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TSR Wiki > Study Help > Exams and Qualifications > A Levels > STEP > STEP I 2007 question 6 solution

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(i)

We are given and also

You can change the into (difference of two squares)

So we now have and we can substitute in the expression for d

We also know that and also which we can substitute into the above equation.

Substituting Y

Substituting X

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We are now told that and also

Let and and also change the range:

We now must try different values for in order to get values for x and y that satisfy our range.

By trial and error, we find that when we get and

This gives us and

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(ii)

As before, we realize that we must reduce the expression on the right. We used the difference of two squares in the previous part and so this time we shall use the different of two cubes.

We now have:

Which is equivalent to

You should, at this point, realize that the left hand side of the above equation should factorize into something with a . We notice this by simply looking at what our end product should be ().

The expansion of is and so we can alter our own equation accordingly:

Which when rearranged gives

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We have also been told that which we can rearrange to and then substitute into the equation we have just worked out.

The above equation, you should notice is now a quadratic equation and we can use the quadratic formula:

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Note that following the same method as above we get

As in part (i) we must simply try values of until we get two positive integers for and .

This occurs when and we get the answer and

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Solution by The Muon