# STEP Maths I, II, III 1988 solutions

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(Updated as far as #68) SimonM - 15.06.2009

Seeing as people seem to have asked for a thread for this for quite a while I thought I might as well make one.

I guess you're all familiar what these threads are for by now; post your solutions to the older STEP papers as there are no solutions available freely on the net - plus it's good revision Feel free to post alternative solutions, and please point out when you see a weakness or mistake in a solution.

If you look in the STEP megathread you will find a link to the papers - if you still have problems accessing them, PM me or someone else who has them.

1: Solution by nota bene

2: Solution by nota bene

3: Solution by Glutamic Acid

4: Solution by brianeverit

5: Solution by Swayum

6: Solution by brianeverit

7: Solution by Glutamic Acid

8: Solution by brianeverit

9: Solution by nota bene

10: Solution by brianeverit

11: Solution by brianeverit

12: Solution by brianeverit

13: Solution by brianeverit

14: Solution by brianeverit

15: Solution by nota bene

16: Solution by brianeverit

1: Solution by kabbers

2: Solution by SimonM

3: Solution by Glutamic Acid

4: Solution by brianeverit

5: Solution by Squeezebox

6: Solution by SimonM

7: Solution by Square

8: Solution by brianeverit

9: Solution by SimonM

10: Solution by brianeverit

11: Solution by brianeverit

12: Solution by brianeverit

13: Solution by brianeverit

14: Solution by brianeverit

15: Solution by brianeverit

16: Solution by brianeverit

1: Solution by kabbers

2: Solution by Squeezebox

3: Solution by mikelbird and brianeverit

4: Solution by mikelbird

5: Solution by kabbers

6: Solution by kabbers

7: Solution by squeezebox

8: Solution by SimonM

9: Solution by SimonM

10: Solution by ben-smith

11: Solution by Jkn

12. Solution by ben-smith

13: Solution by ben-smith

14: Solution by ben-smith

15:

16: Solution by ben-smith

1987 - 1988 - 1989 - 1990 - 1991 - 1992 - 1993 - 1994 - 1995 - 1996 - 1997 - 1998 - 1999 - 2000 - 2001 - 2002 - 2003 - 2004 - 2005 - 2006 - 2007

Seeing as people seem to have asked for a thread for this for quite a while I thought I might as well make one.

I guess you're all familiar what these threads are for by now; post your solutions to the older STEP papers as there are no solutions available freely on the net - plus it's good revision Feel free to post alternative solutions, and please point out when you see a weakness or mistake in a solution.

If you look in the STEP megathread you will find a link to the papers - if you still have problems accessing them, PM me or someone else who has them.

**STEP I:**1: Solution by nota bene

2: Solution by nota bene

3: Solution by Glutamic Acid

4: Solution by brianeverit

5: Solution by Swayum

6: Solution by brianeverit

7: Solution by Glutamic Acid

8: Solution by brianeverit

9: Solution by nota bene

10: Solution by brianeverit

11: Solution by brianeverit

12: Solution by brianeverit

13: Solution by brianeverit

14: Solution by brianeverit

15: Solution by nota bene

16: Solution by brianeverit

**STEP II:**1: Solution by kabbers

2: Solution by SimonM

3: Solution by Glutamic Acid

4: Solution by brianeverit

5: Solution by Squeezebox

6: Solution by SimonM

7: Solution by Square

8: Solution by brianeverit

9: Solution by SimonM

10: Solution by brianeverit

11: Solution by brianeverit

12: Solution by brianeverit

13: Solution by brianeverit

14: Solution by brianeverit

15: Solution by brianeverit

16: Solution by brianeverit

**STEP III**1: Solution by kabbers

2: Solution by Squeezebox

3: Solution by mikelbird and brianeverit

4: Solution by mikelbird

5: Solution by kabbers

6: Solution by kabbers

7: Solution by squeezebox

8: Solution by SimonM

9: Solution by SimonM

10: Solution by ben-smith

11: Solution by Jkn

12. Solution by ben-smith

13: Solution by ben-smith

14: Solution by ben-smith

15:

16: Solution by ben-smith

**Solutions written by TSR members:**1987 - 1988 - 1989 - 1990 - 1991 - 1992 - 1993 - 1994 - 1995 - 1996 - 1997 - 1998 - 1999 - 2000 - 2001 - 2002 - 2003 - 2004 - 2005 - 2006 - 2007

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#2

STEP I Q1

For max/min set h'(x)=0 i.e.

To find the nature of the stationary point, consider e.g. h'(e-0.2)=positive and h'(e+0.2)=negative hence a (global) maximum. For sketching purposes might be worth noting that and . Graph see attached (Mathematica because of lack of scanner).

Solving is equivalent of solving (last step valid as neither m nor n can be 0).

m=n is the trivial solution.

Consider the line f=c (i.e. a constant). For there are two solutions. There's one solution when or . And for there are no solutions.

Now if then ().

Thus and .

(What am I missing here, this looks trivial for a STEP question...)

----

STEP I Q2

()

Let

Now WLOG

Has solutions cos(x)=0 and sin(x)=1/2

i.e.

where

By quadratic formula the solutions are i.e. u=-1, 2

has no real solutions and has the root (which we have already found).

where

Solutions to this are and as sin(x)=u will have no real solutions.

, so a (local)min

, so a (global) min

, so a (global) max

, so a (local) max

For max/min set h'(x)=0 i.e.

To find the nature of the stationary point, consider e.g. h'(e-0.2)=positive and h'(e+0.2)=negative hence a (global) maximum. For sketching purposes might be worth noting that and . Graph see attached (Mathematica because of lack of scanner).

Solving is equivalent of solving (last step valid as neither m nor n can be 0).

m=n is the trivial solution.

Consider the line f=c (i.e. a constant). For there are two solutions. There's one solution when or . And for there are no solutions.

Now if then ().

Thus and .

(What am I missing here, this looks trivial for a STEP question...)

----

STEP I Q2

()

Let

Now WLOG

Has solutions cos(x)=0 and sin(x)=1/2

i.e.

where

By quadratic formula the solutions are i.e. u=-1, 2

has no real solutions and has the root (which we have already found).

where

Solutions to this are and as sin(x)=u will have no real solutions.

, so a (local)min

, so a (global) min

, so a (global) max

, so a (local) max

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#3

I haven't bothered to download the paper, so this may not be a valid objection, but I feel I need to point out that 2^4 = 4^2...

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#5

High, quick question: What programme are you using to write out the Mathematics? I've seen such quite alot here but have no idea to do so myself. :|

Thanks.

Thanks.

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#6

Nota: I think you would be expected to give at least some justification why (2,4) and (4,2) are the only 'non-trivial' solutions. (Without giving you a hard time for the slipup: if you missed (2,4), how do you know you didn't miss anything else?)

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#7

(Original post by

High, quick question: What programme are you using to write out the Mathematics? I've seen such quite alot here but have no idea to do so myself. :|

Thanks.

**The Lyceum**)High, quick question: What programme are you using to write out the Mathematics? I've seen such quite alot here but have no idea to do so myself. :|

Thanks.

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#8

(Original post by

LaTeX. It's installed on the forum software - see the thread in the maths forum called "PLEASE READ BEFORE POSTING !!" for an explanation of how to use it.

**generalebriety**)LaTeX. It's installed on the forum software - see the thread in the maths forum called "PLEASE READ BEFORE POSTING !!" for an explanation of how to use it.

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#9

Im about half way through II/7.

So if someone could refrain from spoiling my fun and posting the solution before I wake up tomorrow and finish the question please!

So if someone could refrain from spoiling my fun and posting the solution before I wake up tomorrow and finish the question please!

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#11

(Original post by

Nota: I think you would be expected to give at least some justification why (2,4) and (4,2) are the only 'non-trivial' solutions. (Without giving you a hard time for the slipup: if you missed (2,4), how do you know you didn't miss anything else?)

**DFranklin**)Nota: I think you would be expected to give at least some justification why (2,4) and (4,2) are the only 'non-trivial' solutions. (Without giving you a hard time for the slipup: if you missed (2,4), how do you know you didn't miss anything else?)

Consider the line f=c (i.e. a constant). For there are two solutions. There's one solution when or . And for there are no solutions.

Now if then ().

Thus and .

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#12

Sorry, but I'm not seeing how the above shows there's only two solutions.

(I would give a very simple solution based on the graph and the location of the turning point).

(I would give a very simple solution based on the graph and the location of the turning point).

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#13

III/1:

Sketch

I'm not going to bother posting most of the differentiating legwork, there arent too many tricks, its just an algebra bash.

Differentiating (quotient rule is your friend), we get

So we have turning points at x = 0, and x =

Differentiating again, we get

Substituting in values of x for the turning points, we find that x = 0 is a minimum, and the other two maximums.

Noticing the denominator of y, we find that there is an asymptote at x = -1.

And because of the exponential properties of e^-x, y tends to zero as

y tends to as since the numerator remains positive and given the exponenential properties of e^x much greater than the denominator, while the denominator becomes negative.

So the graph will look http://www.thestudentroom.co.uk/show...8&postcount=14

Prove

First note that

Hence we may split the integral into

Consider

Now consider

I posit the inequality for x > 0

So our inequality holds.

So:

Thus

And therefore,

Now notice that the graph of is greater than 0 for x > 0, and hence so is the infinite integral.

So,

please point out any mistakes

Sketch

I'm not going to bother posting most of the differentiating legwork, there arent too many tricks, its just an algebra bash.

Differentiating (quotient rule is your friend), we get

So we have turning points at x = 0, and x =

Differentiating again, we get

Substituting in values of x for the turning points, we find that x = 0 is a minimum, and the other two maximums.

Noticing the denominator of y, we find that there is an asymptote at x = -1.

And because of the exponential properties of e^-x, y tends to zero as

y tends to as since the numerator remains positive and given the exponenential properties of e^x much greater than the denominator, while the denominator becomes negative.

So the graph will look http://www.thestudentroom.co.uk/show...8&postcount=14

Prove

First note that

Hence we may split the integral into

Consider

Now consider

I posit the inequality for x > 0

So our inequality holds.

So:

Thus

And therefore,

Now notice that the graph of is greater than 0 for x > 0, and hence so is the infinite integral.

So,

please point out any mistakes

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#14

k here's the graph from the above q. if i've made any mistakes, please feel free to point them out

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#15

You've made a mistake integrating (x-1)e^{-x}. (Still right answer, but some of what you've written is wrong).

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#16

(Original post by

You've made a mistake integrating (x-1)e^{-x}. (Still right answer, but some of what you've written is wrong).

**DFranklin**)You've made a mistake integrating (x-1)e^{-x}. (Still right answer, but some of what you've written is wrong).

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#17

I/9

i) Recall that , now we can use this to integrate by parts.

Putting in the limits gives So

ii) Let

So the integral becomes

Going back to x we have

i) Recall that , now we can use this to integrate by parts.

Putting in the limits gives So

ii) Let

So the integral becomes

Going back to x we have

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#18

I've got III/3 out. Will post later. Suprisingly easy, its more like a FP3 question to be fair. Might need a check over because my complex transformations are more than a little rusty.

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#20

(Original post by

Done rough "workings out" for I/9 again two surprisingly easy integrals.

**insparato**)Done rough "workings out" for I/9 again two surprisingly easy integrals.

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