AS1051CITY UNIVERSITYLondonBSc Honours Degree in Actuarial SciencePart 1Mathematics for Actuarial Science: Paper 22014Time allowed: 2 hoursFull marks may be obtained for correct answers toALL of the SIX questions in Section A andTWO of the THREE questions in Section B.If more than TWO questions from Section B are answered,the best TWO marks will be credited.All necessary working must be shown.1 Tu
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AS1051
CITY UNIVERSITY
London
BSc Honours Degree in Actuarial Science
Part 1
Mathematics for Actuarial Science: Paper 2
2014
Time allowed: 2 hours
Full marks may be obtained for correct answers to
ALL of the SIX questions in Section A and
TWO of the THREE questions in Section B.
If more than TWO questions from Section B are answered,
the best TWO marks will be credited.
All necessary working must be shown.
1 Turn over : : :
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Section A
1. (a) [4 marks] State the definitions of a reflexive and a transitive relation.
(b) [4 marks] Let R be a relation defined on the set M of 2 × 2 matrices
and let A be a specific 2 × 2 matrix. For any two matrices X 2 M,
Y 2 M, XRY if there is some k 2 R such that X - Y = kA.
Determine if R is an equivalence relation.
2. (a) [4 marks] Find the general solution to An+1 + 4An = 2n.
(b) [4 marks] Find the solution to An - An-1 - An-2 = 0: when A0 = 1
and A1 = 1.
3. (a) [2 marks] State the formula for obtaining the area of the surface
formed by rotating y = f(x) between x = a and x = b.
(b) [6 marks] Now consider rotating a curve x = g(y) between y = a
and y = b about the y-axis to form a surface. Write down the appropriate formula for obtaining the area of this surface and use it to
find the surface area of revolution formed by rotating the parabola
x = a cosh(y=a) about the y-axis from y = -a to y = a.
4. [8 marks] Use the integrating factor technique to find the general solution
of
x
dy
dx + 2y = e-2x:
5. (a) [2 marks] Consider f = y sec(x) + y2e-2x + x sin(y). Obtain an expression for fxy.
(b) [6 marks] Consider a function f of two real variables x; y. How can
you determine where the stationary points of such a function are?
What is the test that is used to determine whether a stationary point
is a maximum, minimum or saddle point?
6. [8 marks] Find all complex solutions of z3 = -1.
2 Turn over : : :
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