Oberlin College MATH CALCULUS MasterAlbatross20

IBDP Mathematics: Analysis and Approaches, Higher Level Paper 3 Practice Questions Daniel Hwang [email protected] Suggestions and comments are welcome. Updated September 5, 2020 Question number ➔ 1 2 3 ...

IBDP Mathematics: Analysis and Approaches, Higher Level Paper 3 Practice Questions Daniel Hwang [email protected] Suggestions and comments are welcome. Updated September 5, 2020 Question number ➔ 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 M aximum mark ➔ 3 1 3 0 2 9 24 31 2 9 2 9 31 2 7 26 2 3 31 26 2 8 23 2 6 30 24 23 32 2 4 2 4 29 23 2 7 Number & algebra Sequences & series x x x x x x x x Exponents & logs x x x x x x x x x Binomial expansion x x Counting x x x Partial fractions x x Complex numbers x x Proof by induction x Proof by contradiction x x x x Systems of equations x Functions Domain & range x x x Inverse function x Curve sketching x x x x x Quadratic function x x x Modulus function x Transformations x x Factor & remainder thrms x x Sum & product of roots x x Geo & trig Trigonometry x x x x x x x x x x x x x Arcs & sectors x x Triangle trigonometry x x 3D solids x Vectors x x x Statistics & prob Descriptive statistics x x x Correlation & regression x Probability x x x x x Discrete random variables x x Binomial distribution x Cont. random variables x Normal distribution x x Calculus Limits x x x x x x x x Differentiation x x x x x x x x x x x x x x x Integration x x x x x x x x x Continuous & differentiable x x x L'Hopital's rule x x Differential equations x Maclaurin series x x 1. [Maximum mark: 31] This question asks you to investigate the throwing of darts at a target. Assume that when you throw a dart at a circular target of radius 1, your accuracy is random, i.e. every point on the target is equally likely to be hit. Let � = distance from dart′ s landing point to centre of target. (a) (i) Find the median value of �. (ii) Show that the probability density function of � is �(�) = 2�. (iii) Find E(�). [7] Define "score" as � = 1 � . (b) (i) Find the median value of �. (ii) Show that the probability density function of � is �(�) = 2�−3. (iii) Find E(�). [10] Your IB Higher Level Mathematics teacher, whose accuracy is also random, throws a dart. You then start throwing darts until you get a score that is higher than your teacher’s score. Let � = number of your throws. (c) (i) Show that P(� = 5) = ( 1 6 ) ( 1 5 ). (ii) Verify that the sum of probabilities of all � equals 1. (iii) Show that E(�) = 1 2 + 1 3 + 1 4 + 1 5 + ⋯ . (iv) By comparing E(�) with 1 ( 1 2 ) + 2 ( 1 4 ) + 4 ( 1 8 ) + ⋯ , or otherwise, show that E(�) = ∞. (v) Is the result E(�) = ∞ realistic? Why or why not? [11] (d) Your teacher asks you to design a non-circular target so that � is approximately normally distributed. Is this possible? If yes, draw such a target; if no, explain why not. [3] 2. [Maximum mark: 30] This question asks you to investigate a ladder of length � m sliding down a wall and across the floor. (a) Find the equation of the path of the midpoint of the ladder, in terms of �, � and �. [2] (b) The speed, in m s−1, of the end of the ladder on the wall is equal to the distance, in m, of the other end from the wall. Find the time it takes the ladder to move from vertical to horizontal. [6] (c) Now assume that the end of the ladder on the floor has constant speed 1 m s−1. (i) Find the speed of the other end just before the ladder hits the floor. (ii) Is your answer to (c)(i) realistic? Explain. [7] As the ladder slides, an envelope curve appears. The ladder is always tangent to the envelope curve. (d) (i) On the same set of axes, in the first quadrant, sketch and label �� + �� = � � for � = 1 2 , 1, 2, 3. (ii) Show that the equation of the tangent to �� + �� = � �, � ≠ 0, at (�, �) is � = −��−1�1−�� + �1−�� �. (iii) Find the equation of the envelope curve, in terms of �, � and �