Math Sl Past Paper

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  • Topic: Critical point, IB Diploma Programme, Maxima and minima
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IB DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI

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

22077304

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

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

(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...
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