IB DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI

M07/5/MATME/SP2/ENG/TZ1/XX

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mathematics staNDaRD level PaPeR 2 Tuesday 8 May 2007 (morning) 1 hour 30 minutes

INSTRUcTIONS TO cANDIDATES not open this examination paper until instructed to do so. Do Answer all the questions. Unless otherwise stated in the question, all numerical answers must be given exactly or correct to three significant figures.

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8 pages © IBO 2007

http://www.xtremepapers.net

–2–

M07/5/MATME/SP2/ENG/TZ1/XX

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. [Total mark: 25] Part a [Maximum mark: 10]

The following diagram shows part of the graph of a quadratic function, with equation in the form y = ( x − p) ( x − q) , where p , q ∈ .

(a)

Write down (i) (ii) the value of p and of q ; the equation of the axis of symmetry of the curve. [3 marks] [3 marks]

(b)

Find the equation of the function in the form y = ( x − h) 2 + k , where h , k ∈ . Find dy . dx

(c)

[2 marks]

(d)

Let T be the tangent to the curve at the point (0 , 5) . Find the equation of T.

[2 marks]

(This question continues on the following page)

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–3– (Question 1 continued) Part B [Maximum mark: 15]

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The function f is defined as f ( x) = e x sin x , where x is in radians. Part of the curve of f is shown below.

There is a point of inflexion at A, and a local maximum point at B. The curve of f intersects the x-axis at the point c. (a) (b) Write down the x-coordinate of the point c. (i) (ii) (c) (d) Find f ′( x) . Write down the value of f ′( x) at the point B. [4 marks] [2 marks] [1 mark]

Show that f ′′( x) = 2e x cos x . (i) (ii) Write down the value of f ′′( x) at A, the point of inflexion. Hence, calculate the coordinates of A.

[4 marks]

(e)

Let R be the region enclosed by the curve and the x-axis, between the origin and c. (i) (ii) Write down an expression for the area of R. Find the area of R. [4 marks]

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turn over

–4– 2. [Maximum mark: 14]

M07/5/MATME/SP2/ENG/TZ1/XX

The following diagram shows the triangle AOP, where OP = 2 cm , AP = 4 cm and AO = 3 cm .

diagram not to scale

(a)

calculate AOP , giving your answer in radians.

[3 marks]

The following diagram shows two circles which intersect at the points A and B. The smaller circle C1 has centre O and radius 3 cm, the larger circle C2 has centre P and radius 4 cm , and OP = 2 cm . The point D lies on the circumference of C1 and E on the circumference of C2 . Triangle AOP is the same as triangle AOP in the diagram above.

diagram not to scale

(b)

Find AOB , giving your answer in radians.

[2 marks]

(This question continues on the following page)

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–5– (Question 2 continued) (c)

Given that APB is 1.63 radians, calculate the area of

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(i) (ii) (d)

sector PAEB ; sector OADB . [5 marks]

The area of the quadrilateral AOBP is 5.81 cm 2 . (i) (ii) Find the area of AOBE . Hence find the area of the shaded region AEBD . [4 marks]

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turn over

–6– 3. [Maximum mark: 12] A pair of fair dice is thrown. (a)

M07/5/MATME/SP2/ENG/TZ1/XX

copy and complete the tree diagram below, which shows the possible outcomes.

[3 marks] Let E be the event that exactly one four occurs when the pair of dice is thrown. (b) calculate P ( E ) . [3 marks]

The pair of dice is now...