# Mathematics and Decimal Place

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level

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ADDITIONAL MATHEMATICS

4037/01

Paper 1 May/June 2004

2 hours

Additional Materials: Answer Booklet/Paper Graph paper (3 sheets) Mathematical tables

READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Write your answers on the separate Answer Booklet/Paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. At the end of the examination, fasten all your work securely 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. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers.

This document consists of 5 printed pages and 3 blank pages. MCS UCH206 S53343/5 © UCLES 2004

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2

Mathematical Formulae

1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, –b ± √ b2 – 4ac x = –––––––––––––– . 2a

Binomial Theorem (a + b)n = an +

(n) a 1

n – 1b

+

(n) a 2

n – 2 b2

+…+

( nr ) a

n – r br

+ … + b n,

where n is a positive integer and

–––––––– ( nr ) = (n –n! r! . r)!

2. TRIGONOMETRY Identities sin2 A + cos2 A = 1. sec2 A = 1 + tan2 A. cosec2 A = 1 + cot2 A.

Formulae for ∆ ABC c b a –––– = –––– = –––– . sin A sin B sin C a2 = b 2 + c2 – 2bc cos A. 1 ∆ = – bc sin A. 2

4037/1/M/J/04

3 1 Given that y = 3x − 2 x2 + 5 , find dy , dx

(i)

an expression for

(ii)

the x-coordinates of the stationary points. [4]

2

Find the x-coordinates of the three points of intersection of the curve y # x 3 with the line y # 5x 0 2, expressing non-integer values in the form a & √b, where a and b are integers. Sketch on the same diagram the graphs of y # | 2x ! 3 | and y # 1 0 x. Find the values of x for which x ! | 2x ! 3 | # 1.

[5]

3 (i) (ii)

[3] [3]

4

The function f is defined, for 0° ≤ x ≤ 360°, by f(x) = a sin (bx) + c, where a, b and c are positive integers. Given that the amplitude of f is 2 and the period of f is 120°, (i) state the value of a and of b. [2]

Given further that the minimum value of f is 01, (ii) state the value of c, [1] [3]

(iii) sketch the graph of f.

5

The straight line 5y ! 2x # 1 meets the curve xy ! 24 # 0 at the points A and B. Find the length of AB, correct to one decimal place. [6]

6

The table below shows the daily production, in kilograms, of two types, S 1 and S 2, of sweets from a small company, the percentages of the ingredients A, B and C required to produce S 1 and S 2. Percentage A Type S 1 Type S 2 60 50 B 30 40 C 10 10 Daily production (kg) 300 240

Given that the costs, in dollars per kilogram, of A, B and C are 4, 6 and 8 respectively, use matrix multiplication to calculate the total cost of daily production. [6]

4037/1/M/J/04

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4 7 To a cyclist travelling due south on a straight horizontal road at 7 ms 01, the wind appears to be blowing from the north-east. Given that the wind has a constant speed of 12 ms 01, find the direction from which the wind is blowing. [5]

8

A curve has the equation y # (ax ! 3) ln x, where x p 0 and a is a positive constant. The normal to the curve at the point where the curve crosses the x-axis is parallel to the line 5y ! x # 2. Find the value of a. [7] 18

9

(a) Calculate the term independent of x in the...

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