# Mathematics and Decimal Place

Topics: Mathematics, Indian mathematics, Real number Pages: 7 (1304 words) Published: June 18, 2013
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Ordinary Level

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4037/01

Paper 1 May/June 2004

2 hours

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

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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

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