1. This is a question on the use and application of row reduction methods

a. Use row operations to calculate the inverse of the matrix A given by

A = ╱ 10(3)   7(2) 、:

[5 Marks]

b.  Solve the matrix inversion problem A刘 = b, where

A =  ．(╱)        ．(、) ; b =．(╱)  ．(、)

[10 Marks]

c. Which of the following sets in R3  are linearly independent?

V   =   {v1  = (1; 3; 0);  v2  = (2; ≠3; 4);  v3  = (3; 0; 4)}

U   =   {u1  = (1; 2; 3);  u2  = (2; 3; 1);  u3  = (3; 1; 2)}

[10 Marks]

2. This is a question on calculus

a. Find the equation of the normal to y = x2  ≠ 7x + 3 at the point (x0 ;y0 ) which is parallel to

b. Evaluate the following integral

\0 2 (x + 1) ^x2 + 2xdx

[10 Marks]

c.  Sketch the graph of

y =              ; x  ≠ 1

showing clearly (any) turning points, intercepts on both axes, and behavior at x 与 f如.  [9 Marks]

3. This question is on di§erential equations

a.  Solve the di§erential equation                                    

dy        ^1 + y2   

dx     xy (1 + x2 )

and show that the general solution can be written as

^1 + y2  =  ln ╱ 、 ;

where A is an arbitrary constant. [15 Marks].

b. Find a particular solution of the initial value problem

 ≠  = (x + 1)4 ; y (0) =

and show it can be expressed as y = k (x + 1)X ; k 4 Q; n 4 N:

[10 Marks]

4. BoJo Wa扭es share price is trading at 会100 per share. Each year the stock price will either go up u or go down d; and u = d = 25%. The annual risk-free interest rate is 5% discretely compounded.

a. By constructing a replicating portfolio, determine the price of a two-year European call option with the strike E = 会110. [20 Marks]

b. Use the call price to calculate the price of the put on BoJo Wa扭es with the same strike and expiry. [5 Marks]