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Ib Matme Sl1 2010

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M10/5/MATME/SP1/ENG/TZ1/XX 22107303 mathematics staNDaRD level PaPeR 1 Wednesday 5 May 2010 (afternoon) 1 hour 30 minutes iNSTrucTioNS To cANdidATES ? Write your session number in the boxes above. ? not open this examination paper until instructed to do so. do ? are not permitted access to any calculator for this paper. You ? Section A: answer all of Section A in the spaces provided. ? Section B: answer all of Section B on the answer sheets provided. Write your session number on each answer sheet, and attach them to this examination paper and your cover sheet using the tag provided.

At the end of the examination, indicate the number of sheets used in the appropriate box on your cover sheet. ? unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures. 0 0 candidate session number 2210-7303 11 pages © international Baccalaureate organization 2010 0111 –2– M10/5/MATME/SP1/ENG/TZ1/XX Full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations. Where an answer is incorrect, some marks may be given for a correct method, provided this is shown by written working.

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You are therefore advised to show all working. Section a Answer all the questions in the spaces provided. Working may be continued below the lines, if necessary. 1. [Maximum mark: 7] Let f ( x) = 8 x ? 2 x 2 . Part of the graph of f is shown below. (a) (b) Find the x-intercepts of the graph. (i) (ii) Write down the equation of the axis of symmetry. Find the y-coordinate of the vertex. [4 marks] [3 marks] ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. 2210-7303 0211 –3– 2. [Maximum mark: 6] ? 2? ?1 3 2? ? ? and P = ? 3 ? . Let W = ? 2 0 1 ? ? ? ?1? ? 0 1 3? ? ? ? ? (a) (b) Find WP. ? 26 ? ? ? Given that 2WP + S = ? 12 ? , find S. ? 10 ? ? ? M10/5/MATME/SP1/ENG/TZ1/XX 3 marks] [3 marks] ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. 2210-7303 turn over 0311 –4– 3. [

Maximum mark: 6] (a) (b) Expand (2 + x) 4 and simplify your result. 1 ? ? Hence, find the term in x 2 in (2 + x) 4 ? 1 + 2 ? . ? x ? M10/5/MATME/SP1/ENG/TZ1/XX [3 marks] [3 marks] ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. 210-7303 0411 –5– 4. [Maximum mark: 7] The straight line with equation y = (a) (b) M10/5/MATME/SP1/ENG/TZ1/XX 3 x makes an acute angle ? with the x-axis. 4 [1 mark] Write down the value of tan ? . Find the value of (i) (ii) sin 2? ; cos 2? . [6 marks] ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. 2210-7303 turn over 0511 –6– 5. [Maximum mark: 6] M10/5/MATME/SP1/ENG/TZ1/XX Consider the events A and B, where P (A) = 0. , P (B ) = 0. 7 and P ( A ? B) = 0. 3 .

The Venn diagram below shows the events A and B, and the probabilities p, q and r. A B p q r (a) Write down the value of (i) (ii) p; q; [3 marks] [2 marks] [1 mark] (iii) r. (b) (c) Find the value of P ( A | B? ) . Hence, or otherwise, show that the events A and B are not independent. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. 2210-7303 0611 –7– 6. [Maximum mark: 6] M10/5/MATME/SP1/ENG/TZ1/XX The graph of f ( x) = 16 ? x 2 , for ? 2 ? x ? 2 , is shown below. y 5 4 3 2 1 –3 –2 –1 0 –1 –2 –3 –4 –5 1 2 3 x The region enclosed by the curve of f and the x-axis is rotated 360? about the x-axis. Find the volume of the solid formed. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. 2210-7303 turn over 0711 –8– 7. [Maximum mark: 7] Let f ( x) = log 3 x , for x > 0 . (a) (b) Show that f ? 1 ( x) = 32 x . Write down the range of f ? 1 . M10/5/MATME/SP1/ENG/TZ1/XX [2 marks] [1 mark]

Let g ( x) = log 3 x , for x > 0 . (c) Find the value of ( f ? 1 ? g ) (2) , giving your answer as an integer. [4 marks] ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. …………………………………………………………. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. ………………………………………………………….. 2210-7303 0811 –9– do NOT write on this page. Section B M10/5/MATME/SP1/ENG/TZ1/XX Answer all the questions on the answer sheets provided. Please start each question on a new page.

8. [Maximum mark: 14] 1 Let f ( x) = x3 ? x 2 ? 3 x . Part of the graph of f is shown below. y A x B There is a maximum point at A and a minimum point at B (3 , ? 9) . (a) (b) Find the coordinates of A. Write down the coordinates of (i) (ii) the image of B after reflection in the y-axis; ? ?2 ? the image of B after translation by the vector ? ? ; ? 5? [8 marks] (iii) the image of B after reflection in the x-axis followed by 1 a horizontal stretch with scale factor . 2 [6 marks] 2210-7303 turn over 0911 – 10 – do NOT write on this page. 9. [Maximum mark: 13] Let f ( x) = (a) (b) cos x , for sin x ? 0. sin x ? 1 . sin 2 x M10/5/MATME/SP1/ENG/TZ1/XX Use the quotient rule to show that f ? x) = Find f ?? ( x) . [5 marks] [3 marks] ??? ??? In the following table, f ? ? ? = p and f ?? ? ? = q . The table also gives approximate ? 2? ?2? ? values of f ? ( x) and f ?? ( x) near x = . 2 x f ? ( x) f ?? ( x) (c) (d) ? ? 0. 1 2 –1. 01 0. 203 ? 2 p q ? + 0. 1 2 –1. 01 – 0. 203 [3 marks] Find the value of p and of q. Use information from the table to explain why there is a point of inflexion on the ? graph of f where x = . 2 [2 marks] 2210-7303 1011 – 11 – do NOT write on this page. 10. [Maximum mark: 18] M10/5/MATME/SP1/ENG/TZ1/XX ? ? 3 ? ? 2? ? ? The line L1 is represented by the vector equation r = ? 1 ? + p ? 1 ? . ? ? ? ?25 ? ? ? 8 ? ? ? ? ? A second line L 2 is parallel to L1 and passes through the point B(? 8 , ? 5 , 25) . (a) Write down a vector equation for L 2 in the form r = a + tb . [2 marks] ?5? ? ? 7 ? ? ? ? ? A third line L 3 is perpendicular to L1 and is represented by r = ? 0 ? + q ? ?2 ? . ? 3? ? k ? ? ? ? ? (b) Show that k = ? 2 . [5 marks] The lines L1 and L 3 intersect at the point A. (c) Find the coordinates of A. [6 marks] ? 6 ? ? ? The lines L 2 and L 3 intersect at point C where BC = ? 3 ? . ? ?24 ? ? ? > (d) (i) (ii) Find AB . Hence, find |AC | . > > [5 marks] 2210-7303 1111

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Ib Matme Sl1 2010. (2016, Oct 14). Retrieved from https://graduateway.com/ib-matme-sl1-2010/

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