UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level

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1 UNIVERSITY F CAMBRIDGE INTERNATINAL EXAMINATINS General Certificate of Education rdinar Level * * ADDITINAL MATHEMATICS 4037/22 Paper 2 Ma/June hours Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTINS FIRST Write our Centre number, candidate number and name on all the work ou hand in. Write in dark blue or black pen. You ma use a pencil for an diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. D NT WRITE IN ANY BARCDES. Answer all the questions. Give non-eact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accurac is specified in the question. The use of an electronic calculator is epected, where appropriate. You are reminded of the need for clear presentation in our answers. At the end of the eamination, fasten all our work securel together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 80. This document consists of 15 printed pages and 1 blank page. DC (JF/SW) [Turn over

2 2 Mathematical mulae 1. ALGEBRA Quadratic Equation the equation a 2 + b + c = 0, b b ac = 4 2 a 2. Binomial Theorem (a + b) n = a n + ( n 1 ) an 1 b + ( n 2 ) an 2 b ( n r ) an r b r + + b n, where n is a positive integer and ( n r ) = n! (n r)!r!. 2. TRIGNMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A mulae for ABC a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A = 1 2 bc sin A

3 3 1 (4, 18) (1, 3) 2 Variables and are such that when is plotted against 2 a straight line graph passing through the points (1, 3) and (4, 18) is obtained. Epress in terms of. [4] [Turn over

4 4 2 (a) Solve the equation 3 p+1 = 0.7, giving our answer to 2 decimal places. [3] (b) Epress # (4 ) in the form 2 a b c, where a, b and c are constants. [3]

5 5 3 (a) A B C D E F (i) Write down the letter of each graph which does not represent a function. [2] (ii) Write down the letter of each graph which represents a function that does not have an inverse. [2] (b) 5 = f() 5 The diagram shows the graph of a function = f(). n the same aes sketch the graph of = f 1 (). [2] [Turn over

6 6 4 The position vectors of the points A and B, relative to an origin, are 4i 21j and 22i 30j respectivel. The point C lies on AB such that AB = 3AC. (i) Find the position vector of C relative to. [4] (ii) Find the unit vector in the direction C. [2]

7 7 5 Calculators must not be used in this question. A B 7 D 2 X C The diagram shows a trapezium ABCD in which AD = 7 cm and AB = ^4 + 5h cm. AX is perpendicular to DC with DX = 2 cm and XC = cm. Given that the area of trapezium ABCD is 15^ 5 + 2h cm 2, obtain an epression for in the form a + b 5, where a and b are integers. [6] [Turn over

8 8 6 r A 3 r B The shaded region in the diagram is a segment of a circle with centre and radius r cm. r Angle AB = radians r (i) Show that the perimeter of the segment is r c m cm. [2] 3 (ii) Given that the perimeter of the segment is 26 cm, find the value of r and the area of the segment. [5]

9 9 7 Differentiate, with respect to, (i) (3 5) 12, [2] (ii) 2 sin, [2] (iii) tan 1 + e 2. [4] [Turn over

10 10 8 Solutions to this question b accurate drawing will not be accepted. The points A( 6, 2), B(2, 6) and C are the vertices of a triangle. (i) Find the equation of the line AB in the form = m + c. [2] (ii) Given that angle ABC = 90, find the equation of BC. [2]

11 11 (iii) Given that the length of AC is 10 units, find the coordinates of each of the two possible positions of point C. [4] [Turn over

12 12 9 (a) The graph of = k(3 ) + c passes through the points (0, 14) and ( 2, 6). Find the value of k and of c. [3] (b) The variables and are connected b the equation = e e. (i) Find the value of when = 4. [1] (ii) Find the value of e when = 20 and hence find the corresponding value of. [4]

13 13 10 (a) The function f is defined, for 0 360, b f() = cos 2. (i) Sketch the graph of = f() on the aes below. [4] (ii) State the amplitude of f. [1] (iii) State the period of f. [1] (b) Given that cos = p, where 270 < < 360, find cosec in terms of p. [3] [Turn over

14 A curve has equation = 3 + ( - 4) 2 3. d d (i) Find and. [4] d 2 d (ii) Show that the coordinates of the stationar points of the curve are (5, 16) and (3, 8). [2] (iii) Determine the nature of each of these stationar points. [2]

15 15 1 (iv) Find c3 + 3m d. [2] ( - 4) (v) Hence find the area of the region enclosed b the curve, the line = 5, the -ais and the line = 6. [2]

16 16 BLANK PAGE Permission to reproduce items where third-part owned material protected b copright is included has been sought and cleared where possible. Ever reasonable effort has been made b the publisher (UCLES) to trace copright holders, but if an items requiring clearance have unwittingl been included, the publisher will be pleased to make amends at the earliest possible opportunit. Universit of Cambridge International Eaminations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of Universit of Cambridge Local Eaminations Sndicate (UCLES), which is itself a department of the Universit of Cambridge.

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education *0835058084* ADDITIONAL MATHEMATICS 0606/11 Paper 1 October/November 2012 2 hours Candidates