By Paul Urban, David Martin, Robert Haese, Sandra Haese, Michael Haese

Arithmetic for the foreign pupil: arithmetic HL has been written to include the syllabus for the two-year arithmetic HL direction, to be first tested in 2014. it isn't our purpose to outline the direction. academics are inspired to exploit different assets. now we have constructed this publication independently of the overseas Baccalaureate association (IBO) in session with many skilled lecturers of IB arithmetic. The textual content isn't really recommended by way of the IBO. Syllabus references are given at first of every bankruptcy. the hot variation displays the hot arithmetic HL syllabus. dialogue issues for the speculation of information were integrated during this version. See web page 12 for a precis. in line with the creation of a calculator-free exam paper, the assessment units on the finish of every bankruptcy were categorized as ’calculator’ or ’non-calculator’. additionally, the ultimate bankruptcy includes over 2 hundred examination-style questions, classified as ’calculator’ or ’non-calculator’. those questions should still offer more challenging demanding situations for complex scholars.

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12 ax2 + (3 ¡ a)x ¡ 4 = 0 has roots which are real and positive. What values can a have? 13 a Determine the equation of: i the quadratic function ii the straight line. y -3 b For what values of x is the straight line above the curve? 3 x -27 14 Show that the lines with equations y = ¡5x + k y = x2 ¡ 3x + c if and only if c ¡ k = 1. are tangents to the parabola cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 15 4x2 ¡ 3x ¡ 3 = 0 has roots p, q. Find all quadratic equations with roots p3 and q3 .

Ym a Explain why 9x + 8y = 1800: b Show that the area of each pen is given by A = ¡ 98 x2 + 225x m2 . xm c If the area enclosed is to be maximised, what are the dimensions of each pen? y = x2 - 3x x 3 cyan magenta yellow 95 100 50 75 25 95 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 y = 2x - x2 0 b Find the maximum vertical separation between the curves for 0 6 x 6 2 12 . 5 2 The graphs of y = x2 ¡ 3x and y = 2x ¡ x2 are illustrated. a Without using technology, show that the graphs meet where x = 0 and x = 2 12 .

Find: a f ±g b g ± f. 7 Given f(x) = x+3 x+2 and g(x) = a (f ± g)(x) x+1 , x¡1 find in simplest form: b (g ± f )(x) c (g ± g)(x) cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 In each case, find the domain of the composite function. cdr Monday, 20 February 2012 1:41:56 PM BEN IB_HL-3ed 64 FUNCTIONS (Chapter 2) 8 Given f(x) = p 1 ¡ x and g(x) = x2 , find: a (f ± g)(x) 9 b the domain and range of (f ± g)(x) a If ax + b = cx + d for all values of x, show that a = c and b = d.