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SAMPLE PAPER
Level 3 Mathematics and Statistics
(Mathematics)
91579 (3.7): Apply integration methods in solving
problems
Credits: Six
Check that you have completed ALL parts of the box at the top of this page.
You should answer ALL parts of ALL questions in this booklet.
If you need more room for any answer, use the space provided at the back of this booklet.
Check that this booklet has pages 2–14 in the correct order and that none of these pages is blank.
YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE ALLOTTED TIME.
OVERALL LEVEL OF PERFORMANCE
© New Zealand Qualifications Authority, 2012
All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.
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Mathematics and Statistics 91579 (3.7)
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You are advised to spend 60 minutes answering the questions in this booklet.
QUESTION ONE
(a)
Find the integrals.
You do not need to simplify your answers.
Do not forget the arbitrary constant.
(b)
(i)
x3 + 5
ò x dx
(ii)
ò sec 4x tan 4x dx
An object’s acceleration is given by the formula
a(t) = 0.8 + 0.4e0.2t
where a is the acceleration of the object in m s -2
and t is the time (in seconds) since the start of the object’s motion.
If the object had a velocity of 4 m s -1 after 3 seconds, how far did it travel between t = 5 and
t = 6?
Give the results of any integration needed to solve this problem.
Mathematics and Statistics 91579 (3.7)
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(c)
Find the integral
ò 4x
x 2 + 3 dx
Mathematics and Statistics 91579 (3.7)
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5
(d)
The curves on the graphs below are y = sin 2 kx and y = cos2 kx
y
x
Find the shaded area.
Give the results of any integration needed to solve this problem.
Mathematics and Statistics 91579 (3.7)
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6
QUESTION TWO
(a)
Assessor’s
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Find the integral.
You do not need to simplify your answer.
Do not forget the arbitrary constant.
ò 4e
(b)
3x-5
dx
A curve passes through the co-ordinates given in the table below.
x
y
6
1.2
6.5
4
7
5.7
7.5
5.5
8
4.2
8.5
3.6
9
2
Use the Trapezium rule with h = 0.5 to find the area bounded by the curve, the x -axis and
the lines x = 6 and x = 9.
Mathematics and Statistics 91579 (3.7)
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(c)
Find k such that
k2
ò1
x
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dx = 3
Give the results of any integration needed to solve this problem.
(d)
æ xö
5 cos ç ÷
è 2ø
π
π
dy
If
, and y = 2 when x = , find the value of y when x =
=
dx
6y
3
2
Give the results of any integration needed to solve this problem.
Mathematics and Statistics 91579 (3.7)
8
(e)
A rumour starts that New Zealand is going to become part of Australia. The rate at which
people hear the rumour is proportional to the number of people who have not heard the
rumour. Let P be the population of New Zealand.
(i)
Write the differential equation which expresses this statement.
After 5 hours half the population have heard the rumour.
Assume that no-one had heard the rumour at t = 0 .
(ii)
Solve the differential equation in part (i) to find how long it would take for 90% of the
population to have heard the rumour.
Show all working and give the results of any integration needed to solve this
problem.
Mathematics and Statistics 91579 (3.7)
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QUESTION THREE
(a)
Find the following integral.
You do not need to simplify your answer.
Do not forget the arbitrary constant.
ò (2x - 4)
(b)
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4
dx
Part of the graph y = e2 x is shown below.
y
x
1
k
Find the value of k such that the shaded area is 40.
Give the results of any integration to solve this problem.
Mathematics and Statistics 91579 (3.7)
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(c)
Assume that at any time the population of a colony of mice increases at a rate proportional to
its population.
Initially the population of the mouse colony is 60.
5 months later the population is 220.
How long does it take, from the initial time, for the population to reach 1000?
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(d)
Part of the graph y = 4sin 3x cos x is shown below.
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y
x
π
3
Use integration to find the area enclosed by the curve y = 4sin 3x cos x and the x -axis from
x = 0 to x =
π
3
Show your working.
Give the results of any integration needed to solve this problem.
Mathematics and Statistics 91579 (3.7)
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(e)
The curve on the graph below is defined by the equation y = (e0.2 x )2 . Point R is (5, 7.39) and
RS is the tangent to the curve at R.
y
R
x
S
Find the shaded area.
Give the results of any integration needed to solve this problem.
Mathematics and Statistics 91579 (3.7)
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Extra space if required.
Clearly number the question (if required).
Question
number
Mathematics and Statistics 91579 (3.7)
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Extra space if required.
Clearly number the question (if required).
Question
number
Mathematics and Statistics 91579 (3.7)
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