Hello and Welcome to Advanced GCE Mathematics: Feedback on June 2015 (Unit FP2) Live Online
Event.
Here are the Aims and Objectives for the session. In this training session, you will:
Receive feedback on national performance of candidates in Unit FP2 of the June 2015 examination
series. Consider the variation of candidates' performance
on different questions and possible reasons why.
Discuss the Examiners' reports. Address common issues and FAQs.
Here's the outline that we will be following during this training session:
Introductions, delegate polls and Agenda. Introduction to the FP2 Qualification and
Feedback on the FP2 June 2015 examination. Common issues, further training, any questions.
Finish.
Now we're going to start with the Introduction to FP2.
Here's a screenshot of the Specification. Please download it from the website if you
think it's necessary. The paper was found to be accessible to a
wide range of candidates. The mean score was 58 out of 75.
The work was generally well set out although some handwriting was difficult to read with
some variables being indistinguishable from each other e.g. u, y and x in question 8.
Timing did not appear to be a problem as most candidates seemed to finish the paper. There
was some evidence that candidates spent too long on Question 3.
Some candidates are still attempting to use a formula without quoting it first e.g. the
Taylor Series in question 7. This question tested inequalities.
The most efficient solutions involved either 1: Multiplying through by (x + 3)2 or 2: Collecting
terms to one side and finding a common denominator. Some wasted time with method 1 expanding to
obtain a cubic when there was a common factor of (x + 3).
The most common error was to multiply through by (x + 3) with no consideration of the effect
of the direction of the inequality. The question asked for the use of algebra
- candidates are at liberty to use sketches once they have used algebra to find the critical
values. A graph helped to answer part (b).
Many candidates just gave their answer to part (b) as their answer to part (a).
Efficient solutions involved multiplying through by (x + 3)2
Or collecting terms to one side and finding a common denominator.
Incorrect solutions involved multiplying through by (x + 3) only.
This question tested complex numbers and the application of de Moivre's theorem.
Almost all candidates were able to attempt part (a) and the correct modulus was seen
in the majority of responses. Virtually all candidates gained a mark for using the correct
approach for the argument but obtaining ?/3 was the most common error.
Many candidates could attempt part (b) although a surprising number left the answer un-simplified
as, for example, 4906 (cos 4? + i sin 4?).
Part (c) was answered poorly in relation to (a) and (b). The majority of candidates preferred
to find z3 first and then find the fourth root.
A significant proportion applied de Moivre's theorem and were able to find a single root
but a significant proportion only found 1 further root and so did not gain further marks.
Many candidates were successful in part (a). In part (b) some candidates did not simplify
their answer. In part (c), many candidates stopped after
finding one root. This question tested finding the solution
of a first order differential equation. The first 3 marks were obtained by many candidates
using an appropriate method. The most common error here was obtaining the integrating factor
as cos x. To integrate the RHS, good candidates spotted
that using cos2? = 2cos2? - 1 gave an integrable function
and obtained the required solution quickly and efficiently.
Some correctly integrated by parts once and then used the cos 2? substitution and a smaller
number correctly used a factor formula. A common response was to attempt integration
by parts and to give up after one application. Those who didn't give up battled on usually
obtaining an incorrect integral although many candidates tried several approaches before
abandoning their attempt. This question tested knowledge of the method
of differences. This question really tested the candidates'
mathematical communication skills and many were let down by poor notation and a lack
of reasoning. Most candidates were able to prove the given
result in part (a). The majority opted to expand the two pairs of brackets first and
then either multiply by r2 or simplify. A small number used the difference of two squares.
Many candidates did not gain full marks for part (b). Some candidates treated the identity
as if it were an equation and started by dividing throughout by r2.
Most made some attempt to list terms but some did not list sufficient terms to demonstrate
that cancelling would occur, in particular, candidates had a tendency not to write down
a complete result for r = n. In part (a) the majority opted to expand the
two pairs of brackets then multiply by r2 and simplify.
Many were let down by poor notation and a lack of reasoning and candidates had a tendency
not to write down a complete result for r = n.
The next stage of the process was to extract the terms which had not cancelled out, most
candidates were able to do this correctly although some did not explicitly state n2(n
+ 1)2 and others wasted time expanding this result which was not a necessary step.
Candidates who had been listing terms without considering the LHS of the process often reached
this stage and did not know how to continue, others made mistakes expressing the LHS showing
a general uncertainty about how to deal with the ?.
The final step of the proof proved to be the most problematic as many candidates ended
the proof with the sum of the cubes equals the sum of the squares squared and not the
given result. This was a question on transformations of
the complex plane. Many candidates demonstrated excellent algebraic
skills and worked through part (a) very well, achieving the full 8 marks. Common errors
were forgetting to square either the 2 or the 3 when finding the modulus and these lost
all the subsequent accuracy marks. Some candidates began by using z = x + yi
and found an expression for w, which they then multiplied by the complex conjugate of
their denominator in order to make the denominator real. However, very few got beyond this point
as they were unable to see how to use ?z?= 2 in this context,
In part (b) most got the follow through mark for drawing a circle on an Argand diagram
that was correctly placed for their centre and radius. The last mark required the correct
circle to have been drawn so was not accessible to those who had made errors earlier. For
the shading, a sensible approach was to use the coordinates of the centre of the circle
to find whether it was in the region required. Common errors in part (a) were forgetting
to square either the 2 or the 3 when finding the modulus and these lost all the subsequent
accuracy marks. This was a question on Polar Coordinates.
This question was very well answered. In both parts of the question the majority of candidates
knew the methods needed to solve the problem and were able to apply them accurately and
with good use of mathematical notation. Those who did struggle with part (a) were still
able to achieve some marks in part (b). In part (a) most candidates were aware that
they needed to differentiate rsin? but some struggled with solving the resulting quadratic.
In part (b) the formula for area was used confidently and the majority of candidates
were able to manipulate r2 into an integrable form using the double angle formula for cosine.
The most commonly seen errors in this question were slips such as evaluating r2 as 3a(1 +
cos ?)2 or 3a2(1 + cos ?)2, getting a sign wrong in the double angle formulae, or integrating
cosine to obtain negative sine. This was a question on the Taylor Series.
In part (a) The first derivative was achieved correctly by most candidates using the chain
rule. For the second derivative, most candidates chose to differentiate the first derivative
as it stood rather than to re-write it solely in terms of tan x first.
In part (b) a few candidates gave an expansion in powers of x rather than (x +?/3) and some
did not expand as far as necessary. The final mark was lost by candidates who
otherwise had a perfectly good solution by either writing the answer as
"f(x) =..." where there had been no prior definition of the function f.
The omission of ?3 was sometimes seen in the coefficient of
(x -?/3)3. In part (a) the result was achieved correctly
by many candidates. In part (c) some candidates incorrectly gave
an expansion in powers of x. This question was on second order differential
equations Part (a) was a challenging question for many
candidates although some could achieve the result efficiently.
The first derivative was usually found quickly, easily and correctly; it was the second derivative
that caused the majority of problems. Almost everyone who attempted the second derivative
realised the product rule would be required but some failed to use the chain rule as well
and then struggled to make further progress. There was evidence of many candidates adjusting
their previously incorrect working (scribbling out, over-writing and squeezing in extra variables)
to try to get to the given result. Candidates generally did considerably better
with part (b). A few candidates derived the auxiliary equation rather than simply writing
it down but most examples of this were completely correct.
Forming and solving the auxiliary equation was dealt with confidently and the vast majority
achieved m = 4 and then produced an acceptable expression for the complementary function
which usually appeared in factorised form although occasionally as two separate terms.
When attempting the particular integral most candidates knew what was involved and started
with an appropriate linear expression although there were several examples of careless algebra
and calculation errors which meant that the correct coefficients could not be obtained.
A typical error would be starting with y = au + b, getting to 16a = 2 and stating that a
= 8. A more serious error was candidates' use of y = au or other equivalent linear expressions
without constant terms. This was seen far too often.
Candidates were familiar with the need to combine their complementary function and particular
integral to obtain a general solution and therefore many received the mark here.
A more serious error was candidates' use of y = au or other equivalent linear expressions
without constant terms. This slide shows the distribution of marks
on FP2. Candidates should make sure that they state
formulae before attempting to use them. Candidates should be very careful in the presentation
of their work and make their variables clear. Candidates should be aware when they are spending
too long on a question when they could be scoring marks on other questions.
Candidates can also return to question later if they have time.