Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the math181 examination paper for january 2011, which includes 9 questions from various topics in calculus, vectors, complex numbers, and probability. The paper is intended for first-year students in bachelor of science, master of chemistry, master of earth sciences, and master of physics programs. The paper is divided into two sections, with section a carrying 55% of the available marks and requiring answers to the whole section and three questions from section b.
Typology: Exams
1 / 6
This page cannot be seen from the preview
Don't miss anything!
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.
u = 3 i + 4k , v = 4 i − 5 j − 3 k.
Find the angle between these vectors. [5 marks]
(a) sin(2 + x^2 ) , (b) cosh(x) sin(2x) , (c)
x + 1 x^2 + 1
[6 marks]
y = sin^3 x
have the gradient
y′^ = ±
[8 marks]
(a)
∫ (^4)
0
x x − 5
dx , (b)
∫ (^) ∞
0
(x + 2)^3
dx.
[6 marks]
x^2 + x − 2 x^2 + x − 6
x − 2
x + 3
Use this result to find (^) ∫ x^2 + x − 2 x^2 + x − 6
dx.
[8 marks]
(ii) Use the results of part (i) to sketch the function
f (x) =
x^2 + x − 2 x^2 + x − 6
showing all stationary points and zeroes. [7 marks]
∫ (^2)
0
dx
∫ (^1)
0
dy ln(xy).
[7 marks] (b) Integrate the function f (x, y) = xy
over the region between the lines
y = x and y = x^2 − x.
[8 marks]
1 (1 + i)^2
in the form a + ib. [3 marks]
(b) Find in polar form all the roots of the equation
z^5 = − 32 i ,
and draw a diagram showing their position in the complex plane. [8 marks]
(c) Use Euler’s formula eiθ^ = cos θ + i sin θ to show that
cos^5 θ =
(cos 5θ + 5 cos 3θ + 10 cos θ).
[4 marks]
f (x, y) = y exp(x^2 + 2xy).
[6 marks]
(b) Confirm that
F (x, t) = cosh(3x) cosh(3vt) − cos(x + vt)
is a solution of the wave equation
∂^2 ∂t^2
F (x, t) = v^2
∂x^2
F (x, t).
(v is a constant.) [9 marks]
