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ANDERSON JUNIOR COLLEGEH2 MATHEMATICS (JC1)

2006 MID-YEAR COMMON TEST

Answer all questions

1. Solve the inequality2

52

x

x

, giving your answer in exact form.

Hence deduce the range of values ofxfor which2 5

2x

x

. [5]

2. Sketch the graph of 2

1

x ay

x

where a> 2. Hence find the set of values ofxsuch that

2

1

x a

x ax

. [6]

3. Using the binomial theorem, express 3

3

x

x

3

as a series in ascending powers ofxup to and

including the term inx2. State the range of values ofxfor which the expansion is valid.

By putting1

3x , find an approximation for (10)

1

3 , giving your answer as a fraction in its

lowest terms. [7]

4 i) The equation of the curve C1given by2 24 9 36x y . Sketch the graph of C1, showing

all its axial intercepts clearly. [2]

ii) Find the asymptotes of the curve C2 given by2 24 6x y . Sketch C2 on a separate diagram,

showing its asymptotes and its axial intercepts clearly. [3]

iii) Find the equation of the parabola that passes through the origin and the points of intersection

of the curves C1 and C2wherex> 0. [3]

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5. The functionsfandgare defined by3

: 2 , , 04

xf x x x

: , , 0x ag x e x x where ais a positive constant.

i) Sketch the graph offand find its range. Hence state the set of values ofxfor which

( ) ( )ff x f f x [3]

ii) Sketch the graph ofgand define the inverse function g . [3]

iii) Show thatgfdoes not exist. Find the maximal domain offfor whichgfexists. [3]

6a) A geometric series is given as follows:

12 3

2 211 12 1 ...

2 2 2

r

rxx xx

+.

i) Find the set of values ofxsuch that the sum to infinity of the series does not exist. [2]

ii) When1

3x , find the sum of all the odd-numbered terms ( ie the first, third, fifth, .) of

the series. [3]

b) A sequence of positive integers are arranged in a triangular formation such that the integers in

each row are arranged from left to right and every row (other than the first) contains twointegers more than the previous row as shown below:

1

2 3 4

5 6 7 8 9

10 11 12 13 14 15 16

:

i) Find an expression for the total number of integers in the first nrows. [2]

ii) State, in terms of n, the last integer of the nth

row. [1]

iii) Hence or otherwise, find the sum of all the integers in the nth

row. [3]

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7 a) A sequence 1 2 3, , , ...u u u . is defined by 1ln( 1)

ln( 2)r r

ru u

r

. Given u1= 1, find an expression

for unin terms of n. [3]

b) i) Find the limit of the sequence 1 2 3, , , ...u u u ..when2

2 1r

ru

r

. [2]

ii) Express2

21

r

r in the form

1 1

B CA

r r

where A, B and C are constants to be

determined.

Hence or otherwise, find the sum of the series2

n

r

r

u

in terms of n. [6]

iii) State with a reason, whether2

n

r

r

u

converges as n . [2]

8 a) Prove by induction that

1

1

2 21

1 2 2

r nn

r

r

r r n

for all n . [4]

Hence find, in terms of n,

34

0

3 2

4 5

kn

k

k

k k

. [3]

b) A sequence of numbers 1 2 3, , , ...u u u is defined by 1 1u and

1

2 2 1

2n n

nu u

n

for all

positive integral values n. Prove by induction that

2 !

1 ! !

n

nu

n n

. [4]

END