Hwa Chong Institution H2 Mathematics 2009 C2 Term 2 Revision Package

TABLE OF CONTENTS Partial Fractions and Binomial Expansion .................................2 Graphing Techniques....................................................................3 Differentiation & its Applications..............................................11 Maclaurin’s Series.......................................................................16 AP & GP & Sigma Notation.......................................................18 Mathematical Induction .............................................................20 Methods of Difference/ Recurrence Relations ..........................21 Probability ...................................................................................24 Binomial Distribution and Poisson Distribution ......................28 Normal Distribution and Its Approximations ..........................31 Complex Numbers.......................................................................34

1

Partial Fractions and Binomial Expansion 1. PJC/I/5

Express ( ) ( )( )5f

1 3 2xx

x x−

=+ +

in partial fractions and hence, or otherwise, obtain ( )f x as a

series expansion in ascending powers of x as far as the term in 3x . State the range of values of x for which the expansion is valid. Find also, the coefficient of nx in this expansion, where . [8]

1>n

2. VJC/I/5

Express )21)(1(

1 2

xxx+−

+ in partial fractions. [3]

Hence, or otherwise, find the constant term in the expansion of )21)(1(3

1 2

xxxx

+−−+ in ascending

powers of x. [3] 3. HCI/I/6

Expand 11

nxx

−⎛⎜ +⎝ ⎠

⎞⎟ in ascending powers of x up to and including the term in x2. [3]

State the set of values of x for which the series expansion is valid. [1]

Hence find an approximation to the fourth root of 1921

, in the form pq

, where p and q are positive

integers. [3] 4. JJC/I/2 (i) Expand 4− x in ascending powers of x up to and including the term in 2x . Find the range of values of x for which the expansion is valid.

[4]

(ii) Let 7

=ax in the above expansion of 4− x , where a is an integer such that 1 7≤ ≤a .

Choose a suitable value of a and show that 1568075927

≈ . [2]

5. SAJC/I/2

Find the binomial expansion of x

x21

1 2

++ in ascending powers of x up to and including the term

in x2. State the set of values of x for which the expansion is valid. [3] By giving x a suitable value, use the expansion to find 120 in exact form. [3]

2

Answers

1. ( ) 6 17f1 3 2

xx x

= −+ +

; 2 35 27 105 3632 4 8 16

x x x− + − + +… ; valid for 2 23 3

x− < < ;

( ) 17 31 62 2

nn ⎡ ⎤⎛ ⎞− −⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

2. 1 3 5 ;2 2(1 ) 6(1 2 ) 3x x

− + +− +

1

3. , 2 21 2 2nx n x− + 1x < , 31213200

4. (i) 2

24 64x x

− − +… ; 4<x , (ii) 3=a

5. { }1 12 2: ,x x x∈ − < < ;

37404

Graphing Techniques 1. AJC/I/9 2

y

x

2

A sketch of the curve 2f ( )2ax bxx

x k+

=−

, where a, b and k are non-zero constants, is shown above.

Explain why a = 4 and k < 0. [2] Using the given graph,

(i) sketch the graph of kx

bxaxy−−

= 2

2

2 for all values of x. [2]

(ii) determine the number of stationary points of the curve whose equation is [ ]2f( )y x= . [2]

3

2. AJC/II/4 (a) The graphs of f ' ( )y x= and are shown below: )(f2 xy =

)(f2 xy = Sketch, with clear labeling, the graph of y = f(x) for ℜ∈x . [3]

(b) Find the values of the constants A and B such that 22

2

)12(BA

)12(1212

++=

++

xxxx for all values

of x except 21

−=x . Hence state precisely a sequence of transformations by which the

graph of 2

2

)12(1212

++

=x

xxy may be obtained from the graph of 2

3x

y = . [4]

3. ACJC/I/8

A curve has the equation 13axyx b−

=+

for some constants a, b. Given that the asymptotes are

13

x = , , find a, b and sketch the curve showing the asymptotes and the coordinates of the

points of intersection with the axes. [3]

2y =

Sketch, on a separate diagram , the curve with the equation 2 13axyx b−

=+

, showing the asymptotes

and the coordinates of the points of intersection with the axes. [2]

State the sequence of transformations which transform the graph of 1yx

= to the graph of

13axyx b−

=+

. [3]

f ' ( )y x=

-2 2 x 2 x -1

3

-3

-2

4

4. ACJC/I/12

Given that 24( ) , 2, 2 32

af x x a x ax

= + + ≠ < <−

. In terms of a,

(i) find the asymptotes of f ( )y x= . [2] (ii) find the coordinates of the stationary points. [3] (iii) sketch the graph of f ( )y x= , labeling clearly the asymptotes, turning points and axial

intercepts. [3] 5. DHS/I/5 (a) The curve C has equation . Sketch C, indicating clearly the equations

of any asymptotes and the coordinates of any axial intersections and turning points. [4]

2 22 4 4y x x− + − = 0

2 0(b) Show that the curve with equation ( )228y kx y− + = , where k is a positive constant, lies

entirely in the region 2 2xk

≥ . [3]

6. HCI/II/2 The diagram shows the graph of ( )fy x= . The equation of the oblique asymptote, also shown

in the diagram, is . There is a minimum at the point A(a, 0) and a maximum at the

point B(– a, – 4a) where .

2y x a= −

2a >

On separate diagrams, sketch the graphs of

(i) ( )1

fy

x= , [3]

(ii) , [3] ( )f 'y = x

(iii) . [3] ( )f 1y x= −

State the equations of any horizontal and/or vertical asymptotes, the coordinates of the points

corresponding to A, B and any points of intersection with the x-axis.

5

7. IJC/I/4 The curve C has equation . ( ) ( )2 24 1 1x y+ − − =1

)

(i) Sketch the curve C, stating the equations of the asymptotes clearly. [4] (ii) Find the greatest value of k, where k is a positive integer, for which the curve

(lny x k= + cuts C at only one point. [2]

8. IJC/I/7

The diagram below shows the graph of ( )fy x= . There is a minimum at the point 2 ,2aa⎛ ⎞

⎜ ⎟ , a

maximum at the point

− −⎝ ⎠

,2⎝ ⎠

0x =

a a ⎞− ⎟ )⎛⎜ and the curve cuts the x-axis at the point ( . The curve has

asymptotes ,

,0a−

x a= 0y =

( )fy x a= +

and . Sketch, on separate diagrams, the graphs of

(i) , [2]

(ii) ( )1

fy

x= , [3]

(iii) . [3] ( )f 'y = x

y = f(x)

,2a a⎛ ⎞−⎜ ⎟

⎝ ⎠2 ,

2aa⎛ ⎞− −⎜ ⎟

⎝ ⎠

a−O

a x

y

6

9. MJC/I/5

The curve C has equation 2

2

xyxλ

λ=

+,

where λ is a non-zero constant. In separate diagrams, sketch C for the cases where (i)

0λ > , [2]

(ii) 0λ < . [3]

Sketch also, the derivative curve of C for the case where 0λ > . [2] 10. NJC/II/3

(i) The diagram below shows the curve given by y = f(x) having x-intercepts at −2, 1 and 6. y

Sketch the graph C1 given by y2 = f(x), indicating clearly the behaviour of the graph along the x-axis. [2]

(ii) Give a full description of the graph C2 given by the equation , where a > 0 and 2 2 2 22 16 32 2x a y x a+ − + − = 0 2a . Sketch C2 on a separate diagram,

indicating any axial intercepts. [3] ≠

(iii) Determine the range of values of a such that C1 and C2 intersect at exactly two distinct points. [2]

x O 1 −2 6

7

11. PJC/II/1 The given sketches show the graphs of f '( )y x= and f ( )y x= .

Sketch the graph of , showing clearly the stationary points and intercepts. [3] f ( )y x= 12. RJC/II/4 The graph of has a minimum turning point at and passes through the origin. The lines and are asymptotes to the graph, as shown in the diagram below.

f ( )y x=2y =

(4,0)2x =

(i) State the range of values of x for which the graph of 1f( )

yx

= is decreasing. [1]

(ii) State the range of values of x for which the graph of f '( )y x= is below the x-axis. [2] (iii) Sketch the graph of 2 f ( )y x= , showing clearly the equations of all asymptotes and the

shape of the graph at the origin. [3] (iv) Sketch the graph of , showing clearly the equations of all asymptotes and the

coordinates of the stationary points. [3] f (| |) 2y x= +

x

y

0 2− 4 9

f '( )y x= 52

x

y ( 2,5)−

0

( )4, 2

(9,4) f ( )y x=

4− 6 1212

f ( )y x=

2

2

0 4

y

x

8

Answers 1.(i) (ii) 4

y

2 x

2. A = 3 and B = -3 Translation (-1) unit in the x-direction Scaling, parallel to the x-axis, factor ½. Reflection about the x-axis. Translation of 3 units in the y-direction 3. 6, 1a b= = −Translation of 1 unit in direction of the positive x – axis, followed by

)(xfy =

2 x -1

9

-2

scaling parallel to the x – axis with scale factor 13 unit, followed by

translation of 2 unit in direction of the positive y – axis

16

y

13

2

x16

y

6 13 1

x

0

1

yx−

=−

13

x 0 1

x2 6 1−y =x3 1−

2

-12−

9

4. (i) , 2y x a x= + =(ii) , (2 2 ,2 5 )a a+ + (2 2 ,2 3 )a a− −

y x a= +)(2 2 ,2 5a a+ +

22a a− (2 2 ,2 3 )a a− −

2x =

5. 10. (i) (ii) (iii) 2 < a < 3

O 1 −2 6

y

x

y

xO 4 − a 4 + a 4

√2

10

y 11. f ( )y x=

12. i) The graph of 1

f( )y

x= is decreasing for (4, )x∈ ∞ .

(ii) The graph of f '( )y x= is below the x -axis for ( , 2) (2,4x )∈ −∞ ∪ . (iii) (iv)

Differentiation & its Applications 1. ACJC/I/Q6

The parametric equations of a curve are ln(cos ), ln(sin ), 02

x y πθ θ θ= = < < . Find the equation

of the tangent to the curve at the point where 4πθ = , leaving your answer in the form of

where a and b are exact values to be found. [4] y ax b= + Explain, using an algebraic method, why the tangent will not meet the curve again. [2]

x 0

( 2, 5)− −

(4, 2)

(9, 4)−

52

− 4− 6 12 1

2

2 f ( )y x=

2

√2

0 4

y

x

−√2

x = 2

y = √2

y = −√2

f ( ) 2y x= +

2

4

0

(4,2)

y

x

(−4,2)

−2

x=−2 x=2

y=4 2

11

2. DH/I/Q12

The diagram shows a sketch of the curve exxy = .

(i) Find, by differentiation, the exact maximum value of y. [3]

exxy =

y

x

(ii) Hence show that ln 1x x≤ − for all positive values of x. [3]

(iii) Determine the range of values of x for which the graph of exxy = is concave upwards.

[3] (iv) Another curve has equation 2 ( 1y k x )= − , 0.k ≠ Find the range of values of k such that the

equation 2

( 1exx k x )= −

⎝ ⎠⎛ ⎞⎜ ⎟ has (a) 1 real root, (b) 2 real roots. [3]

3. HCI/I/Q11 The parametric equations of a curve C are

21x t= − − and ln(2 )y t= − , t 2< .

(i) Sketch the curve C, showing clearly the axial intercepts. [2]

(ii) Find the equations of the normal to the curve at ( 1, ln 2)− and the tangent to the

curve at ( . [6] 5, 2 ln 2)−

(iii) Find exactly the coordinates of the point of intersection of the tangent and the

normal. [2]

(iv) Find, in radians, the acute angle between the tangent and the normal. [2]

4. JJC/I/Q4

Consider the function ( ) ( )( )1f ' 1

6 1x

x x= +

− −, , 1.x x∈ >

(i) Sketch the graph of showing all asymptotes and intercepts clearly. [4] ( )f 'y = x

(ii) State the x-coordinate of any stationary points of ( )f x . [1]

(iii) Deduce the range of values of x for which ( )' 0xf ' < . [2]

12

5. JJC/I/Q12 A curve has parametric equations

1 2sin ,x θ= + 4 3 cosy .θ= +

Determine the rate of change of xy at 6πθ = if x increases at a constant rate of 0.1 units per

second. [4] 6. MI/I/Q10

(a) Express f(x) in partial fractions where f(x) = ( )( )xxx

213417

+−+ .

Given that, when x = 0.25, x increases at a constant rate of 1.5 unit/s, find the rate of change of f(x) at this instant. [6]

B C (b)

x

x

x

E D

A

The diagram above shows a pentagon ABCDE of fixed perimeter P cm. Its shape is such that ABE is an equilateral triangle and BCDE is a rectangle. If the length of AB is x cm,

(i) show that the area of ABCDE denoted by S is Pxx21

23

43 2 +⎟

⎟⎠

⎞⎜⎜ , ⎝

⎛−

(ii) find the value of xP for which S is a maximum, leaving your answer

in surd form. [6] 7. NYJC/I/Q8

(a) A petrol tanker is damaged in a road accident, and petrol leaks onto a flat section of a

motorway. The leaking petrol begins to spread in a circle of thickness 2 mm. Petrol is

leaking from the tanker at a rate of 0.0084 m3s−1. Find the rate at which the radius of the

circle of petrol is increasing at the instant when the radius of the circle is 3 m, giving your

answer in m s−1 to 2 decimal places. [4]

13

(b) A curve is defined parametrically by 22 ,

1 1tx y

t t= =

t+ +

. Find the equation of the normal to

the curve at the point P 11,2

⎛⎜⎝ ⎠

⎞⎟ . The normal at P meets the curve again at Q. Find the exact

coordinates of Q. [8] 8. NJC/I/Q2

• A

35 m

• P

• C

50 m In the above diagram, not drawn to scale, a man in a boat at B is 35 m from A, the nearest point on a straight shore AC. He intends to disembark from his boat at the point P and runs along the shore to point C which is 50 m from A. He can row at 3 1ms− and run at 4 along straight paths.

1ms−

(i) If x denote the distance, in metres, between A and P, and t denote the total time, in seconds,

required to travel from B to C, show that 21225 50

3 4x xt + −

= + . [2]

(ii) Find the exact value of x such that the man is able to travel from B to C in the shortest

possible time. You may assume that the time taken to disembark from the boat is negligible. [3]

9. RJC/I/Q11 (a) The curve C has the equation 2 y x− = . The point A on C has x-coordinate a where a > 0.

Show that dd lyx a= −

1n 2

at A and find the equation of the tangent to C at A. [3]

Hence find the equation of the tangent to C which passes through the origin. [2] The straight line y mx= intersects C at 2 distinct points. Write down the range of values of m. [1] (b) A movie theatre screen which is 5 m high, has its lower edge 1 m above an observer’s

eye. The visual angle θ of the observer seated x m away is as shown in the diagram below.

14

5

1 θ

x

(i) Show that 1 16 1tan tanx x

θ − −⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

. [1]

(ii) Find the exact distance the observer should sit to obtain the largest visual angle. [You need not establish that the distance gives the largest visual angle.] (iii) Suppose that the observer is situated between 2 m and 15 m from the screen. Find,

to the nearest degree, the smallest visual angle. [2] 10. VJC/I/Q10 A curve is defined by the parametric equations

22x at= , 3y at= , where a is a non–zero constant. Show that the tangent to the curve at any point with parameter t

has equation 2

23

43 atxyt += . [3]

Find the range of values of p such that the line l with equation 2x py a 0− + = intersects the curve at two distinct points. [3] Given that l is a tangent to the curve, show that , where the values of s are to be determined. [2]

014 2 =++ stt

11. YJC/II/Q4 The parametric equations of a curve are ttx ln+= and , . tety += 0>t

(i) Sketch the curve, indicating clearly all intercepts and asymptotes. [3]

(ii) Show that, for all the points on the curve, 1

)1(dd

++

=t

etxy t

. Hence, deduce that the curve

does not have any turning points. [3]

15

Answers 1. ln 2y x= − −

2. (i) Max y = 1e

(iii) 2x ≥(iv) (a) k > 0; (b) k < 0

3. (ii) ; ln 2y = 5 2ln 216 16xy = − − + (iii) (16ln 2 5, ln 2)− (iv) 0.0624θ =

4. (ii) 1.21 , 5.79(iii) or 3.50 6x< < 6x >

5. 920

6. (a) xx 213

345

+− + , -1.87 unit/s (b)(ii) 6 - 3

7. (a) 0.22ms−1 (b) ; 6 4y x+ = 7 4914,

6⎛ ⎞−⎜ ⎟⎝ ⎠

8. (ii) 15 7x =

9. a) 1 1 lnln 2 ln 2 ln 2

ay xa

= − + − ; 1e ln 2

y x= − ; 1 0e ln 2

m− < < b) ii) 6 m iii) 18 °

10. 4 4,3 3

p p> < − , 4s = ±

Maclaurin’s Series 1. ACJC/I/9

Given that 1siny x−= , prove that ( )3 2

23 21 3d y d y dyx x

dxdx dx0− − − = . Hence, find the Maclaurin’s

series for y up to and including the term in 5x . [7]

Deduce the expansion for 2

1

1 x−. [2]

Hence estimate the value for 3 . [2] 2. CJC/II/1

(a) Given that x is small, show that 1 + sin x ≈ 1 + 12 x −

18 x 2 [2]

(b) Determine the value of a such that 3x − e x ln a = 0.

Given that ex ≈ 1 + x + x2

2 + x3

3! , find the first four terms in the expansion of 3x as a series in

ascending powers of x up to and including x3. [4]

16

(c) A curve passes through the point (0, 1) and satisfies the relation )(xfy =

2sin12 x

dxdyy +

= .

By further differentiation of this result, or otherwise, find the Maclaurin’s series for y as far as the term in . [6] 3x

3. HCI/II/1

If 1 22 tan3

xyx

− ⎛ ⎞= ⎜⎜ +⎝ ⎠

⎟⎟ , show that ( )2 d2 3 2dyx xx

.+ + = [1]

By further differentiation of the above result, find the Maclaurin’s series expansion for y in

ascending powers of x up to and including the term in 3.x

Hence find the first three non-zero terms in the expansion of 2

12 3x x+ +

. [7]

4. NYJC/II/3

A curve C is defined by the equation 2d2dyy yx

1= − and (0 , 2) is a point on C.

(i) Find the equation of the normal to the curve at the point (0 , 2). [2]

(ii) Show that at x = 0, 3

3

d 5d 256

yx

=7 .

Find the Maclaurin’s series of y up to and including the term in 3x . [6] (iii) Hence find the series expansion of , up to and including the term in ye 2x . [3] 5. VJC/II/1 Given that y = ln (2 + e x ), show that

22

2d d d .

d ddy y y

x xx⎛ ⎞+ =⎜ ⎟⎝ ⎠

[2]

By further differentiation of this result, or otherwise, find the Maclaurin series for y in ascending powers of x, up to and including the term in x3. [3]

Deduce the Maclaurin series for e2 e

x

x+ in ascending powers of x, up to and including the term in

x2. [1] Answers

1. 3 53 ...

6 40x xy x= + + + , 2563

147=

2. (b) a = 3, 1 + xln3 + (xln3)2

2 + (xln3)3

3! , (c) ...241

211 3 +−+ xx

3. 2 32 2 23 9 81

y x x x≈ − + , 21 2 13 9 27

x x− +

17

4. (i) 234

+−= xy , (ii) ...51219

6415

432 32 ++++ xxx , (iii) ⎟

⎠⎞

⎜⎝⎛ ++ 22

6433

431 xxe

5. ...811

91

313ln 32 ++++ xxx

AP & GP & Sigma Notation 1. ACJC/I/4 An ant of negligible size walks a distance of 10 units from the origin in the x-y plane along the x-axis. It then turns left and goes up 5 units from its current point. If the ant continues turning left and going half the distance it had previously walked, repeating the pattern, find the coordinates of the point where the ant will eventually end up. [4] 2. CJC/I/3 The product of the first three terms in a convergent geometric series is 1728. When the third term is decreased by 2, the three numbers will then form an arithmetic series. (i) Find the first three terms of the geometric series. [4] (ii) State the condition for convergence and find the sum to infinity, S, of the geometric series.

[2] (iii) The sum of the first n terms of the arithmetic series is denoted by Sn. Find the values of n

for which 5Sn exceeds S. [4] 3. NYJC/I/4 A geometric sequence { } has first term a and common ratio r. The sequence of numbers { } satisfy the relation for

naln(

nb)n nb a= n +∈ .

(i) Show that { } is an arithmetic sequence and determine the value of the common difference in terms of r. [3]

nb

(ii) Find an expression for in terms of a, 1

1nn

N

b+

=∑ 1Na + and N. [2]

Hence, obtain an expression for 1 2 Na aa 1+× × × in terms of a, and N. [2] 1Na +

4. PJC/I/6 An arithmetic progression has 889 terms. The sum of all the even-numbered terms of the progression is 408480. The 1st term, 9th term, and the 21st term of the progression are three consecutive terms of a geometric progression. Find the first term and the common difference of the arithmetic progression. [8] 5. TJC/I/11 (a) The terms u1, u2, u3, … form an arithmetic sequence with first term a and having non-

zero common difference d. (i) Given that the sum of the first 8 terms of the sequence is 98 more than u29, find

the first term of the sequence. [2] (ii) If u15 is the first term in the sequence greater than 196, show that 13 14d< ≤

[3]

18

(b) The terms v1, v2, v3, … form a geometric sequence with common ratio r. Another

sequence {wn} is then defined by wn = v2n − 1 + v2n for all positive integers n.

(i) Show that wn is a geometric sequence with common ratio r2. [2] (ii) Given that v1 = 4 and the sum of all the odd-numbered terms w1, w3, w5, … in the

sequence {wn} is 3215 , find the value of r and

1r

rv

∞

=∑ . [4]

6. SAJC/I/8 (a) (i) Given that a, l and Sn are the first term, n th term and the sum of the first n terms of an arithmetic progression respectively. Express n in terms of a, l and Sn . [1] A roll of adhesive tape is wound round a circular cylinder of diameter 83 mm. The external diameter of the complete roll is 110 mm. The total length of the tape is 65800 mm. (ii) Let x be the thickness of the tape. By considering the length of the tape in the layer,

show that [2] thn

2( 1) 27n x− =(iii) Find x, giving your answer correct to three significant figures. [3] (b)The first term of a geometric progression is 8. The sum of its first ten terms is one-eighth of

the sum of the reciprocals of these terms. Show that the sum of the first seven terms of the original geometric progression is the same as the sum of the reciprocals of the seven terms.

[5] 7. HCI/II/3 John took a bank loan of $200 000 to buy a flat. The bank charges an annual interest rate of 3% on the outstanding loan at the end of each year. John pays $1000 at the beginning of each month until he finishes paying for his loan. Let denote the amount owed by John at the end of nth year, where n .

nu+∈

(i) Show that , where k is a constant to be determined. [1] 1( 12000)n nu k u −= −

(ii) Express in the form of , where a and b are constants to be determined. [3]

nu (1.03 )na + b

(iii) Find the minimum number of years required for John to pay up the bank loan. [2] (iv) Suppose John decides to terminate his loan after 15 years by paying the remaining sum

by cash. However, there is a penalty of 5% of the remaining loan for early termination. If John had not terminated his loan earlier, find the total interest he has to pay after 15 years.

Determine, with justification, if it is to John’s benefit to make an early termination. [4]

19

8. RJC/I/4 A gardener needs to spread 1500 kg of sand over his garden. He spreads 5 kg during the first day, and increases the amount he spreads each subsequent day by 2 kg. (i) Find an expression for the mass of sand the gardener has spread by the end of the nth day.

[2] (ii) Deduce the minimum number of days required for him to spread the 1500 kg of sand.

[3] The gardener’s neighbour also needs to spread 1500 kg of sand over his garden. He decides to spread 75 kg each day, but discovers that during each subsequent day, the amount of sand he can spread is 5% less than that of the previous day. (iii) Find an expression for the mass of sand the neighbour has spread by the end of the nth

day. [2] (iv) Comment on the practicality of the approach taken by the neighbour. [1] Answers 1. (8, 4)2. (i)18, 12, 8 (ii) 54 (iii)n = 1,2,3,4,5,6

3. (i) ln (ii)( )r 11 ln( )

2 NN aa ++ , ( )

12

1

N

Naa+

+

4. 32, 2a d= =

5. (a) (i) 14 (b)(ii) − 12 ;

83

6. (a)(i) n = la

Sn+

2 (a)(iii) 0.063 mm

7. (i) (ii) 412000 (iii)23 years 11.03( 12000)n nu u −= − 212000(1.03 )n−(iv)8317.42 , It’s to John’s benefit to terminate his loan early 8. (i) (ii)37 (iii)1500( (iv)Not practical as (4 )n n+ 1 0.95 )n− 1500S∞ = Mathematical Induction 1. HCI/I/9(a) A sequence of real numbers is defined by = 5 and 1 2 3, , ,....u u u 1u 1 8n nu u n+ 8= + +

) 4

for all .

Prove by induction that for every positive integer n,

1n ≥

( 22 1nu n= + − . [4] Show that is a product of two odd numbers. [1] nu 2. RJC/I/1

Prove by induction that, for , 2n ≥1

2

(3 2 )3 39( 1)

n r n

r

rr r n

+

=

−= −

−∑ . [6]

20

3. YJC/II/2(a) The sequence of real numbers is an arithmetic progression. …,,, 321 aaa

Prove, by mathematical induction, that 6

)2)(1( 1

1

nn

rr

aannra

++=∑

= for all . [4] n +∈

4. NJC/I/8 The sequence is defined by the recurrence relation nu

1 1u = and 1 2n

nn

uuu+ =

+ for . 1n ≥

(i) Write down the values of and . Hence, make a conjecture for in 2 3,u u 4u nu terms of n. [2] (ii) Use mathematical induction to prove that your conjecture is true for all positive integers n.

[5] Answers

4. 13

; 17

; 115

; 12 1n nu∴ =−

Methods of Difference/ Recurrence Relations 1. AJC/I/8(a)

Let 1 1ln1 1n

nun n

⎛ ⎞= +⎜ ⎟+ +⎝ ⎠ n−

N

,

(i) Find in terms of N, an expression for , where NS 1 2 ...NS u u u= + + + , simplifying your result as far as possible. [3]

(ii) Show that for all N ≥ 1. [2] 0NS < 2. HCI/I/9(b) A sequence of real numbers is defined by 1 2, ,....u u

( )( )( )

12 1 2 3 2 5ru

r r r=

+ + + for all . 1r ≥

By taking ( ) ( )( )1f

2 3 2 5r

r r=

+ +, express as a difference between and ru ( )f r (f 1r )− . [2]

Evaluate where nS 1 1 ........3 5 7 5 7 9nS = + +⋅ ⋅ ⋅ ⋅

. to n terms. [3]

21

Find the limit of as n → ∞. [1] nS 3. PJC/I/4

A sequence , , … is such that 1,u 2u 3u 1 30eu = and 1 2 14n n n

eu u − −− = , for all . 2n ≥

Use the method of difference to show that 11 11

20 3 16

N

Neu

−⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

for all . [4] 1N ≥

Hence or otherwise, determine if

(i) is a convergent or divergent sequence, [2] nu

(ii) is a convergent or divergent series. [2] 1

N

nn

u=∑

4. SRJC/I/7

Let f(r) [ ]sin (2 1)cos

r θθ+

= , . r +∈

(i) Show that f ( ) f ( 1) A cos(B ) tan( )r r rθ θ− − = , where A and B are to be determined. [3]

(ii) By using the result in part (i), show that

[ ]sin (2 1)1cos 2 cos 4 ... cos 2 12 sin

NN

θθ θ θ

θ⎡ ⎤+

+ + + = −⎢ ⎥⎣ ⎦

. [5]

5. NYJC/II/4 A sequence of numbers, nx , satisfy the relation

11

4n

nnx

x x+

−=

−, for .n +∈

(i) If the sequence converges to a number L, show that L satisfies the equation . [2] 2 3 1LL − − = 0

(ii) Obtain the exact value of the roots α and β of the equation 2 3 1 0xx − − = , where α β< . [2]

(iii) If , find the value of L. [2] 1 0x <

(iv) Show that 1( 1)(

4)n

nn

L Lx xLx+

+ −− =

−. Deduce that 1

1| |5

| n nx L x L+ |− < − if . nx L<

[3]

22

(v) Find a value N such that 20| 10| nx L −− < for all where n N≥ 1 1x = − . [2]

Answers

1. (i) 11 ln( 1) 1NS N

N= − − + +

+

2. ( ) ( ){ }1 14ru f r f r= − − − ;

( )( )1 1 14 15 2 3 2 5n n⎧ ⎫⎪ ⎪−⎨ ⎬+ +⎪ ⎪⎩ ⎭

; 1lim60nn

S→∞

=

3. (i) convergent (ii) divergent 4. (i) A = 2, B = 2

5. (ii) 3 132

α −= and 3 13

2β += (iii) 3 13

2L −= (v) 30, a possible value of 30n n≥ =

23

Probability 1. CJC/II/7 (a) Ten guests are to sit at a round dinner table. Three of the guests in the list are not

comfortable sitting beside either of the other two. If seating is done randomly by a computer, what is the probability that all the three guests are seated separately? [3]

(b) The probability that John goes to school by MRT, by Bus and by Car are respectively 0.4,

0.5 and 0.1. John is late for school 5% of the time if he takes the MRT. The corresponding figures for Bus and Car are 6% and 1%. Find the probability that John does not take the MRT given that he is late for school. ? [4]

During a 5-day week, find the probability that John is late for the second time on the last day. [3]

2 HCI/II/9

A bag initially contains 2 white and 6 black balls. A game is played by drawing 3 balls, one

at a time, from the bag. Each time a ball is drawn, its colour is noted and it is replaced in the

bag along with 2 other balls of the same colour.

To win the grand prize, a player has to draw 3 white balls but if he obtains only 2 white balls

in his 3 draws, he gets a consolation prize. Otherwise, he will walk away empty-handed.

(i) Show that the probability that a player will win the grand prize is 120

. [1]

Find the probability that

(ii) a player wins a consolation prize, [3]

(iii) a player wins a consolation prize, given that his first draw is a white ball. [3]

A player plays the game four times. Find the probability that he wins at least 2 grand prizes

in his 4 attempts at the game. [3]

24

3 NJC/II/13 (a) Two events A and B are independent and defined in the same finite sample space.

Given that 1P( )4

A = and 2P( )3

A B∪ = , find

(i) P( )B , and justify if A and B are mutually exclusive. [3] (ii) ( )P ( ) ( )A B A B∩ ∪ . [2]

(b) When Mrs Ong visits a Spa centre, she will do either Javanese massage or Swedish

massage. During her first visit, the probability that she does Javanese massage is 0.8. Thereafter, the probability that she will do Javanese massage is 0.6 if she did one in her previous visit, and 0.7 if she did not do one in her previous visit.

(i) Show that the probability that Mrs Ong does Javanese massage in the first three

consecutive visits is 0.288. [1]

(ii) Find the probability that she does exactly one Javanese massage in the first three consecutive visits. [2]

(iii) Find the probability that the fourth visit is the second time she does Javanese massage. [2]

4. RJC/II/10

The probability that a hockey team wins any match is 12

and the probability that it loses any

match is 16

. Three points are awarded for a win, one point for a draw and no point for a defeat.

The team plays four matches and the outcome of each match is independent of the other matches.

(i) Show that the probability that the team has exactly one draw and exactly one defeat is 16

.

[1] Find the probability that the team (ii) wins the first match and goes on to win exactly one other match, [3] (iii) wins exactly one match, given that it obtains four points. [4]

25

5. JJC/II/7 In a game played by two people A and B, each player is given three cards with three different animal pictures printed on them. The players flash their cards simultaneously to indicate one of the three animals, ‘elephant’, ‘cat’ and ‘mouse’. ‘Elephant’ defeats ‘cat’, ‘cat’ defeats ‘mouse’ and ‘mouse’ defeats ‘elephant’. If the animals are the same, the game is a draw. At each game, A indicates ‘elephant’, ‘cat’ and ‘mouse’ with probabilities 0.3, 0.3 and 0.4 respectively, while the corresponding probabilities for B are 0.2, 0.5 and 0.3. The game continues until a winner is found. Find the probability that (i) a game ends in a draw; [2] (ii) A is the winner in the first game; [2] (iii) B is the winner in the first game given that the first game does not end in a draw; [3] (iv) B will be the winner in the contest. [2] 6. NYJC/II/6 Five girls Mary, June, April, Alice and Deborah, and five boys Matthew, Mark, Luke, John and

Paul, stand in a queue in random order to enter the ballroom for Dinner and Dance 2008 –

Splendour. Find the probability that

(i) Matthew is first and Mary is sixth in the queue, [1]

(ii) either Alice is first or Mark is second (or both) in the queue, [2]

(iii) there is only one girl chosen if five people are chosen at random from this group,

[2]

(iv) all five girls are next to each other given that at least four boys stand next to each

other. [3]

7. VJC/II/8 Four marbles are randomly chosen, without replacement, from a bag of 18 marbles of which 3

are red, 6 are green and 9 are blue.

Find the probability that

(i) the marbles chosen are all green, [2]

(ii) the marbles chosen consist of at least one of each colour, [3]

(iii) there is 1 green marble given that 3 red marbles are chosen. [3]

26

8 DH/II/11 Three cards are drawn from an ordinary pack of 52 cards, at random and without replacement. Cards drawn have the following values:

Aces score 1 point, Tens, Jacks, Queens and Kings score 10 points, and Cards from two to nine score as many points as the numbers they carry (i.e. twos

score 2 points, threes score 3 points, and so on).

Find the probabilities that

(i) exactly two cards each scoring more than 5 points are drawn, [2]

(ii) all three cards are of different suits, [3]

(iii) total score of the three cards is more than 28 points given that all three cards are of

different suits. [4]

Answers:

1. (a) 5/12, (b) 0.051, 0.00889 2. (i)1/20, (ii) 3/20, (iii) 2/5, 0.0140 3. (a) (ii) 5/24, (b) (i) 0.288, (ii) 0.194, (iii) 0.132 4. (ii) 3/16, (iii) 9/11 5. (i) 0.330, (ii) 0.320, (iii) 0.522, (iv) 0.522 6. (i) 1/90, (ii) 17/90, (iii) 25/252, (iv) 1/9 7. (i) 1/204, (ii) 27/68, (iii)2/5 8. (i) 496/1105 or 0.449, (ii) 169/425 or 0.398, (iii) 112/2197 or 0.0510

27

Binomial Distribution and Poisson Distribution 1. AJC/II/11 In a small company, the employees send an average of 1.2 print jobs to the colour printer and α print jobs to the laser printer per day. It is assumed that these are the only two printers in the company and the print jobs are independent. (i) Given that on 1 in 100 working days there are no print jobs for both printers. Show that

α = 3.41 correct to 3 significant figures. [2] (ii) Find the probability that a total of 3 print jobs were sent in on a working day. [1] (iii) A typical working day consists of 8 hours of work. Find the probability that more than

half of the total print jobs sent during a typical working day occurs within the first hour of work, given that there was a total of 3 print jobs for the day. [4]

2 CJC/II/8 In the production of plastic sheets, small air bubbles occur at random at an average of 1 air bubble in every 2 plastic sheets. (i) Show that, in a randomly chosen plastic sheet, the probability that there are at least three

bubbles is 0.0144, correct to three significant figures. [2] (ii) Find the most likely number of air bubbles occurring in 5 plastic sheets. [2] These plastic sheets are delivered to customers in crates, with each crate containing 15 plastic sheets. A crate is rejected by a customer if it contains at least 2 plastic sheets with at least 3 air bubbles. (iii) Find the probability that a randomly chosen crate is rejected. [3] 100 randomly chosen crates are delivered to a customer. Find, by using a suitable approximation, the probability that more than 98 crates are not rejected. [3] 3 RJC/II/11 In a certain factory, it is known that 5% of all articles produced are defective. (i) Show that the probability of a randomly chosen sample of size 10 containing exactly one

defective article is 0.315, correct to 3 significant figures. [3]

(ii) 80 random samples, each of size 10, are drawn. Using a suitable approximation, find the probability that at least 25 but fewer than 35 samples will each contain exactly one defective article. [5]

28

4 MJC/II/11 A small company has 3 trucks which can each be hired out for a day at a time. The total number

of requests for the hire of a truck, for any particular day, follows a Poisson distribution with

mean 2. Find the probability that, for a particular day,

(i) at most one truck is not hired out, [2] (ii) all requests for the trucks can be met. [2] If, for any day, fewer than 3 requests are received for the hire of a truck, the trucks used are

chosen at random. Find the probability that a particular truck is not used on a given day.

[3]

Find the least number of trucks that the company should have so that, for each day, the probability that a request for the hire of a truck for that day has to be refused is less than 0.01. [2] 5 TJC/II/7 A carpet shop in the remote town of Kikiboo receives an average of λ customers per day. Given that it is equally likely that the shop will receive either 3 or 4 customers, show that λ = 4. [2] Show that the probability of the number of visitors to the shop exceeding 30 in a given week of 7 days is 0.30965, correct to 5 significant figures. [2] Assuming that there are 52 weeks in a year, use a suitable approximation to find the probability that there are at least 8 but not more than 12 weeks during which this happens. [4] 6 NYJC/II/8 A petrol station at Serangoon Gardens is opened 24 hours daily. On average, the number of cars

arriving at the station is 7 in every 20 mins.

(i) Find the probability that at most 5 cars will arrive at the petrol station during a 10- minute

period; [2]

(ii) Using a suitable approximation, find the probability that there are at least 500 cars arriving

at the petrol station on a particular day. [3]

The petrol station will employ more staff if the probability that there are more than k 10-minute intervals in a day where more than 5 cars arrive at the petrol station is less than 0.05. Find the least value of k. [3]

29

7 VJC/II/12 An airline company has a ticket reservation hotline which receives calls at a rate of 6 calls per hour and a baggage services hotline which receives calls at a rate of 3.6 calls per hour. (i) Show that the probability that the two hotlines receive a total of more than 5 calls in 1

hour is 0.916. State the assumption made in your calculation. [3] (ii) Calculate, using a suitable approximation, the probability that at least 15 but fewer than

25 calls are received on the ticket reservation hotline in a 3-hour period. [3]

(iii) Both hotlines are open 8 hours daily from 0900 to 1700h, 7 days a week. By using a

suitable approximation, find the probability that in a particular week, there will be at least 50 1-hour periods in which the two hotlines receive a total of more than 5 calls per hour. [4]

(iv) The only officer manning the baggage services hotline wishes to take a short tea break. Find the longest break, in minutes, he can take so that the probability of him missing any calls is less than 0.4. [4]

8 NJC/II/9 In a highly competitive organization, employees are graded according to their performance levels annually as shown in the following probability distribution:

Performance Level Excellent Good Average Poor Probability 0.04 0.26 0.43 0.27

The performance of employees is independent of one another.

(i) Find the greatest value of n if, in a random sample of n employees, the probability of at least 1 employee with good performance is at most 0.96. [3]

(ii) In a particular year, there are 120 employees in the organization. Using a suitable approximation, find the probability that there are more than 7 employees with excellent performance. [2]

(iii) In a division consisting of 20 employees, find the probability that there are at least 12 employees with average or poor performance, given that there are less than 17 employees with average or poor performance. [3]

Answers: 1. (ii) 0.163 (iii) 0.0430 2. (ii) 2 , (iii) 0.0192, (iv) 0.427 3. 0.554 4. (i) 0.594 (ii) 0.857 , 0.406 , 65. 3.90, 0.665, 0.784 6. (i) 0.858 (ii) 0.579; 28 7. 0.916, 0.733, 0.806, 8.51min 8. 10n = ; 0.113 ; 0.873]

30

Normal Distribution and Its Approximations 1 AJC/II/10 The lifespan of a halogen bulb is normally distributed with mean 160 hours with standard deviation 10 hours, while the lifespan of a fluorescent bulb is normally distributed with mean 240 hours and standard deviation 12 hours. The lifespan of any bulb is independent of one another. (i) A halogen bulb is randomly chosen. Find the greatest value of a, correct to three

significant figures, if the probability that its lifespan lies in the range of (160-a, 160+a) is at most 0.4. [3]

(ii) Find the probability that the difference between the average lifespan of two fluorescent bulbs and twice the lifespan of a halogen bulb does not exceed 70 hours. [3]

(iii) The halogen bulbs are packed in boxes of n bulbs, where n is large. If there are more than 10 bulbs that have lifespans of less than 150 hours, the box will be rejected. Using a suitable approximation, find the greatest value of n so that the probability that a box will be rejected is less than 0.2. [5]

2 ACJC/II/11 A fruit grower grows both red and green apples which have masses that are normally distributed. The mass of a randomly chosen red apple has mean 75g and standard deviation 12.5g. The mass of a randomly chosen green apple has mean of 55g and standard deviation 10.5g. (i) Find the probability that the total mass of 3 randomly chosen green apples exceeds twice

the mean mass of 3 randomly chosen red apples. [3] (ii) A red apple is considered “underweight” if it weighs less than 70g. Red apples are packed

into bags of 10 for transportation to a supermarket. A bag is considered to have passed the quality test if it contains less than 2 “underweight” apples. Calculate the probability that in a randomly chosen batch of 20 bags of red apples, all the bags fail the quality test. [4]

3 DH/II/10 A confectionary produces durian and mango cakes with individual weights X g and Y g, distributed normally with means μx = 800 and μy = 600, and variances σx

2 = 250 and σy2 = 200

respectively.

(i) Ms Marian ordered one durian cake and one mango cake. Write down the probability that the durian cake weighs more than 800 g and the mango cake weighs less than 600 g. [1]

(ii) Find the probability that three randomly chosen durian cakes weigh more than four randomly chosen mango cakes by more than 50 g. [4]

(iii) State the condition you have used in your calculations. [1]

31

4 HCI/II/11 A customer can order a regular 500 ml cup drink or a double 1000 ml cup drink from a vending machine. To make a regular cup drink, the vending machine dispenses once, while to make a double cup drink, the vending machine actually dispenses twice. Drinks which dispense from the vending machine follow a normal distribution with mean μ ml and standard deviation 20 ml. The amounts that are successively dispensed are independent. (a) Find the set of values for μ so that at most 1% of regular cups overflow. [3] (b) It is later found that . Find the probability that 470μ =

(i) a regular cup will overflow, [2] (ii) no overflow occurs if the customer orders 2 regular size drinks, [2] (iii) for a double cup, no overflow occurs, [3] (iv) more than 20 cups overflowed if 300 cups of 500 ml are dispensed in one day, by using a suitable approximation. [4] 5 MI/II/12 The heights, in metres, of female and male students in a particular school follow normal distributions with the following data

Mean Height (metres) Variance (metres2) Female Students 1.6 0.04 Male Students μ σ 2

It is known that 20% of the males are shorter than 1.6m, while 75% of them are below 1.8m. Show that μ and σ are 1.71m and 0.132m respectively. [3] Find the probability that (i) the heights of two randomly selected female students both exceed 1.55m, [2] (ii) the total height of two randomly selected female students is more than 3m, [3] (iii) a randomly selected male student is taller than a randomly selected female

student by at least 20cm. [3] A random sample of 20 female students is chosen. Find the probability that at least 15 of these female students are shorter than 1.7m. [4] 6 PJC/II/10 The travelling time for a randomly chosen ferry trip from Mersing to Pulau Tioman may be assumed to have a normal distribution with mean 90 minutes and standard deviation 3.6 minutes. A ferry trip is considered to be well-timed if its travelling time differs from the mean travelling time by at most 5 minutes. Show that the probability of a randomly chosen ferry trip being well-timed is 0.835, correct to 3 significant figures. [2] A total of 10 ferry trips are made in a day. Assuming independence between the ferry trips, find the least possible value of m such that the probability of at most m trips being well-timed in a day exceeds 0.8. [3] A random sample of 55 ferry trips is taken. By means of a suitable approximation, find the probability that less than 10 trips are not well-timed. [4] 7 RJC/II/12 A factory manufactures a large number of containers with press-on lids. The diameter of

32

a randomly chosen lid is X cm, where X follows a normal distribution with mean 12.1 and standard deviation 0.03. The diameter of the top of a randomly chosen container into which the lid is pressed is Y cm, where Y follows an independent normal distribution with mean 12.0 and standard deviation 0.03. (a) Find the probability that the sum of the diameters of 3 randomly chosen lids and twice the

diameter of the top of a randomly chosen container is less than 60.5 cm. [4] (b) A lid and a container are chosen randomly and paired. A pairing is accepted if 0.02 and discarded otherwise. 0.17X Y< − <

(i) Find the probability that a pairing is accepted. [3] (ii) Estimate the number of trials required to obtain 2000 accepted pairings. [2]

Answers: 1. (i) (ii) (iii) 52 5.24 (3 . )a s= f 0.3232. (i) 0.741 (ii) 0.147 3. (i) 0.25 (ii) 0.102 4. 453 (3 s.f.)μ ≤ 0.0668; 0.871; ; 0.458 0.9835. (i) 0.358 (ii) 0.760 (iii) 0.354 ; 0.384 6. Least m = 9, 0.562 7. 0.994 ; 0.921; or 2170 2172

33

Complex Numbers 1. AJC/I/6 If z = i is a root of the equation ( ) ( )3 21 3 2 3 2 0z i z i z+ − − + − = , determine the other roots.

Hence find the roots of the equation ( ) ( )3 21 3 3 2w i w i 2 0w+ + + − − = . [7] 2. IJC/I/10 (a) Given that find the exact values of the real 2(4 i) (8 i)(3 i) 8i 43,λ μ− + + − + =

numbers andλ μ . [4]

(b) Express the complex number ( )1 1 i 32

− + in exponential form. [2]

Solve the equation

( ) ( )4 12 1 i2

w+ = − + 3 ,

giving your answers in form ia b+ , where a and b are real values. [4] Show that the points representing the roots of the given equation in an Argand diagram lie on a circle. Write down the centre and radius of this circle. [2] 3. CJC/I/9

(a) The complex number z is such that it has modulus 2 and argument 2π . If 2

1 1az

i i= +

+ −b , find the

values of a and b, where . . [3] ,a b∈(b) The cubic equation P(z) = 0 has real coefficients. If two of the roots are 1 and i, state the third root and find the equation in the form Az3 + Bz2 + Cz + D = 0. [3] (c) Solve the equation , expressing the solutions in the form 4 81iz = − ire θ , where 0r > and π θ π− < ≤ . [4] 4. HCI/1/5

The complex number z is given by cos i sinz θ θ= + , where π θ π− < ≤ . Show that the equation ( ) ( )i 334 i 4 1 ez π θ++ − = can be reduced to 3z i= . Hence express each root of

( ) ( )i 334 i 4 1 ez π θ++ − = in the form ix y+ , where x and y are real numbers. [6]

34

5. RJC/I/9 (a) Given that z is a non-zero solution of the equation ( ) ( )8 i 4 7i *z+ = − z , find the possible

values of arg( correct to 3 decimal places. [5] )z (b) Find the exact roots of the equation 5z = − (16√2 )(1 i)+ , expressing them in the form ier θ ,

where and 0r > π θ π− < ≤ . [4]

On an Argand diagram the points which represent the above roots are rotated 110

π radian

anti-clockwise about the origin to obtain points , A B , C , and . D E Find the equation whose roots are represented by , A B , , and , giving your answer in the form

(16√2C D E

5z = − )( i )p q+ where p and are real constants to be determined. [2] q 6. TPJC/I/6 Given that is a root of the quadratic equation 1 1 2iz = − 2 0z az b+ + =

2P

1 2,z z

where a and b are real, find the values of a and b. Mark on an Argand diagram and , the points representing the roots and of the quadratic equation . A third point is on the real axis such that , and form a triangle which encloses the origin with an area of 8 square units. Find represented by the point , h e n c e f o r m a c u b i c e q u a t i o n h a v i n g r o o t s . [ 7 ]

1P 1z 2z2 0z az b+ + = 3P 1P 2P 3P

3z and

3P

3z 7. VJC/II/3 Let 1 3z = − i .

(i) Find z and the exact value of . [2] arg( )z

(ii) Given that 4 1 3w = − i , find the complex numbers w in the form ier θ , where and 0r > π θ π− < ≤ .

[3]

(iii) Given that (1 3 in

− ) is real and n is positive, use de Moivre’s Theorem to show that the values

of n are terms in an arithmetic progression. [3]

35

8. SRJC/I/4 Sketch in an Argand diagram, the set of points representing all complex numbers z satisfying the following inequalities

( )0 arg 14

z π≤ − ≤ and 1 5z i z i+ − ≤ − − . [3]

Hence, find

(i) the maximum value of , [1] ( )arg z

(ii) the exact range of values of z . [2] 9. DHS/1/10 (a) Two complex numbers w and z are such that w* = z − 2i , |w|2 = z + 6. By eliminating z or otherwise, find w in the form ia b+ , where a and b are real and

positive. [4] (b) The point P in an argand diagram represents the variable complex number z, and the

point A in the first quadrant represents the fixed complex number a. Draw on the same diagram, the point A and the locus of P for the following cases,

making clear the relationship between the locus and the point A: (i) z a a− = ,

(ii) arg (z − a) = arg (a) + 2π .

Hence, find a value of a such that the complex number z satisfying both equations in parts (i) and (ii) is purely imaginary. [6]

10. HCI/II/4

(a) Consider the complex number

2

3

cos i sin4 4

cos i sin3 3

z

π π

π π

⎛ ⎞−⎜ ⎟⎝ ⎠=⎛ ⎞+⎜ ⎟⎝ ⎠

.

(i)Find the modulus and the exact value of the argument of z. [3]

(ii)Find the set of values of n such that is purely imaginary. [3] nz

36

(b) Show clearly on an Argand diagram the locus given by

( )3 iarg arg 3 i2 2

w⎛ ⎞

+ − = −⎜ ⎟⎜ ⎟⎝ ⎠

. [3]

A complex number u satisfies 3 i 2u − + = . Sketch on the same Argand diagram, the locus of

the point which represents u. Hence find the greatest possible value of 3 i2 2

u + − . [4]

11. AJC/II/5 Sketch, on an Argand diagram, the locus given by 1 3z i 1− − = . Find the minimum value of

arg ( )452

zi

⎡ ⎤+⎢ ⎥⎢ ⎥⎣ ⎦

. [5]

Given that is a complex number with a 1a = and arg ( )a θ= , where 02πθ< < , determine the

value of θ such that the locus given by 2w a w a= − is the y axis. [3] −

Find the minimum value of 3z iw− + in exact form. [3] 12. ACJC/I/11 On an Argand diagram, sketch the set of points representing the complex number z satisfying the equation 2

3arg( 2)iz π+ = . [2] Hence find the least value of 3z − + i . [3] If there is exactly one complex number w that satisfies both conditions 2

3arg( 2)iw π+ = and 3w i− + = m , find the range of values of m. [2]

13. JJC/I/10 In an Argand diagram, the point P represents the complex number z such that

2 4 4z i− + ≤ and arg( 2 ) 04

z iπ− ≤ + < .

(i) Sketch the locus of P. [4] (ii) Hence, find the exact range of values for 2z + . [4]

37

14. NJC/I/6 (i) The set of points P in an Argand diagram represents the complex number z that satisfies

2 i 2.z − − = Sketch the locus of P. [1] (ii) The set of points Q represents another complex number w given by arg( 2 3i)w θ− + = where π πθ− < ≤ . Give a geometrical description of the locus of Q. [1]

(a) Find the range of values of θ such that the locus of Q meets the locus of P more than once. [2]

(b) In the case where 1 3tan4

θ − ⎛ ⎞= ⎜ ⎟⎝ ⎠

, find the least value of z w− in exact form. [3]

15. NYJC/I/10 (a) In the Argand diagram, the point P represents the complex number z. Draw a clearly

labelled diagram to show the locus of P when z satisfies (i) | [1] 3 4i | 5z − − =(ii) | [2] 2 3 | 1i | i |0 3z z− + − −=

Hence find the greatest and least possible values of when | |z | 3 4i | 5z − − ≤ and . [3] | 2 3 | 1i | i |0 3z z− + − −≥

(b) Find in the form ire θ , where π θ π− < ≤ , all complex numbers z satisfying the equation

6 52 3

izi

+=

+. [3]

(c) If iz e θ= , where , show that 1z ≠

( )( )122 112

sin1

sinnz

nzz

θθ

−+ + =+ + . [3]

16. YJC/II/5 (i) Sketch, on an Argand diagram, the set of points representing the complex

number z which satisfy both conditions: 2 i 4 2 iz z+ ≤ − − and

0 ≤ arg( ) . [3] 2 iz + 1tan 2−<

Hence, find the greatest value of 4 iz − which satisfy the above conditions,

giving your answer in exact form. [1]

(ii) With the help of your sketch in (i), sketch, on another Argand diagram, the set

38

of points representing the complex number which satisfy both conditions: z

2 i 4 2 iz z∗ ∗+ ≤ − − and 0 ≤ arg( 2 iz∗ + ) . [2] 1tan 2−<

Answers 1. z = 2i, z = -1; w = -i, -2i and -1.

2. (a) 3μ = ± , 3 38

λ = ± (b) 3 1 3 3 3 1 52 i, + i, 2 i,2 2 2 2 2 2 2

⎛ ⎞ ⎛ ⎞− − − − − + − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

3 i2

The points representing the roots of the given equation in an Argand diagram lie on a circle with centre (-2,0) and radius 1 unit.

3. (a) a = 2, b = –1 (b) The other root is –i; 3 2( ) 1P z z z z= − + − (c) 1,0,1,2,3)

82(

−−==+

nezin ππ

4. , i− 3 1+ i2 2

, 3 1+ i2 2

−

5. ( )8 3 i

200.588,2.554;2e , 0, 1, 2; 16 2 1 ik

kπ−⎛ ⎞

⎜ ⎟⎝ ⎠− = ± ± − − +

6. a = -2, b = 5; = -3; 3z 3 2 15 0z z z+ − + =

7. 1 1 5 1 11 1 7i i i i4 12 4 12 4 12 4 122, ;2 e ,2 e ,2 e ,2 e ; 3

3n k

π π π ππ − −− =

8. 0.464 radθ ≈ ,1 5z< ≤ 9. (a) w = 2 + 2i (b) (ii) a = 1 + I (or any other correct ans)

10. 1z = ; 2π ; { }2 1,n m m= + ∈ 3 i 5

2 2u + − =

11. - 62.4°;3πθ = ; 3 1−

12. 3 2m ≥ 13. (ii) 2 2 2 4(1 2)z< + ≤ + 14. (ii) Locus of Q is a half line from the point (2 , -3) that makes an angle of θ with the positive

real axis (a) π 2π3 3

θ< < . (b) 65

15. (a) 10; 245

(b) 1

12(8 1)

242 0, 1,ik

z keπ

−= = ± , 2,3±

16. (i) 2 13

39