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. Ink Drawings           Ink Drawings Ink Drawings Ink Drawings        Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings

 

Untitled picture.png Machine generated alternative text: ii) (12 y dy Find the general solution to + 2— + 2y — 20e dc Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings  Ink Drawings Ink Drawings Ink Drawings Ink Drawings              Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings  
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; — Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings    Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings  Ink Drawings Ink Drawings Ink Drawings   Ink Drawings Ink Drawings Ink Drawings Ink Drawings  
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. Ink Drawings Ink Drawings Ink Drawings Ink Drawings          
       
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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