IB DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI 22077204 PROGRAMA DEL DIPLOMA...

IB DIPLOMA PROGRAMMEPROGRAMME DU DIPLÔME DU BIPROGRAMA DEL DIPLOMA DEL BI

MATHL/HP1/ENG/TZ1/XX

mathematicshigher levelPaPer 1

INsTRUcTIONs TO cANDIDATEs

Write your session number in the boxes above.Do not open this examination paper until instructed to do so.Answer all the questions in the spaces provided.Unless otherwise stated in the question, all numerical answers must be given exactly or correct to

three significant figures.

55 minutes0 0

0121

22077204

Thursday, April 17 Mr. Surowski (for Mr. Russell)

Mock Paper 1—No calculator (55 minutes; 54 marks total) Full marks arenot necessarily awarded for a correct answer with no working. Answers must besupported by working and/or explanations.

Section A—Short-response questions. Each question is worth 6 points.

1. Let � be the function defined for � � ��

�by setting ��

(a) Find � ��.

(b) Find the equation of the normal to the curve � � �� at the pointwhere � � . Give your answer in the form � � ��, where �� .

2. Bag 1 contains 4 red cubes and 5 blue cubes. Bag 2 contains 7 red cubesand 2 blue cubes. Two cubes are drawn at random, the first from Bag 1and the second from Bag 2.

(a) Find the probability that the cubes are of the same color.

(b) Given that the cubes selected are of different colors, find the proba-bility that the red cube was selected from Bag 1.

3. Let � and � be complex numbers. Solve the simultaneous equations

� � ��

Give your answers in the form � �� , where �� .

4. The lines �� and �� have parametric equations

��

��

Find the angle between �� and �� at their common point of intersection.

5. Solve the differential equation

��

��

given that � � � when � � ��.

6. The region enclosed by the curves �� and �� , where � � �, isdenoted by �. Given that the area of � is 12, find the value of �.

Section B, Extended-response questions

7. (8 marks) Use mathematical induction to prove that

�� where � � ��

8. The continuous random variable � has probability density function

��

��

�

� � ��for � � � � �

�� otherwise.

(a) (5 marks) Find the exact value of �.

(b) (2 marks) Find the mode of � .

(c) (3 marks) Calculate � ��

End of Mock Paper 1


MATHL/HP1/ENG/TZ1/X





55 minutes0 0

0121

22077204M

Thursday, April 17 Mr. Surowski (for Mr. Russell)

55 minutes

Mr. Surowski (for Mr. Russell)

Mock Paper 2—Calculators allowed (55 minutes; 56 marks total) Un-less otherwise stated, all numerical answers must be exact or correct to threesignificant figures.

9. The random variable � follows a Poisson distribution. Given that� �� , find

(a) (5 marks) the mean of the distribution;

(b) (3 marks) � �� .

10. (a) (5 marks) The line �� passes through the point �� and is per-pendicular to the plane �� . Find the Cartesian equationsof ��.

(b) (3 marks) The line �� is parallel to �� and passes through the point� �� . Find the vector equation of the line ��.

(c) (i) (5 marks) The point � is on the line �� such that��

�� is perpendic-ular to �� and ��. Find the coordinates of �.

(ii) (2 marks) Hence find the distance between �� and ��.

11. Consider the system of equations

��

��

��

(a) (5 marks) Find the set of values of � for which this system of equa-tions has a unique solution.

(b) (6 marks) For each value of � that results in a non-unique solution,find the solution set.

12. Let � � � �� and � � � � where � � ��.

(a) (i) (2 marks) Show that�

��

��

�

��

�

(ii) (3 marks) By expressing both � and � in modulus-argument formshow that

�

��

�

��

�

�

��

(iii) (3 marks) Hence find the exact value of ��

��in the form ��

��

where �� .

(b) (4 marks) Let �

��

, and show that �� .

13. Let � be the region enclosed by the functions �� and �� for� �

��

��

�.

(a) (4 marks) Find the exact values of the �-values of the points of inter-section.

(b) (6 marks) Sketch the functions � and � and clearly shade the region�.Find the exact volume of the solid obtained by revolving � about the

�-axis. (Use the identity ��

.)


MATHL/HP1/ENG/TZ1/X





2207-7204 21 pages

60 minutes0 0

© IBO 2007

0121

22077204M

Monday, April 21 Mr. Surowski and Ms Zhu

Mock Paper 3—Calculators allowed (60 minutes; 64 points total) Full marksare not necessarily awarded for a correct answer with no working. Answers must be supported byworking and/or explanations.

1. (a) Use l’Hopital’s Rule to find

(i) (4 points) ��

��

�� .

(ii) (4 points) ��

��

�� .

(iii) (4 points) Giving a reason, state whether the following argument is correct orincorrect.

“Using l’Hopital’s Rule, ��

��

��

��

�

��

�

��”

2. (9 points) Given that the Maclaurin series for �� is � � �� , find thevalues of �� and �.

3. Consider the infinite series��

�

��

(a) (5 points) Show that the series is convergent.

(b) (i) (5 points) Express�

� � �in partial fractions.

(ii) (5 points) Hence, compute��

�

� � �explicitly.

4. (a) (6 points) Use integration by parts to show that�

��

Consider the differential equation��

��

(b) (6 points) Find an integrating factor.

(c) (6 points) Solve the differential equation, given that � � � when � � �. Givenyour answer in the form � � ��.

(d) (10 points) Find the interval of convergence of the series��

��

��.

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