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Mathematics HL - May 2001 - P1

Mathematics HL - May 2001 - P1 - INTERNATIONAL...

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Unformatted text preview: INTERNATIONAL BACCALAUREATE MOI/510mm BACCALAUREAT INTERNATIONAL BACHILLERATO INTERNACIONAL MATHEMATICS Name HIGHER LEVEL PAPER 1 Number Monday 7 May 2001 (afternoon) 2 hours INSTRUCTIONS TO CANDIDATES - Write your name and candidate 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 to three significant figures, as appropriate. - Write the make and model of your calculator in the box below 6. g. Casio fx-9750G, Sharp EL—9400, Texas Instruments TI—85. Calculator Make Model EXAMINER TEAM LEADER TOTAL 2217236 15 pages — 2 — M01/510/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. Letf(z)=z5{1—i;]. Finde(t)dt. 2t3 Ir Working: Answer: . 1: TC 2. Solve 2s1nx=tanx, where ——2— < x < E Working: Answers: 2217236 — 3 — M01/510/H(1) 3. Give a full geometric description of the transformation represented by the matrix mlwmlh VII-bull!» Working: Answer: 221—236 Turn over —4— M01/510/H(1) 4. Find the gradient of the tangent to the curve 3x2 + 4y2 2 7 at the point where x = 1 and y > 0 . Working: (a) the set of real values of x for which f is real and finite ; (b) the range of f . Working: Answers: (a) (b) 221-236 — 5 — M01/510/H(1) 6. A machine produces packets of sugar. The weights in grams of thirty packets chosen at random are shown below. mm memmmmm Find unbiased estimates of (a) the mean of the population from which this sample is taken; (b) the variance of the population from which this sample is taken. Working: Answers: (a) (b) _—.— 221—236 . Turn over _ 6 _ M01/510/H(1) 7. The nth term, u" , of a geometric sequence is given by un = 3(4)"+1, n e Z+ . (a) Find the common ratio r . (b) Hence, or otherwise, find Sn , the sum of the first n terms of this sequence. Working: Answers: (a) (b) sin x 8. Let f:xi—> , 7: S x S 37:. Find the area enclosed by the graph off and the x-axis. x Working: Answer: 221—236 — 7— M01/510/H(1) 9. Find the equation of the line of intersection of the two planes —4x+ y+z=—2 and 3x— y + 22:—1 . Working: 2217236 Turn over — 8 — MOI/510/H(1) 10. (z + 21) is a factor of 223 — 322 + 82 — 12 . Find the other two factors. Working: Answers: 221—236 —9— MOI/510/H(1) 11. Given that P(X) = % P(Y| X) = % and P(Y| X’) = i, find (a) P(Y’); (b) P(X’UY’). Working: Answers: (a) (b) 12. Find an equation of the plane containing the two lines x—1=1;Z=z—2andx+1—2Ay 2+2. 2 3’3 5 221~236 Turn over — 10— M01/510/H(1) 13. Z is the standardised normal random variable with mean 0 and variance 1. Find the value of a such that P( | Z| S a)=0.75. Working: Answer: 14. Given that z = (b + i)2 , where b is real and positive, find the exact value of b when arg Z = 60° . 221—236 — 11 — M01/510/H(1) 15. X is a binomial random variable, where the number of trials is 5 and the probability of success of each trial is p . Find the values of pif P(X=4)=0.12. Working: 221—236 Turn over — 12— M01/510/H(1) d 16. Find the general solution of the differential equation d—): = kx(5 — x), where 0 < x < 5 , and k is a constant. Working: Answer: 221—236 - 13 — M01/510/H(1) 17. An astronaut on the moon throws a ball vertically upwards. The height, s metres, of the ball, after t seconds, is given by the, equation s=401+ 0.5at2, where a is a constant. If the ball reaches its maximum height when t: 25 , find the value of a . Working: Answer: 18. The equation kx2 — 3x + (k + 2) = 0 has two distinct real roots. Find the set of possible values of k . Working: Answers: 221—236 Turn over — 14— M01/510/H(1) 19. The diagram shows the graph of the functions yl and y2 . yl On the same axes sketch the graph of 21—. Indicate clearly Where the x-intercepts and y2 asymptotes occur. Working: 221—236 — 15 — M01/510/H(1) 20. The function f is given by f: x I—> e(1+5i““x) , x 2 0 . (a) Find f ’(x) . Let xn be the value of x where the (n +1)‘h maximum or minimum point occurs, n e N. (i. e. x0 is the value of x Where the first maximum or minimum occurs, x1 is the value of x where the second maximum or minimum occurs, etc). (b) Find xn in terms of n . Working: Answers: (a) (b) 221—236 ...
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