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Second Semester Examination, Sample Exam 1, 2002
MATH1040, SAMPLE 1
Basic Mathematics
(Unit Courses)
Time:
TWO Hours for working
Ten minutes for perusal before examination begins
Check that this examination paper has 17 printed pages!
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THIS EXAMINATION PAPER!
There are 120 marks on the paper. All questions carry the indicated number of
marks, and students should aim to complete all questions. Page 17 contains
formulae. You may write on the front cover only of this booklet during
perusal. Pocket calculators allowed.
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2
MATH1040, SAMPLE 1 — Basic Mathematics
Second Semester Examination, Sample Exam 1, 2002 (continued)
1. Find R if
2
X
2
1
=
.
R i=−1 | 2i | −1
(4 marks)
2. (a) Find all x such that |2x + 1| < 5.
(3 marks)
(b) Write your answer to part (a) in interval format.
(1 mark)
(c) Mark your answer to part (a) on the real line.
(1 mark)
3. Let A = {a, b, d, e}, B = {a, d, f, g} and C = {a, c, e, f }. For parts (b), (c), (d) and
(e), either write the elements of the answer set, or (if appropriate) state that the
answer is the empty set.
(a) Mark the three sets A, B and C on the following Venn diagram, with the
elements of each set written in appropriate places on the diagram. (2 marks)
(b) Write the set (A ∩ C) \ B.
(2 marks)
(c) Write the set B \ (B ∩ B).
(2 marks)
(d) Write the set (A ∪ B) ∩ (C \ A).
(2 marks)
(e) Write the set (A \ ∅) ∩ (B \ ∅) ∩ (C \ ∅).
(2 marks)
4. Simplify
2n+1 − 2n
.
2n
(3 marks)
TURN OVER
3
MATH1040, SAMPLE 1 — Basic Mathematics
Second Semester Examination, Sample Exam 1, 2002 (continued)
5. Solve the following system of two simultaneous equations.
3y − x = 6
2y + 3x = 15
(4 marks)
6. Find the straight line distance between the points (4, 2) and (1, 5), expressing your
answer as a surd in simplest form.
(2 marks)
7. I roll two fair, six-sided dice, one coloured green and the other blue. Let g denote
the number shown on the green die, and b denote the number shown on the blue
die. What is the probability that:
(a) g + b = 7?
Answer:
(2 marks)
(b) g is odd and b is even?
Answer:
(2 marks)
(c) g < b?
Answer:
(2 marks)
8. Find from first principles, using limits, the slope of the following function at any
point x:
1
y= .
x
(4 marks)
√
1
. (You may
4
use any rules for differentiation that you like: there is no need to use limits.)
(3 marks)
9. (a) Show that the slope of the curve y =
x at the point (4, 2) is
(b) Hence, or √
otherwise, find the equation of the line which is tangential to the
curve y = x at the point (4, 2).
(2 marks)
10. If y = sin(e2x ), find
11. If y =
dy
.
dx
(4 marks)
dy
(2x + 1)
, find
.
(2x − 1)
dx
(4 marks)

−1 2
0 1
−1
2
12. Let A =
, B = (1 − 1 2) , C =
, D =  0 1 .
1 0
2 −1
−2 0
If possible, evaluate the following expressions. Where not possible, indicate why.

TURN OVER
4
MATH1040, SAMPLE 1 — Basic Mathematics
Second Semester Examination, Sample Exam 1, 2002 (continued)
(i) A2 + 2C
(2 marks)
(ii) D × B
(2 marks)
(iii) B × D
(2 marks)
13. Let f (x) = −x3 + 9x2 − 24x − 3.
(a) Find the coordinates of all critical points of f (x).
(5 marks)
(b) Classify each critical point as a local maximum or minimum.
(3 marks)
(c) Find the absolute maximum and minimum values attained by f (x) on the
interval x ∈ [1, 3].
(3 marks)
14. The following two sets of axes each contain a graph of y = sin x, for x ∈ [−2π, 2π].
On the first set of axes, sketch a graph of y = 2 sin x.
(2 marks)
1
sin 2x.
(2 marks)
2
There will be a second set of axes here, just like the set above.
On the second set of axes, sketch a graph of y =
15. The following set of axes contains the graph of a function and the graph of the
derivative of the function. The two graphs have been labelled Graph A and Graph
TURN OVER
5
MATH1040, SAMPLE 1 — Basic Mathematics
Second Semester Examination, Sample Exam 1, 2002 (continued)
B. Clearly identify which graph represents the function, and which represents the
derivative. Explain your answer. (Most marks will be assigned to your explanation.)
(3 marks)
Graph B
Graph A
The function is Graph
.
The derivative is Graph
.
Explanation:
16. On the following page there are eight equations, with their graphs drawn below the
equations, in random order, with eight extra (unused) graphs included. Match each
equation with its graph, by writing the letter of the corresponding graph next to
each equation. Note that the graphs are labelled A to P.
(8 marks)
Equation
Graph
Equation
Graph
y 2 + 2y + 2 = (y + 1)2 − x.
y = −e−x .
y = −(x − 1)(x − 3).
y = − | x |.
3x − 3 + 2y = 0.
x = (−1)4 y.
y=
r
1
.
16
x−
√
5 = −2.
TURN OVER
6
MATH1040, SAMPLE 1 — Basic Mathematics
Second Semester Examination, Sample Exam 1, 2002 (continued)
17. A rocket takes off vertically at time t = 0, with velocity v(t) = 6t metres/second.
(i) Find an expression for the displacement of the rocket at any time t. Assume
the rocket launching pad is 6 metres below ground level, so the displacement
at t = 0 is −6.
(4 marks)
(ii) At what time t is the rocket 42 metres above ground (so it has travelled 48
metres in total).
(3 marks)
TURN OVER
7
MATH1040, SAMPLE 1 — Basic Mathematics
Second Semester Examination, Sample Exam 1, 2002 (continued)
18. Evaluate
Z
1
(4x3 + ex + 1) dx.
(3 marks)
−1
19. Find
Z 2
− sin x + e2x
x
dx.
(4 marks)
20. For parts (a), (b), (c) and (d) below and on the next page, let y = e−x −x2 + x − 1 .
dy
= e−x (x2 − 3x + 2).
dx
(4 marks)
(a) Using the product rule (or otherwise), show that
(b) Find all values of x for which
dy
= 0.
dx
(2 marks)
(c) Find the second derivative of y.
(5 marks)
(d) Hence, or otherwise, find and classify all local maxima and minima of the
function y.
(2 marks)
21. Let I be the 2 × 2 identity matrix and define the matrices X and Y as follows:
X=
Prove that
a b
c d
1
Y =
(ad − bc)
,
d −b
−c a
XY − I 2 = 0.
.
(5 marks)
22. Find all values of x for which



4
0
7
−2
0 1 x  2 2 0   −1  = 0.
1 −1 1
x
(5 marks)
TURN OVER FOR FORMULAE SHEET
8
MATH1040, SAMPLE 1 — Basic Mathematics
Second Semester Examination, Sample Exam 1, 2002 (continued)
Formulae Sheet:
Roots of ax2 + bx + c = 0 are:
x=
−b ±
√
b2 − 4ac
2a
Limits and derivatives:
f (x + h) − f (x)
h→0
h
f 0 (x) = lim
Product rule:
(uv)0 = u0 · v + u · v 0
Chain rule:
dy
dy du
=
.
dx
du dx
Quotient rule:
u 0
v
=
u0 · v − u · v 0
v2
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