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Expand the function f(x)=P′₃(x) in a series of the form: ∞
∑ A_kP_k(x)
k=0
Obtain the harmonic conjugate v of the function u=2x(1-y)
Obtain the Taylor series expansion of cos^2zabout z = 0.
b) Using the method of residues, evaluate the following integral:
∫0 to 2π{dθ/](3+2cosθ)}
Locate and name the singularities of the following functions in the finite z-plane:
1. ln(z+3i)/z^2

2. z^2-2z/(z^2+2z+2)
KPMG, an auditing firm has noticed that of the companies it audits, 85% show no inventory shortages, 10% show small inventory shortages and 5% show large inventory shortages. KK firm has devised a new accounting test for which it believes the following probabilities hold:

P (Company will pass test/no shortage) = 0.90
P (Company will pass test/small shortage) = 0.50
P (Company will pass test/large shortage) = 0.20

Required:
i) Determine the probability if a company being audited fails this test has large or small inventory shortage.

ii) If a company being audited passes this test, what is the probability of no inventory shortage?
Describe how quadratic equations can be used in decision-making.
Explain the assumptions of the linear regression model
Calculate the regression equation of X on Y (5 mks)
X 1 2 3 4 5
Y 2 5 3 8 7
A batch of 5000 electric lamps has a mean life of 1000 hours and a standard deviation of 75 hours.
Assume a normal distribution.
i) How many lamps will fail before 900 hours?
ii) How many lamps will fail between 950 and 1000 hours?
iii) What proportion of lamps will fail before 925 hours?
iv) Given the same mean life, what would the standard deviation have to be to ensure that no more than 20% of lamps fail before 916 hours
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