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Unformatted text preview: Paper 1 Candidate name: QUESTIONS ANSWERED M99/510/HU) INTERNATIONAL BACCALAUREATE
BACCALAUR EAT INTERNATIONAL
BACHILLERATO INTERNACIONAL
MATHEMATICS Higher Level Tuesday 4 May 1999 (afternoon) 2 hours Candidate category & number: _
IIIIll.. This examination paper consists of 20 questions.
The maximum mark for each question is 4.
The maximum mark for this paper is 80. INSTRUCTIONS TO CANDIDATES
Write your candidate name and number in the boxes above.
Do NOT open this examination paper until instructed to do 50. Answer ALL questions in the spaces provided. Unless otherwise stated in the questiou, all numerical answers must be given exactly or to three
signiﬁcant ﬁgures as appropriate. TEAM LEADER EXAMINER
TOTAL
ISO EXAMINATION MATERIALS Required: IB Statistical Tables Calculator Ruler and compasses Allowed: A simple translating dictionary for candidates not working in their own language
Millimetre square graph paper 229281 17 pages Trigonometrical identities: Integration by parts: Standard integrals: Statistics: FORIVIULAE sin(a + [3) = sin a 00513 + cosa sin B cos(a + L?) = cos 0: cos B — sin a sin [3 tanm+m= tana+tanﬁ
l—tanatanf)’
sina+sinﬁ=2sina+Bcosawﬁ
2 2
sina—sinﬁ=2cosa+ﬁsina_ﬁ
' 2 2
cosa+cosf3=2cosa+ﬂcosaﬂ
2 2
cosa—cosﬁ=25ina:ﬁsinﬁ;a c0526=200526—1=1—25in26=c0526—sin29 Iftan2 = I thensin9 2 2t 2 andcosﬂ =
2 1+:
uﬂdx = uv— vglidx
dx dx dx 1 x
l 2 =—arctan—+c
x +a a a dx . x
= arcsm—+c (x< a)
Qaz—x2 ‘1 If (x1 , x2 , . . . , x”) occur with frequencies (fl , f2 , . standard deviation 5 are given by m Binomial distribution: 229281 __ Ef'xi . S = 2m —m)2
21: £1? ’ px 2F] px(l—p)”'x, x =0, l, 2,..., x 1—:2
l+t2 M99/510/H(l) ..,fn) then the mean m and M 1,2,... — 3 — M99ISIOIH(I) 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, necessary, or an extra sheets of paper provided these are securer fastened to this
examination paperl 1. When the function f(x) = 6x4 + 1 1x3  22):2 + ax + 6 is divided by (x + l) the remainder is —20 .
Find the value of a . Working: 2. A bag contains 2 red balls, 3 blue balls and 4 green balls. A ball is chosen at random from the
bag and is not replaced. A second ball is chosen. Find the probability of choosing one green
ball and one blue ball in any order. 229281 Turn over _ 4 — M99/510/H(l) 3. The second term of an arithmetic sequence is 7. The sum of the ﬁrst four terms of the
arithmetic sequence is 12. Find the ﬁrst term, a, and the common difference, at, of the
sequence. Working: Answers: 4. Find the coordinates of the point where the line given by the parametric equations x = 2/1 + 4 ,
y = A 2 , z = 3/1 + 2 , intersects the plane with equation 2x + 3y — z = 2 . Working: 22928i _ 5 _ M99/510/H(l) 5. Let z=x +yi. Find the values ofx and y if (1 i)z= 1 3i. 'Answers.‘ 6. Find the value of a for which the following system of equations does not have a unique
solution. 4x— y+22 =1
2x+3y =—6 7
25—2 +az=—
y 2 Working: 229231 Tum over 6_ M99/510/H(l) 7. The diagram below shows the graph of yl =f(x) . The xaxis is a tangent to f(x) at x: m and
f(x) crosses the xaxis at x = n . y1=f(x) On the same diagram sketch the graph of y2 nﬂxk), where 0 < k< n—m and indicate the
coordinates of the points of intersection of y2 with the xaxis. Working: 229281 —7— M99/510/H(1) 8. In a bilingual school there is a class of 21 pupils. In this class, 15 of the pupils speak Spanish
as their ﬁrst language and 12 of these 15 pupils are Argentine. The other 6 pupils in the class
speak English as their ﬁrst language and 3 of these 6 pupils are Argentine. A pupil is selected at random from the class and is found to be Argentine. Find the probability
that the pupil speaks Spanish as his/her ﬁrst language. Working:
9. If 2):2 — 3y2 = 2 , ﬁnd the two values of 2}: when x = 5 .
dx Working: Answers: 229281 Turn over — 8 — M99/510/H(1) 10. (3) Find a vector perpendicular to the two vectors: —) —> —) —)
0P: 1' —3j+2k _) ) —) D
0Q=2i+j—k —> —>
(b) If 01’ and 0Q are position vectors for the points P and Q, use your answer to part (a), or otherwise, to ﬁnd the area of the triangle OPQ . Answers: (a)
(b) 11. Differentiate y = arccos (1 — 2x2) with respect to x , and simplify your answer. Working: 229281 9 M99/510/H(1) 12. Given f(x)=x2 + x(2—k) +k2, ﬁnd the range of values of k for which f(x) > 0 for all real
values of x . 229281 Turn over — 10 — M99/510/H(I) 13. The area of the enclosed region shown in the diagram is deﬁned by yaic2+2,y$ax+2,wherea>0. This region is rotated 360° about the xaxis to form a solid of revolution. Find, in terms of a ,
the volume of this solid of revolution. 229281 — ll — M99/510/H(1) . . . 1 . .
14. Usmg the substntuuon u = 3x + l, or otherwzse, ﬁnd the Integral Jlxqiix+l dx.
2 Working: 229281 Turn over _ 12.. M99/510/H(l) 15. On the diagram beiow, draw the locus of the point P(x , y), representing the complex number
z=x+yi, given thatiz—4—3i1=z—2+i. Working: 229281 —13— M99/510/H(1) 16. Given that
(1+x)5(1+ax)5sl+bx+10x2+ . _ . . . ..+a6x”, ﬁnd the values of a , b e 2* . Working: Answers: 229281 Turn over — 14— M99/510/H(I) 17. A biased die with four faces is used in a game. A player pays 10 counters to roll the die. The
table below shows the possible scores on the die, the probability of each score and the number
of counters the player receives in return for each score. Probability Number of counters player receives Find the value of n in order for the player to get an expected return of 9 counters per roll. Working: 229281 — 15— M99/510/H(1) 18. A factory has a machine designed to produce 1 kg bags of sugar. It is found that the average
weight of sugar in the bags is 1.02 kg. Assuming that the weights of the bags are normally
distributed, ﬁnd the standard deviation if 1.7% of the bags weigh below 1kg. Give your
answer correct to the nearest 0.1 gram. 229281 Turn over ~16— M99/510/H(1) 19. When air is released from an inﬂated balloon it is found that the rate of decrease of the volume
of the balloon is proportional to the volume of the balloon. This can be represented by the . . . dv . . . . .
differential equation — = — kv, where v 15 the volume, I IS the time and k is the constant of Cl!
preportionality. (a) If the initial volume of the balloon is 1:0, ﬁnd an expression, in terms of k, for the volume
of the balloon at time t . . . . . . I}
(b) Find an expressmn, in terms of k , for the time when the volume IS —23 . Answers: 229281 — 17 — M99/510/H(l) 20. A particle moves along a straight line. When it is a distance 5 from a ﬁxed point, where s > 1 , (3s + 2)
(2s — l) . Find the acceleration when s = 2 . the velocity v is given by v = Working: 229281 ...
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 Winter '10
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 Math, Calculus

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