UNIVERSITY OF CAMBRIDGE UNINTENTIONAL EXAMINATIONS General Certificate of Education
Advanced Subsidiary Level and Advanced Level
MATHEMATICS A2 9709/03
Paper 3 Pure Mathematics 3 (P3) November 2008
Up to 1.5 hours
Additional Materials: Answer Booklet/Paper
READ THESE INSTRUCTIONS FIRST
If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your English Name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid (forbidden examination materials)
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the test, hand in ALL the papers you received; both question paper and your worked answers.
Make sure your name is on every answer sheet.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 60.
This document consists of 2 printed pages
[Turn over]
1. Differentiate the following expressions with respect to x:
a) tan² (4x³ – 5) [4]
b) 3x e2x-5 [3]
c) (2x - 5) / (3x² + 7) [3]
2. Solve:
a) tan² x – 5 tan x + 4 = 0 [3]
b) 9 e2x+4 – 7 ex+2 – 16 = 0 [4]
c) |3x – 2| > |x| [3]
d) 3⁴x + 4. 32x - 3 = 0 [3]
3a) Develop an expression for cos 3ø only in terms of cos ø [3]
b) Find all the values of ø that satisfy the equation 4 sin 3x = sin x, -2π ≤ ø ≤ 2π [5]
4.Integrate these expressions
a) e2x-5 dx [1]
b) 3x e2x-5 dx [4]
c) ( (2x - 5) / (3x² + 7) ) dx [4]
5. Write the first four terms of each of these expansions:
a) (p + q) 12 [1]
b) (1 + 3x) 1.5 [3]
c) (1 – x ) -1 [2]
d) Integrate your answer use to part c) and state what expression this is an approximation for.
e) Estimate the % relative error when x is 0.01. [5]
6. The expression x⁴ - x³ – 22 x² - 5x + 75 is to be factorised. Use the factor theorem to find at least one factor and hence find all the factors.[5]
Hence solve e4t - e3t – 22 e2t – 5et + 75 = 0, giving answers to 4 sig,fig. [4]
END OF TEST
Q1 Chain Rule, Product Rule, Quotient Rule
Q2 Three of these are qaudratics (often not seen, but practised in class). The other, (c) , has the solution as the excluded interval, proving too hard for some. They got this wrong in October, too.
Q3 a) Too hard, despite being set as a homework and with repeated instructions to learn these b) was rarely solved with the radians implicit in the question; many missed the sin x factor—and some that found it simply cancelled it from both sides as their first move.
Q4 (a) was the easy mark to be reused in (b), where By Parts is needed (c) needs division—not a nice one, either.
Q5 Again, (a) meant to be easy with integer factors. (b) has a change of sign on the fourth term. (c) is assumed to be easy so that teh log expansion is obvious - it was not seen as part of the question. The majority found an error value without identifying any logs at all—no marks for that.
Q6 Factor theorem must be used, or only 3/5 in first part. Trying ±1,±3, ±5 must produce digits on the page for the marks, not just f(5) =0. Most tried to do division. Quadratic root (predictably) is left over and only the two positive roots can be transferred to the et version of the equation.
One A, one C, two Ds. The other thirty E and U. Oh dear. The A convinces me I haven’t set it all too hard, unfortunately for the others.