3a. Use Gaussian elimination to reduce to upper triangular form the system of equations

\begin{aligned}x − z & = 1 \\ 3x + y + az & = 1 \\ 4x + ay − 6z & = 8 \end{aligned}

where a is a parameter. Hence find all solutions to these equations, paying particular attention to any special values of a which may arise.

3b. Find the plane through the points (0, 0, 0), (1, 3, 4) and (1, 1, 8).

For what value of the parameter b do the points (0, 1, b) and (−1, b, −6) also lie on this plane?

Parts a and b carry respectively 65% and 35% of the marks 4a. Show that the only stationary point of the function

f(x, y, z) = x^{2} + 4xy + y^{2} + exp(−3z^{2})

is x = y = z = 0.

4b. Calculate the matrix of second partial derivatives of f evaluated at the stationary point. 4c. Find the eigen-values and eigen-vectors of this matrix. 4d. What kind of stationary point is (0, 0, 0)?

Parts a, b, c, d carry respectively 20%, 20%, 40%, 20% of the marks.

/Turn over… 5a. Every complex number z = x + iy can be represented by a 2-vector \underline{x} = ( \begin{matrix}x \\ y \end{matrix} ). Find a

2 × 2 matrix A such that A\underline{x} represents (a + ib)z, for given constants a and b and for all x and y.

Find the inverse of this matrix, A^{−1}, assuming it exists.

What operation on the complex number z is the equivalent of pre-multiplying \underline{x} by A^{−1}? Justify your answer.

5b. Find all complex numbers z satisfying z^{6} − 7z^{3} − 8 = 0.

Indicate the position of these numbers in the Argand diagram.

6a. The sequence of numbers (Y_{n}), for n = 0, 1, 2 … obeys the recurrence relation

Y_{n+1} − 2Y_{n} + Y_{n−1} = h^{3}n + 2h^{2} where h is constant.

Find the general solution for Y_{n}. 6b. The function y(x) obeys the differential equation

\frac{d^{2}y}{dx^{2}} = x + 2 .

Write down four terms of the Taylor series for y(x ± h) where h is small, and explain why the recurrence relation can be considered as an approximation to the differential equation.

6c. Find the exact solution for y(x), and compare it with the approximation Y_{n}.

Parts a, b, c carry respectively 60%, 20%, 20% of the marks