2009 TJC P1

TEMASEK JUNIOR COLLEGE, SINGAPORE Preliminary Examination 2009 Higher 2

MATHEMATICS

9740/01

Paper 1 September 2009 Additional Materials:

3 hours

Answer Paper List of Formulae (MF 15)

READ THESE INSTRUCTIONS FIRST Write your Civics Group and Name on all the work that 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, graphs, music or rough working. 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 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, 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 4 printed pages.

© TJC/Prelim Exam 2009/MA H2 9740

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2 Answer all questions

1

Find the real numbers a and b such that Hence, find

2

5 x 2  3x  1 ax  1 b = 2 .  2 x 1 x  2 ( x  1)( x  2)

5 x 2  3x  1  ( x2  1)( x  2) dx.

[2] [3]

Without the use of a calculator, solve the inequality

3x  2. x  6 x  11

[5]

Hence, find the range of values of x which satisfies

3e x  1. 4e2 x  12e x  11

[2]

2

n

3

The sequence u1, u2, u3, … is defined by un   r  r ! for all positive integers n. r 1

(i) Calculate the values of 1  un for n =1, 2, 3 and 4. (ii) Make a conjecture for un in terms of n. (iii) Prove your conjecture in part (ii) by mathematical induction.

4

5

A radioactive substance decays at a rate which is proportional to the square of the mass remaining. Initially, the mass is 100 mg and after 9 days, the mass remaining is 25 mg. Letting x be the mass of radioactive substance remaining and t be the time in days, find x in terms of t. Determine the total time (to the nearest day) after which the mass remaining is 7 mg.

(a) Find the roots of the equation z 4  1  i in the form rei where r   and –  <   . (b) In an Argand diagram, O represents the origin and P represents the complex number 4 + 3i. Given that O, P, Q and R are the vertices of a square described in an anti-clockwise orientation, find the complex numbers represented by Q and R.

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[2] [1] [4]

[5] [2]

[3]

[4]

Referred to the origin O, the position vectors of A and B are i + 3j + 2k and 3i + 2j – k respectively. (i) Find the position vector of M, where M is the mid-point of A and B. (ii) Find the cosine of angle AOB. (iii) OM is extended such that OX = 2OM. Using the result in (ii), find the exact area of OAXB. What is the geometrical shape of OAXB?

TJC/Prelim Exam 2009/MA H2 9740

[2] [2] [4]

3

Answer all questions

7

8

A sequence of positive real numbers x0 , x1 , x2 , x3 , ... satisfies the recurrence relation 2x 1 xn  n 1 , for positive integers n. xn 1  3 (i) As n   , xn   . Determine the exact value of  . (ii) Describe the behaviour of the sequence when x0  0.5 . (iii) Show that xn1  xn when xn   . The functions f and g are defined by

f:x g:x

  x  2   4, x  , x  2 , 2

1

 x  7

2

4

,

x  , x  k, x ≠ 5, x ≠ 9.

(a) (i) Sketch the graphs of f and f −1 on the same axes, showing the relationship between the two graphs. (ii) If f    f 1   , show that  2  3  0 . (b) State the maximum value of k such that the inverse of g exists. Hence find (i) the inverse of g, stating its domain, (ii) the range of gf.

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(a)

(b)

Find the sum of the arithmetic series (m + 1) + (m + 3) + (m + 5) + . . . . + (3m − 3) where m is a positive integer.

[3] [1] [1] [3] [1]

[4]

A customer purchases a new 52-inch LCD television set from Counts Hypermarket for $7000 and decides to pay the entire amount by loan instalment. He takes the loan at the beginning of June during the Mid-Year Sale and he repays $p at the end of each month where p < 7000. The Hypermarket will then charge a 5% interest on the outstanding balance after each monthly repayment. Show that the outstanding amount [3] owes at the end of the nth month is 7000(1.05)n  21 p 1.05n 1 . Deduce the least monthly repayment amount p, rounded off to the nearest dollar, required to pay off the entire loan by the end of the 12th month.

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[3] [1] [4]

[3]

Let y  tan  ln(1  x)  .

dy  1 y2 . dx Find Maclaurin’s series for y, up to and including the term in x 3 . Prove that (1  x)

Hence, find the Maclaurin’s series expansion of y = sec2  ln(1  x)  up to and including the term in x2. TJC/Prelim Exam 2009/MA H2 9740

[2] [6] [2]

4

Answer all questions

11

The region R shown in the figure below is bounded by the graph of y  2 x and the ellipse

y2 x   1. 4 2

(i) Express the area of R in the form



a 0

f ( x) dx  b where a and b are constants to be

determined and f(x) is a function of x. Hence find the exact area of R by means of the [7] substitution x = sin  . (ii) Find the volume of the solid generated when R is rotated  radians about the y-axis. y

[3]

2

y2 x R −1

1

O

y2 x  1 4

x

2

12

−2

The lines l and m have the equations   15  1      r =  4      1 and r =  2  1     

 2  1       7     3  3  2      2    respectively. The plane  has the equation r .  0   20 .  1   (i) Find the position vector of the point of intersection between l and m.

[3]

(ii)

Find the position vector of the point of intersection, A, of l and .

[3]

(iii)

Show that m and  have no common point.

[2]

(iv)

Find, in the form r.n = p, an equation of the plane containing m and A.

[4]

   End of Paper   

TJC/Prelim Exam 2009/MA H2 9740