# STEP Maths I, II, III 1989 solutions

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**STEP I (Mathematics)**

1:

2: Solution by kabbers

3: Solution by Dystopia

4: Solution by Dystopia

5: Solution by Dystopia

6: Solution by Swayam

7: Solution by squeezebox

8: Solution by squeezebox

9: Solution by nota bene

10:

11: Solution by Glutamic Acid

12:

13:

14:

15:

16:

**STEP II (F.Maths A)**

1: Solution by squeezebox

2: Solution by kabbers

3: Solution by squeezebox

4:

5: Solution by Dystopia

6: Solution by *bobo*

7:

8:

9:

10:

11:Solution by *bobo*

12:

13:

14:Solution by squeezebox

15:

16:

**STEP III (F.Maths B)**

1: Solution by squeezebox

2: Solution by *bobo*

3:

4:

5:

6:

7:

8: Solution by Dystopia

9: Solution by squeezebox

10: Solution by squeezebox

11:

12:

13:

14:

15:

16:

**Solutions written by TSR members:**

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I am no expert at STEP, so don't hesitate to correct me, it is more than likely made that I have made a mistake.

Using good old de Moivre's theorem;

(using

and similary;

Now, consider:

Where c is .

Now let ,

so:

Which gives;

().

These values of give distinct values of

or

()

Hence the roots of the equation are:

and .

__STEP I - Question 8__Using good old de Moivre's theorem;

(using

and similary;

Now, consider:

Where c is .

Now let ,

so:

Which gives;

().

These values of give distinct values of

or

()

Hence the roots of the equation are:

and .

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__STEP III - Question 10__

Lets assume that the result is true for n = k;

(*)

This is the same as (*) except k+1 replaces k. Hence if the result is true for n=k, its true for n= k+1.

When n=1,

LHS of (*) = 1x2x3x4x5 = 120

RHS of (*) = (1/6)x1x2x3x4x5x6 = 120.

So (*) is true for n=1.

Hence, by induction;

.

Since;

.

Using (*);

In this case,

From the previous parts, we know that:

and clearly, also:

as ,

and

So,

Using a similar arguement, we can show that:

We have shown that the limits exist and are equal to .

Hence;

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#4

**STEP I, Q2**The

**For , find**

Using integration by parts ():

**By approximating the area corresponding to by n rectangles of equal width and with their top right hand vertices on the curve , show that, as ,**

Firstly, note that

(by the above indefinite integral)

The sum of the areas of our rectangles is going to be

Claim:

So for k + 1, we must prove that is the equation we arrive at.

Basis case (let k=1):

Inductive step:

So,

As , our sum tends to the integral . So,

is this ok? have i missed anything out or not explained each step in enough detail?

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#5

Question 4, STEP I

With six points, each point is attached to five other points, so at least three lines radiating from A must be of the same colour (gold, say). The points which are joined to A must then be joined to each other by silver lines, or an entirely gold triangle would be made. However, then a triangle with entirely silver edges can be made from these three points.

Diagram showing five points is attached.

With six points, each point is attached to five other points, so at least three lines radiating from A must be of the same colour (gold, say). The points which are joined to A must then be joined to each other by silver lines, or an entirely gold triangle would be made. However, then a triangle with entirely silver edges can be made from these three points.

Diagram showing five points is attached.

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#6

Question 5, STEP I

a)

Let x = 1.

Let x = -1

As n is even. So

As their sum is

b) Suppose that

Then

Note that and that

So, upon adding these two expressions, we get

Also,

So true by induction.

a)

Let x = 1.

Let x = -1

As n is even. So

As their sum is

b) Suppose that

Then

Note that and that

So, upon adding these two expressions, we get

Also,

So true by induction.

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__STEP II Q1__

Substituting x= y-a into the equation:

Notice that the reduced form shown in the question has no term, so we need to find the value of a which will get rid of the term.

Expanding out and collectiing terms we get:

So the value of a to get rid of the is -1.

So we now have:

(*)

Substituting y=z/b into (*);

dividing both sides by 2 and multiplying by ;

.

And so to make this equal to;

we make b = .

Lets take b = and let z =

so;

and (these give distinct values of , and hence, three distinct solutions.)

we know that; x = y - a = (z/b) - a.

Hence the solutions of the equation are:

and

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#8

Question 11, STEP I.

Vertical component =

Using s = ut + 1/2at^2.

Quadratic in t. Ignoring the negative root.

R = horizontal component x time.

Horizontal component =

Multiply both numerator and denominator by 2.

Using

We can simplify it.

Splitting up Mr. Big Fraction.

Ignore the first fraction for the moment, we'll work on the second one.

We can use our old friend Mr. Double Angle Formula to simplify it:

The whole fraction looks like:

And lo and behold, there's a common factor to both fractions. Let's take it out:

And that's what we're looking for.

*I'll finish off the last small bit in a few minutes*

Vertical component =

Using s = ut + 1/2at^2.

Quadratic in t. Ignoring the negative root.

R = horizontal component x time.

Horizontal component =

Multiply both numerator and denominator by 2.

Using

We can simplify it.

Splitting up Mr. Big Fraction.

Ignore the first fraction for the moment, we'll work on the second one.

Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.

`\dfrac{2v^2 \cos \theta \sin \theta}{2g} \times \sqrt{1 + \dfrac{2hg}{v^2 \sin^2 \theta}`

We can use our old friend Mr. Double Angle Formula to simplify it:

The whole fraction looks like:

Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.

`R = (\dfrac{v^2\sin 2\theta}{2g}) + (\dfrac{v^2 \sin 2\theta}{2g}) \sqrt{1 + \dfrac{2hg}{v^2 \sin^2 \theta}`

And lo and behold, there's a common factor to both fractions. Let's take it out:

And that's what we're looking for.

*I'll finish off the last small bit in a few minutes*

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#9

STEP I Question 6

The normal to the curve will therefore have gradient -1/f'(x). The equation of the normal would be

(where x' is the general x coordinate of the normal).

At the point Q, x' = 0 as it cuts the y axis.

The distance PQ can be worked out using Pythagoras' theorem.

It's given that

So

(use a substitution of u = x^2 if you can't see why)

The normal to the curve will therefore have gradient -1/f'(x). The equation of the normal would be

(where x' is the general x coordinate of the normal).

At the point Q, x' = 0 as it cuts the y axis.

The distance PQ can be worked out using Pythagoras' theorem.

It's given that

So

(use a substitution of u = x^2 if you can't see why)

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#10

STEP III, Q8

Let

Then

Using an integrating factor, , we get

Applying boundary conditions:

---

I think that I shall restrict myself to a maximum of three solutions per day (at least at the start), so that everyone has a chance to answer some if they want.

Let

Then

Using an integrating factor, , we get

Applying boundary conditions:

---

I think that I shall restrict myself to a maximum of three solutions per day (at least at the start), so that everyone has a chance to answer some if they want.

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#11

Hang on a sec, mine was mislabelled, should be Step I not step II.. sorry about that!

edit: thanks.. now typing up II/2

edit: thanks.. now typing up II/2

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#12

__STEP II/2__Therefore, by (*),

we can also use (*) to express 2cot2x as

Therefore,

we can compare the coefficients of the respective summations and say

for the second part, note that

we have to find an identity for 2csc2x

(using partial fractions)

Also,

, so:

Comparing coefficients of the polynomials (in x) of the expansion,

Rearranging and substituting in the earlier definition of ,

any corrections welcome !

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__STEP II Question 3__

Equating real and imaginary parts:

(*)

(**)

if x=a, then:

(1)

(2)

Now, subbing (1) and (2) into:

We get:

As required.

Substituting y=b into (*) and (**):

(3)

(4).

This time, using , we end up with:

.

Since and , . Hence, () corresponds to a single hyeprbola branch, which is above the line v=0.

One point of intersection is: and .

Differentiating implicitly we get:

.

Which is equal to: at and .

Differentiating implicitly we get:

.

Which is, when and .

X

Hence the curves intersect and right angles at this point.

Regarding the sketch, I think the elipse becomes a vertical straight line from v = -1 to v = 1, and the hyperbola becomes the line v = 1.

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#17

STEP 1, Q3

(By the ratio theorem.)

These vectors are perpendicular, so the scalar product is zero.

(Note that the scalar product is commutative and distributive.)

Let

Minimum and maximum values occur at the endpoint of a range or at a turning point. There is a turning point when x = 1/2.

So

Note that

Where

The maximum value is occurs when

Suppose that

Then

So

By the cosine rule,

Let F be the midpoint of AB. By Pythagoras,

(By the ratio theorem.)

These vectors are perpendicular, so the scalar product is zero.

(Note that the scalar product is commutative and distributive.)

Let

Minimum and maximum values occur at the endpoint of a range or at a turning point. There is a turning point when x = 1/2.

So

Note that

Where

The maximum value is occurs when

Suppose that

Then

So

By the cosine rule,

Let F be the midpoint of AB. By Pythagoras,

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*bump* (just to remind people that this thread is still here ) - I have solutions that I can type up if no one else wants to, but I really don't want to take over the thread..

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__STEP I - Question 7__

*Graph Attached*

If we work out the area of a quarter of the rectangle, which is in the top right hand bit of the axes, and maximise this, it make calculation soooo much eaiser. (Since we can forget about modulus signs and y being negative) :

Let the area of the rectangle be .

Base of smaller rectangle = 1-x

height =

so

which we want to maximise.

Which is zero when

Finding the second derivative and subsituting x = in shows is a maximum at this point.

So .

________

We can do the second bit in the same way, since its a closed curve that is symmetrical in both axes.

Let the area of the second rectangle be .

Working in the top right quadrant again;

Base = 1-x

height = .

So .

Which is zero when x = 0 or x = .

Clearly the area is a minimum when x= 0, and so is a maximum when x =.

.

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#20

I've got a solution to I question 9 , but don't have a scanner available so will add graphs when I get the opportunity.

Question 9 STEP I

and Setting y'=0

f(2)=-2, f(-2)=2, also for graphing purposes it may be worth to note that f(0)=0 and f(4)=2 and f(-4)=-2. y=0 has solutions x=0 or f''(-2)=- i.e. local max f''(2)=+ i.e. local min

Graph, see attachment

(a) i.e. therefore the equation of y in this X-Y plane is

setting it equal to 0 gives f(1)=-2 ans f(-1)=2

For graphing it can also be good to see that f(2)=2, f(-2)=-2 and

Graph, see attachment

(b) i.e. therefore the equation of y in this X-Y plane is

Setting this equal to 0 gives . f(2)=-1 and f(-2)=1

For graphing, also see that , f(4)=1 and f(-4)=-1

Graph, see attachment.

(c) i.e. therefore the equation in the X-Y plane is

Setting this equal to 0 gives f(0)=2, f(2)=-2.

For graphing, also see that and f(-1)=-2, f(3)=2

(d) i.e. therefore the equation in the X-Y plane is

setting this equal to 0 gives f(2)=0 f(-2)=2

For graphing I'll check f(4) and f(-4) too, which give 2 and 0 respectively.

For the last part, we're looking for a graph in the X-Y plane with local min (0,0) and local max (1,1). To obtain this on the form X=ax+b and Y=cy+d, first observe that for a graph of a cubic, changing the sign means a reflection in the x-axis i.e. local max and min swap places, this is what we need to do. That means either a or c is negative (but not both!). a squeezes the graph along the X-axis and c squeezes it along the Y-axis, c moves the graph to the left/right and d moves the graph up/down. Knowing these we can see that and will produce the desired graph.

Okay, so I hope I've interpreted everything correct in the question. Seems pretty nice as a question.

Question 9 STEP I

and Setting y'=0

f(2)=-2, f(-2)=2, also for graphing purposes it may be worth to note that f(0)=0 and f(4)=2 and f(-4)=-2. y=0 has solutions x=0 or f''(-2)=- i.e. local max f''(2)=+ i.e. local min

Graph, see attachment

(a) i.e. therefore the equation of y in this X-Y plane is

setting it equal to 0 gives f(1)=-2 ans f(-1)=2

For graphing it can also be good to see that f(2)=2, f(-2)=-2 and

Graph, see attachment

(b) i.e. therefore the equation of y in this X-Y plane is

Setting this equal to 0 gives . f(2)=-1 and f(-2)=1

For graphing, also see that , f(4)=1 and f(-4)=-1

Graph, see attachment.

(c) i.e. therefore the equation in the X-Y plane is

Setting this equal to 0 gives f(0)=2, f(2)=-2.

For graphing, also see that and f(-1)=-2, f(3)=2

(d) i.e. therefore the equation in the X-Y plane is

setting this equal to 0 gives f(2)=0 f(-2)=2

For graphing I'll check f(4) and f(-4) too, which give 2 and 0 respectively.

For the last part, we're looking for a graph in the X-Y plane with local min (0,0) and local max (1,1). To obtain this on the form X=ax+b and Y=cy+d, first observe that for a graph of a cubic, changing the sign means a reflection in the x-axis i.e. local max and min swap places, this is what we need to do. That means either a or c is negative (but not both!). a squeezes the graph along the X-axis and c squeezes it along the Y-axis, c moves the graph to the left/right and d moves the graph up/down. Knowing these we can see that and will produce the desired graph.

Okay, so I hope I've interpreted everything correct in the question. Seems pretty nice as a question.

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