This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ere f(x) is not defined.
(3) (d) Find the exact value of the x-coordinate of the local maximum of f(x), for 0 < x < 1.5.
(You may assume that there is no point of inflexion.)
(3) (e) Write down the definite integral that represents the area of the region enclosed by f(x)
and the x-axis. (Do not evaluate the integral.)
(Total 16 marks) 112 163. A machine is set to produce bags of salt, whose weights are distributed normally, with a mean
of 110 g and standard deviation of 1.142 g. If the weight of a bag of salt is less than 108 g, the
bag is rejected. With these settings, 4% of the bags are rejected.
The settings of the machine are altered and it is found that 7% of the bags are rejected. (a) (i) If the mean has not changed, find the new standard deviation, correct to three
(4) The machine is adjusted to operate with this new value of the standard deviation.
(ii) Find the value, correct to two decimal places, at which the mean should be set so
that only 4% of the bags are rejected.
(4) (b) With the new settings from part (a), it is found that 80% of the bags of salt have a weight
which lies between Ag and Bg, where A and B are symmetric about the mean. Find the
values of A and B, giving your answers correct to two decimal places.
(Total 12 marks) 164. Differentiate from first principles f(x) = cos x.
(Total 8 marks) 165. Prove by mathematical induction that d (xn) = nxn–1, for all positive integer values of n.
(Total 10 marks) 166. A computer manufacturing company buys large quantities of hard discs from several suppliers.
Hard disc quality is checked by a process called RTT which gives results on a continuous scale
from 0 to 100. Based on previous experience the company assumes that the results are normally
distributed with a mean of 68 and standard deviation of 3.
Shipments arrive from suppliers on a daily basis. A sample of 10 hard discs is taken from each
shipment at random and tested. If the mean of the sample is more than 67, the shipment is
accepted, otherwise it is rejected. 113 (a) What is the probability that a hard disc selected at random has a result less than 67?
(2) (b) Find the probability that a shipment is rejected.
(4) (c) There is a $1000 penalty each day that a shipment is rejected. A particular supplier’s
hard discs have a mean of 67.5 and a standard deviation of 2.8.
(i) What is the probability that this supplier’s shipment is accepted?
(3) (ii) What is the expected penalty per 6-day week for this supplier?
(6) (d) The company’s own production of hard discs has a mean of 68 and a standard deviation
of 3. However, to keep the production within the acceptable limits, the company samples
10 hard discs every hour and examines whether the sample is accepted or rejected.
During a particular hour, the following results were recorded for a sample that was
randomly chosen for testing:
68.1747, 68.0473, 66.3189, 66.2735, 66.957, 66.9738,
66.1438, 67.0744, 66.1875, 67.8804
At the 5% level of signi...
View Full Document
- Fall '13
- The Land