STEP 2025 Solutions and Reconstruction

10

Please discuss the content of questions here so we can reconstruct papers to analyse for the correct solutions. Also, post your solutions if you can remember them.

STEP 2 Reconstruction:
Q1 by R. Reed
Q3 by R. Reed
Q4 by username7694270
Q5 by R. Reed
Q7 by R. Reed

STEP 2 Solutions:
Q1 by R. Reed
Q3 by R. Reed
Q4 by username7694270
Q5 by fu6tf87g

(edited 8 months ago)

Reply 1

AlfieThomson

5

I have stuff in my notes app about what i did but how did everyone find it? I did Questions 1,2,4,5,11 and 12. I got about 20 on 1 2 and 5, 4 and 11 were okay partials i think and 12 i only did the first part :frown:

. I think I got low 70s/High 60s so i hope the 1 boundary is where its been the past few years. I can't tell if the paper was easy or hard :/

Reply 2

username7312206

OP

10

Original post

by AlfieThomson

I have stuff in my notes app about what i did but how did everyone find it? I did Questions 1,2,4,5,11 and 12. I got about 20 on 1 2 and 5, 4 and 11 were okay partials i think and 12 i only did the first part :frown:

. I think I got low 70s/High 60s so i hope the 1 boundary is where its been the past few years. I can't tell if the paper was easy or hard :/

I found it pretty good! I got full marks on Q1 and Q5, around 17 for the lnx/x question, and around 15 for Q7 which was the differential equations question. I also did a little bit of the floor function question, wish I'd done it earlier because it was so easy. Overall my mark is roughly 76, minimum 72ish. I think the grade boundaries will go up a bit tho, it felt a bit easier than last year unfortunately. There were a few weird tricks in it though, so who knows

Reply 3

username7312206

OP

10

Q1
i) Sketch the graph of $Min(x^2,2x)$
ii) Solve the equation $Min(x^2,2x)+Min(x^3,4x)=6x$
iii) Show that the function $f(x)=Min(x^2,x^3)-5x$ has a local maximum at the point (1,-3). Hence find the remaining three local minima and maxima of the function.
iv) Sketch f(x)

Reply 4

AlfieThomson

5

Original post

by R. Reed

I found it pretty good! I got full marks on Q1 and Q5, around 17 for the lnx/x question, and around 15 for Q7 which was the differential equations question. I also did a little bit of the floor function question, wish I'd done it earlier because it was so easy. Overall my mark is roughly 76, minimum 72ish. I think the grade boundaries will go up a bit tho, it felt a bit easier than last year unfortunately. There were a few weird tricks in it though, so who knows

I think I got around what you did, I really hope people didnt find it too easy (selfish but yk). I think 5 was similar to the 2023 integral question maybe a little harder because the algebra and u^-k sub. I thought question 2 was very straightfoward so i hope people were scared of the complex numbers (first part was probably hardest). Question one was pretty accessible but i missed 6x=6x had infinite solutions like a goober so i think i got 18 or 19 on that. I thought the stats were hard, but im glad i got to use the differentiate (1+x)^n trick because ive practiced questions like that quite a lot. Question 4 i wrote that you could use x + 1/n to get from between 0 and 1/n to any real number and that was the last thing i did (had no idea how to prove the next bit). It depends how they mark partials for me, my probability in 11 was wrong but i applied the right ideas so idk what thatll get me. Id say Q1~18 Q2~19 Q5~20 Q4~7? Q11~7 Q12~3 so 74 ig?

Reply 5

username7312206

OP

10

Q3
i) Sketch the graph of $\frac{\ln(x)}{x}$ . Hence show that
a) $\pi^3<3^\pi$
b) Something very similar
ii) Determine, for 1<x<2, if (x+2)^x or x^(x+2) is greater.
iii) Show that the inequalities $9^{\sqrt2}>\sqrt2^9$ and $3^{2\sqrt2}>(2\sqrt2)^3$ are equivalent. Determine which of $9^{\sqrt2}$ and $\sqrt2^9$ is greater.
iv) Determine, with justification, which of $8^{\sqrt[3]{3}}$ and $\sqrt[3]{3}^8$ is greater.

Reply 6

username7312206

OP

10

Q5
i) By considering the substitution $x=u^{-1}$ , show that
$\displaystyle\int_0^\infty \frac{\sqrt{x}-1}{\sqrt{x(x^3+1)}}dx = 0$ .
ii) By considering the substitution $x=u^{-2}$ , show that
$\displaystyle\int_0^\infty \frac{1}{\sqrt{x^3+1}}dx = 2\int_0^\infty \frac{1}{\sqrt{x^6+1}}dx$
iii) Find, for fixed p and s, r such that
$\displaystyle\int_0^\infty\frac{x^r-1}{\sqrt{x^s(x^p+1)}}dx=0$
iv) Show that, for any given positive k, it is possible to find p and q such that
$\displaystyle\int_0^\infty\frac{1}{\sqrt{x^p+1}}dx =k\int_0^\infty\frac{1}{\sqrt{x^q+1}}dx$

Reply 7

username7312206

OP

10

Q7
The movement of a particle is described by the differential equation
$\displaystyle\frac{d^2x}{dt^2}=2x\frac{dx}{dt}$
and, for t=0, x=a, where a is a positive constant.
i) Solve the differential equation in the case that $\displaystyle\frac{dx}{dt}=a^2$ when t=0. Describe the motion of the particle in this case as t increases from zero.
ii) Solve the differential equation in the case that $\displaystyle\frac{dx}{dt}=a^2+p^2$ when t=0, where p>0. Describe the motion of the particle in this case as t increases from zero.
iii) Solve the differential equation in the case that $\displaystyle\frac{dx}{dt}=a^2-q^2$ when t=0, where q>0. Describe the motion of the particle as t increases from zero, giving different answers based on conditions on q and a which arise.

Reply 8

AlfieThomson

5

Original post

by R. Reed

Q5
i) By considering the substitution $x=u^{-1}$ , show that
$\displaystyle\int_0^\infty \frac{\sqrt{x}-1}{\sqrt{x(x^3+1)}}dx = 0$ .
ii) By considering the substitution $x=u^{-2}$ , show that
$\displaystyle\int_0^\infty \frac{1}{\sqrt{x^3+1}}dx = 2\int_0^\infty \frac{1}{\sqrt{x^6+1}}dx$
iii) Find, for fixed p and s, r such that
$\displaystyle\int_0^\infty\frac{x^r-1}{\sqrt{x^s(x^p+1)}}dx=0$
iv) Show that, for any given positive k, it is possible to find p and q such that
$\displaystyle\int_0^\infty\frac{1}{\sqrt{x^p+1}}dx =k\int_0^\infty\frac{1}{\sqrt{x^q+1}}dx$

I think for the last two i got 2s + p + 2/2 or something like that and 2k + 2 and 2k+2/k i think?

Reply 9

username7312206

OP

10

Original post

by AlfieThomson

I think for the last two i got 2s + p + 2/2 or something like that and 2k + 2 and 2k+2/k i think?

Yep it’s correct, I got the same

Reply 10

AlfieThomson

5

Original post

by R. Reed

Yep it’s correct, I got the same

What are you thinking for grade boundaries, its kind of eerie that there's no exam discussion for STEP how there is for fm and physics etc :frown:

Reply 11

AlfieThomson

5

The last part of question to was solving z^3 + |z+2|^2 = 4 i dont remember the exact other parts but it was about purely real and purely imaginary solutions

Reply 12

username7312206

OP

10

Original post

by R. Reed

Q1
i) Sketch the graph of $Min(x^2,2x)$
ii) Solve the equation $Min(x^2,2x)+Min(x^3,4x)=6x$
iii) Show that the function $f(x)=Min(x^2,x^3)-5x$ has a local maximum at the point (1,-3). Hence find the remaining three local minima and maxima of the function.
iv) Sketch f(x)

Q1 Solution:
Graphs
ii) It’s just a lot of case work to type out, the solution for 6x ends up being $x\in[-2, 0]\cup[2,\infty)$ , and the solution for 2x is x=0, x=1.
iii) After investigating the function, the behaviour of the minimum switches at x=1, so for $x<1$ , $f(x)=x^3-5x$ , and for $x\geq1$ , $f(x)=x^2 - 5x$ . Taking left and right handed derivatives about the point x=1 gives opposite signs, and hence there is a local maximum there (cusp). The other turning points are at $x=\pm \sqrt{5/6}, 5/4$ .

Reply 13

username7312206

OP

10

Original post

by AlfieThomson

What are you thinking for grade boundaries, its kind of eerie that there's no exam discussion for STEP how there is for fm and physics etc :frown:

I’ve got a different post for it. This one’s meant to mainly be about getting questions and solutions rather than general discussion. I think grade boundaries should be slightly higher than last year, so I’m expecting around 71, 72ish for a 1 I’d say.

Reply 14

shinyduck21

9

Hmmm I found every question was either suspiciously easy or rlly rlly hard. I think I got near full marks on 1, 2, and 5. Then maybe 7 on 3? I drew the graph and proved the 3/π one and whatever the next one was, and did the 9/root2 inequality but I couldn’t do the x and x+2 one (anyone know how to do it? I got that x was less than e and x+2 more than e but I got stuck there). Then maybe 4 on 6 (did the first 2 r^2 parts but I realised I missed out a 2 on the second one) and 2 on 4 (didn’t click w the question but was desperate for marks so wrote anything I could think of). So hopefully 73 ish… as long as boundaries don’t go up that’s fine but I rlly can’t gage how other people found it.

Reply 15

shinyduck21

9

Original post

by R. Reed

Q7
The movement of a particle is described by the differential equation
$\displaystyle\frac{d^2x}{dt^2}=2x\frac{dx}{dt}$
and, for t=0, x=a, where a is a positive constant.
i) Solve the differential equation in the case that $\displaystyle\frac{dx}{dt}=a^2$ when t=0. Describe the motion of the particle in this case as t increases from zero.
ii) Solve the differential equation in the case that $\displaystyle\frac{dx}{dt}=a^2+p^2$ when t=0, where p>0. Describe the motion of the particle in this case as t increases from zero.
iii) Solve the differential equation in the case that $\displaystyle\frac{dx}{dt}=a^2-q^2$ when t=0, where q>0. Describe the motion of the particle as t increases from zero, giving different answers based on conditions on q and a which arise.

Do you mind sharing how u did this? I got that dx/dt = e^x^2 but didn’t know how to go from there

Reply 16

username5832319

1

What do you think the grade boundary for an S will be?

Original post

by R. Reed

I’ve got a different post for it. This one’s meant to mainly be about getting questions and solutions rather than general discussion. I think grade boundaries should be slightly higher than last year, so I’m expecting around 71, 72ish for a 1 I’d say.

Reply 17

username5832319

1

Original post

by AlfieThomson

I think for the last two i got 2s + p + 2/2 or something like that and 2k + 2 and 2k+2/k i think?

yeah same

Reply 18

username6380546

4

Original post

by shinyduck21

Do you mind sharing how u did this? I got that dx/dt = e^x^2 but didn’t know how to go from there

Im not OP, but I used integration by parts on the RHS to get dx/dt.. got some weird fractions, not e^x^2. Same procedure for all parts of the question. I see e^x^2 works too, but it fails to meet the initial conditions. I doubt im right tho, I couldnt concentrate at all :frown:

Reply 19

AlfieThomson

5

How many marks do you think the part of Q4 before the second proof and series is worth?

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