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Additional Mathematics Paper 1 (2008) Preliminary Examination 3

1 Find all the angles between 0o and 360o which satisfy the equationxx sin432cos2 =

Find all the angles between 0 and 4 which satisfy the equation

xx cos33

sin =

[4]

[4]

2 Find the range of values ofp for which the expression 222 ++ ppxxx is

never negative for all real values ofx. [4]

3 Given that 13 += xx eey , find the rate of change ofx at the instant when 1=y ,

given thaty is changing at the rate of 4.0 units per second at this instant. [4]

4A curve is such that

( ) 312

8

d

d

=

xx

y. The tangent to the curve, at a certain point,

cuts the curve at the point (1, 2). Find the equation of the curve. [4]

5 The diagram shows part of the curve xxy 2sin3cos2 += .

Find the area of the shaded region, bounded by the curve, the coordinate axes and

the line6

5=x .

[6]

6 If 14323 ++= xxxy , show thaty is an increasing function for all real values of

x.

Hence, state the minimum value of the gradient of this function. [5]

Preliminary Examination 3 Additional Mathematics Paper 1 (2008)

y

x

2

6

50

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Additional Mathematics Paper 1 (2008) Preliminary Examination 3

[4]

7Find thex-coordinate of the stationary point of the curve

( )

1

13

+

=

x

xy ,x > 0.

By considering the sign ofdx

dy, or otherwise, determine the nature of the stationary

point.[6]

8 Given that 322 + xx is a factor of the polynomial bbxaxxx 65232 234 +++ ,

find(a)

(b)

the value of a and ofb,

the other quadratic factor of the polynomial.

[5]

[2]

9 (a)

(b)

Given that 32 =p , expressp

p32 in the form 3ba + , where a and

b are integers.

Solve the equation

81log)12(log6)13(log2 382 = xx

[4]

[5]

10The first three terms in the expansion of ( ) ( ) npxx

+11 , where p and n are

constants, are 2951 xx + . Find the value of a and of n.

[7]

11 Given that the equation of a circle is 0156222 =++ yxyx , find the

(a)

(b)

coordinates of the centre and the radius of the circle,

equation of another circle such that the new circle is a reflection of

0156222 =++ yxyx in they-axis.

[3]

[2]

Preliminary Examination 3 Additional Mathematics Paper 1 (2008)

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Additional Mathematics Paper 1 (2008) Preliminary Examination 3

12 The diagram shows a roof in the shape of a right inverted circular cone whose

radius is rm and its slanted height lm. The sloping surface of the roof is covered

with a sheet of thin metal whose area is 34 m2.

[Curved Surface Area of Cone = rl where l represents the slanted height of thecone]

(a)

(b)

Express lin terms ofrand show that

the volume of the cone, V cm3 is given by

4483

rrV =

.

Given that rcan vary, find

[3]

(i)

(ii)

an expression fordrdV ,

the value of rfor which the Vhas a stationary value.[2]

[3]

13 Variables x and yare related by the equation xaby = , where a and b areconstants. The table below shows measured values ofx and y.

x 1 1.5 2 2.5 3

y 10.0 14.9 22.2 33.1 49.4

(i) On graph paper, plot yln against x , and draw a straight line graph to

represent the equationxaby =

[2]

(ii) Use your graph to estimate the value of a and of b. [3]

(iii) By drawing a suitable line on your graph, solve the equation xx eab 21+= [3]

End of Paper

Preliminary Examination 3 Additional Mathematics Paper 1 (2008)

l

r