Click here for the paper and the mark scheme and read on for comments.

**Comments**

1. You did much better than in the exam at the start of term. I’m not allowed to give you your marks, but I am allowed to give you back your papers without the marks on, and you can see there what’s been marked wrong and what’s been marked right.

2. There were no parabola and hyperbola questions in the paper you sat. That is because it was a “Silver” paper, a practice paper designed by Edexcel to be only moderately difficult; and on the whole parabola and hyperbola questions are the ones which cause most difficulty to FP1 students. It makes more even more pleased with the results than I would be otherwise because we have spent so much time in recent weeks on getting up to the mark on parabola and hyperbola questions. Thus the test was mostly about things you have had less (though some) practice with recently. The real exam will have parabola and hyperbola questions.

3. The other reason to be pleased was that many of the lost marks, even for those of you who did less well, were lost through little slips – “losing” a minus between one line of working and the next, having your calculator in degrees when it should have been in radians, not reading the question carefully – rather than through misunderstanding the mathematical method. With practice, you can all learn to avoid those slips, and improve your marks a lot.

4. It looked to me as if you all did the paper starting with question 1 and then going through the questions in order. It’s better, I think, to take out five or ten minutes to read the paper and rank the questions in the order of which you find easiest, and do those you find easiest first. In this paper, for example, a lot of you obviously found question 1 harder than others, and it’s best to build your confidence by starting with a question you find easy.

Question 1. Series

Everyone had the basically the right method. The most common way of losing marks was a slip in expanding r(r+1)(r+3). It’s r^3+4r^2+3r, not e.g. r^3+4r^2+3

If you make a slip like that, then you see you’re not getting the result you want. And that’s good. You have a warning that you made a slip somewhere. Look back and find it.

Some who made that slip in part (a) didn’t attempt (b). That’s sort of understandable, since u didn’t know what k was.

But think: you could find k just by putting n=1

so 1 x 2 x 3 = (1/12) (1) (2) (3) (3+k)

so you could have found k=13 even if you couldn’t find your slip in part (a)

and then got your two easy marks for part (b).

Question 2. Newton-Raphson

Everyone was good on the basic method. Marks were lost only through slips.

Question 3. Matrices

Almost everybody was good with finding A^2 and describing it (a couple of people made slips such as losing the minus sign in the bottom right element of A).

Almost everyone was good on identifying the geometrical transformation.

Absolutely everyone was good on (c), finding k for which C is singular.

Question 4. Complex numbers

This question is actually easier than the usual ones about solving quartics or cubics, but it’s a little unusual, so some of you got tangled up.

Sum-of-roots and product-of-roots are quick and efficient methods if you already have one root, and if the equation isn’t factorised.

But this equation is factorised! Or, at least, half-factorised.

If you have an equation (x+2)(x-3)=0, you don’t multiply it out in order to solve it. You just say x is -2 or 3. Factorising equations is essentially the same thing as solving them.

So the answer here is to solve

4x^2+9 = 0 to get two roots

And x^2-2x+5=0 to get the other two.

Question 5. Matrices

Almost everyone was good on working out R^2.

A few of you had enlargement with scale 15 as

0 15

15 0

But it’s

15 0

0 15

Rough rule of thumb: the leading diagonal (top left to bottom right) tells you what the matrix does in terms of scaling and enlargement; the other diagonal (top right and bottom left) tells you what it does in terms of rotating and shearing.

Question 6. Linear interpolation

Almost everyone had the method right. Where students lost marks, it was through having their calculators in degrees rather than radians.

When the question tells you –π<x<π

Then that tells it’s working in radians.

Numerical methods questions with sin, cos, or tan in them are always working in radians.

If you get an answer which looks weird in a numerical methods question, always check whether your calculator is set correctly to radians.

Question 7. Matrices

Apart from one of you who didn’t attempt this question, and one who made a slip to get a wrong sign in the determinant, everyone found the value of a for A to be singular correctly.

One of you wrote that he had forgotten what the inverse matrix B^(-1) is! With that embarrassing exception all the lost marks in part (b) were slips in working.

A few of you got tangled up in part c.

You can either work out B^(-1) Q, with Q as a column vector, and show it comes out as the column vector with k and k+3

Or put P as the column vector with x and y, equate BP = Q, and eliminate k.

Questions 8 and 9.

Almost everyone was good with the induction questions, with the odd slip here and there.

We need a bit more work in being careful about what words you write to explain your work in induction questions, or otherwise you could lose marks with a grumpy marker.