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ACS(Independent)Math Dept/Y4EXP_GEP/AM1/2008/Prelim

1 Write down the inverse of the matrix2 2

3 2

and use it to solve the simultaneous equations

15x y+ = ,

2 3 5.y x = [4]

2 In triangle ABC, ( )3 2AB = + cm and4

BAC

= radians. Given that the area of triangle ABC is

( ) 214 3 2 cm , find ,C leaving your answer in the form 2a b+ , where a and b are rational. [4]

3 The roots of the equation 23 2 1 0x kx k + = are and . If 2 2 109

+ = , find the possible values

ofk. [5]

4 A curve has the equation 5 3ln2 5

xy

x

+=

.

(i) Find the gradient of the curve at the point where the curve meets thex-axis. [4]

(ii) Show that5 3

ln2 5

xy

x

+=

has no stationary point for all real values ofx. [2]

5 A circle, C, whose equation is given by 2 2 2 10 2 0x y ax y+ + = , where 0>a , has radius 6 units.(i) Find the value of a. [2]

(ii) Show that the line 3 5x= passes through the centre ofC. [2]

(iii) Find the equation of another circle which passes through the point (7, 2) and has the same centre as

C. [3]

6 (a) Find the range of values of m for which )3(2)3(2 +>++ mxmx for all real values ofx. [4]

(b) Find the range of the values of k for which the line 2x k= + intersects the curve 22xy y k = at

two distinct points. [4]

7 (i) Express2

16

2 9 5

x

x x

in partial fractions. [4]

(ii) Hence evaluate8

26

16

2 9 5

xdx

x x

. [4]

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8 (a) Evaluate

22

0

1

x

x

edx

e

+ . [3]

(b) Sketch the graph of 3ln( 2)y x= + , showing clearly the asymptote and thex-intercept.Draw a suitable straight line in your sketch to illustrate how the graphical solution of the equation

1 23( 2)

x

e e+

+ = could be obtained. [5]

9 Solve the following equations

(a) 1)2(72 32 =++ xx , [4]

(b) 3 33log ( 5) log ( 1) log 2x x+ = . [5]

10 In the figure,PQR andRBTare tangents to the circle at Q andB respectively.PSA andPCB are secants andAB is a chord of the circle. Given that ABC is an equilateral triangle and PACAPC = ,

(a) find, with explanation, an angle equal to ABT , [2]

(b) explain why a circle can be drawn, using Cas centre, passing throughA andP, [2]

(c) prove that PBPCPAPS = , and hence show that PAPSAB =2

12. [5]

11 Solve forx, between o0 and o180 , which satisfy the following equations(a) 6sin cos 1 0x x = , [4]

(b) xx cos12cos2 = , [4]

(c) sin 4 cos3 sin 2 0x x x = . [4]

End of Paper

B

R

C

Q

S

T

AP

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Answers

Q1. 5, 10x y= =

Q2.92 96

27 7

+

Q3.1

or 22k k= =

Q4 (i).1

50

Q5. (i) 3a = (iii) 2 2 6 10 9 0x y x y+ + + =

Q6. (a) 53