IB Math HL 11 - Summer Assignment 2013

IB Math HL 12Name:SUMMER REVIEW 2013401. The second term of a geometric sequence is . The sum to infinity of the corresponding3geometric series is 12. Find the first term and the common ratio.2. Find the sum of all the integers between 200 and 500 which are divisible by 7.3. The first, second and the nth terms of an arithmetic sequence are 2, 6, and 58, respectively.(a) Find the value of n.(b) For that value of n, find the exact value of the sum of n terms of a geometric sequences1whose first term is 2 and common ratio is . 24. In the diagram below, each successive parallelogram is formed by joining the midpoints of thesides of the previous parallelogram. The largest parallelogram has sides of length 4.If the pattern shown is infinite, find the sum of the areas of the parallelogram.

5.6. Given the function ( )3 x2 2f x e x 4, find the domain and range of f (x).27. If f ( x) ln6x 5x 6(a) the exact domain of f (x);(b) the range of f (x)., find18. Let f ( x) and g ( x) x 2 1. Findx 1(a) f 1 ( x ) ;(b) ( f g)(x).2

9. The graphs given below are those of the same function f (x)for a x b .Sketch, on the given axes, the graphs of(a) f (x);(b)1.f ( x)Indicate clearly the positions of any asymptotes.3

11.By observing the above graph,(a) what can be said about the solutions of the equation mx 2 + nx = –p?(b) what can be said about the value of n 2 – 4mp?12. Find all the possible values of k if x = k is a solution of the equationx 3 + kx 2 – x – k = 0.213. Find all values of m such that the equation mx 2( m 2) x m 2 0 has(a) two real roots;(b) two real roots, one positive and one negative.3 214. Let P(x) x 3x 4x c .(a) Calculate P (2).(b) If the remainder when P (x)is divided by (x + 2) is -23, find the value of c.15. Calculate x and y exactly, as rational numbers, if15 3 x y 25 5x 2 yand 7 49 1.(b) Hence, or otherwise, solve the equation in part (a).5

16. (a) Given thatlogcb3logab , find the real numbers k and m such that log9x k log3x andlog aclog27 512 mlog38 .3(b) Find all values of x for which log9x log3x log27512 .1217. Solve the system of simultaneous equations:x 2y 54x 8y18. Find all angles , 0 360, correct to the nearest degree, such that 3sin 4cos 5.2219. Find all values of , so that 3sin 7sin 5 3cos ,0 90.20. Given that2cos and sin 0 , find the exact values of sin , tan and sec .521. Find all values of x in the interval 0 x 2, so that sin x tan x sin x .22. The diagram below shows part of the graph of the function y asinbx for x > 0. Sketch on thea bxsame axes the graph of y sin .2 26

23. A triangle ABC is such that angle A = 58º, AB = 10 cm and AC = 5.1 cm.(a) Find BC, correct to three significant figures.(b) Find the size of Ĉ , correct to the nearest tenth of a degree.24. A triangle ABC has sides whose lengths are AB = 5 cm, BC = 6 cm and CA = 9 cm. Find theangles of the triangle.25. A reconnaissance airplane A, flying at a height of three thousand metres above the point R on thesurface of the sea, spots a freighter B at an angle of depression 37º and a tanker C at an angle ofdepression 21º, shown in the figure. The angle BAC is 110º.Find, to the nearest metre,(a) the distance CA between the plane and the tanker;(b) the distance BC between the two ships.26.7

27.28.29. Given A(2, -1, 4), B(6, 0, 5), C(1, 3, -3), find cos ABˆ C .30. Given two vectors u and v , with u 4, find the value of u 2vin the following cases: (a) v 3u;(b) u and v are perpendicular and v 3u.31. The coordinates of P and Q are (3, -1) and ( , 4 ), respectively, where is a constant. If O isthe origin, find all values of for which OP is perpendicular to OQ .8

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