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FOR EDEXCEL
GCE ExaminationsAdvanced Subsidiary
Core Mathematics C3
Paper E
Time: 1 hour 30 minutes
Instructions and Information
Candidates may use any calculator EXCEPT those with the facility for symbolic
algebra, differentiation and/or integration.
Full marks may be obtained for answers to ALL questions.
Mathematical formulae and statistical tables are available.
This paper has seven questions.
Advice to Candidates
You must show sufficient working to make your methods clear to an examiner.
Answers without working may gain no credit.
Written by Shaun Armstrong
Solomon Press
These sheets may be copied for use solely by the purchasers institute.
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1. Express
3 2
2
2
4
x
x
+
2
2
2 5 3
x
x x
as a single fraction in its simplest form. (5)
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2. (a) Prove that, for cosx 0,
sin 2x tanx tanx cos 2x. (5)
(b) Hence, or otherwise, solve the equation
sin 2x tanx = 2cos 2x,
forx in the interval 0 x 180. (5)
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3. f(x) =x2 + 5x 2 secx, x , 2
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3. continued
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4. (a) Differentiate each of the following with respect tox and simplify
your answers.
(i) 1 cosx
(ii) x3
lnx (6)
(b) Given that
x =1
3 2
y
y
+
,
find and simplify an expression ford
d
yin terms ofy. (5)
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4. continued
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5. (a) Express 3 sin + cos in the form Rsin ( + ) where R > 0
and 0 < < 2
. (4)
(b) State the maximum value of 3 sin + cos and the smallest positive
value of for which this maximum value occurs. (3)
(c) Solve the equation
3 sin + cos + 3 = 0,
for in the interval , giving your answers in terms of. (5)
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5. continued
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6. The function f is defined by
f(x) 3 x2, x , x 0.
(a) State the range of f. (1)
(b) Sketch the graphs ofy = f(x) and y = f1(x) on the same diagram in the
space provided. (3)
(c) Find an expression for f1(x) and state its domain. (4)
The function g is defined by
g(x) 8
3 , x , x 3.
(d) Evaluate fg(3). (2)
(e) Solve the equation
f1(x) = g(x). (3)
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6. continued
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7. T
18
12
O 10 60 70 120 t
Figure 1
Figure 1 shows a graph of the temperature of a room, TC, at time tminutes.
The temperature is controlled by a thermostat such that when the temperature falls
to 12C, a heater is turned on until the temperature reaches 18C. The room then
cools until the temperature again falls to 12C.
Fortin the interval 10 t 60, Tis given by
T= 5 +Aekt,
whereA and kare constants.
Given that T= 18 when t= 10 and that T= 12 when t= 60,
(a) show that k= 0.0124 to 3 significant figures and find the value ofA, (6)
(b) find the rate at which the temperature of the room is decreasing when t= 20. (4)
The temperature again reaches 18C when t= 70 and the graph for 70 t 120
is a translation of the graph for 10 t 60.
(c) Find the value of the constantB such that for 70 t 120
T= 5 +Bekt. (3)
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7. continued
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7. continued
END
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