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Unformatted text preview: INTERNATIONAL BACCALAUREATE MOO/510m“)
BACCALAUREAT INTERNATIONAL
BACHILLERATO INTERNACIONAL MATHEMATICS Name
HIGHER LEVEL PAPER 1 F ? Wednesday 3 May 2000 (afternoon) Number 2 hours M INSTRUCTIONS TO CANDIDATES  Write your name and candidate number in the boxes above.  Do not open this 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
to three signiﬁcant ﬁgures as appropriate. ° Write the make and model of your calculator in the box below e. g. Casio fx7400G,
Sharp EL9400, Texas Instruments TI—80 Calculator Make Model EXANHNER TEANILEADER IBCA
TOTAL
M0 TOTAL TOTAL
/60 /60 220281 16 pages , 2 — M00/510/H(1) Maximum marks will be given for correct answers. Where an answer is wrong some marks may be
given for a correct method provided this is shown by written working. Working may be continued below
the box, if necessary. Where graphs from a graphic display calculator are being used to ﬁnd solutions, you should sketch these graphs as part of your answer. 1. (a) Sketch the graph of f (x) = sin 3x + sin 6x , 0 S x S 2112 . (b) Write down the exact period of the function f. Working: 220—281 — 3 ~ M00/510/H(1) 0—1 2. The transformation T1 is represented by the matrix {
—1 0 )and the transformation T2 is represented by the matrix [ (I) 3]. (a) Calculate the matrix (T 1 Tz)‘l . (b) Describe the transformation represented by the matrix (T 1 T 2)’1 . Answers: (a) (b) 220281 Turn over —4— M00/510/H(1) 3. Letzl=a cos£+isinE and22=b cos£+isin£.
4 4 3 3 ll 3
Express (—1) in the form 2 = x + yi . :7 ‘V Working: Answer: 220—28 1 4. A sample of 70 batteries was tested to see how long they last. The results were: Find Time (hours) Number of batteries
(frequency) OSt<10 2 4
_—
——
— 
 (a) the sample standard deviation; M00/510/H(1) (b) an unbiased estimate of the standard deviation of the population from which this sample is taken. Working: Answers: (a) (b) 220281 Turn over —6r M00/510/H(1) 5. Find the coefﬁcient of x7 in the expansion of (2 + 3x)10 , giving your answer as a whole number. 6. The system of equations represented by the following matrix equation has an inﬁnite number of
solutions. 2 ~1 —9 x 7
l 2 3 y = l
2 l —3 z k Find the value of k . 220281 —7— M00/510/H(1) 7. In a game a player rolls a biased tetrahedral (fourfaced) die. The probability of each possible score is shown below. Find the probability of a total score of six after two rolls. Probability Working: 220281 Turn over — 8 — M00/510/H(1) 8. Find a vector that is normal to the plane containing the lines LI and L2 , whose equations are: L]: r=i+k+l(2i+j—2k)
szr:3i+2j+ 2k +u(j+3k) Working: Answer: 9. The sum of the ﬁrst n terms of an arithmetic sequence is S" = 3n2 , 2n . Find the nth term u" . Working: Answer: 220—281 — 9 — M00/510/H(1) y+1_z—3
2 l . Findl. 10. The plane 6x — 2y + z = 11 contains the line x — 1 = Working: J Answer: 220281 Turn over , 10 _ M00/510/H(1) 11. The points 21 =—l + 21 and 22: 2 + 4i and the line segment [21 22] are shown in the complex
plane below. Let 23:421 and z4=—izz.
(a) Plot Z3 and 24 on the complex plane and draw the line segment [23 24] . (b) Write down the transformation that maps the line segment [21 22] onto the line segment
[Z3 Z4] . __
Working: Answer: 0)) 220281 — 11 _ M00/510/H(1) 12. The probability distribution of a discrete random variable X is given by P(X=x)=k(§] ,forx=0,1,2, ...... Find the value of k . 13. The velocity, v, of an object, at a time t, is given by v = keiE in m 8'1 . Find the distance travelled between t: 0 and t: a . , where t is in seconds and v is Working: 220281 Turn over — 12~ M00/510/H(1) 14. Mr Blue, Mr Black, Mr Green, Mrs White, Mrs Yellow and Mrs Red sit around a circular
table for a meeting. Mr Black and Mrs White must not sit together. Calculate the number of different ways these six people can sit at the table without Mr Black
and Mrs White sitting together. Working: A nswer: 15. Find the coordinates of the point which is nearest to the origin on the line L:x=l—l,y=2—3/l,z= . Working: Answer:
L 220—281 ~13 , M00/510/H(1) 16. Given that x > 0 , ﬁnd the solution of the following system of equations: Work ing.‘ Answers: 220281 Turn over — 14— M00/510/H(l) 17. A rectangle is drawn so that its lower vertices are on the x—axis and its upper vertices are on the
curvey=sinx, where OSxSn. (a) Write down an expression for the area of the rectangle. (b) Find the maximum area of the rectangle Working: A nswers: (a)
(b)
[12
18. Find the values of a > 0 , such that] 1 1 2 dx = 0.22.
a + X Working: Answers: 220281 ~15— M00/510/H(1) l9. Letf: xr—>e5i“".
(a) Find f’(x). There is a point of inﬂexion on the graph of f , for O < x < 1 . (b) Write down, but do not solve, an equation in terms of x , that would allow you to ﬁnd the
value of x at this point of inﬂexion. Answers: (a)
(13) 220281 Turn over — 16— M00/510/H(1) 20. The diagram shows the graph of y = f ’(x) . Indicate, and label clearly, on the graph
(a) the points where y =f(x) has minimum points;
(b) the points where y = f (x) has maximum points; (0) the points where y =f(x) has points of inﬂexion. Working: 220281 ...
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 Winter '10
 Chang
 Math, Calculus

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