- Solve the following equations:
- Log (x – 1) + log(x + 8) = 2 log (x + 2) [4 marks]

- 9
^{x-1 }= 81^{2x+1 }[2 marks]

- 2
^{3x-1 }= 128 [2 marks]

- Evaluate:

- If f(x) = 4x
^{3 }+ 3x^{2}– 1, find*f’(x)*and*f’’(x)*hence calculate*f’(6)*and*f’’(-3)*[5 marks]

- Two aircraft leave an airfield at the same time. One travels due north at an average speed of 300 km/h and the other due west at an average speed of 220 km/h. Calculate their distance apart after 5 hours. [3 marks]

- In two dimensional space, the position of an object is indicated by the Cartesian coordinates
*x=*20 and*y=*10. What is the position of the object expressed in polar coordinates? [2 marks]

- The sag S at the centre of a wire is given by the formula: . Make l the subject of the formula and evaluate l when d = 1.75 and S= 0.80 [5 marks]

- Use the chain, product or quotient rule to differentiate the following functions:

- f(x) = ;
- g(x) = ;
- h(x) = . [9 marks]

- The surface area s of rust on a plate is believed to be related to time t by Use the values in the table below to estimate a and b using the method of least squares; hence find the formula( the law) for s.

t 2 3 4 5 6 7 8

s 2.34 2.75 3.16 3.63 4.16 4.79 5.62 [8 Marks]

- A rectangular sheet of metal having dimensions 20 cm by 12 cm has squares removed from each of the four corners and the sides bent upwards to form an open box. Determine the maximum possible volume of the box. [8 marks]

- A fabric cutter needs to find the area of a particular piece of leather used to make a section of the interior trim of a business jet. The area can be described by the shape of the function: y =x
^{2 }-1 and y = x + 1. What is the area? [9 marks]

- Electric wire is produced in lengths of 300 metres. On average there is 1 defect in every 3000 metres of wire produced. (Use your knowledge of the Poisson Distribution to answer these questions)

a) Explain why a Poisson distribution may be a suitable model for the number of defects per 300 metre length. [1 mark]

b) Find the probability that in a 300 metre length there will be at least one defect.

[3 marks]

c) Find the probability that out of ten 300 metre lengths, more than one contains one or more defects. [5 marks]

A wire of length N metres is produced. The probability of at least one defect in this length of wire is 0.95.

d) Find the value of N, giving your answer to the nearest metre. [3 marks]

- Five races are to be held between the same eight athletes, who are assumed to be “equally good”.
- Find the probability that the athlete A will have:

i. No wins [2 marks]

ii. Two wins [2 marks]

iii. At least two wins [2 marks]

iv. Not more than two wins [2 marks]

- How many races must be held so that A has at least a 95% chance of winning at least one race? [4 marks]

- Determine the area enclosed by the two curves If this area is rotated 360° about the x axis determine the volume of the solid of revolution produced. [10 marks]

- Solve the following:
- [4 marks]
- [5 marks]

Subject | Mathematics |

Due By (Pacific Time) | 08/05/2014 04:00 pm |

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