June07P2

 IB DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI M07/5/MATHL/HP2/ENG/TZ1/XX mathematics Higher level Paper 2 Tuesday 8 May 2007 (morning) instructions to candidate s  Do not open this examination paper until instructed to do so.  Answer all the questions.  Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures. 2207-7205 5 pages 2 hours © IBO 2007 22077205M07/5/MATHL/HP2/ENG/TZ1/XX 2207-7205 – – Please start each question on a new page. Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. In particular, 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. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working. You are therefore advised to show all working. 1. [Maximum mark: 23] Two planes π 1 and π 2 are represented by the equations π λ μ 1 315 2 23 210 : r =  +  −  +   π 2 : 2x − y − 2z = . (a) (i) Find  −  ×   2 23 210 . (ii) S how that the equation of π 1 can be written as x − 2y + 2z =11. [4 marks] (b) S how that π 1 is perpendicular to π 2 . [4 marks] (c) The line l1 is the line of intersection of π 1 and π 2 . Find the vector equation of l1 , giving the answer in parametric form. [5 marks] (d) The line l2 is parallel to both π 1 and π 2 , and passes through P (3 , – 5 , – 1) . Find an equation for l2 in Cartesian form. [3 marks] (e) Let Q be the foot of the perpendicular from P to the plane π 2 . (i) Find the coordinates of Q. (ii) Find PQ. [7 marks]M07/5/MATHL/HP2/ENG/TZ1/XX 2207-7205 – – Turn over 2. [Maximum mark: 24] (a) Using the formula for cos (A+ B) prove that cos2 cos 2 1 2 θ θ = + . [3 marks] (b) Hence, find ∫cos2 x dx . [4 marks] Let f (x) = cos x and g (x) = sec x for x ∈  −   π π 2 2 , . Let R be the region enclosed by the two functions. (c) Find the exact values of the x-coordinates of the points of intersection. [4 marks] (d) Sketch the functions f and g and clearly shade the region R . [3 marks] The region R is rotated through 2π about the x-axis to generate a solid. (e) (i) Write down an integral which represents the volume of this solid. (ii) Hence find the exact value of the volume. [10 marks]M07/5/MATHL/HP2/ENG/TZ1/XX 2207-7205 – – 3. [Total Mark: 26] Part Maximum mark: 18] The time, T minutes, spent each day by students in Amy’s school sending text messages may be modelled by a normal distribution. 30 % of the students spend less than 10 minutes per day. 35 % spend more than 15 minutes per day. (a) Find the mean and standard deviation of T . [6 marks] The number of text messages received by Amy during a fixed time interval may be modelled by a Poisson distribution with a mean of 6 messages per hour. (b) Find the probability that Amy will receive exactly 8 messages between 16:00 and 18:00 on a random day. [3 marks] (c) Given that Amy has received at least 10 messages between 16:00 and 18:00 on a random day, find the probability that she received 13 messages during that time. [5 marks] (d) During a 5-day week, find the probability that there are exactly 3 days when Amy receives no messages between 17:45 and 18:00 . [4 marks] Part B [Maximum mark: 8] Twenty candidates sat an examination in French. The sum of their marks was 826 and the sum of the squares of their marks was 34 132 . Two candidates sat the examination late and their marks were a and b . The new mean and variance were calculated, giving the following results: mean = 2 and variance = 32 . Find a set of possible values of a and b . [8 marks]M07/5/MATHL/HP2/ENG/TZ1/XX 2207-7205 – – 4. [Total Mark: 21] Part Maximum mark: 11] (a) Find the probability that a number, chosen at random between 200 and 800 inclusive, will be a multiple of 9. [3 marks] (b) Find the sum of the numbers between 200 and 800 inclusive, which are multiples of 6, but not multiples of 9. [8 marks] Part B [Maximum mark: 10] Prove by induction that 12n + 2(5n−1) is a multiple of 7 for n ∈ +  . [10 marks] 5. [Maximum mark: 26] (a) (i) Factorize t3 − 3t2 − 3t +1, giving your answer as a product of a linear factor and a quadratic factor. (ii) Hence find all the exact solutions to the equation t3 − 3t2 − 3t +1 = 0 . [5 marks] (b) Using de Moivre’s theorem and the binomial expansion (i) show that cos3θ = cos3θ − 3cosθ sin2 θ ; (ii) write down a similar expression for sin 3θ . [7 marks] (c) (i) Hence show that tan tan tan tan 3 3 1 3 3 2 θ θ θ θ = − − . (ii) Find the values of θ , 0 ≤θ ≤180, for which this identity is not valid. [7 marks] (d) Using the results from parts (a) and (c), find the exact values of tan15 and tan 75 . [7 marks]