckin stupid
10. For Differential Equation, remember to give your final answer with units in context
11. Then there’s the light ray question, which is hard but I got full marks for it,
i’ll probably get A for A Levels either way
2019 P2
1. My brother in Christ it goes in the denominator
2. as a general rule
l
3. Comment on your Maclaurin Estimation:
[range] gives small values of x, and
extremely small)
Therefore
can be ignored (because they are
and estimation is accurate for small x
4.What is the definition of “Random”? Why is it good?
5. Probability is not independent, an event is independent. A probability is also a fraction/decimal value,
not a percentage
6. Just in case, even if they ask you to find an ‘expression’ for an event happening in a binomial
distribution, you still want to convert 6C2 into 45, always just in case
7.
I don't know why my brain put white horse and rider and yellow and orange
bird as the only possible combinations. You slot each 4 with both 5s, creating
2x2 scenarios
8. I'm not sure how dead i am if i continue doing this, but differentiate between population variance and
sample variance
9.
one is that CLT bcos we know nothing
two is that sample variance estimate, if necessary to find, it's better if sample size is larger
10.
Mixed up variance values when I went to the next page. It never hurts to transpose the information of the
population distribution from the previous page onto the next page
11.
Missing a square index in your variance when you were trying to find the variance of a different
expression…because the question gave it in the form of 𝜎 square
MISSING: 10
2021 P1
Didn't bother 97%
2021 P2
1. f (x) is always y which is always height in some capacity
2. Phrasing
3. Possible method
4. What you did
5. What the possible method implies
5. blank
6. Forgot to subbed in established value of b, because it is “given that…” and according to your
calculation, b would have to be -1
7. I hate stats so fucking much
8. [REMAINS] [FAULTY]
9.
You will lose the presentation mark here if you don’t do this
10. Hopefully you’ll (Diego) remember this
E(X) can be found in terms of p and q
E(X^2) can be found in terms of p and q
Var(X) can be found in terms of p and q, but due to introduction of second degree things get very
complicated, very quickly
Instead, since 2p+q=0.5 according to probability table
You can convert all the qs in E(X) and E(X^2) into expressions of p
11. We do not conclude H0 is false, hence there is insufficient evidence to conclude that H1(in
context form) is true
12. Oh rip r is not r enough
13. Again, more Regression memory work
14. In binomial, if all that happens is 5 observations are taken instead of 1, only sample size changes,
but probability remains constant
Prelim B P1 88/100
1. (Yeah) Don’t forget to label between end points of the same graph
2. Possible solution to doing this fast enough is to just graph everything at the same time
and expedites
3. AGAINST THE TIME
Previous part→Diagram→Ratio theorem to relate the two unknowns together
(when you dot product two vectors together,r you get the square of their magnitudes)
Similar triangles method
4. REMEMBER: if you would like to quickly get rid of the volume of a cone, the volume for it is
V = ⅓ 丌h x r^2
5. Replace 1/w with z, and hence 1/w is = to everything z was…
(im leaving a sticky note here)
6. Intercept with axis was missing
7. Using the equation to identify asymptotes was convenient, effective
8. How to lock certain discussion of roots away
9. Always pay attention to the impossible points
10. Additive tools are available to you
So you can solve it way, way faster
11. Ignored when it should not have been
12. Don;t forget that total area may be multiplied by 2
13. Don’t forget that the coordinates are meant to be fixed
14. Overly contrived, clue should have been the sine and cosine meaning that it can be directly
Differentiated
15. Sometimes, when you use the GC to check the validity of a given relationship, it can be wrong
because you will end up substituting the wrong value and get fucked endlessly.
They could give rate and there isn’t a way for you to check that.
16. This was the careless mistake
all your careless mistakes tend to be about adding and
subtracting
17. Orange is the first term
That first term is the outcome of a GP earlier, which is not the weird summation one
If you need to break it down, write it all out, one by one…
18.
Prelim B P2 81/100
1. Remember to actually differentiate when you jot expressions from the previous page
2. You have a tendency to ignore the coefficient once you’ve narrowed f’(x) = 0, because for the
solution to that equation, of course the unknown becomes irrelevant
3.
4.
this is a + k(b-1) ≠ 16, k ∈ℝ
5. Yea good the direction of the vector matters in searching out the angle, as it can affect whether or
not you end up with obtuse or accurate angles
6.
“Conditional” considerations, if we know for sure that we made it 4 balls in without any of the 3
repeating, then there is no way that they repeated for any of the first 3, so those first 3 all have to be
distinct and hence all events have a ⅓ probability of occurring
7. My feel when they ask why i don't need to know either distribution or variance
8. At worst, 2 variables to define
9. I don't have anything else in this world
10. (m-2)C2 = (m-2)(m-3)/2! ≠ (m-2)(m-1)/2!
(This error is about how you are dense as fuck)
Prelim C P1 82/100
1,
This intercept was misidentified as ‘a’ at the heat of the moment, because
-diego subs in a=2
-finds the intercept at 2
-little does he know, it actually also satisfies the real equation
Moral: If you want to cheat the process of subbing in one by one, after you’re doing, still go back to check
if you were right
2. Differentials can be substituted directly into GC to find the 0 intercept
3. Maximum points can be checked with the usage of the graphing calculator
4. “The simplest form”
How about go fuck yourself? How is that for the simplest form?
Wait
Wait what the fuck
Oh this is a^2 - b^2 = (a-b)(a+b)
Which is now the simplest…form…
5. This was terrible
6. Simplest form
versus
7. WHY YOU FORGET THE i YOU DENSE AHHHHH
(the i in 3/2 + (⇃3)i/2
8. Remember this method for finding power of required integers:
9. Here, just change all x coordinates to y coordinates, all y coordinates to x coordinates
10. EXCUSE ME NOBODY TOLD ME THIS
11. Hence stands to reason that
12. Ok, so you shouldn’t assume that cartesian form will always yield the answer faster. The parametric
method took very little time, whereas the cartesian form was a convoluted mess.
Cartesian equation doesn’t enable an expression in terms of a^2, the parametric version also allows you to
just use the GC to get the 2dp faster. See which version of the expression allows you to keep the
coefficient constant outside the integrand expression, see which version of the expression will lock it
inside
And most importantly, which one is less complicated (terms being brought up to the power of ⅔ should
never be trusted)
13. If the coordinates are irritating to produce in each step, you’re allowed to just keep it as algebras
representing them, for the time being
(16.3, 49.7) is strange
14. Another hack the question from 13 uses is looking for the area of a rectangle using the height x width.
If you can bring in a geometric shape, a formula, then bring it in to at least get expression marks
16. (Oh lord jesus christ please save my soul from perdition) Highest level of Algebraic Manipulation
a)Use the final expression to create a unique expression (then we get stuck)
b)Ok, so next we need to use expressions of the new unknown. We don’t necessarily need to
know what those unknowns are, so long as we can get some kind of a comparison
c)We can create this comparison with the usage of division, when we “divide throughout”
17.
Prelim C P2
Statistics
1.
apparently this is curvilinear…?
2.
Model way of answering, making sure to state that there are other factors at play
3.
Wrote a 5sf answer for 3sf (though, this is more accurate and better, just make sure not to write (3sf) next
to your answer and you will be ok)
4.
Explaining abnormal distribution:
Simply state that there would be more kids outside the average range than the average range itself, instead
of searching out some strange percentile fact
5.
Misread “will not be” as “will be”
In the next part, I forgot to square standard deviation for variance…fuckin hell. Those two cost me 4
marks
6. Don’t treat Y as a normal distribution, because Y is a random discrete variable
I lost so many marks to stats already. More errors to be shown later!
8.
If you want to prevent the negative value, you can preemptively flip it around without needing to go
through the trouble of justifying yourself
9.
“When x = 0” no need to use the GC for this conversation
If you do use the GC and find everything in terms of z, it’s more troublesome than just setting x=0
10.
Don't assume k = 2n, and so they would sum into one another (saves time)
11. “That was, terrible”
Transformations:
Use calculus
Rearrange terms
(For complex expressions) Compare real and imaginary parts
12. What I did here was actually slightly different, but the error was that I forgot to cube the common
difference below, my mind kind of processed that there were 3n terms. It helps to separate the
actual common difference (1/27) from the number of terms.
Also, for this question, I wasted time because I confused this with an overlap summation
expression, where
A+B+
B + C+
C + D +...
This is not what’s happening, and it’s me being influenced by a question that I have done before.
Instead, it makes more sense to list out the pattern before assuming anything.
(ok anywhere they’re additive so this should have been very obvious)
13.
Concept was correct, in fact I did something kind of clever where
dA/dt = dV/dt x dA/dV
But dA/dV requires more implicit differentiation and more derivatives to find
so i shortened it into…
dA/dt = dV/dt x dA/dr x dr/dV
Naturally, this is less troublesome
Still i uh, i subbed in 72 as 2…
14. 2➗3/10 is not 0.6….
15. Insertion method is required here
😞
Please don’t assume that they are locked into a 3-person unit
16.
v is not s.d..
2018 P2 88/100
1. Once again, cross multiplying with a fraction in the denominator is kind of fucky for you, and you
need to be more aware of those complications
2. As well, be aware that when (y)^⅔ is transformed into y^3, indeed there is a negative root you
will have to account for but CAN IGNORE if it is above the curve
3. They are negative you NING NONG
4. Parallelogram ≠ Trapezium
5. The one thing that is impossible to check for in a vector question is the carrying over before cross
product. Even if you sub points back into the equation, you won’t have a way of knowing that
you got it wrong. Be very wary of this, because that means your checks will make your answer
appear correct even if it is wrong…
6. Always, whenever the route to the answer doesn’t seem clear (example, bug question with right
or left forks), do a Computer Science style listing of the various pathways to gain some insight as
to how it should work
7. “And” statement used for inequalities that do not contradict one another
8.
9. Maclaurin trick for expanding the thing inside first
10. Has it all
11. This entire problem, verbatim
12. The only reason that you couldn’t do the “impossible” 6 mark question, was that [E(S)]^2 was
represented incorrectly
It was originally (10n+36)/(n+5), but your mind made the shortcut and assumed they had the
same denominator
They don’t, the entire thign which makes the expression of Expectation is unique is that it
managed to get rid of n+4. Identify as many of these assumptions as you “flip” to the next page.
Be careful, open eye big big
13. Use the appearance again, even if you had specified so in the previous question
14. If question never delves into one being dependent on the other, you can use the same relationship
to derive them from either side of the lake of iodine
15. Explicitly mention how you are replacing certain variables
16. Always, before the normal question even begins, state distribution
Prelim Paper 1
1. Careless
2. Couldn’t see it
3. Trick value
4. Stationary is important
5. Don’t make stationary point too blunt, emphasise the sharp turn
6. Parametric, undelayed, immediate searching out of value
6b. No need to prove anything more
7. Distinguish between theta and x
8. Remember centre of any conics graph
9. TV dinners
10. Wake up and smell the ashes
11.