STEP marking scheme and solutions
Regarding the STEP marking schemes
1) If a subproblem does not explicitly state that I need to use a result derived previously in the question: provided that I use only mathematics from A-levels, does solving with an alternative method cost any marks?
2) The solutions usually require that for proofs I need to solve in the 'right direction': Prove that if B is true, then A holds: am required to rearrange from B to get A. However, if I claim that A is true if and only if B is true by demonstrating every line of working is reversible, am I able to write 'from A to get B' without losing marks?
1) If a subproblem does not explicitly state that I need to use a result derived previously in the question: provided that I use only mathematics from A-levels, does solving with an alternative method cost any marks?
2) The solutions usually require that for proofs I need to solve in the 'right direction': Prove that if B is true, then A holds: am required to rearrange from B to get A. However, if I claim that A is true if and only if B is true by demonstrating every line of working is reversible, am I able to write 'from A to get B' without losing marks?
Original post
by fortniteballz69Regarding the STEP marking schemes
1) If a subproblem does not explicitly state that I need to use a result derived previously in the question: provided that I use only mathematics from A-levels, does solving with an alternative method cost any marks?
2) The solutions usually require that for proofs I need to solve in the 'right direction': Prove that if B is true, then A holds: am required to rearrange from B to get A. However, if I claim that A is true if and only if B is true by demonstrating every line of working is reversible, am I able to write 'from A to get B' without losing marks?
1) If a subproblem does not explicitly state that I need to use a result derived previously in the question: provided that I use only mathematics from A-levels, does solving with an alternative method cost any marks?
2) The solutions usually require that for proofs I need to solve in the 'right direction': Prove that if B is true, then A holds: am required to rearrange from B to get A. However, if I claim that A is true if and only if B is true by demonstrating every line of working is reversible, am I able to write 'from A to get B' without losing marks?
1) Unless specifically directed to use a method, you may use any valid method, whether from A-Level Maths or not. STEP is marked by Cambridge Part III grads and PhD students so it's not like in A-Level where the marker may not be familiar with your method.
The only caveat to this is that you can't use an advanced method / theorem that makes the question trivial without proof. For example, there was a question that got you to derive the area of a triangle given its sides alone using vector methods (it worked you through it) so if you were to just quote Heron's formula* without explanation, then you wouldn't get the marks, since deriving that formula was the crux of the question. This caveat is fairly minor and should be fairly obvious if you try to flout it. The questions are rarely written such that a formula from A-Levels can be quoted and make the whole question trivial.
2) If you show A <=> [something] <=> [something] <=> [something] <=> [something] <=> [something] <=> B, then you have necessarily shown that B => A. Just note that you would need to be careful and justify some things in the reverse direction that you may not need to justify in the other. For example, x = y => x^2 = y^2 wouldn't require much additional justification, whereas x^2 = y^2 => x = y would need you to justify that x =/= -y for x,y =/= 0. Also note that not every logical implication is reversible.
One thing you may want to consider is that 'B => A' and 'not A => not B' are logically equivalent. If you want to read more into this, the term is 'proof by contrapositive' and this may allow you to work in the 'reverse direction'. It is often lumped in with proof by contradiction.
*Heron's formula is the formula that gives the area of a triangle with side lengths a,b,c. Specifically, it says that the area is sqrt[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2.
Reply 2
Thanks for the help, I'm still not quite sure what 'make the whole question trivial' refers to. I'll refer to this example:
In STEP 3 2021, Question 2, last line of part (ii), would it be too trivial to prove x+y+z >= 3 > 2 given xyz+xy+yz+zx >=0 using the AM-GM inequality on xyz and xy+yz+zx? Considering the solution used a neat trick, I feel like this isn't in line with what the question is asking.
-----
Furthermore I'm assuming that to avoid losing marks, any advanced tech I use must be in line with what the question wants: e.g. if the question is a vector proof, I can use euclidean geometry to help speed up small parts of my answer but I can't use it to finish an entire subquestion etc.?
In STEP 3 2021, Question 2, last line of part (ii), would it be too trivial to prove x+y+z >= 3 > 2 given xyz+xy+yz+zx >=0 using the AM-GM inequality on xyz and xy+yz+zx? Considering the solution used a neat trick, I feel like this isn't in line with what the question is asking.
-----
Furthermore I'm assuming that to avoid losing marks, any advanced tech I use must be in line with what the question wants: e.g. if the question is a vector proof, I can use euclidean geometry to help speed up small parts of my answer but I can't use it to finish an entire subquestion etc.?
(edited 8 months ago)
Reply 3
Original post
by fortniteballz69Thanks for the help, I'm still not quite sure what 'make the whole question trivial' refers to. I'll refer to this example:
In STEP 3 2021, Question 2, last line of part (ii), would it be too trivial to prove x+y+z >= 3 > 2 given xyz+xy+yz+zx >=0 using the AM-GM inequality on xyz and xy+yz+zx? Considering the solution used a neat trick, I feel like this isn't in line with what the question is asking.
-----
Furthermore I'm assuming that to avoid losing marks, any advanced tech I use must be in line with what the question wants: e.g. if the question is a vector proof, I can use euclidean geometry to help speed up small parts of my answer but I can't use it to finish an entire subquestion etc.?
In STEP 3 2021, Question 2, last line of part (ii), would it be too trivial to prove x+y+z >= 3 > 2 given xyz+xy+yz+zx >=0 using the AM-GM inequality on xyz and xy+yz+zx? Considering the solution used a neat trick, I feel like this isn't in line with what the question is asking.
-----
Furthermore I'm assuming that to avoid losing marks, any advanced tech I use must be in line with what the question wants: e.g. if the question is a vector proof, I can use euclidean geometry to help speed up small parts of my answer but I can't use it to finish an entire subquestion etc.?
Examiner's report said some used AM GM correctly so yes you can use it
Original post
by fortniteballz69Thanks for the help, I'm still not quite sure what 'make the whole question trivial' refers to. I'll refer to this example:
In STEP 3 2021, Question 2, last line of part (ii), would it be too trivial to prove x+y+z >= 3 > 2 given xyz+xy+yz+zx >=0 using the AM-GM inequality on xyz and xy+yz+zx? Considering the solution used a neat trick, I feel like this isn't in line with what the question is asking.
-----
Furthermore I'm assuming that to avoid losing marks, any advanced tech I use must be in line with what the question wants: e.g. if the question is a vector proof, I can use euclidean geometry to help speed up small parts of my answer but I can't use it to finish an entire subquestion etc.?
In STEP 3 2021, Question 2, last line of part (ii), would it be too trivial to prove x+y+z >= 3 > 2 given xyz+xy+yz+zx >=0 using the AM-GM inequality on xyz and xy+yz+zx? Considering the solution used a neat trick, I feel like this isn't in line with what the question is asking.
-----
Furthermore I'm assuming that to avoid losing marks, any advanced tech I use must be in line with what the question wants: e.g. if the question is a vector proof, I can use euclidean geometry to help speed up small parts of my answer but I can't use it to finish an entire subquestion etc.?
I think in that case it would be fine.
I wouldn't really worry about the 'too trivial' thing. Essentially it just means that if a question is, in essence, getting you to prove some formula or theorem that you may have not yet encountered, then quoting that formula or theorem without proof may not be accepted. It's unlikely it will be an issue and it should be fairly obvious.
As a similar example, suppose the whole question was to prove that (x_1 + x_2 + ... + x_n)/n > (x_1 * x_2 * ... * x_n)^(1/n) for each n and you just said "AM-GM inequality" and nothing else, then you wouldn't get the marks because that is what the question is asking you to prove, even if they haven't worded the question with that exact term. They may be very clear and say "[method / theorem] cannot be used without proof".
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