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
A math exam paper from may 2010, covering topics such as integration, partial fractions, differential equations, and motion. It includes various types of questions, including indefinite and definite integrals, partial fraction decomposition, and solving differential equations. The exam also includes problems related to motion under gravity and a simple pendulum.
Typology: Exams
1 / 5
This page cannot be seen from the preview
Don't miss anything!
Examiner: Dr. T.M. Mohaupt, Extension 9-55177.
Time allowed: Three hours
ALL answers to Section A and the best THREE answers to Section B will be counted. Section A carries 55 % of the available marks. The marks shown against questions, or parts of questions, indicate their relative weight. Your attention is drawn to the Formulae Sheet which accompanies this exam paper.
(a)
∫ (7x − 4)^25 dx , [2 marks]
(b)
∫ cos(3x + 2) dx , [2 marks]
(c)
∫ (x + 5)−^3 /^2 dx , [3 marks]
(d)
∫ (^) √ (5x − 9)^5 dx. [3 marks]
(a)
∫ (^2)
3 / 2
3 e^2 x−^3 dx , [2 marks]
(b)
∫ (^1)
− 1
x^7 (3x^2 − 5) dx , [2 marks]
(c)
∫ (^3) π/ 2
π
sin(2x − 3 π) dx , [3 marks]
(d)
∫ √ 2
1
x
x^2 − 1 dx. [3 marks]
(a)
− 3 x + 34 (x − 4)(x + 7)
x − 4
x + 7
(b)
3 x^2 − 10 x + 18 (x + 2)(x − 3)^2
x + 2
x − 3
(x − 3)^2
Compute the constants A, B, C, D, E. [5 marks]
(ii) Hence evaluate the following integrals:
(a)
∫ (^) − 3 x + 34
(x − 4)(x + 7)
dx , [2 marks]
(b)
∫ (^6)
4
3 x^2 − 10 x + 18 (x + 2)(x − 3)^2
dx. [3 marks]
(i) Derive the equation of motion
d^2 y dt^2
= −g
of the particle, where y(t) is the vertical position, t the time and g the accelera- tion due to gravity. [4 marks]
(ii) Find the general solution of this differential equation. Give explicit expres- sions for the vertical position and the vertical velocity as functions of time. Ex- press the integration constants in terms of the initial position and the initial velocity. [4 marks]
(b) (i) A particle is thrown upwards with initial velocity v 0 = 1ms−^1 from the initial height y 0 = 5m. Find its maximal height. The approximate value of the gravitational acceleration is g ≈ 10 ms−^2. [3 marks]
(ii) A particle falls down from the initial height y 0 = 10m. What is the velocity with which it hits the ground? Assuming that the particle is reflected at the ground, how long does it take until it is back at its initial position? [3 marks]
(iii) A particle is thrown downward from the initial height y 0 = 5m with initial velocity v 0 = 2ms−^1. Assuming that it is reflected at the ground, what is the maximal height it will reach? [1 marks]
and find the particular solution where
y(
π 20
) = 0 and
dy dx
π 20
[8 marks]
(b) Evaluate the following integral: ∫ √ 5 √ 10 / 2
dx √ 5 − x^2
Hint: Use the substitution x =
5 sin(u). [7 marks]
d^2 θ dt^2
where t is time and k^2 = 36.
(i) Find the general solution of this differential equation. [3 marks] (ii) Find the particular solution where
θ
( π 24
2 and
dθ dt
( π 24
) = 0.
[4 marks] (iii) Show that the solution you obtained in part (ii) is identical to
θ =
sin(6t +
π 4
[3 marks] (iv) Specify the amplitude and the period of the particular solution. [2 marks] (v) Plot the particular solution on a graph, where the horizontal axis is t and the vertical axis is θ. The plot should display at least one full period of the function, and it should clearly indicate the positions of the maxima, minima and zeros of the solution. [3 marks]
Paper Code MATH 014 May 2010 Page 5 of 5 END
