PJC/I/5. Express ( ). (. )(. ) 5 f. 1 3. 2 x x x x. −. = +. + in partial fractions and hence,
or otherwise, obtain ( ) f x as a series expansion in ascending powers of x as ...
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 f ( x ) =
x−5 in partial fractions and hence, or otherwise, obtain f ( x ) as a ( x + 1)( 3x + 2 )
series expansion in ascending powers of x as far as the term in x3 . State the range of values of x for which the expansion is valid. Find also, the coefficient of x n in this expansion, where n > 1 . [8] 2. VJC/I/5 Express
1+ x2 in partial fractions. (1 − x)(1 + 2 x)
[3]
Hence, or otherwise, find the constant term in the expansion of powers of x.
1+ x2 in ascending − 3 x(1 − x)(1 + 2 x) [3]
3. HCI/I/6 n
⎛ 1− x ⎞ 2 Expand ⎜ ⎟ in ascending powers of x up to and including the term in x . ⎝ 1+ x ⎠ State the set of values of x for which the series expansion is valid. Hence find an approximation to the fourth root of integers.
[3] [1]
19 p , in the form , where p and q are positive 21 q [3]
4. JJC/I/2 (i) Expand 4 − x in ascending powers of x up to and including the term in x 2 . Find the range of values of x for which the expansion is valid. [4] (ii)
a Let x = in the above expansion of 7
4 − x , where a is an integer such that 1 ≤ a ≤ 7 .
Choose a suitable value of a and show that
7≈
15680 . 5927
[2]
5. SAJC/I/2 Find the binomial expansion of
1 + x2
in ascending powers of x up to and including the term 1 + 2x [3] in x2. State the set of values of x for which the expansion is valid. By giving x a suitable value, use the expansion to find 120 in exact form. [3]
2
Answers
6 17 5 27 105 2 363 3 2 2 − ;− + x− x + x +… ; valid for − < x < ; x + 1 3x + 2 2 4 8 16 3 3 n 17 3 ⎤ n ⎡ ( −1) ⎢6 − ⎛⎜ ⎞⎟ ⎥ 2 ⎝ 2 ⎠ ⎥⎦ ⎢⎣
1. f ( x ) =
1 1 3 5 + ; 2. − + 2 2(1 − x) 6(1 + 2 x) 3 3121 3. 1 − 2nx + 2n 2 x 2 , x < 1 , 3200 2 x x 4. (i) 2 − − + … ; x < 4 , (ii) a = 3 4 64 5. {x : x ∈ , − 12 < x
2 .
On separate diagrams, sketch the graphs of
1 , f ( x)
[3]
(ii) y = f ' ( x ) ,
[3]
(iii) y = f (1 − x ) .
[3]
(i) y =
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 2
2
The curve C has equation 4 ( x + 1) − ( y − 1) = 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 [2] y = ln ( x + k ) cuts C at only one point. 8. IJC/I/7 a⎞ ⎛ The diagram below shows the graph of y = f ( x ) . There is a minimum at the point ⎜ −2a, − ⎟ , a 2⎠ ⎝ ⎛a ⎞ maximum at the point ⎜ , −a ⎟ and the curve cuts the x-axis at the point ( −a, 0 ) . The curve has ⎝2 ⎠ y asymptotes x = 0 , x = a and y = 0 . Sketch, on separate diagrams, the graphs of
y = f(x)
−a a⎞ ⎛ ⎜ −2a, − ⎟ 2⎠ ⎝
O
a
x
⎛a ⎞ ⎜ , −a ⎟ ⎝2 ⎠
(i)
y = f ( x + a) ,
[2]
(ii)
y=
1 , f ( x)
[3]
(iii)
y = f '( x) .
[3]
6
9. MJC/I/5 The curve C has equation y =
λ x2 , x2 + λ
where λ is a non-zero constant. In separate diagrams, sketch C for the cases where
(i)
λ >0,
[2]
(ii) λ < 0 .
[3]
Sketch also, the derivative curve of C for the case where λ > 0 .
[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
−2
O
1
6
x
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 2 x 2 + a 2 y 2 − 16 x + 32 − 2a 2 = 0 , where a > 0 and a ≠ 2 . 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]
7
11. PJC/II/1 The given sketches show the graphs of y = f '( x) and y = f ( x) . y y (−2,5) y = f ( x)
(9, 4)
y = f '( x) 5 2
−2
0
4
9
( 4, 2 )
x
−4
0
6
1 2
Sketch the graph of y = f ( x) , showing clearly the stationary points and intercepts.
x
12
[3]
12. RJC/II/4 The graph of y = f ( x) has a minimum turning point at (4, 0) and passes through the origin. The lines x = 2 and y = 2 are asymptotes to the graph, as shown in the diagram below. y
y = f ( x) 2 0
2
4
x
1 is decreasing. [1] f(x) (ii) State the range of values of x for which the graph of y = f '( x) is below the x-axis. [2] (iii) Sketch the graph of y 2 = f ( x) , showing clearly the equations of all asymptotes and the shape of the graph at the origin. [3] (iv) Sketch the graph of y = f (| x |) + 2 , showing clearly the equations of all asymptotes and the coordinates of the stationary points. [3] (i)
State the range of values of x for which the graph of y =
8
Answers 1.(i)
y (ii) 4
2
x y = f (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
9
-2
-1
x
2
3. a = 6, b = −1 Translation of 1 unit in direction of the positive x – axis, followed by scaling parallel to the x – axis with scale factor 13 unit, followed by translation of 2 unit in direction of the positive y – axis y y
y=
6x −1 3x − 1
y2 =
2 1 0
1 6
1 3
x
2 1 0 1 1 -1 6 3 − 2
6x −1 3x − 1 x
9
4. (i) y = x + a, x = 2 (ii) (2 + 2a, 2 + 5a) , (2 − 2a, 2 − 3a)
(2 + 2a, 2 + 5a )
y = x+a
a − 2a 2 (2 − 2a, 2 − 3a )
x=2
5.
10. (i)
y
−2
O
1
x
6
(ii)
(iii) 2 < a < 3 y
√2 x O
4−a
4
4+a
10
y 11.
y = f ( x)
(4, 2)
−4
0 −
5 2
1 2
6
12
x
(9, −4) (−2, −5) 1 is decreasing for x ∈ (4, ∞) . f(x) The graph of y = f '( x) is below the x -axis for x ∈ (−∞, 2) ∪ (2, 4) . y
12. i) The graph of y = (ii)
y
(iii)
(iv)
y 2 = f ( x)
√2
y = f( x)+ 2
y = √2 0
2
4
4 (−4,2)
x
−2
y = −√2
−√2
x=2
y=4
2
(4,2) 2
0
x=−2
x
x=2
Differentiation & its Applications 1. ACJC/I/Q6 The parametric equations of a curve are x = ln(cos θ ), y = ln(sin θ ), 0 < θ < of the tangent to the curve at the point where θ =
π
π 2
. Find the equation
, leaving your answer in the form of 4 y = ax + b where a and b are exact values to be found. [4] Explain, using an algebraic method, why the tangent will not meet the curve again. [2]
11
2. DH/I/Q12 x The diagram shows a sketch of the curve y = x . e y
y=
x ex
x
(i) (ii)
Find, by differentiation, the exact maximum value of y. Hence show that ln x ≤ x − 1 for all positive values of x.
(iii) Determine the range of values of x for which the graph of y =
[3] [3] x is concave upwards. ex
[3] (iv) Another curve has equation y 2 = k ( x − 1) , k ≠ 0. Find the range of values of k such that the 2
⎛ x ⎞ equation ⎜ x ⎟ = k ( x − 1) has (a) 1 real root, (b) 2 real roots. ⎝e ⎠ 3. HCI/I/Q11 The parametric equations of a curve C are x = −1 − t 2 and y = ln(2 − t ) , t < 2 .
[3]
(i)
Sketch the curve C, showing clearly the axial intercepts.
(ii)
Find the equations of the normal to the curve at (−1, ln 2) and the tangent to the curve at (−5, 2 ln 2) .
(iii)
[6]
Find exactly the coordinates of the point of intersection of the tangent and the normal.
(iv)
[2]
[2]
Find, in radians, the acute angle between the tangent and the normal.
[2]
4. JJC/I/Q4 Consider the function f ' ( x ) =
1
( x − 6 )( x − 1)
+ 1 , x ∈ , x > 1.
(i)
Sketch the graph of y = f ' ( x ) showing all asymptotes and intercepts clearly. [4]
(ii)
State the x-coordinate of any stationary points of f ( x ) .
[1]
(iii)
Deduce the range of values of x for which f '' ( x ) < 0 .
[2]
12
5. JJC/I/Q12 A curve has parametric equations x = 1 + 2sin θ ,
y = 4 + 3 cos θ .
Determine the rate of change of xy at θ =
π 6
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) =
17 + x . (4 − 3 x )(1 + 2x )
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
(b)
C
x x
A x
D
E
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)
⎛ 3 3⎞ 1 show that the area of ABCDE denoted by S is ⎜⎜ − ⎟ x 2 + Px , ⎟
(ii)
find the value of
⎝ 4
2⎠
2
P for which S is a maximum, leaving your answer x
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 x =
2t t2 ,y= . Find the equation of the normal to t +1 t +1
⎛ 1⎞ the curve at the point P ⎜ 1, ⎟ . The normal at P meets the curve again at Q. Find the exact ⎝ 2⎠ coordinates of Q. [8] 8. NJC/I/Q2
35 m
• A
•
•
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 m s −1 and run at 4 m s −1 along straight paths. (i)
If x denote the distance, in metres, between A and P, and t denote the total time, in seconds, 1225 + x 2 50 − x + . [2] required to travel from B to C, show that t = 3 4
(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. dy 1 =− at A and find the equation of the tangent to C at A. [3] Show that dx a ln 2 Hence find the equation of the tangent to C which passes through the origin.
(b)
[2]
The straight line y = mx intersects C at 2 distinct points. [1] Write down the range of values of m. 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)
(ii) (iii)
⎛6⎞ ⎛1⎞ Show that θ = tan −1 ⎜ ⎟ − tan −1 ⎜ ⎟ . [1] ⎝x⎠ ⎝x⎠ 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.] 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 y = 3at , x = 2at 2 , where a is a non–zero constant. Show that the tangent to the curve at any point with parameter t has equation yt =
3 4
3
x + at 2 . 2
[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] 2 Given that l is a tangent to the curve, show that 4t + st + 1 = 0 , where the values of s are to be determined. [2] 11. YJC/II/Q4 The parametric equations of a curve are x = t + ln t and y = t + e t , t > 0 . (i)
Sketch the curve, indicating clearly all intercepts and asymptotes.
(ii)
Show that, for all the points on the curve,
does not have any turning points.
[3]
dy t (1 + e t ) = . Hence, deduce that the curve t +1 dx [3]
15
Answers 1. y = − x − ln 2 1 2. (i) Max y = e (iii) x ≥ 2 (iv) (a) k > 0; (b) k < 0 x 5 3. (ii) y = ln 2 ; y = − − + 2ln 2 (iii) (16ln 2 − 5, ln 2) (iv) θ = 0.0624 16 16 4. (ii) 1.21 , 5.79(iii) 3.50 < x < 6 or x > 6 9 5. 20 6. (a) 4 −53 x + 1 +32 x , -1.87 unit/s (b)(ii) 6 - 3
7. (a) 0.22ms−1
49 ⎞ ⎛ (b) 6 y + 4 x = 7 ; ⎜14, − ⎟ 6 ⎠ ⎝
8. (ii) x = 15 7 1 1 ln a 1 1 x+ − ;y=− x;− < m < 0 b) ii) 6 m iii) 18° 9. a) y = − a ln 2 e ln 2 ln 2 ln 2 e ln 2 4 4 10. p > , p < − , s = ±4 3 3
Maclaurin’s Series 1. ACJC/I/9
(
Given that y = sin −1 x , prove that 1 − x 2
3
2
= 0 . Hence, find the Maclaurin’s ) ddxy − 3x ddx y − dy dx 3
2
series for y up to and including the term in x5 . 1 Deduce the expansion for . 1 − x2 Hence estimate the value for
3.
[7] [2] [2]
2. CJC/II/1 1 1 (a) Given that x is small, show that 1 + sin x ≈ 1 + 2 x − 8 x 2 x
[2]
x ln a
(b) Determine the value of a such that 3 − e = 0. 2 3 x x x x Given that e ≈ 1 + x + 2 + 3! , find the first four terms in the expansion of 3 as a series in ascending powers of x up to and including x3. [4]
16
(c) A curve y = f (x ) passes through the point (0, 1) and satisfies the relation dy 1 + sin x . y2 = dx 2 By further differentiation of this result, or otherwise, find the Maclaurin’s series for y as far as the term in x 3 . [6]
3. HCI/II/1 ⎛x 2⎞ If y = 2 tan −1 ⎜⎜ ⎟⎟ , show that x 3 + ⎝ ⎠
(x
2
+ 2x + 3
) ddyx = 2.
[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 x3 . Hence find the first three non-zero terms in the expansion of
1 . x + 2x + 3 2
[7]
4. NYJC/II/3 dy = y 2 − 1 and (0 , 2) is a point on C. dx (i) Find the equation of the normal to the curve at the point (0 , 2). d 3 y 57 (ii) Show that at x = 0, 3 = . dx 256 Find the Maclaurin’s series of y up to and including the term in x3 . (iii) Hence find the series expansion of e y , up to and including the term in x 2 . A curve C is defined by the equation 2 y
[2]
[6] [3]
5. VJC/II/1 Given that y = ln (2 + e x ), show that 2
d2 y ⎛ d y ⎞ dy +⎜ [2] ⎟ = . 2 dx dx ⎝dx⎠ 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] x e Deduce the Maclaurin series for in ascending powers of x, up to and including the term in 2 + ex x2. [1] Answers
1. y = x +
x3 3x5 + + ... , 6 40
2. (b) a = 3, 1 + xln3 + 3. y ≈
3=
256 147
(xln3)2 (xln3)3 1 1 3 2 + 3! , (c) 1 + 2 x − 24 x + ...
2 2 2 1 2 1 2 x − x 2 + x3 , − x+ x 3 9 81 3 9 27
17
4 3
4. (i) y = − x + 2 , (ii) 2 + 1 3
1 9
5. ln 3 + x + x 2 +
3 15 2 19 3 3 33 2 ⎞ ⎛ x ⎟ x+ x + x + ... , (iii) e 2 ⎜1 + x + 4 64 ⎠ 4 64 512 ⎝
1 3 x + ... 81
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 xaxis. 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 { an } has first term a and common ratio r. The sequence of numbers { bn } satisfy the relation bn = ln(an ) for n ∈
+
.
(i)
Show that { bn } is an arithmetic sequence and determine the value of the common difference in terms of r. [3]
(ii)
Find an expression for
N +1
∑b n =1
n
in terms of a, aN + 1 and N.
Hence, obtain an expression for a1 × a2 ×
× aN +1 in terms of a, aN + 1 and N.
[2] [2]
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 nonzero 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 < d ≤ 14 [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) (ii)
Show that wn is a geometric sequence with common ratio r2. [2] Given that v1 = 4 and the sum of all the odd-numbered terms w1, w3, w5, … in the ∞ 32 [4] sequence {wn} is 15 , find the value of r and ∑ vr . r =1
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 n th layer, show that 2(n − 1) x = 27 [2] (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 un denote the amount owed by John at the end of nth year, where n ∈ + . (i) Show that un = k (un −1 − 12000) , where k is a constant to be determined. (ii) (iii) (iv)
[1]
Express un in the form of a + (1.03n )b , where a and b are constants to be determined. [3] Find the minimum number of years required for John to pay up the bank loan. [2] 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] Comment on the practicality of the approach taken by the neighbour. [1] (iv)
Answers 1. ( 8, 4 )
2. (i)18, 12, 8 (ii) 54 (iii)n = 1,2,3,4,5,6 N +1 N +1 ln(aaN +1 ) , ( aaN +1 ) 2 3. (i) ln(r ) (ii) 2 4. a = 32, d = 2 1 8 5. (a) (i) 14 (b)(ii) − 2 ;3 6. (a)(i) n =
2 Sn (a)(iii) 0.063 mm a+l
7. (i) un = 1.03(un−1 − 12000) (ii) 412000 − 212000(1.03n ) (iii)23 years (iv) 8317.42 , It’s to John’s benefit to terminate his loan early 8. (i) n(4 + n) (ii)37 (iii)1500(1 − 0.95n ) (iv)Not practical as S∞ = 1500
Mathematical Induction 1. HCI/I/9(a) A sequence of real numbers u1 , u2 , u3 ,.... is defined by u1 = 5 and un + 1 = un + 8n + 8 for all n ≥ 1 . Prove by induction that for every positive integer n, un = ( 2n + 1) − 4 . 2
Show that un is a product of two odd numbers.
[4] [1]
2. RJC/I/1 n
Prove by induction that, for n ≥ 2 ,
∑ r =2
(3 − 2r )3r 3n +1 . =9− r (r − 1) n
[6]
20
3. YJC/II/2(a) The sequence of real numbers a1 , a 2 , a3 , … is an arithmetic progression. n
Prove, by mathematical induction, that
∑ ra r = r =1
n(n + 1)(a1 + 2a n ) for all n ∈ 6
+
.
[4]
4. NJC/I/8 The sequence un is defined by the recurrence relation u u1 = 1 and un +1 = n for n ≥ 1 . un + 2 Write down the values of u2 , u3 and u4 . Hence, make a conjecture for un in terms of n.
(i)
(ii)
[2]
Use mathematical induction to prove that your conjecture is true for all positive integers n. [5]
Answers
4.
1 1 1 1 ; ; ; ∴ un = n 3 7 15 2 −1
Methods of Difference/ Recurrence Relations 1. AJC/I/8(a) 1 1 ⎛ n ⎞ Let un = ln ⎜ − , ⎟+ n ⎝ n +1⎠ n +1 (i) Find in terms of N, an expression for S N , where S N = u1 + u2 + ... + u N , simplifying your result as far as possible. [3] (ii) Show that S N < 0 for all N ≥ 1. [2] 2. HCI/I/9(b) A sequence of real numbers u1 , u2 ,.... is defined by ur =
1 for all r ≥ 1 . ( 2r + 1)( 2r + 3)( 2r + 5)
By taking f ( r ) =
1
( 2r + 3)( 2r + 5)
Evaluate Sn where S n =
, express ur as a difference between f ( r ) and f ( r − 1) . [2]
1 1 + + ........ . to n terms. 3⋅5⋅ 7 5⋅ 7 ⋅9
[3]
21
Find the limit of Sn as n → ∞.
[1]
3. PJC/I/4 e e and un − un −1 = 2 n −1 , for all n ≥ 2 . 30 4 N −1 e ⎡ 1⎛ 1 ⎞ ⎤ Use the method of difference to show that u N = ⎢1 − ⎜ ⎟ ⎥ for all N ≥ 1 . 20 ⎢⎣ 3 ⎝ 16 ⎠ ⎥⎦ Hence or otherwise, determine if
A sequence u1 , u2 , u3 , … is such that u1 =
(i)
un is a convergent or divergent sequence,
[4]
[2]
N
(ii)
∑u n =1
n
is a convergent or divergent series.
[2]
4. SRJC/I/7 Let f(r) =
sin [ (2r + 1)θ ] , r∈ cos θ
+
.
(i)
Show that f (r ) − f (r − 1) = A cos(Brθ ) tan(θ ) , where A and B are to be determined. [3]
(ii)
By using the result in part (i), show that cos 2θ + cos 4θ + ... + cos 2 Nθ =
1 ⎡ sin [ (2 N + 1)θ ] ⎤ − 1⎥ . ⎢ 2⎣ sin θ ⎦
[5]
5. NYJC/II/4 A sequence of numbers, xn , satisfy the relation 1 − xn , for n ∈ + . xn +1 = xn − 4 (i) If the sequence converges to a number L, show that L satisfies the equation L2 − 3L − 1 = 0 . [2] 2 (ii) Obtain the exact value of the roots α and β of the equation x − 3x − 1 = 0 , where α 0 and −π < θ ≤ π .
[3] [4]
4. HCI/1/5
The complex number z is given by z = cos θ + i sin θ , where −π < θ ≤ π . Show that the equation i π + 3θ 4 + i ( 4 z 3 − 1) = e ( ) can be reduced to z 3 = i . Hence express each root of 4 + i ( 4 z 3 − 1) = e (
i π + 3θ )
in the form x + iy , 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 ) z = ( 4 − 7i ) z * , find the possible values of arg( z ) correct to 3 decimal places.
(b)
[5]
Find the exact roots of the equation z 5 = − (16√2 )(1 + i) , expressing them in the form r eiθ , [4] where r > 0 and −π < θ ≤ π . 1 On an Argand diagram the points which represent the above roots are rotated π radian 10 anti-clockwise about the origin to obtain points A , B , C , D and E .
Find the equation whose roots are represented by A , B , C , D and E , giving your answer in the form z 5 = − (16√2 )( p + i q ) where p and q are real constants to be determined. [2]
6. TPJC/I/6 Given that z1 = 1 − 2i is a root of the quadratic equation z 2 + az + b = 0 where a and b are real, find the values of a and b. Mark on an Argand diagram P1 and P2 , the points representing the roots z1 and z2 of the quadratic equation z 2 + az + b = 0 . A third point P3 is on the real axis such that P1 , P2 and P3 form a triangle which encloses the origin with an area of 8 square units. Find z3 represented by the point P3 , 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 z1 , z2 and z3 . [7]
7. VJC/II/3
Let z = 1 − 3 i . (i)
Find z and the exact value of arg( z ) .
[2]
(ii) Given that w4 = 1 − 3 i , find the complex numbers w in the form reiθ , where r > 0 and −π < θ ≤ π . [3] (iii)
(
Given that 1 − 3 i
)
n
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 ( z − 1) ≤
π 4
and z + 1 − i ≤ z − 5 − i .
[3]
Hence, find (i)
the maximum value of arg ( z ) ,
[1]
(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 a + ib , 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 , arg (z − a) = arg (a) +
π
. 2 Hence, find a value of a such that the complex number z satisfying both equations in parts (i) and (ii) is purely imaginary. [6] (ii)
10. HCI/II/4 2
π π⎞ ⎛ ⎜ cos − i sin ⎟ 4 4⎠ (a) Consider the complex number z = ⎝ . 3 π π⎞ ⎛ ⎜ cos + i sin ⎟ 3 3⎠ ⎝ (i)Find the modulus and the exact value of the argument of z.
[3]
(ii)Find the set of values of n such that z n is purely imaginary.
[3]
36
(b) Show clearly on an Argand diagram the locus given by
⎛ 3 i⎞ − ⎟ = arg arg ⎜⎜ w + 2 2 ⎟⎠ ⎝
(
)
3 −i .
[3]
A complex number u satisfies u − 3 + i = 2 . Sketch on the same Argand diagram, the locus of the point which represents u. Hence find the greatest possible value of u +
3 i − . 2 2
[4]
11. AJC/II/5
Sketch, on an Argand diagram, the locus given by z − 1 − 3i = 1 . Find the minimum value of ⎡ ( z + 5)4 ⎤ arg ⎢ ⎥. ⎢⎣ 2i ⎥⎦
[5]
Given that a is a complex number with a = 1 and arg ( a ) = θ , where 0 < θ
