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The question paper for the math181 resit examination held in 2007 for students pursuing bachelor of science and masters in chemistry, earth sciences, and physics. The exam covers topics such as functions, differentiation, integration, infinite series, complex numbers, and calculus. Candidates are required to answer all questions in section a and three questions from section b. Section a carries 55% of the available marks.
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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
INSTRUCTIONS TO CANDIDATES
Answer ALL questions in Section A and THREE questions from Section B. Sec- tion A carries 55% of the available marks.
for 0 ≤ x ≤ 4 π. Sketch this function. [3 marks]
y = f (x) =
2 x + 3 2 x − 5
What is the domain and range? Find f −^1 (x) given that the function f (x) is one-to-one. [5 marks]
(i) 3 x^7 e−x
2 , (ii)
2 x^3 (x + 5)^2
, (iii) cos(3x) sinh^4 (2x).
[9 marks]
x^4 − y^3 x +
x
sin(y) = 8.
Find implicitly dy/dx in terms of y and x. [5 marks]
(i)
∫ ( x −
x
) dx , (iii)
∫ sinh^2 (5x)dx.
[5 marks]
(i)
∫ (^) π 0
x sin^2 xdx , (iii)
∫ (^) ∞ −∞
xe−^4 x
2 dx.
[5 marks]
∫ (^4)
3
x^2 (x − 2)(x + 2)
dx.
[6 marks]
(ii) Evaluate the integral ∫ (x^2 + y^2 )
√ dx^2 + dy^2
on the curve x = cos θ and y = sin θ between θ = 0 and θ = π. [3 marks]
(iii) Using a suitable substitution or otherwise evaluate the indefinite integral
∫ (^1)
(e^2 x^ + 5ex)
dx.
[6 marks]
f (x, y) = (x^2 + y^2 )^5 /^2
over the area enclosed by the curve x^2 + y^2 = 4. [6 marks]
(ii) Evaluate the integral ∫
A
( y^4 sin x + y^2 sin 2x
) dxdy
where the area A is bounded by the lines y = 0, x = 0 and y = cos x. [9 marks]
[6 marks]
(ii) Use Eulers formula to show that
cos(α ± β) = cos α cos β ∓ sin α sin β.
[6 marks]
(iii) Using this result determine ∫ cos 5θ cos 3θdθ.
[3 marks]
y =
x + 2 x − 1
in detail. [6 marks]
(ii) Using the Maclaurin series expansion to the first three terms of sin(x^3 ), com- pute the approximate value of
∫ (^1)
0
sin(x^3 )dx.
[5 marks]
(iii) Evaluate the limit
lim x→ 0
3 − 2 cos x − 1 x^2
[4 marks]
