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Highlighted questions are not tested in 2015 C1 Block Test

1. Amanda sells fresh seafood at a market stall, and every day she has a kg of fish, b kg of

crabs and c kg of prawns to sell. She sells fish at $8 per kg, crabs at $12 per kg and

prawns at $10 per kg. However, the cost price of the seafood is different each day, and

the table below shows the cost price of each of the seafood for 3 particular days, and

the profit that Amanda made for each day:

Cost Price of

Fish per kg

Cost Price of

Crab per kg

Cost Price of

Prawn per kg

Profit made

Day 1 $3.10 $5 $4.80 $1874.50

Day 2 $3.30 $5.50 $4.40 $1833.50

Day 3 $3.70 $5.90 $4 $1670.50

Amanda sold all her seafood on Day 1 and 2, but was left with 5 kg of fish and 15 kg of

crabs unsold on Day 3. Ignoring the loss incurred by the leftover stock, calculate the

values of a, b and c respectively. [3]

2. Show the sum ( 4) ( 2) ... ( 2 )m m m m n , where , , 2m n n , can be

expressed as (n + 3)(m + n 2). [2]

Hence find the sum of all odd integers between 100 and 500 which are NOT divisible

by 5. [4]

3. Expand 4 5

3 1 2x x as a series in ascending powers of x, up to and including the

term in 2x , simplifying the coefficients. [3]

Find the coefficient of 8x in the expansion of 4 5

3 1 2x x and state the range of

values of x for which the expansion is valid. [3]

4. Given that 0 1x , write down the first three non-zero terms in the expansion

of2

11

x , in ascending powers of x. Hence, by putting

1x

n in your result, where n is

an integer to be determined, obtain an approximate value of 37 , correct to 3 decimal

places. [6]

5. The nth term of a sequence is 3 1

3( 1)

2

n

n

. Prove that the sequence follows a geometric

progression and state its first term and common ratio. Find the sum, nS , of the first n

terms of the sequence, leaving your answer in terms of n. [4]

Find the sum to infinity, S , of the progression and the least value of n for which the

difference between nS and S is less than 1210 . [4]

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6. The functions f, g and h are defined as follows:

f : 3

2

xx

x

, x ,

g : ln(4 )x x , 4x ,

h : 1

2,ex

x x b where b > 0.

(i) Show that f is not a one-to-one function. If the domain of f is restricted to [ a, ), find the least value of a for which f is one-to-one. [2]

(ii) Define, in a similar form, the function f 1 corresponding to the new domain

found in part (i). [3]

(iii) Given that the composite function gh is well defined and the range of gh

is (ln 6 , ln12] , find the exact value of b. [3]

7. (i) Show algebraically, for any real values of x, 2 3 4x x is always negative. Hence, without the use of a graphic calculator, solve the inequality

2

2 3 1

7 10 2

x

x x

. [5]

(ii) Using the result in part (i), solve the inequality 2

2 4 2

2 3 1

7 10 2

ax

a x ax

, where

0a , leaving your answer in exact form in terms of a. [3]

8. Prove by the method of mathematical induction that

12

2

sin1cos cos 2 ... cos

2 2sin

nn

for n , 0 < < . [5]

Hence find an expression for 10

cosN

r

r

, N , 10N , in the form 2

sin cos

sin

a b

,

where a and b are constants to be determined in terms of N . [3]

9. (a) Given that f is a one-to-one function, determine if 1ff exists. Justify your

answer. [2]

(b) The functions g and h are defined as follows:

g : e 1, 0,xx x

h : ln 2 , 3.x x x (i) Show that the function gh exists. Define gh in a similar form and find its

range. [4]

(ii) Sketch, on the same diagram, the graphs of g, 1g and 1gg . [3]

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(iii) State the range of values of x satisfying the equation 1 1g g( ) gg ( ).x x [1]

10. The curve C has parametric equations

3cosec 2x , 3cot 3y , , 0 .

(i) Determine the exact values of where C cuts the axes. Hence find the coordinates of the axial intercepts of C, leaving your answer in exact form. [4]

(ii) Determine the Cartesian equation of C, and find the equations of the asymptotes

of C, leaving your answer in exact form. [4]

(iii) An ellipse has its centre at 2, 3 , width h and height k. Find the equation of the ellipse in terms of h and k. Given that the ellipse does not intersect C, give the

range of possible values of h and k. [3]

11. Let f ( ) ( 1)( 2)r r r r . By considering f ( ) f ( 1)r r , show that

13

1

( 1) ( 1)( 2)n

r

r r n n n

. [5]

Deduce that 2 16

1

( 1)(2 1)n

r

r n n n

. [3]

Hence find the sum of the series

2 2 2 2 2 2 2 21 2 2 3 2 4 5 2 6 ... 2( 1)n n ,

where n is an odd integer. [4]

12. The curve C has equation

2 29 18

f( ) ,2

x kx kx

x k

where k is a real constant such that 0k .

(i) Determine the equation(s) of the asymptote(s) of C. [3]

(ii) Determine the coordinates of the axial intercept(s) of C, leaving your answer(s) in

terms of k. [3]

(iii) Given that C has 0, 9k and 4 , k k as its stationary points, sketch the graph

of 1

f( )y

x , stating clearly the coordinates of the axial intercept(s) and the

stationary point(s), and the equation(s) of the asymptote(s) in terms of k where

applicable. [4]

The curve D has equation 2 24 22 28

g( )2 3

x kx kx

x k

. By considering g( ) f( )x ax b ,

determine the values of a and b, and hence determine the sequence of transformations

that will transform C to D. [4]

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