VICTORIA JUNIOR COLLEGEPreliminary Examination

Higher 2MATHEMATICS 9740/ 01

(Paper 1)

September 2010

3 hoursAdditional materials: Answer paper

Graph PaperList of Formulae (MF15)

READ THESE INSTRUCTIONS FIRST

Write your name and CT group 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.Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise.Where unsupported answers from a graphic calculator are not allowed, 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.

The number of marks is given in brackets [ ] at the end of each question or part question.At the end of the examination, fasten all your work securely together.

This document consists of 5 printed pages

© VJC 2010 VICTORIA JUNIOR COLLEGE [Turn over

1 The equation of a curve C is . Show that every line parallel to the x-axis cuts C at two distinct points. [3]

Without differentiating, explain, giving a reason, if there is any point on the curve at which the tangent is parallel to the x-axis.[1]

2 In the game of TapFarm, a player is given plots of land to grow tomatoes, pumpkins and

cherries. The player can only plant a type of fruits on a plot of land each time and each plot of land produces 1 kg of fruits. The time taken from planting to harvesting the fruits, cost price and selling price for 1kg of fruits are as follows:

Time required (Hours) Cost price ($) Selling price ($)Tomatoes 3 15 35Pumpkin 5 a 50Cherries 16 45 100

Tommy has only 1 plot of land. He used $545 and spent 154 hours playing the game before harvesting a total of 23 kg of fruits and earned a profit of $735. Showing your working clearly, find the value of a.

[4]

[You are to assume that Tommy harvests the fruits once they are ready and there is no time lapse between harvesting and planting new fruits.]

3 Find the exact value of by using the substitution .

[5]

4 (i) Given that , using the substitution show algebraically that

is always positive,[2]

(ii) Hence, solve the inequality

where k and m are real and 0 < k < m.[4]

5 (a) Solve the equation

giving the roots in the form , where p is a real number, r > 0 and .

[4]

2

(b) Given that is a root of the equation ,

find the values of the real numbers a and b.[3]

6 (i) Express in partial fractions.

[2]

(ii) Hence, find . (There is no need to express your answer as a single

algebraic fraction.)[4]

(iii) Given that , for all , find the least value of k, showing your

working clearly.[2]

7 Positive odd integers, starting at 1, are grouped into sets containing 1, 3, 9, … odd integers , as indicated below, so that the number of odd integers in each set after the first is thrice the number of odd integers in the previous set.

{1}, {3, 5, 7}, {9, 11, 13, 15, 17, 19, 21, 23, 25}, …

Find, in terms of k, (i) the number of terms in the kth set,

[1] (ii) the number of terms in the first k sets,

[2](iii) the first integer in the kth set,

[1](iv) the last integer in the kth set,

[2] (v) the sum of all integers in the first k sets.

[3]

8 The planes and have equations and

respectively. The point A has position vector .

(i) Find the position vector of the foot of perpendicular from A to .[3]

(ii) Find a vector equation of the line of intersection of and [2]

The plane has equation

,

where are real parameters and b is a constant.

3

Given that have no point in common, find the value of b.

[3] meets and in lines and respectively. Without finding the equations of

and , describe the relationship between and , giving a reason. [2]

9 A sequence of real positive numbers satisfies the recurrence relation

and .

(i) Prove by induction that for all .

[4](ii) Determine, giving your reasons, if this is a converging or diverging sequence.

[2]

The sequence is modified to

and .

As .(iii) Find the value of correct to 3 decimal places.

[2](iv) Show, graphically or otherwise, that if , .

[3][Turn over

10 (a) Sketch the curve given by the equation

, [2]

Given that m is a constant and the equation

has 2 real roots, use your sketch above to find, in terms of k, an inequality satisfied by . [3]

(b) A curve C has equation .

(i) By considering , show that the gradient of C is always positive.

[3] (ii) Find the equations of the asymptotes of C.[2]

(iii) Sketch C. [1]

11 (a)

4

y

(i) The diagram shows the curve C with equation The region R is

bounded by C, the positive x-axis and the positive y-axis. Find the exact area of R.

[3]

(ii) Denoting the answer you have obtained in part (i) by q, write down, in terms of q, the area of R if C is scaled by factor 2 parallel to the y-axis. [1]

(iii) Sketch the graph of , indicating clearly the equations of the

asymptotes and the shape of the curve for points near y = 0.[3]

11 (b)

A girl intends to design a bowl by rotating a section AB of the curve completely about the y-axis. She wants the bowl to hold 300 cm3 of fluid.

Given that the diameter of the rim is twice that of the base, write down, in terms of a, the coordinates of the point B. Hence find the exact value of a.

[5]

5

x

1

O

y = 1

y = 0R

Ox

yy = ln x

12 Verify that the general solution of the differential equation, is

. [1]

(i) Sketch three members of the family of solution curves, one for positive A, one for negative A and one for A = 0.

[3]

(ii) It is given that A = 3 and k is a positive constant.

Find in terms of k. Hence, state the value of .

[4]Indicate, on a clearly labeled diagram, the region whose area is given by

.

[1]

(iii) A point P is conditioned to move along the curve such that the x-coordinate of P increases at a constant rate of 2 units per second.

(a) State the range of values of x for which .

[1](b) Find the x-coordinate of the point on the curve at which P is moving such

that .

[3]

6