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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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