2010 year 6 h2 maths prelim exam qp1

Name ( ) Class

RIVER VALLEY HIGH SCHOOL 2010 Year 6 Preliminary Examinations Higher 2

MATHEMATICSPaper 1

Additional Materials: Answer Paper List of Formulae (MF15)

Cover Page

9740/01 14 September 2010

3 hours

READ THESE INSTRUCTIONS FIRST

Do not open this booklet until you are told to do so.Write your name, class and index number in the space at the top of this page.

Write your name and class on all the work you hand in.Write in dark blue or black pen on both sides of the paper.You may use a soft pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.

Answer all the questions.Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.You are expected to use a graphic calculator.Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands.You are reminded of the need for clear presentation in your answers.

At the end of the examination, place the cover page on top of your answer paper and fasten all your work securely together.

The number of marks is given in brackets [ ] at the end of each question or part question.

This document consists of 6 printed pages.

1 A shampoo manufacturing company produces shampoo that comes in three different sizes of bottles (small, medium and large). The amount of plastic required to manufacture the bottles, and the profit made from the sale of each bottle of shampoo are shown in the table below:

Small Medium LargeVolume of shampoo (cm3) 200 450 600Amount of plastic (cm2) 150 335 475Profit ($) per unit sold 2.50 3.80 4.20

On a particular round of production, the volume of shampoo and the amount of plastic used were 370700cm3 and 280400cm2 respectively. When all the bottles of shampoo from this round of production were sold, the profit made from the sale of the medium bottles was twice the total profit made from the sale of the small and large bottles. Determine the number of each size of shampoo produced and state an assumption made about the bottles manufactured. [6]

2 The complex number z satisfies the relations and .

(i) Illustrate both of these relations on a single Argand diagram. [3]

(ii) Find the greatest and least possible values of arg , giving your answers in radians correct to 3 decimal places. [3]

3 Given that is sufficiently small for and higher powers of to be neglected,

express as a quadratic expression in .

[4]

By letting , show that . [2]

RIVER VALLEY HIGH SCHOOL 9740/01/2010

2

4 (i) Find . [1]

(ii) Evaluate the integral numerically.

[1]

(iii) Given that and

,

show that , where a and b are constants to be

determined. [4]

5 The sequence of numbers u u u1 2 3, , , is given by and for all positive

integers.

(i) By writing down the terms and , make a conjecture for in terms of . [2]

(ii) Prove your conjecture by mathematical induction. [4]

(iii) Write down the limit of as n tends to infinity. [1]

6 (i) Verify that [1]

(ii) By considering , show that

. [4]

(iii) Evaluate . [2]


3

7 The diagram below shows a sketch of the graph of where , , and

are constants. The asymptotes, also shown in the diagram, are and .

(i) Write down the value of and find the values of and . [4]

(ii) Given that the curve passes through the point , find the value of . [1]

(iii) By sketching the graph of , determine the range of values of

such that the equation , where , , and are

the values found above, has at least one negative root. [3]

8 (i) Solve the equation

,

giving the roots in the form , where and . [4]

(ii) Express as a product of three linear factors. [2]

(iii) Hence, find the values of the real numbers a and b in the equation where the roots are the complex numbers found in (i). [2]

(iv) Explain why the roots in (i) lie on a circle centre (2,0) and radius 2. [1]


4

y

xO24

4

9 It is given that , where .

(i) Show that . [1]

(ii) By further differentiation of this result, find the Maclaurin series for , up to

and including the term in . Hence, write down the equation of the tangent of

at . [5]

Denote the Maclaurin series of in (ii) by .

(iii) On the same diagram, sketch the graphs of and for

.

[2]

(iv) Find, for , the set of values of for which the value of is

within of the value of .

[2]

10 (a) Given that the first, third and fourth terms of an arithmetic progression are three consecutive terms of a geometric progression, and that the sum of the first twenty even-numbered terms of the arithmetic progression is 960, find the common difference of the arithmetic progression. [4]

(b) Annie puts $x on 1 January 2010 into a bank account which pays compound

interest at a rate of 3% per month on the last day of each month. On the first day of each subsequent month, she puts in an amount which is $x more than the amount she puts in the previous month. For example, she puts in $x on 1 January, $2x on 1 February and so on.

(i) Show that the amount of money in the bank account on the last day of

March is . [4]

(ii) Find the least integer value of x so that the amount of interest earned for the first three months of the year 2010 exceeds $100. [2]


5

11 The lines and meet at P. The line lies on the same plane containing and , and

is perpendicular to . Given that and are parallel to the vectors a and b respectively,

show that is parallel to the vector . [3]

The lines and have equations and

respectively. The line lies on the plane containing

and and is perpendicular to . Given that the three lines intersect, find the equation of

. Find also the acute angle between and . [3]

The plane contains the lines , and , and the plane has equation

. Given that is parallel to , show that . Hence, find the exact

distance between the two planes. [6]

12 (i) Sketch the graph of .

[2]

(ii) The region R is bounded by the curve, the axes and the line . Using the substitution , find the exact area of R. [5]

(iii) Find the exact volume of revolution when R is rotated completely about the y-axis. [6]

End of Paper


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1

Name ( ) Class

Tutor Calculator Model

RIVER VALLEY HIGH SCHOOL 2010 Year 6 Preliminary Examinations Higher 2

MATHEMATICSPaper 1

Cover Page

9740/01 14 September 2010

INSTRUCTIONS TO CANDIDATES

Attach this cover page on top of your answer paper.Circle the questions you have attempted and arrange your answers in NUMERICAL ORDER.

Question Mark Max. Mark Question Mark Max. Mark

1 6 7 8

2 6 8 9

3 6 9 10

4 6 10 10

5 7 11 12

6 7 12 13

Total 100 Grade


2010 year 6 h2 maths prelim exam qp1

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