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This is the Past Exam of Mathematics which includes Rational Functions, Definite Integrals, Indefinite Integrals, Partial Fractions, Constants, Evaluate, Integration, Parts, First Order Differential etc. Key important points are: Rational Functions, Definite Integrals, Indefinite Integrals, Partial Fractions, Constants, Evaluate, Integration, Parts, First Order Differential, Separation
Typology: Exams
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Examiner: Dr. T. M. Mohaupt, Extension 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.
(i)
∫ 5 x−^4 dx [2 marks] , (ii)
∫ (^) dx √ 5 x − 1
[2 marks] ,
(iii)
∫ x(x + 1)^8 dx [3 marks] , (iv)
∫ e−^5 x+2^ dx [3 marks].
(i)
∫ (^1)
− 1
x^2 (4x + 3) dx [2 marks] , (ii)
∫ (^27)
3
dx √ 3 x
[2 marks] ,
(iii)
∫ (^3)
0
(4x^2 − 7 x + 1) dx [3 marks] , (iv)
∫ (^) π π 2 cos(5x)^ dx^ [3 marks]^.
(i)
5 x − 13 (x − 7)(x + 4)
x − 7
x + 4
(ii)
4 x^2 − 3 x − 1 (x − 2)(x + 1)^2
x − 2
x + 1
(x + 1)^2
Compute the constants A, B, C, D, E. [5 marks]
Hence evaluate the following integrals:
(iii)
∫ (^5) x − 13
(x − 7)(x + 4)
dx ,
[2 marks]
(iv)
∫ (^5)
3
4 x^2 − 3 x − 1 (x − 2)(x + 1)^2
dx.
[3 marks]
(i) A car drives with a speed of 40m s. How long does it take to travel a distance of 4km? [1 marks]
(ii) A speed limit forces the driver to reduce his speed from 40 m s to 20m s. If this takes 4 seconds, what is the deceleration during this period? And what is the distance traveled during this period? [6 marks]
(iii) The car continues with a speed of 20m s for 6 minutes. What is the distance traveled during this period? [1 marks]
(iv) The driver applies the same deceleration as under (ii) to bring the car from the speed 20 m s to complete rest. How long does this take and what is the distance traveled during this period? [5 marks]
(v) What is the total distance traveled in the trip described in (i)-(iv)? How much time does the trip take? [2 marks]
(i) Solve the following second order differential equation
d^2 y dx^2
and find the particular solution where y = −2 and dy dx =
3 when x = π 6
[8 marks]
(ii) Evaluate the following integral ∫ (^0)
− π 8
sin^3 (4x) cos^2 (4x) dx
Hint: You may use the substitution u = cos(4x). [7 marks]
(i) Using the notation that the vertically upwards direction is denoted by y, with the origin y = 0 at ground level, show that the equation of motion for the stone is d^2 y dt^2
= −g ,
where g ≈ 10 m s 2 is the acceleration due to gravity, and t is time. [3 marks]
(ii) Find the general solution of this differential equation. [3 marks]
(iii) Compute the initial velocity v 0 and the maximal height reached by the stone. Hint: the maximal height is reached after one half of the total time. [6 marks]
(iv) If you double the initial velocity, how does this change the maximal height and the total time of the flight? [3 marks]
d^2 θ dt^2
where t is time and k^2 = 499.
(i) Given that θ = 3 and dθ dt = 7 at t = 0, solve the above differential equation. [5 marks]
(ii) Show that your solution is identical to the function
θ(t) = 3
2 sin(^73 t + π 4 ). (2)
You may either use trigonometric identities or show that the function (2) solves the differential equation (1) and satisfies the initial conditions spec- ified in part (i). [5 marks]
(iii) Plot this function on a graph, where the horizontal axis is t and the vertical axis is θ, such that you display at least one full period of the function. Specify explicitly the period of the function. [5 marks]
Paper Code MATH 014 Page 5 of 5 END
f (x) : cxn^
c x
c xn^
cex^ c ln x sin x cos x tan x
df dx
: c nxn−^1 −
c x^2
nc xn+^
cex^
c x
cos x − sin x
cos^2 x
= sec^2 x
f (x)
∫ f (x)dx f (x)
∫ f (x)dx
(ax + b)n^
(ax + b)n+ a(n + 1)
, (n 6 = −1)
ax + b
a
ln |ax + b|
(ax + b)^2
a(ax + b)
(ax + b)(cx + d)
ad − bc
ln
∣∣ ∣∣ ∣
ax + b cx + d
∣∣ ∣∣ ∣ ,^ (ad^ −^ bc^6 = 0)
ax^2 + b
ab
arctan
(√ a b
x
) , (a, b > 0)
ax^2 − b
ab
ln
∣∣ ∣∣ ∣
ax −
b √ ax +
b
∣∣ ∣∣ ∣, (a, b >^ 0)
x ax^2 + b
2 a
ln |ax^2 + b| eax^
a
eax
sin(ax) −
a
cos(ax) cos(ax)
a
sin(ax)
∫ f (x) dx =
∫ ( h(u)
dx du
) du , where f (x) = h(u(x)).
∫ (^) b
a
f (x)dx =
∫ (^) u(b)
u(a)
( h(u)
dx du
) du , where f (x) = h(u(x)).
∫ u(x)
dv dx
dx = u(x)v(x) −
∫ (^) du
dx
v(x) dx + C.
∫ (^) b
a
u(x)
dv dx
dx = [u(x)v(x)]ba −
∫ (^) b
a
du dx
v(x) dx.
Separation of Variables
Equation Solution
dy dx
= f (x) y =
∫ f (x)dx
dy dx
= ay y = eax+C^ , for y > 0.
Second order, linear, unforced (homogeneous) differential equations
Standard form:
a
d^2 y dx^2
dy dx
Characteristic equation: aα^2 + bα + c = 0. (2)
Solution of characteristic equation (2) for a 6 = 0:
α 1 , 2 =
−b ±
b^2 − 4 ac 2 a
Discriminant: D = b^2 − 4 ac.
