Question Two

Friday, 2 November 2018

12:35 AM

Untitled picture.png Machine generated alternative text:
2. 
2 
i) Consider the initial value problem — + (2 + —)y — —, 
da; 
with y(l) — 0, defined for a; > 0. 
a) Show that an integrating factor for this equation is xe 
b) Hence solve the initial value problem. 
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Untitled picture.png Machine generated alternative text:
ii) 
(12 y dy 
Find the general solution to + 2— + 2y — 20e 
dc 
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Untitled picture.png Machine generated alternative text:
iii) Consider the MAPLE session: 
> a(n+l) ; 
nn(x - 
> /a(n) ,n=infinity) ; 
ex — e 
Using MAPLE session above, or otherwise, find the open interval of convergence I = 
(a, b) for the power series 
Tin (a; — 
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Untitled picture.png Machine generated alternative text:
iv) Consider the set S consisting of the vectors VI = 
1 
1 
2 
2 
3 
—1 
from IR3 and 
let u — 
1 
—1 
12 
a) Find scalars and ,u such u /\VI + ,uv2. 
b) A linear transformation T : IR3 + IR3 has VI, v2 as eigenvectors with eigenvalues 2 
and —1, respectively. 
a) Find T (u) as a linear combination of VI, v2. 
[3) Denote T (T (u)) by T 2 (u), T(T(T(u))) by T 3 (u), and so on. Express T n (u) as 
a linear combination of VI, v2, where n is a positive integer. 
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Machine generated alternative text:
v) The two most popular soft drinks in Old South Wales are AppleAde and BananAde. 
Assume that no-one in Old South Wales likes both of these drinks equally (that is, 
everyone has a preference for one or the other). Past statistics show that 50 % of the 
population prefer AppleAde. 
Last month the manufacturer advertised AppleAde on television for a week. After that, 
a survey was conducted by taking a random sample of 100 people. Of the 100 people 
sampled, 60 preferred AppleAde and 40 preferred BananAde. 
a) Assuming that the advertising had no effect on people's preferences, write down 
an expression for the tail probability that 60 or more people preferred AppleAde in 
a sample of 100. 
b) Use the normal approximation to the binomial to calculate the tail probability in 
(a), giving your answer to 3 decimal places. 
c) Giving reasons, is there evidence that the advertising campaign increased the per- 
centage of the population that prefer AppleAde?

 

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