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A math exam paper from may 2009, focusing on integration, differential equations, and motion. It includes various types of questions such as evaluating definite and indefinite integrals, solving differential equations, and deriving equations of motion. The exam covers topics like partial fractions, integration by parts, and separation of variables.
Typology: Exams
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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)
∫ 9 x^12 dx , [2 marks]
(b)
∫ (^) dx √ 3 x − 7
, [2 marks]
(c)
∫ 3 e^4 x+2^ dx , [2 marks]
(d)
∫ 3 xe^4 x
(^2) + dx. [4 marks]
(a)
∫ (^) − 1 / 3
− 4 / 9
(9x + 4)^11 dx , [2 marks]
(b)
∫ (^1)
− 1
x^5 (2x + 7) dx , [2 marks]
(c)
∫ (^2) π
−π
cos(x 2 + π) dx , [2 marks]
(d)
∫ (^3)
4 / √ 3
x dx √ 3 x^2 + 9
. [4 marks]
(a)
7 x − 1 (x + 2)(x − 3)
x + 2
x − 3
(b)
5 x^2 + 2x + 9 (x − 3)(x^2 + 3)
x − 3
Dx + E x^2 + 3
Compute the constants A, B, C, D, E. [5 marks]
(ii) Hence evaluate the following integrals:
(a)
∫ (^7) x − 1
(x + 2)(x − 3)
dx , [2 marks]
(b)
∫ √ 3
0
5 x^2 + 2x + 9 (x − 3)(x^2 + 3)
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. [3 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]
(iii) The particle is thrown downwards from the height h and reaches the ground after 2 seconds with a speed of 21 ms−^1. Find the initial speed and the height h. The approximate value of the gravitational acceleration is g ≈ 10 ms−^2. [5 marks]
(iv) The particle is reflected off the ground and now moves upwards with an initial speed of 21 ms−^1. Find the maximal height. How long does it take the particle from the ground to its maximal height? [3 marks]
d^2 y dx^2
dy dx
and find the particular solution where
y(1) = 3e−^3 and
dy dx
(1) = − 7 e−^3.
[8 marks]
(b) Evaluate the following integral:
∫ (^) π
0
cos^3 (x 2 ) sin(x 2 ) dx.
Hint: Use a substitution involving a trigonometric function. [7 marks]
(i) Show that the motion of the particle is governed by the differential equation
M
dv dt
= −kv
where v is the velocity of the particle, M is the mass of the particle, and k is a positive constant. [3 marks]
(ii) Find the general solution of this differential equation. In your final formula for v(t), express the integration constant in terms of the initial velocity. [3 marks]
(iii) Using your previous result, derive an expression for the distance travelled by the particle since the time where it had its initial velocity. [3 marks]
(iv) Given that the initial speed is 5 ms−^1 , the mass is 50 kg and k = 3kgs−^1 , how much time does it take until the particle slows down to 1 ms−^1? What is the distance which the particle travels during this time? [5 marks]
(v) Compute the maximal distance the particle can travel until it comes to a complete stop. [1 marks]
d^2 θ dt^2
where t is time and k^2 = 25.
(i) Find the general solution of this differential equation. [3 marks] (ii) Find the particular solution where
θ
( (^) π
20
) = 1 and
dθ dt
( (^) π
20
) = 0.
[4 marks] (iii) Show that the solution you obtained in part (ii) is identical to
θ = sin(5t + 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 2009 Page 5 of 5 END
