IB DIPLOMA PROGRAMME
PROGRAMME DU DIPLÔME DU BI
PROGRAMA DEL DIPLOMA DEL BI
N05/5/MATHL/HP2/ENG/TZ0/XX
MATHEMATICS
HIGHER LEVEL
PAPER 2
Friday 4 November 2005 (morning)
INSTRUCTIONS TO CANDIDATES
Do not open this examination paper until instructed to do so.
Answer all five questions from Section A and one question from Section B.
Unless otherwise stated in the question, all numerical answers must be given exactly or to three significant figures.
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11 pages
3 hours
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Please start each question on a new page. You are advised to show all working, where possible. Where an answer is wrong, some marks may be given for correct method, provided this is shown by written working. Solutions found from a graphic display calculator should be supported by suitable working, e.g. if graphs are used to find a solution, you should sketch these as part of your answer.
SECTION A
Answer all five questions from this section.
1. [Maximum mark: 12]
(a) Given that xxxaxbxcx2221111()()()()++≡++++, calculate the value of a,
of b and of c.
[5 marks]
(b) (i) Hence, find Ixxxx=++∫2211()()d.
(ii) If I=π4 when x=1, calculate the value of the constant of integration
giving your answer in the form pqr+ln where pqr,,∈¡.
[7 marks]
2. [Maximum mark: 16]
(i) (a) Let M=−−−1345111kk. Find detM.
[2 marks]
(b) Find the values of k for which the following system of equations does not have a unique solution.
−−+=−++=−+=xkyzxyzxykz314521
[3 marks]
(ii) The plane π contains the line xyz−=−=−121356 and the point (1, −2, 3).
(a) Show that the equation of π is 6237xyz+−=−.
[7 marks]
(b) Calculate the distance of the plane π from the origin.
[4 marks]
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Turn over
3. [Maximum mark: 17]
(i) In a game a player pays an entrance fee of $ n. He then selects one number from 1, 2, 3, 4, 5, 6 and rolls three standard dice.
If his chosen number appears on all three dice he wins four times his
entrance fee.
If his number appears on exactly two of the dice he wins three times the entrance fee.
If his number appears on exactly one die he wins twice the entrance fee.
If his number does not appear on any of the dice he wins nothing.
(a) Copy and complete the probability table below.
[4 marks]
Profit ($)
−n
n
2n
3n
Probability
75216
(b) Show that the player’s expected profit is $ −17216n.
[2 marks]
(c) What should the entrance fee be so that the player’s expected loss per game
is 34 cents?
[2 marks]
(ii) (a) Use mathematical induction to prove that
12121211()(),rrnnnrn−+=+∈=+Σ¢.
[6 marks]
(b) Hence show that the sum of the first ()n+1 terms of the series 13115135163++++... is ()()nn++123.
[3 marks]
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4. [Maximum mark: 10]
(a) Write down the term in xr in the expansion of ()xhn+, where 0≤≤∈+rnn,¢.
[1 mark]
(b) Hence differentiate xnn,∈+¢, from first principles.
[5 marks]
(c) Starting from the result xxnn×=−1, deduce the derivative of xnn−+∈,¢.
[4 marks]
5. [Maximum mark: 15]
(i) The complex numbers zz12 and are z12=+i, z23=+i.
(a) Find zz12, giving your answer in the form abab+∈i,,¡.
[1 mark]
(b) The polar form of z1 may be written as 512,arctan.
(i) Express the polar form of zzz212, in a similar way.
(ii) Hence show that π41213=+arctanarctan.
[5 marks]
(ii) A man PF is standing on horizontal ground at F at a distance x from the bottom of a vertical wall GE. He looks at the picture AB on the wall. The angle BPA is θ.
Let DA=a, DB=b, where angle PDE is a right angle. Find the value of x for which tanθ is a maximum, giving your answer in terms of a and b. Justify that this value of x does give a maximum value of tanθ.
[9 marks]
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Turn over
SECTION B
Answer one question from this section.
Statistics
6. [Maximum mark: 30]
(i) Let X and Y be two independent variables with E()X=5, Var()X=3, E()Y=4, Var()Y=2. Find
(a) E()2X;
(b) Var()2X;
(c) E()32XY−;
(d) Var()32XY−.
[4 marks]
(ii) (a) Two samples are drawn from a normal population that has an unknown mean μ and unknown variance σ2.
The results of the first sample are given in the following table.
x9.19.29.39.49.5Frequency1216192315
The second sample of 72 items gave the following results
xx==ΣΣ669662282. and .
Use the two samples to calculate an estimate for μ and for σ2.
[7 marks]
(b) Based on the combined sample data find a 95 % confidence interval for μ.
[4 marks]
(This question continues on the following page)
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(Question 6 continued)
(iii) (a) A sample of size n is drawn from a population which is normally distributed with mean μ and variance σ2. Describe fully how the sample mean
is distributed.
[2 marks]
(b) A machine shop manufactures steel rods for use in a car production plant. The lengths in metres for a sample of 8 rods are given below.
0.999, 1.001, 1.005, 1.011, 1.005, 1.001, 0.998, 1.004
Previous observations have shown that the machine settings gave rods with lengths that are normally distributed with standard deviation 0.0028 m.
Stating the type of test used, determine at the 1 % significance level if the mean length of the rods produced is 1.005 m.
[7 marks]
(iv) A six-sided die is thrown 300 times and the outcomes recorded in the
following table.
Score123456Frequency455751564744
Perform a suitable test at the 5 % significance level to determine if the die
is fair.
[6 marks]
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Sets, Relations and Groups
7. [Maximum mark: 30]
(i) Use Venn diagrams to show that
(a) ABAAB∪∩′′=∪′();
[2 marks]
(b) ()ABB∩′∪()′=∅.
[2 marks]
(ii) Let M be the set of all matrices of the form 101x where x∈R.
(a) Show that (M, +) is not a group.
[1 mark]
(b) Show that M forms an abelian group under matrix multiplication.
(You may assume that matrix multiplication is associative).
[5 marks]
(iii) The set Sabcd={,,,} forms a group under each of two operations # and ∗, as shown in the following group tables.
#abcdaabcdbbcdaccdabddabc
(a) Copy and complete the second table.
[4 marks]
(b) Solve the following equations for x.
(i) (bxcd#)∗=.
(ii) axbcb∗()∗=(#).
[5 marks]
(iv) Let max,xy() be equal to the largest of x and y. Define the relation R on the xy plane by
(,)(,)max,max,abRpqabpq⇔()=().
(a) Show that the relation R is an equivalence relation.
[6 marks]
(b) (i) Find the equivalence classes.
(ii) Hence describe the equivalence classes.
[5 marks]
∗
a
b
c
d
a
b
a
b
d
b
c
c
d
a
b
d
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Discrete Mathematics
8. [Maximum mark: 30]
(i) Using Euclid’s algorithm, show that 64 and 33 are relatively prime.
[3 marks]
(ii) Explain why Z is not well-ordered.
[2 marks]
(iii) Consider the following graph.
(a) Show that this graph has
(i) an Eulerian circuit;
(ii) a Hamiltonian cycle.
[3 marks]
(b) The edge joining V2 and V6 is removed. Does the graph still have an Eulerian circuit and a Hamiltonian cycle? Give reasons for
your answers.
[3 marks]
(c) Replace the edge joining V2 and V6, and remove the edge joining V1
and V2.
(i) Find an Eulerian trail.
(ii) Find a Hamiltonian path.
[4 marks]
(This question continues on the following page)
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Turn over
(Question 8 continued)
(iv) The Fibonacci sequence is defined by the recurrence relation
uuunnn=+−−12, for n≥2 and uu011==.
(a) Write out the first eight terms of the sequence.
[1 mark]
(b) Solve the recurrence relation to obtain the formula
unnn=+−−++1515215211, for n≥0.
[8 marks]
(c) (i) Hence show that un is also equal to the closest integer to 151521++n.
(ii) Given that un=102334155, find the value of n.
[6 marks]
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Analysis and Approximation
9. [Maximum mark: 30]
(i) (a) State the mean value theorem and illustrate it with the aid of a sketch.
[2 marks]
(b) Use the mean value theorem to prove that if ′=fx()0 for all x in a closed interval then f is constant on that interval.
[2 marks]
(ii) (a) Find 3502xxd∫.
[1 mark]
(b) Use Simpson’s Rule with four sub-intervals to approximate 3502xxd∫.
[4 marks]
(c) What is the error in this approximation?
[1 mark]
(d) How many sub-intervals are necessary for the error to be less than 0.0001?
[5 marks]
(iii) (a) (i) Prove that the alternating series given by ()()!−−−=∞Σ112111nnn
converges.
(ii) Approximate the series by finding the 4th partial sum. Give your answer to six decimal places.
(iii) What is the upper bound for the error in this approximation?
[7 marks]
(b) (i) Find the first four non-zero terms of the Maclaurin series for sinx.
(ii) Deduce the nth term of this series.
(iii) Use the ratio test to show that the series is convergent for all values of x.
(iv) Use your series for sinx to find the first four non-zero terms of the Maclaurin series for cosx.
[8 marks]
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Euclidean Geometry and Conic Sections
10. [Maximum mark: 30]
(i) Consider the triangle ABC, where C is a right angle. Let D be the point on [AB] such that (CD) is perpendicular to (AB).
(a) Show that triangle ADC is similar to triangle CDB.
[2 marks]
(b) Hence show that CD=ADBD2×.
[2 marks]
(ii) If S is the midpoint of the base [QR] of a triangle PQR, prove Apollonius’ theorem
PQPRPSQS2222+=+2().
[8 marks]
(iii) (a) In the xy plane a particle moves such that xtt=−1263 and yt=122,
where t is time in seconds. Sketch the path of the particle over the interval 04≤≤t. Indicate clearly on the sketch the direction of motion of
the particle.
[5 marks]
(b) Show that the equation of the tangent to the curve at tt=1 can be expressed in the form −+−=+464121121412txyttt().
[7 marks]
(iv) The harmonic mean of p and q is given by 2pqpq+.
Let the points C and D divide the line [AB] such that ACCB=ADBD, as shown in
the diagram below.
Show that AB is the harmonic mean of AC and AD.
[6 marks]