# STEP Maths I, II, III 1994 Solutions

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#81

Insparato:This may be an artifact of your LaTeXing more than anything else, but I'm really struggling to follow what you're doing with each of the substitutions. Mainly because you don't actually link the integral before the substitution with the one after. It's pretty obvious what the "before and after" must be for the first substitution, but in the 2nd one I really can't tell.

If it's just the LaTeX, don't worry about it, but in the actual exam, try to make it clearer. If nothing else, start off by writing and then later you can write . Makes the examiner's life a lot easier. And a happy examiner is a generous examiner...

As far as the end (guessed) bit goes, an important observation is .

If it's just the LaTeX, don't worry about it, but in the actual exam, try to make it clearer. If nothing else, start off by writing and then later you can write . Makes the examiner's life a lot easier. And a happy examiner is a generous examiner...

As far as the end (guessed) bit goes, an important observation is .

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#82

I'm a bit behind, but I did that last bit differently still! (In that ancient STEP I question...)

and of course

So by log rules, the integral becomes

And dividing through by cos and taking the sqrt2 outside (as a -ln2) via the log(a/b) rule gives your 1+tan½x, and so it's been rewritten as the problem a few inches above it.

---

As for the current challenge...Perhaps sin2t = 2sintcost?

That would certainly evaluate the integral (ln2 + lnsint + lncost which means I=-pi ln2/2 ?) but only given that result...hm.

and of course

So by log rules, the integral becomes

And dividing through by cos and taking the sqrt2 outside (as a -ln2) via the log(a/b) rule gives your 1+tan½x, and so it's been rewritten as the problem a few inches above it.

---

As for the current challenge...Perhaps sin2t = 2sintcost?

That would certainly evaluate the integral (ln2 + lnsint + lncost which means I=-pi ln2/2 ?) but only given that result...hm.

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#83

(Original post by

Insparato:This may be an artifact of your LaTeXing more than anything else, but I'm really struggling to follow what you're doing with each of the substitutions. Mainly because you don't actually link the integral before the substitution with the one after. It's pretty obvious what the "before and after" must be for the first substitution, but in the 2nd one I really can't tell.

If it's just the LaTeX, don't worry about it, but in the actual exam, try to make it clearer. If nothing else, start off by writing and then later you can write . Makes the examiner's life a lot easier. And a happy examiner is a generous examiner...

As far as the end (guessed) bit goes, an important observation is .

**DFranklin**)Insparato:This may be an artifact of your LaTeXing more than anything else, but I'm really struggling to follow what you're doing with each of the substitutions. Mainly because you don't actually link the integral before the substitution with the one after. It's pretty obvious what the "before and after" must be for the first substitution, but in the 2nd one I really can't tell.

If it's just the LaTeX, don't worry about it, but in the actual exam, try to make it clearer. If nothing else, start off by writing and then later you can write . Makes the examiner's life a lot easier. And a happy examiner is a generous examiner...

As far as the end (guessed) bit goes, an important observation is .

I do try to make it as clear as possible in exams. Although i think i can see that slipping after June considering i probably wont be taking any formal exams ever again.

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#84

Oh wait I think I did it.

I had some integral from 0 to pie, but you can split that up to an integral over (0,½pi) and (½pi,pi) and use u=pi-x on the second bit.

Let me just make sure I'm not missing anything though...

--

Let

If u=pi-2x then the limits are (top) 0 and (bottom) pi

du = -2dx so...

Sorta gave the game away, but I think I'd have gotten it - sin(pi-u)=sinu.

And for the second bit there, let's try

t = pi-u

So the limits are (top) 0 and (bottom) ½pi

dt = -du so we can just flip the limits upside down

Which are both the same thing, so the half goes and we have what we want.

I had some integral from 0 to pie, but you can split that up to an integral over (0,½pi) and (½pi,pi) and use u=pi-x on the second bit.

Let me just make sure I'm not missing anything though...

--

Let

If u=pi-2x then the limits are (top) 0 and (bottom) pi

du = -2dx so...

Sorta gave the game away, but I think I'd have gotten it - sin(pi-u)=sinu.

And for the second bit there, let's try

t = pi-u

So the limits are (top) 0 and (bottom) ½pi

dt = -du so we can just flip the limits upside down

Which are both the same thing, so the half goes and we have what we want.

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#86

STEP I question 9

Let (x,y) be a point on the cannon-ball's trajectory, then:

Now if the distance from the origin is , then we have the point (x,h), so writing the x and y coordinates with the trig function the subject:

But , so:

, as required.

Since this equation arose from considering a point we assumed to be on the trajectory, then there has to exist a real value of t at which the cannon-ball is at this point. For a real value of t, we need a real value of t^2, so the solutions to the quadratic must be real, so using the discriminant:

, as required.

Okay, now we need to show that there is an angle of firing such that we have when y=h, so going back to the equations for x and y we get:

and

substitute in the time from the x into the y:

For there to exist an angle of firing as described, we again need real roots to this quadratic, , which is of course true (and it really is

Let (x,y) be a point on the cannon-ball's trajectory, then:

Now if the distance from the origin is , then we have the point (x,h), so writing the x and y coordinates with the trig function the subject:

But , so:

, as required.

Since this equation arose from considering a point we assumed to be on the trajectory, then there has to exist a real value of t at which the cannon-ball is at this point. For a real value of t, we need a real value of t^2, so the solutions to the quadratic must be real, so using the discriminant:

, as required.

Okay, now we need to show that there is an angle of firing such that we have when y=h, so going back to the equations for x and y we get:

and

substitute in the time from the x into the y:

For there to exist an angle of firing as described, we again need real roots to this quadratic, , which is of course true (and it really is

*an*angle of firing).
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#87

Are these going on the wiki? If not, does anyone think it might be a nice idea to put them there?

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#88

__Step I Q1__

Since all planes are at the same angle \theta to the horizontal, the roof ridge is central.

The height (h) of the roof (see image 1) is equal to . Therefore (see diagram) = . Since the ridge is central, this will be the same at the other end of the roof.

So R (ridge length) = 2q - 2p, since is not present in this expression, the ridge length is independent of .

__Volume of the roof:__

Slicing up the roof, we can join the two ends to make a pyramid and leave a prism. (see image 2)

The volume of a pyramid =

Area of base =

So its volume =

Volume of a prism =

Area of face =

So the volume of the prism =

So the total volume of the roof = volume of prism + volume of pyramid =

there probably is a much simpler way to do this

EDIT: actually now i'm not sure, should theta be in the volume?

*surface area is on its way *

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#89

STEP III Question 8

The determinant of this matrix is

(In hindsight it would have been less algebra to use |AB|=|A||B|)

Now, chose e.g.

and the determinant of the matrix product will be the desired LHS, i.e.

.

(Provided , because then there is a zero-determinant, which we don't want)

Now, to get the RHS of the equality we are to prove, take a look at the matrix again.

You can see that if we let and we have:

'

This matrix also has a real determinant (i.e. ).

Now, if we set and this means the matrix has the determinant , which is the RHS in the equality.

Now to the messy algebra of determining , , and .

We have and

Therefore follows that

So

So

We also have and

Therefore follows that

So

I'd be grateful if someone checked my interpretation of "linear in a, b, c and d and also linear in p, q, r and s" (David?)

Overall a surprisingly nice question I think (latexing it was pain and there is likely a few silly things:/).

The determinant of this matrix is

(In hindsight it would have been less algebra to use |AB|=|A||B|)

Now, chose e.g.

and the determinant of the matrix product will be the desired LHS, i.e.

.

(Provided , because then there is a zero-determinant, which we don't want)

Now, to get the RHS of the equality we are to prove, take a look at the matrix again.

You can see that if we let and we have:

'

This matrix also has a real determinant (i.e. ).

Now, if we set and this means the matrix has the determinant , which is the RHS in the equality.

Now to the messy algebra of determining , , and .

We have and

Therefore follows that

So

So

We also have and

Therefore follows that

So

I'd be grateful if someone checked my interpretation of "linear in a, b, c and d and also linear in p, q, r and s" (David?)

Overall a surprisingly nice question I think (latexing it was pain and there is likely a few silly things:/).

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#90

(Original post by

I'd be grateful if someone checked my interpretation of "linear in a, b, c and d and also linear in p, q, r and s" (David?)

**nota bene**)I'd be grateful if someone checked my interpretation of "linear in a, b, c and d and also linear in p, q, r and s" (David?)

Overall a surprisingly nice question I think (latexing it was pain and there is likely a few silly things:/).

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#91

Paper II number 12.

I am not too sure about the last part but I think most of it is correct.

Solution by Geriatric

I am not too sure about the last part but I think most of it is correct.

Solution by Geriatric

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#94

I've only been glancing at your answers, and up 'til now I've not seen any mistakes I could spot on cursory inspection.

But with 1994 STEP II, Q14, I think you need to be a little more careful at the end: just because m = 2 gives better results than m=1 and m=3 doesn't

But with 1994 STEP II, Q14, I think you need to be a little more careful at the end: just because m = 2 gives better results than m=1 and m=3 doesn't

**prove**m=2 is optimum.
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#96

(Original post by

STEP III Question 8

The determinant of this matrix is

(In hindsight it would have been less algebra to use |AB|=|A||B|)

Now, chose e.g.

and the determinant of the matrix product will be the desired LHS, i.e.

.

(Provided , because then there is a zero-determinant, which we don't want)

Now, to get the RHS of the equality we are to prove, take a look at the matrix again.

You can see that if we let and we have:

'

This matrix also has a real determinant (i.e. ).

Now, if we set and this means the matrix has the determinant , which is the RHS in the equality.

Now to the messy algebra of determining , , and .

We have and

Therefore follows that

So

So

We also have and

Therefore follows that

So

I'd be grateful if someone checked my interpretation of "linear in a, b, c and d and also linear in p, q, r and s" (David?)

Overall a surprisingly nice question I think (latexing it was pain and there is likely a few silly things:/).

**nota bene**)STEP III Question 8

The determinant of this matrix is

(In hindsight it would have been less algebra to use |AB|=|A||B|)

Now, chose e.g.

and the determinant of the matrix product will be the desired LHS, i.e.

.

(Provided , because then there is a zero-determinant, which we don't want)

Now, to get the RHS of the equality we are to prove, take a look at the matrix again.

You can see that if we let and we have:

'

This matrix also has a real determinant (i.e. ).

Now, if we set and this means the matrix has the determinant , which is the RHS in the equality.

Now to the messy algebra of determining , , and .

We have and

Therefore follows that

So

So

We also have and

Therefore follows that

So

I'd be grateful if someone checked my interpretation of "linear in a, b, c and d and also linear in p, q, r and s" (David?)

Overall a surprisingly nice question I think (latexing it was pain and there is likely a few silly things:/).

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#97

(Original post by

if the particles are perfectly elastic then no energy should be lost in the collision, hence the total energy of the particles must be equal to 2ghk(1 + alpha^2). This was deduced by forming the energy equations of each particle and the constants.

***bobo***)if the particles are perfectly elastic then no energy should be lost in the collision, hence the total energy of the particles must be equal to 2ghk(1 + alpha^2). This was deduced by forming the energy equations of each particle and the constants.

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#98

(Original post by

This is hardly a description of the subsequent motion of the particles.

**brianeverit**)This is hardly a description of the subsequent motion of the particles.

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#99

(Original post by

It is good that you are pointing out mistakes in some of the solutions, but if you are going to you could atleast suggest improvement. This comment is not the slightest bit helpful.

**squeezebox**)It is good that you are pointing out mistakes in some of the solutions, but if you are going to you could atleast suggest improvement. This comment is not the slightest bit helpful.

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#100

Need help with STEP I question 5.

I've found that the coordinates of R are ((p-q)/2, (p^2 - q^2)/2)

So x = (p-q)/2

y = (p-q)(p+q)/2 = x(p+q)

Also, using the information about POQ being a right angle, I found pq = -1 (I won't bother posting my working as it won't really help anything - it's just using Pythagoras several times to find the distance PQ). But I can't deduce from this what the locus of R would be.

For the second part, I got

T (x, y)

x = (p+q)/2

y = 2pq

xy = pq(p+q) = -(p+q)

And again I don't know what to do from here.

I've found that the coordinates of R are ((p-q)/2, (p^2 - q^2)/2)

So x = (p-q)/2

y = (p-q)(p+q)/2 = x(p+q)

Also, using the information about POQ being a right angle, I found pq = -1 (I won't bother posting my working as it won't really help anything - it's just using Pythagoras several times to find the distance PQ). But I can't deduce from this what the locus of R would be.

For the second part, I got

T (x, y)

x = (p+q)/2

y = 2pq

xy = pq(p+q) = -(p+q)

And again I don't know what to do from here.

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