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The january 2010 examination questions for year 1 students in the bachelor of science programs for master of chemistry, master of earth sciences, and master of physics. The exam covers various mathematical topics including calculus, integration, complex numbers, and vectors. Candidates are required to answer the entire section a and three questions from section b. 15 questions in total.
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PAPER CODE NO. MATH
Bachelor of Science : Year 1 Master of Chemistry : Year 1 Master of Earth Sciences : Year 1 Master of Physics : Year 1
TIME ALLOWED : Two Hours and a Half
Candidates should answer the WHOLE of Section A and THREE questions from Section B. Section A carries 55% of the available marks.
f (x) =
x + 2
Find the inverse of this function. [5 marks]
n=
3 n^.
[5 marks]
y =
2 ex ex^ + 4
What is the gradient at this point? [6 marks]
(i)
∫ (^4)
0
dx √ x
, (ii)
∫ (^) ∞
4
dx √ x
Say which integral diverges, and calculate the value of the well-defined integral. [6 marks]
f (x) = 6x^2 − 8 x^3 + 3x^4
and classify them. [7 marks]
(b) Calculate from first principles the derivative of
g(x) =
3 x 1 + 2x
[8 marks]
x + 1 x^3 + 2x^2
x^2
x
x + 2
Use this result to find (^) ∫ x + 1 x^3 + 2x^2
dx.
[8 marks]
(b) Use the substitution x = 3 sinh u to calculate the definite integral ∫ (^1)
0
dx √ 9 + x^2
[7 marks]
f (x, y, z) = exp
( xy^2 z^3
) .
[5 marks]
(b) Confirm that F (x, y) = cosh(x + y) cos(x − y)
is a solution of Laplace’s equation
∂^2 ∂x^2
F (x, y) +
∂y^2
F (x, y) = 0.
[10 marks]
∫ (^) π
0
dx
∫ (^) π
0
dy (x + cos y) sin x.
[7 marks]
(b) Integrate the function
F (x, y) = exp(x − y)
over the triangular region
x > 0 , y > 0 , x + y < 2.
[8 marks]
(3 + i)(1 + i) 2 − i
in the form a + ib. [3 marks]
(b) Find in polar form all the roots of the equation
z^3 = −i ,
and draw a diagram showing their position in the complex plane. [8 marks]
(c) Use the result eiθ^ = cos θ + i sin θ to show that
cos A cos B =
cos(A + B) +
cos(A − B).
[4 marks]
PAPER CODE MATH181 Page 5 of 5 END.
