30. miscellaneous questions

Miscellaneousquestions

30Chapter

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Y:\HAESE\IB_HL-2ed\IB_HL-2ed_30\833IB_HL-2_30.CDR Thursday, 24 January 2008 10:39:36 AM PETERDELL

834 MISCELLANEOUS QUESTIONS (Chapter 30)

1 a Simplify (¡1 + ip

2)3:

b Write 5 + ip

2 in the form a3cis µ stating the exact values of a and µ.

c Find the exact solutions of z3 = 5 + ip

2.

d Hence, show that arctan³p

25

´+ 2¼ = 3 arccos

³¡1p3

´.

2 a Find the cube roots of ¡2 ¡ 2i.

b Display the cube roots of ¡2 ¡ 2i on an Argand diagram.

c If the cube roots are ®1, ®2 and ®3, show that ®1 + ®2 + ®3 = 0:

d Using an algebraic argument, prove that if ¯ is any complex number then the sum

of the zeros of zn = ¯ is 0.

3 a Evaluate (1 ¡ i)2 and simplify (1 ¡ i)4n.

b Hence, evaluate (1 ¡ i)16.

c Use your answers above to find two solutions of z16 = 256. Give clear reasons

for your answers.

4 Let z =¡1 + i

p3

4and w =

p2 + i

p2

4.

a Write z and w in the form r(cos µ + i sin µ) where 0 6 µ 6 ¼.

b Show that zw = 14

¡cos 11¼

12 + i sin 11¼12

¢.

c Evaluate zw in the form a + ib and hence find the exact values of

cos 11¼12 and sin 11¼

12 .

5 The sum of the first n terms of a series is given by Sn = n3 + 2n¡ 1.

Find un, the nth term of the series.

6 The tangent to the curve y = f(x) at the point A(x, y) meets the x-axis at the point

B(x¡ 12 , 0). The curve meets the y-axis at the point C(0, 1

e). Find the equation of the

curve.

7 The diagram shows a sector POR of a

circle of radius 1 unit and centre O. The

angle PbOR = µ, and the line segments

[PQ], [P1Q1], [P2Q2], [P3Q3], ...... are

all perpendicular to [OR].

Calculate, in terms of µ, the sum to

infinity of the lengths

PQ + P1Q1 + P2Q2 + P3Q3 + .......

8 Use the method of integration by parts to findRx arctanxdx:

Check that your answer is correct using differentiation.

9 Solve the following equations:

a log2¡x2 ¡ 2x + 1

¢= 1 + log2(x¡ 1) b 32x+1 = 5(3x) + 2

EXERCISE 30

O

P

RQ

P1

P2

Q2 Q1


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Y:\HAESE\IB_HL-2ed\IB_HL-2ed_30\834IB_HL-2_30.CDR Monday, 21 January 2008 9:44:57 AM PETERDELL

MISCELLANEOUS QUESTIONS (Chapter 30) 835

10 Solve exactly for x:3x¡ 1

jx + 1 j > 2.

11 Find the exact values of x for which sin2 x + sinx¡ 2 = 0 and ¡2¼ 6 x 6 2¼.

12 If f : x 7! lnx and g : x 7! 3 + x find: a f¡1(2)£ g¡1(2) b (f ± g)¡1(2).

13 Given an angle µ where sin µ = ¡ 725 and ¡¼

2 < µ < 0, find the exact values of:

a cos µ b tan µ c sin 2µ d sec 2µ:

14 Solvep

3 cosx cscx + 1 = 0 for 0 6 x 6 2¼.

15 The number of snails in a garden plot follows a Poisson distribution with standard

deviation d. Find d if the chance of finding exactly 8 snails is half that of finding

exactly 7 snails in this plot.

16 Find the coordinates of the point on the line L that is nearest to the origin if the equation

of L is r = 2i ¡ 3j + k + ¸(¡i + j ¡ k), ¸ 2 R .

17 The function f(x) satisfies the following criteria: f 0(x) > 0 and f 00(x) < 0 for all x,

f(2) = 1, and f 0(2) = 2.

a Find the equation of the tangent to f(x) at x = 2 and sketch it on a graph.

b Hence, sketch a graph of f(x) on the same axes.

c Explain why f(x) has exactly one zero.

d Estimate an interval in which the zero of f(x) lies.

18 If n 2 Z , n > ¡2, prove by induction that 2n3 ¡ 3n2 + n + 31 > 0.

19 Prove by induction that

nXr=1

r3r = 34 [(2n¡ 1)3n + 1] for all n 2 Z +.

20 Prove by induction that for all n 2 Z +,

1

a(a + 1)+

1

(a + 1)(a + 2)+

1

(a + 2)(a + 3)+ ::::::+

1

(a + n¡ 1)(a + n)=

n

a(a + n).

21 Prove by induction that xn ¡ yn has a factor of x¡ y for all n 2 Z +.

22 Prove that 3¡52n+1

¢+ 23n+1 is divisible by 17 for all n 2 Z +.

23 Assuming Pascal’s rule¡nr

¢+³

nr+1

´=³

n+1r+1

´,

a prove that (1 + x)n

= 1 +¡n1

¢x +

¡n2

¢x2 + :::::: +

¡nn

¢xn for all n 2 Z +.

b Establish the binomial expansion for (a + b)n by letting x =b

ain a.

24 Prove that1

sin 2x+

1

sin 4x+ :::::: +

1

sin(2nx)= cotx¡ cot(2nx) for all n 2 Z +.

25 Show that if y = mx + c is a tangent to y2 = 4x then c =1

mand the coordinates

of the point of contact are

µ1

m2,

2

m

¶:


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26 The illustrated ellipse has equationx2

a2+

y2

b2= 1:

a Show that the shaded region has area given by

b

a

Z a

0

pa2 ¡ x2 dx .

b Find the area of the ellipse in terms of a and b.

c An ellipsoid is obtained by rotating the ellipse about the x-axis through 360o.

Prove that the volume of the ellipsoid is given by V = 43¼ab

2.

27 Consider the following quadratic function where ai, bi 2 R :

f(x) = (a1x¡ b1)2 + (a2x¡ b2)

2 + (a3x¡ b3)2 + :::::: + (anx¡ bn)2.

Use quadratic theory to prove the ‘Cauchy-Schwartz inequality’:µnP

i=1a2i

¶µnP

i=1b2i

¶>

µnP

i=1aibi

¶2

.

28 Show that the equation of the tangent to the ellipse with equation

x2

a2+

y2

b2= 1 at the point P(x1, y1) is

³x1

a2

´x +

³y1b2

´y = 1.

29 Prove that in any triangle with angles A, B and C:

a sin 2A + sin 2B + sin 2C = 4 sinA sinB sinC

b tanA + tanB + tanC = tanA tanB tanC.

30 a A circle has radius r and the acute angled triangle ABC has vertices on the circle.

Show that the area of the triangle is given byabc

4r.

b In triangle ABC it is known that sinA = cosB + cosC.

Show that the triangle is right angled.

31 [AB] is a thin metal rod of fixed length

and P is its centre. A is free to move on

the x-axis and B is free to move on the

y-axis.

What path or locus is traced out by point

P as the rod moves to all possible places?

32 a Show thatp

14 ¡ 4p

6 cannot be written in the form a + bp

6 where a, b 2 Z .

b Canp

14 ¡ 4p

6 be written in the form apm + b

pn where a, b, m, n 2 Z ?

33 (1 + x)n =¡n0

¢+¡n1

¢x +

¡n2

¢x2 +

¡n3

¢x3 + :::::: +

¡nn

¢xn for all n 2 Z +.

Prove that:

a¡n1

¢+ 2

¡n2

¢+ 3

¡n3

¢+ :::::: + n

¡nn

¢= n2n¡1

b¡n0

¢+ 2

¡n1

¢+ 3

¡n2

¢+ :::::: + (n + 1)

¡nn

¢= (n + 2)2n¡1

c¡n0

¢+ 1

2

¡n1

¢+ 1

3

¡n2

¢+ :::::: +

1

n + 1

¡nn

¢=

2n+1 ¡ 1

n + 1.

a

b

�a

�b

B

y

x

A

P


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34 a Ifx + 5

(x2 + 5)(1¡ x)=

Ax + B

x2 + 5+

C

x¡ 1, find A, B and C.

b Hence, find the exact value of

Z 4

2

x + 5

(x2 + 5)(1¡ x)dx:

35 Consider the series1

1 £ 3+

1

2 £ 4+

1

3 £ 5+ :::::: +

1

n(n + 2):

a By writing1

n(n + 2)in the form

A

n+

B

n + 2, find the values of A and B.

b Use a to show that the sum of the series is 34 ¡ 1

2n + 2¡ 1

2n + 4.

c Find

1Xr=1

1

r(r + 2). d Check b using mathematical induction.

36 Find: a

Zxp

1 ¡ x2dx b

Z1 + x

1 + x2dx c

Z1p

1 ¡ x2dx

37 A flagpole is erected at A and its top is B. At C, due west of A, the angle of elevation

to B is ®. At D, due south of A, the angle of elevation to B is ¯.

Point E is due south of C and due west of D. Show that at E, the angle of elevation to

B is arccot³p

cot2 ® + cot2 ¯´

.

38 a Use complex number methods to show that tan 4µ =4 tan µ ¡ 4 tan3 µ

1 ¡ 6 tan2 µ + tan4 µ.

b Hence, find the roots of the equation x4 + 4x3 ¡ 6x2 ¡ 4x + 1 = 0.

39 Find the sum of the series:

a 1 + a cos µ + a2 cos 2µ + a3 cos 3µ + :::::: + an cosnµ

b a sin µ + a2 sin 2µ + a3 sin 3µ + :::::: + an sinnµ for n 2 Z +.

40 Suppose ex can be written as the infinite series ex = a0+a1x+a2x2+::::::+anx

n+::::::

a Show that a0 = 1, a1 = 1, a2 = 12! , a3 = 1

3! , ......

b Hence conjecture an infinite geometric series representation for ex.

c Check your answer to b using the substitution x = 1.

41 a By considering1

a2 ¡ x2=

P

a¡ x+

Q

a + x,

b Use a to show that

Z1

a2 ¡ x2dx =

1

2aln

¯̄̄̄a + x

a¡ x

¯̄̄̄+ c.

c Check b using differentiation.

42 a Assuming that

cosS+cosD = 2 cos¡S+D

2

¢cos¡S¡D

2

¢and sinS+sinD = 2 sin

¡S+D

2

¢cos¡S¡D

2

¢,

prove that cis µ + cis Á = 2 cos³

µ¡Á2

´cis³

µ+Á2

´.

find and .P Q


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Y:\HAESE\IB_HL-2ed\IB_HL-2ed_30\837IB_HL-2_30.CDR Thursday, 24 January 2008 10:42:19 AM PETERDELL


b From a, what is the modulus and argument of cis µ + cis Á?

c Show that your answers in b are correct using a geometrical argument.

d Prove that the solutions of

µz + 1

z ¡ 1

¶5

= 1 are z = ¡i cot

µk¼

5

¶, k = 1, 2, 3, 4.

43 a If z +1

zis real, prove that either jzj = 1 or z is real.

b If jz + wj = jz ¡ wj prove that arg z and argw differ by ¼2 .

c If z = r cis µ, write z4,1

zand iz¤ in a similar form.

44 x2 + ax+ bc = 0 and x2 + bx+ ca = 0 where a 6= 0, b 6= 0, c 6= 0, have a single

common root. Prove that the other roots satisfy x2 + cx + ab = 0.

45 If x = a13 + b

13 , show that x3 = 3(ab)

13 x + (a + b).

Hence, find all real solutions of the equation x3 = 6x + 6.

46 Solve simultaneously: x = 16y and logy x¡ logx y = 83 .

47 Find all values of m for which the quartic equation x4 ¡ (3m + 2)x2 + m2 = 0has 4 real roots in arithmetic progression.

48 ® and ¯ are two of the roots of x3 + ax2 + bx + c = 0.

Prove that ®¯ is a root of x3 ¡ bx2 + acx¡ c2 = 0.

49 x and y satisfy the equations x2 + 3xy + 9 = 0 and y2 + x¡ 1 = 0.

Solve these equations simultaneously for x given that x is real.

50 a Find the value

of the sum:

1

1 +p

2+

1p2 +

p3

+1p

3 +p

4+ ..... +

1p99 +

p100

.

b Can you make any generalizations from a?

51 The three numbers x, y and z are such that x > y > z > 0.

Show that if 1x

, 1y

and 1z

are in arithmetic progression, then x¡ z, y and x¡ y + z

are the lengths of the sides of a right angled triangle.

52 Each summer, 10% of the trees on a certain plantation die out, and each winter, workmen

plant 100 new trees. At the end of the winter in 1980 there were 1200 trees in the

plantation.

a How many living trees were there at the end of winter in 1970?

b What will happen to the number of trees in the plantation during the 21st century

providing the conditions remain unchanged?

53 a I wish to borrow $20000 for 10 years at 12% p.a. where the interest is compounded

quarterly. I intend to pay off the loan in quarterly instalments. How much do I need

to pay back each quarter?

b Find a formula for calculating the repayments R if the total amount borrowed is

$P , for n years, at r% p.a., and there are to be m equal payments at equal intervals

each year.


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54 A rectangle is divided by m lines parallel to one pair of opposite sides and n lines

parallel to the other pair. How many rectangles are there in the figure obtained?

55 a Schools A and B each preselect 11 members for a team to be sent interstate.

However, circumstances allow only a combined team of 11 to be sent away.

In how many ways can a team of 11 be selected and a captain be chosen if the

captain must come from A?

b Use a to show that: 1¡n1

¢2+ 2

¡n2

¢2+ 3

¡n3

¢2+ ...... + n

¡nn

¢2= n

³2n¡1n¡1

´.

56 Two different numbers are randomly chosen out of the set f1; 2; 3; 4; 5; ...... , ng,

where n is a multiple of four. Determine the probability that one of the numbers is four

times larger than the other.

57 A hundred seeds are planted in ten rows of ten seeds per row. Assuming that each seed

independently germinates with probability 12 , find the probability that the row with the

maximum number of germinations contains at least 8 seedlings.

58 Consider a randomly chosen n child family, where n > 1. Let A be the event that the

family has at most one boy, and B be the event that every child in the family is of the

same sex. For what values of n are the events A and B independent?

59 Two marksmen, A and B, fire simultaneously at a target. If A is twice as likely to hit

the target as B, and if the probability that the target does get hit is 12 , find the probability

of A hitting the target.

60 A quadratic equation ax2 + bx+ c = 0 is copied by a typist. However, the numbers

standing for a, b and c are blurred and she can only see that they are integers of one

digit. What is the probability that the equation she types has real roots?

61 Two people agree to meet each other at the corner of two city streets between 1 pm and

2 pm, but neither will wait for the other for more than 30 minutes. If each person is

equally likely to arrive at any time during the one hour period, determine the probability

that they will in fact meet.

62 Use the figure alongside to show that cos 36o =1 +

p5

4.

63 Find ®:

64 For A, B, C not necessarily the angles of a triangle, what can be deduced about

A + B + C if tanA + tanB + tanC = tanA tanB tanC ?

65 Without using a calculator, show how to find arctan(17 ) + 2 arctan(13):

� ��

�

A

B

C

D

E

F�°

10°

40°

30°

60°


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66

67 An A cm by B cm rectangular refrigerator leans

at an angle of µ to the floor against a wall.

a Find H in terms of A, B and µ:

b Explain how the figure can be used to prove

that A sin µ + B cos µ 6pA2 + B2, with

equality when tan µ =A

B.

68 Over 2000 years ago, Heron or Hero discovered a formula for finding the area of a

triangle with sides a, b and c. It is A =ps(s¡ a)(s¡ b)(s¡ c) where 2s = a+b+c.

Prove that this formula is correct.

69 a Given that y = ln(tanx), x 2 ] 0, ¼2 [ , show that

dy

dx= k csc(2x) for some

constant k.

b The graph of y = csc(2x) is illustrated

on the interval ] 0, ¼2 [.

Find the area of the shaded region.

Give your answer in the form a ln bwhere a 2 Q and b 2 Z +.

70 P is a point and line l, with direction vector v, passes

through points A and Q.

a Prove that PQ =j ¡!AP £ v j

j v j .

b Hence, find the shortest distance from (2, ¡1, 3)

to the line

Ãxyz

!=

Ã¡112

!+ ¸

Ã3¡11

!.

71 Find a given that the shaded region has

area 516 units2.

X

A cmH cm

floor

wall

B cm

x

y

3�

6�

2��x

v

l

P

Q

A

a a�

y x x�� '��X,

x

y

B

CA

2 km

northernslope

3 km

A mountain is perfectly conical in shape. The base isa circle of radius km, and the steepest slopesleading up to the top are km long.

From the southernmost point A on the base, a pathleads up on the side of the mountain to B, a point onthe northern slope which is km up the slope fromC. A and C are diametrically opposite.

If the path leading from A to B is the shortest possibledistance from A to B along the mountainside, find thelength of this path.

23

1 5:


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Y:\HAESE\IB_HL-2ed\IB_HL-2ed_30\840IB_HL-2_30.CDR Monday, 21 January 2008 4:44:31 PM PETERDELL


72 Ifdy

dx= x csc y and y(2) = 0, find y as a function of x.

73 By consideringd

dx(tan3 x), find

Rsec4 xdx:

74 What can be deduced if A \B and A [B are independent events?

75 For a continuous function defined on the interval [a, b ], the length of the curve can be

found using L =

Z b

a

q1 + [f 0(x)]2 dx. Find the length of:

a y = x2 on the interval [ 0, 1 ] b y = sinx on the interval [ 0, ¼ ].

76 a Simplify: i (A [B) \A0 ii (A \B) [ (A0 \B).

b Verify that (A \B) [ C = (A [ C) \ (B [ C).

c Prove that if A and B are independent events then so are:

i A0 and B0 ii A and B0.

77 Write (3 ¡ ip

2)4 in the form x + yp

2i where x, y 2 Z .

78 Solve the equation sin µ cos µ = 14 for the interval µ 2 [¡¼, ¼ ].

79 z and w are two complex numbers such that 2z + w = i and z ¡ 3w = 7 ¡ 10i.Find z + w in the form a + bi, where a and b 2 Z .

80 Solve the differential equation (x + 1)2dy

dx= 2xy, x > ¡1 given that y(1) = 4.

81 f is defined by x 7! ln (x(x¡ 2)).

a State the domain of f . b Find f 0(x).

c Find the equation of the tangent to f at the point where x = 3.

82 Hat 1 contains three green and four blue tickets. Hat 2 contains four green and three

blue tickets. One ticket is randomly selected from each hat.

a What is the probability that the tickets are the same colour?

b Given that the tickets are different colours, what is the probability that the green

ticket came from Hat 2?

83 If A3 = A, what can be said about: a jA j b A¡1?

84 If P (x) is divided by (x¡a)2, prove that the remainder is P 0(a)(x¡a)+P (a) where

P 0(x) is the derivative of P (x):

85 A lampshade is a truncated cone open

at the bottom.

Find the pattern needed to make this

lampshade from a flat sheet of material.

32 cm

15 cm

20 cm


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86 [AB] represents a painting on a wall.

AB = 2 m and BC = 1 m.

The angle of view observed by a girl

between the top and bottom of the

painting is 30o.

How far is the girl from the wall?

87 A circle is centred at the origin O. A

second circle has half the diameter of the

original circle and touches it internally. P

is a fixed point on the smaller circle as

shown, and lies on the x-axis.

The smaller circle now rolls around the

inside of the larger one without slipping.

Show that for all positions of the smaller

circle, P remains on the x-axis.

88 a Write ¡8i in polar form.

b Hence find the three cube roots of ¡8i, calling them z1, z2 and z3.

c Illustrate the roots from b on an Argand diagram.

d Show that z 21 = z2z3 where z1 is any one of the three cube roots.

e Find the product of the three cube roots.

89 a Complex number z has an argument of µ. Show that iz has an argument of µ + ¼2 .

b In an Argand plane, points P, Q and R represent the complex numbers z1, z2 and z3respectively. If i(z3 ¡ z2) = z1 ¡ z2, what can be deduced about triangle PQR?

90 z = reiµ, r > 0, is a non-zero complex number such that z +1

z= a + bi, a, b 2 R .

a Find expressions for a and b in terms of r and µ.

b Hence, find all complex numbers z such that z +1

zis real.

91 The diagram shows a simple electrical network.

Each symbol represents a switch.

All four switches operate independently, and the

probability of each one of them being closed is p.

a In terms of p, find the probability that the current flows from A to B.

b Find the least value of p for which the probability of current flow is more than 0:5.

92 If A =³2 10 2

´:

a find A2 and A3.

b prove using mathematical induction that An =³2n n2n¡1

0 2n

´for all n 2 Z +.

93 By considering the identity (1 + i)n = (1 + i)2 (1 + i)n¡2, deduce that¡nr

¢=¡n¡2r

¢+ 2¡n¡2r¡1

¢+¡n¡2r¡2

¢.

2 m

1 m30°

A

B

side view

eye levelG C

P

A B


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94 While driving, Bernard passes through n intersections which are independently controlled

by traffic lights. Each set of lights has probability p of stopping him.

a What is the probability that Bernard will be stopped at least once.

b Suppose Ak is the event that Bernard is stopped at exactly k intersections and Bk

is the event that Bernard is stopped at at least k intersections.

Write down the conditional probability P(Ak j Bk).

c If A1 and B1 are independent, find p.

d Find p if P(A2 j B2) = P(A1) and n = 2.

95 A club has n female members and n male members. A committee of three members is

to be randomly chosen, and must contain more females than males,

a How many committees consist of 2 females and 1 male?

b How many committees consist of 3 females?

c Use a and b to deduce that n¡n2

¢+¡n3

¢= 1

2

¡2n3

¢.

d Suppose the club consists of 12 people, and that Mr and Mrs Jones are both members.

Find the probability that a randomly selected committee contains:

i Mrs Jones ii Mr Jones given that it contains Mrs Jones.

96 In triangle ABC, the angle at A is double the angle at B.

If AC = 5 cm and BC = 6 cm, find:

a the cosine of the angle at B b the length of [AB] using the cosine rule.

c Are both solutions in b valid?

97 x2 + b1x + c1 = 0 and x2 + b2x + c2 = 0 are two quadratic equations where

b1b2 = 2(c1 + c2).

Prove that at least one of the equations has real roots.

98 Suppose that for all n 2 Z +, (2 ¡p3)n = an ¡ bn

p3 where an and bn are integers.

a Show that an+1 = 2an + 3bn and bn+1 = an + 2bn.

b Calculate a 2n ¡ 3b 2

n for n = 1, 2 and 3.

c What do you propose from b?

d Prove your proposition from c.

99 A sequence un is defined by u1 = u2 = 1 and un+2 = un+1 + un for all n 2 Z +.

Prove by induction that un 6 2n for all n 2 Z +.

100 Use the Principle of mathematical induction to prove that, if n > 2, n 2 Z +,

then (1 ¡ 1

22)(1 ¡ 1

32)(1 ¡ 1

42) :::::: (1 ¡ 1

n2) =

n + 1

2n.

101 a Graph y = x3 ¡ 12x2 + 45x and on the graph mark the coordinates of its turning

points.

b If x3 ¡ 12x2 + 45x = k has three real roots, what values can k have?

102 a Use complex number methods to prove that cos3 µ = 34 cos µ + 1

4 cos 3µ.

b Solve the equation x3 ¡ 3x + 1 = 0 by letting y = mx.


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103 Triangle ABC has perimeter 20 cm.

a Find y2 in terms of x and µ and hence

find cos µ in terms of x only.

b If the triangle has area A, show that

A2 = ¡20(x2 ¡ 12x + 20).c Hence, without calculus, find the maximum area of the triangle and comment on

the triangle’s shape when its area is a maximum.

104 a If A =³2 10 1

´, predict the form of An.

b Use mathematical induction to prove your conjecture in a correct.

c If Sn = A + A2 + A3 + ...... + An, find Sn in simplest form and hence find S20.

105 If

mXn=1

f(n) = m3 + 3m, find f(n).

106 ABC is an equilateral triangle with sides 10 cm long. P is a point within the triangle

which is 5 cm from A and 6 cm from B. How far is it from C?

107 A normally distributed random variable X has a mean of 90. Given that the probability

P(X < 85) ¼ 0:16 :

a find the proportion of scores between 90 and 95, i.e., find P(90 < X < 95)

b find an estimate of the standard deviation for the random variable X.

108 A normally distributed random variable X has a mean of 90. Given that the probability

P(X < 88) ¼ 0:28925, find the:

a standard deviation of X to 5 decimal places

b probability that a randomly chosen score is either greater than 91 or less than 89.

109 In an International school there are 78 students preparing for the IB Diploma. Of these

students, 38 are male and 17 of these males are studying Mathematics at the higher level.

Of the female students, 25 are not studying Mathematics at the higher level.

A student is selected at random and found to be studying Mathematics at the higher

level. Find the probability that this student is male.

110 A company manufactures computer chips, and it is known that 3% of them are faulty. In

a batch of 500 such chips, find the probability that between 1 and 2 percent (inclusive)

of the chips are faulty.

111 A factory manufactures rope, and the rope has an average of 0:7 flaws per metre. It is

known that the number of flaws produced in the rope follows a Poisson distribution.

a Determine the probability that there will be exactly 2 flaws in 2 metres of rope.

b Find the probability that there will be at least 2 flaws in 4 metres of rope.

112 A random variable X is known to be distributed normally with standard deviation 2:83 .

Find the probability that a randomly selected score from X will differ from the mean by

less than 4.

113 A discrete random variable X has a probability function given by the rule

P(X = x) = a¡25

¢x, x = 0, 1, 2, 3, ...... Find the value of a.

A

B

C

x cm

8 cm

y cm


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114 Given that events A and B are independent with P(A j B) = 14 and P(B j A) = 2

5 ,

find P(A [B0).

115 In a game, a player rolls a biased tetrahedral

(four-faced) die. The probability of each possible

score is shown alongside in the table.

Score 1 2 3 4

Probability 112 k 1

413

a Find the value of k.

b Let the random variable X denote the number of 2s that occur when the die is rolled

2400 times. Calculate the exact mean and standard deviation of X.

116 The lifetime n (in years) of a particular component of a solar cell is given by the

probability density function f (n) =

½0:6e¡0:6n, n > 0

0, otherwise.

a What is the chance that a randomly chosen component will last for at least one

year?

b A solar cell has 8 components, each of which operates independently of each other.

The solar cell will continue to operate provided at least one of the components are

operating. Find the probability that a randomly chosen solar cell fails within one

year.

117 The random variable X has a Poisson distribution with standard deviation ¾ such that

P(X = 2) ¡ P(X = 1) = 3P(X = 0). Find the exact value of ¾ in surd form.

118 A machine produces soft drink in bottles. The volumes in millilitres (mL) of a sample

of drinks chosen at random are shown below.

Volume (mL) 374:7 374:8 374:9 375:0 375:1 375:2 375:3 375:4

Frequency 6 12 15 16 9 11 8 3

Find unbiased estimates of:

a the mean of the population from which this sample is taken

b the variance of the population from which this sample is taken.

119 In a particular year, a randomly chosen Year 12 group completed a calculus test with the

following results:25Xi=1

xi = 1650 and

25Xi=1

x 2i = 115492, where xi denotes the percentage result of the

i th student in the class. Calculate an unbiased estimate of:

a the mean percentage result of all Year 12 students in the calculus test

b the variance of the percentage result of all Year 12 students in the calculus test.

120 a Using integration by parts, findR

lnxdx. Show how to check that your answer

is correct.

b The continuous random variable X has probability density function defined by

f (x) =

½lnx, 1 6 x 6 k

0, otherwise.Find the exact value of k.

c Write down an equation that you would need to solve to find the median value of

the random variable X. Do not attempt to solve this equation.


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121 Use the cosine rule and the given kite

to show that sin2 µ = 12 ¡ 1

2 cos 2µ

and cos2 µ = 12 + 1

2 cos 2µ.

122 Show that tan µ = 3 tan®.

123 Points P and Q are free to move on the coordinate axes.

N is the foot of the perpendicular from the origin to the

line segment [PQ]. [PQ] makes an angle of µ with the

y-axis.

a Show that N is at (3 sin µ cos2 µ, 3 sin2 µ cos µ):

b Use technology to sketch the graph of the curve

defined by: x = 3 sin µ cos2 µ, y = 3 sin2 µ cos µ:

124 a Find the general term un of the sequence:

1

sin µ¡ sin µ, cos µ, sin µ,

1

cos µ¡ cos µ, ......

b Find an equation connecting consecutive terms of the sequence:

1, cos µ, cos3 µ, cos7 µ, cos15 µ, ......

125 1k

, k, k2 + 1 where k 2 Q are the 3rd, 4th and 6th terms of an arithmetic sequence

respectively.

a Find k. b Find the general term un.

126 ABC is an equilateral triangle with sides of length 2k. P is any point within the triangle.

[PX], [PY] and [PZ] are altitudes from P to the sides [AB], [BC] and [CA] respectively.

a By letting PbCZ be µ, find PX + PY + PZ in terms of µ, and hence show that

PX + PY + PZ is constant for all positions of P.

b Check that your solution to a is correct when P is at A.

c Prove that the result in a is true using areas of triangles only.

127 R and Q are two fixed points on either side of line

segment [AB]. P is free to move on the line segment

so that the angles µ and Á vary. a and b are the

distances of Q and R respectively from [AB].

a Show that for all positions of P,

dÁ

dµ=

¡a cos2 Á

b cos2 µ.

b A particle moves from R to P with constant speed

v1 and from P to Q with constant speed v2.

Deduce that the time taken to go from R to P to Q is a minimum whensin µ

sinÁ=

v1v2

.

A CB

D

�

y

x

P

Q

N

3 m

A B

N

M

Q

R

P

"a

b


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128 At A on the surface of the Earth, a rocket is

launched vertically upwards. After t hours it is

at R, h km above the surface. B is the horizon

seen from R.

Suppose BbOR is µ and arc AB is y km long.

a If the Earth’s radius is r km, show that

dy

dt=

cos2 µ

sin µ

dh

dt.

b If the velocity of the rocket after t hours is

given by r sin t for any t 2 [ 0, ¼ ], find the height of the rocket at t = ¼2 hours.

c If r ¼ 6000, find the rate at which arc AB is changing at the instant when t = ¼2 .

129 Prove that the roots of (m ¡ 1)x2 + x ¡ m = 0 are always real and positive for

0 < m < 1.

130 a Show that sin 15o =p6¡

p2

4 using sin 45o = 1p2

and sin 30o = 12 , together

with a suitable trigonometric formula.

b Hence, find the exact value of cos2 165o + cos2 285o.

131 For ¡¼ 6 x 6 ¼, find the exact solutions to 3 sec 2x = cot 2x + 3 tan2x:

132 Solve exactly for x if 4 sinx =p

3 cscx + 2 ¡ 2p

3 where 0 6 x 6 2¼.

133 The first 3 terms of a geometric sequence have a sum of 39. If the middle term is

increased by 6623%, the first three terms now form an arithmetic sequence. Find the

smallest possible value of the first term.

134 Show algebraically that the equation log3(x¡ k) + log3(x+ 2) = 1 has a real solution

for every real value of k.

135 Solve the following equations, giving exact answers:

a 82x+3 = 4 3p

2 b 32x+1 + 8(3x) = 3

c ln (ln x) = 1 d log 19x = log9 5

136 Solve the following inequalities, giving exact answers:

a (0:5)x+1 > 0:125 b (23)x > (32)x¡1 c 4x + 2x+3 < 48

137 If x2 + y2 = 52xy, show that log

µx¡ y

5

¶= 1

2 (log x + log 2y).

138 If z = cos µ + i sin µ where 0 < µ < ¼4 , find the modulus and argument of 1 ¡ z2:

139 Find z in the form a + bi if z2 = 1 + i +58

9(3 ¡ 7i).

140 Solve the following equations simultaneously: 4x = 8y and 9y =243

3x.

141 Given w =z ¡ 1

z¤ + 1where z = a+bi and z¤ is the complex conjugate of z, write w

in the form x+ yi. Hence determine the conditions under which w is purely imaginary.

h km

A

R

B

r km

y km


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142 Given that x = log3 y2, express logy 81 in terms of x.

143 An infinite number of circles are

drawn in a sector of a circle of radius

10 cm and angle ® = ¼3 as shown.

a What is the total area of this

infinite series of circles?

b Find an expression for the total

area of all circles for a general

angle ® such that 0 6 ® 6 ¼2 :

144 The ratio of the zeros of x2 + ax + b is 2 : 1. Find a relationship between a and b.

145 Find real numbers a and b if the polynomial z3+az2+bz+15 = 0 has a root 2+i.

146 If xn + ax2 ¡ 6 leaves a remainder of ¡3 when divided by (x¡ 1) and a remainder

of ¡15 when divided by (x + 3), find the values of a and n.

147 When a cubic polynomial P (x) is divided by x(2x ¡ 3), the remainder is ax + bwhere a and b are real.

a If the quotient is the same as the remainder, write down an expression for P (x).

b Prove that (2x¡ 1) and (x¡ 1) are both factors of P (x).

c Find the equation of P (x) given that it has a y-intercept (0, 7) and passes through

the point (2, 39).

148 Factorise f(x) = 2x3¡x2¡8x¡5, and hence find the values of x for which f(x) > 0.

149 The graph of a quartic polynomial y = f(x) cuts the x-axis at x = ¡3 and at x = ¡14 ,

and touches it at x = 32 . The y-intercept is 9. Find f(x).

150 The polynomial p(x) = x3 + (5 + 4a)x + 5a where a is real, has a zero ¡2 + i.

a Find a real quadratic factor of p(x).

b Hence, find the value of a and the real zero of p(x).

151 Let h(x) = x3 ¡ 6tx2 + 11t2x¡ 6t3 where t is real.

a Show that t is a zero of h(x).

b Factorise h(x) as a product of linear factors.

c Hence or otherwise, find the coordinates of the points where the graphs of

y = x3 + 6x2 and y = ¡6 ¡ 11x meet.

152 A real polynomial P (x) = x4 + ax3 + bx2 + cx¡ 10 has two integer zeros p and q.

a If P (x) also has a complex zero 1 + ki, where k is an integer:

i use this zero to write an expression for a real quadratic factor of P (x)ii state all possible values of k.

�


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b Using p and q, write another expression for a real quadratic factor of P (x). Hence

list all possible values of pq.

c Given that p + q = ¡1, show that there is only one possible value for pq. Hence

find all zeros of P (x).

153 The real polynomial P (z) of degree 4 has one complex zero of the form 1 ¡ 2i, and

another of the form ai, where a 6= 0 and a is real.

Find P (z) if P (0) = 10 and the coefficient of z4 is 1. Leave the answer in factorised

form.

154 The point A(¡2, 3) lies on the graph of y = f (x). Give the coordinates of the point

that A moves to under the following transformations:

a y = f(x¡ 2) + 1 b y = 2f(x¡ 2) c y = ¡ j f(x) j ¡2

d y = f(2x¡ 3) e y =1

f (x)f y = f¡1(x)

155 The points A(¡1, 0), B(1, 0) and C(0, ¡0:5) are

the x- and y-intercepts of y = f (x).On the same set of axes, sketch the following

graphs. For each case, explain what happens to

the points A, B and C.

a y = f(x + 1) ¡ 1 b y = ¡2f(x¡ 1)

c y = jf(x)j d y =1

f (x)

156 The real quadratic function f (x) has a zero of 3 + 2i, and a y-intercept of ¡13. Write

the function in the form:

a f (x) = ax2 + bx + c b f (x) = a(x¡ h)2 + k.

157 Find a trigonometric equation of the form y = a sin(b(x+ c)) + d that represents the

following graph with the information given below.

You may assume that (3, ¡5) is a minimum point and (6, ¡1) lies on the principal axis.

158 Solve the system using an inverse matrix:

8<: x + 3y ¡ z = 152x + y + z = 7x¡ y ¡ 2z = 0

.

159 The ferris wheel at the Royal Show turns one full circle every minute. The lowest point

is 1 metre from the ground, whilst the highest point is 25 metres above the ground.

a The height of the ferris wheel above ground level after t seconds is given by the

model h(t) = a + b sin(c(t¡ d)). Find the values of a, b, c and d given that you

start your ride after entering your seat at the lowest point.

b If the motor driving the ferris wheel breaks down after 91 seconds, how high up

would you be while waiting to be rescued?

��

�

��

y

x

A

C

B

( )��,

( ) ��,


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160 The equations of two lines are:

l1: r =

Ã¡42¡1

!+ ¸

Ã312

!, ¸ 2 R , and l2: x =

y ¡ 5

2=

¡z ¡ 1

2.

a Determine the point of intersection of l1 and the plane 2x + y ¡ z = 2.

b Clearly explain why l1 and l2 are not parallel.

c Find the point of intersection of l1 and l2.

d Find the equation of the plane that contains l1 and l2.

161 Find the acute angle between the plane 2x + 2y ¡ z = 3 and the line

x = ¸¡ 1, y = ¡2¸ + 4, z = ¡¸ + 3.

162 Consider the following system of

linear equations in which p and qare constants:

8<: x¡ 2y + 3z = 1x + py + 2z = 0

¡2x + p2y ¡ 4z = q.

a Write this system of equations in augmented matrix form.

b Show, using clearly defined row operations, that

this augmented matrix can be reduced to:

Ã1 ¡2 30 p+ 2 ¡10 0 p

¯̄̄̄¯ 1

¡1p+ q

!.

c What values can p and q take when the system has

i a unique solution ii no solutions iii infinite solutions?

d Specify the infinite solutions in parametric form.

163 a Show that the plane 2x + y + z = 5 contains the line l1: x = ¡2t + 2, y = t,z = 3t + 1, t 2 R .

b For what values of k does the plane x + ky + z = 3 contain l1?

c Without using row operations, find the values of p and qfor which the following system of equations has an infinite

number of solutions. Clearly explain your reasoning.

8<:2x + y + z = 5x¡ y + z = 32x + py + 2z = q

d Check your result using row operations.

164 For A =

Ã2 1 ¡1¡1 2 10 6 1

!and B =

Ã4 7 ¡3¡1 ¡2 16 12 ¡5

!, calculate AB and

hence solve the system of equations

8<: 4a + 7b¡ 3c = ¡8¡a¡ 2b + c = 36a + 12b¡ 5c = ¡15.

165 Use vector methods to prove that joining the midpoints of the sides of a rhombus gives

a rectangle.

166 a Given a = i + j ¡ 3k and b = j + 2k, find a £ b.

b Find a vector of length 5 units which is perpendicular to both a and b.

167 Let r = 2i ¡ 2j + k, s = 3i + j + 2k and t = i + 2j ¡ k be the position

vectors of the points R, S, and T respectively. Find the area of the triangle RST.


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168 In the given figure, ABCD is a parallelogram.

X is the midpoint of [BC], and Y is on [AX]

such that AY : YX = 2 : 1. The coordinates

of A, B and C are (1, 3, ¡4), (4, 4, ¡2) and

(10, 2, 0) respectively.

a Find the coordinates of D, X and Y.

b Prove that B, D and Y are collinear.

169 Let a = 3i + 2j ¡ k, b = i + j ¡ k and c = 2i ¡ j + k.

a Show that b £ c = ¡3j – 3k.

b Verify for the given vectors that a £ (b £ c) = b (a ² c) ¡ c (a ² b).

170 Given the vectors p =

Ã12¡2

!and q =

Ã¡t

1 + t2t

!, find t such that:

a p and q are perpendicular b p and q are parallel.

171 Suppose A and B are events such P(A) = 0:3 + x, P(B) = 0:2 + x and

P(A\B) = x.

a Find x if A and B are mutually exclusive events.

b Calculate the possible values of x if A and B are independent events.

172 Find exact solutions for the following:

a j 1 ¡ 4x j > 13 j 2x¡ 1 j b

x¡ 2

6 ¡ 5x¡ x26 0

173 The average number of amoebas in 50 mL of pond water is 20.

a Assuming that the number of amoebas in pond water follows a Poisson distribution,

find the probability that no more than 5 amoebas are present in 10 mL of randomly

sampled pond water.

b If a researcher collected 10 mL of pond water each weekday over 4 weeks (20 days

in all), find the probability that the researcher collected no more than 5 amoebas on

more than 10 occasions in that 4 week period.

174 Solvedy

dx= cos2 x given that y(0) = 4.

175 Solve the differential equation xydy

dx= 1 + y2 given that y = 0 when x = 2.

176 A current of I amperes flows through a coil of inductance L henrys and resistance

R ohms with electromotive force E = LdI

dt+ RI volts.

Assuming that E , L and R are constants, show by separating the variables that

I =E

R

³1 ¡ e¡

RLt´

, given that I = 0 when t = 0.

177 A pair of guinea pigs was released onto an island in early January. Infrared scans of the

island in early May showed the guinea pig population to be 180. Given that the rate of

increase in such a population is proportional to the population at that time, estimate the

island’s guinea pig population in early October.

A

B

C

D

Y

X


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178 a Show algebraically that1

y+

1

P ¡ y=

P

y(P ¡ y).

b Using part a, solve the differential equation y0 = k y³1 ¡ y

P

´given that P = 624, y = 2 when t = 0, and that y = 12 when t = 1.

c The differential equation above describes how a rumour is spread at Beijing College

by 2 people starting at 12 noon. 12 people have heard the rumour by 1 pm.

i Find the number of people at Beijing College, giving a reason for your answer.

ii How many people have heard the rumour by 2 pm?

iii At what time have 90% of the people heard the rumour?

179 a Express 1 + i andp

3 ¡ i in the form reiµ.

Hence write z =¡1 ¡ ip

3 ¡ iin the form re iµ.

b What is the smallest positive integer n such that zn is a real number?

180 There are 12 students in a school’s Hungarian class. Being well-mannered, they line up

in a single file to enter the class.

a How many orders are possible?

b How many orders are there if:

i Irena and Eva are among the last four in the line

ii Istvan is between Paul and Laszlo and they are all together

iii Istvan is between Paul and Laszlo but they are not necessarily together

iv there are exactly three students between Annabelle and Holly?

c Once inside, the class is split into 3 groups of four students each for a vocabulary

quiz. How many ways can this be done:

i if there are no restrictions

ii if Ben and Marton must be in the same group?

181 The velocity of a particle travelling in a straight line is given by v = cos(13 t) cm s¡1.

Find the distance travelled by this particle in the first 10¼ seconds of motion.

182 Year 12 students at a government school can choose from 16 subjects for their Certificate.

Seven of these subjects are in group I, six are in group II, and the other three are in

group III. Students must study six subjects to qualify for the Certificate. How many

combinations of subjects are possible if:

a there are no restrictions

b students must choose 2 subjects from groups I and II and the remaining subjects

could be from any group

c French (a group I subject) is compulsory, and they must choose at least one subject

from group III?

183 Solve the equation:¡n3

¢= 3

¡n¡12

¢¡ ¡n¡11

¢.

184 Find the coefficient of:

a x12 in the expansion of

µ2x3 ¡ 1

2x

¶8

b x2 in the expansion of (1+2x)5(2¡x)6

c x3 in the expansion of (1 + 2x¡ 3x2)4:


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185 The function f is defined by f : x 7! esin2 x , x 2 [ 0,¼ ].

a Use calculus to find the exact value(s) of x for which f (x) has a maximum value.

b Find f 00 (x) and write down an equation that will enable you to find any points of

c Find the point(s) of

186 Solve for x: logx 4 + log2 x = 3

187 Find the exact value of a if a > 0 and

Z a

0

x

x2 + 1dx = 3.

188 Given that A is an acute angle and tan 2A = 32 , find the exact value of tanA.

189 The scores a, b, 6, 13 and 7 where b > a have a mean and variance of 8. Find the

values of a and b.

190 The graph of y = f(x) for ¡9 6 x 6 9is shown alongside.

The function has vertical asymptotes at

x = 2 and x = ¡3 and a horizontal

asymptote at y = 2.

Copy and sketch the graph of

y =1

f(x), indicating clearly the axes

intercepts and all asymptotes.

191 For what values of x is the matrix A =

Ãx¡ 1 ¡2 5¡4 3¡ x ¡2¡2 5 ¡8

!singular?

192 Find a and b if the matrix A =³a ¡1b 2

´is its own inverse. Hence find A11.

193 If z = x + 2i and u = 3 + iy where x, y 2 R , find the smallest positive value of x

for whichz + u

z ¡ uis purely imaginary.

194 If 1 ¡ 2i is a zero of P (x) = x4 + 11x2 ¡ 10x + 50, find all the other zeros.

195 Find the exact value of the volume of the solid formed when the region enclosed by

y = xex3

, the x-axis, and the line x = 1, is rotated through 360o about the x-axis.

196 Determine the sequence of transformations which transform the function

f(x) = 3x2 ¡ 12x + 5 to g(x) = ¡3x2 + 18x¡ 10.

197 Find the area of the region bounded by the curve y = tan2 x+ 2 sin2 x, the x-axis, and

the line x = ¼4 .

198 Simplify sin(2 arcsinx) and hence find

Z 1

0

sin(2 arcsinx) dx.

inflection in the given domain.

inflection in the given domain.

��

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y x��( )


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199 For the system of linear equations:

8<:2x¡ y + 3z = 4

2x + y + (a + 3)z = 10 ¡ a

4x + 6y + (a2 + 6)z = a2,

find the value(s) of a for which the system has:

a no solutions

b infinitely many solutions, and find the form of these solutions

c a unique solution, and find the solution in the case where a = 2:

200 A function f is defined by f(x) =x2 + 1

(x + 1)2.

a Write down the equations of the asymptotes of the graph of y = f(x):

b Find f 0(x) and hence find the position and nature of any stationary points.

c Find f 00(x) and hence find the coordinates of all

d Sketch the graph of y = f(x) showing all the above features.

201 A particle moves in a straight line such that its displacement from point O is s.

The acceleration of the particle is a and its velocity is v where a = 12v

2.

a Find v(t) given that v(0) = ¡1.

b Find the distance travelled in the first 2 seconds of motion.

202 Determine the domain of f(x) = arccos(1 + x¡ x2), and find f 0(x).

203 Find the area of the region enclosed by the graph of y =tanx

cos(2x) + 1, the x-axis, and

the line x = ¼3 .

204 Find the equations of all asymptotes of the graph of the function y =tanx

sin(2x) + 1where ¡¼ 6 x 6 ¼.

205 Find the exact coordinates of the stationary points on the curve y =sinx

tanx + 1where ¡¼ 6 x 6 ¼

2 .

206 The sum of an infinite geometric series is 49 and the second term of the series is 10.

Find the possible values for the sum of the first three terms of the series.

207 If f : x 7! 2x + 1 and g : x 7! x + 1

x¡ 2, find: a (f ± g)(x) b g¡1(x).

208 a Finddy

dxif x2 ¡ 3xy + y2 = 7:

b Hence find the coordinates of all points on the curve for which the gradient is 23 .

209 A and B are two events such that P(A) = 13 and P(B) = 2

7 .

a Find P(A [B) if A and B are: i mutually exclusive ii independent.

b Find P(A j B) if P(A [B) = 37 .

points of inflection.


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210 a Find the coordinates of A, the point of intersection of l1 and l2, where l1 is given

by r =

Ã8

¡13¡3

!+ ¸

Ã3¡5¡2

!and l2 is given by

x + 10

6=

y ¡ 7

¡5=

z ¡ 11

¡5.

b Find the coordinates of B, where l1 meets the plane 3x + 2y ¡ z = ¡2.

c The point C(p, 0, q) lies on the plane in b.

Find the possible values of p if the area of triangle ABC isp32 units2.

211 Find x in terms of a if a > 1 and loga(x + 2) = loga x + 2.

212 Solve for y: (x2 + 1)dy

dx= y + 1 given that y = 2 when x = 0.

213 FindRx2 sinxdx.

214 If f(2x + 3) = 5x¡ 7, find f¡1(x).

215 Find, to 3 significant figures, the area of the region enclosed by the graphs of

y = xesinx and y = x2 ¡ 4x + 6:

216 Finddy

dxif exy + xy2 ¡ sin y = 2.

217 Solve for x:j 2x¡ 1 j +3

j x + 3 j ¡2< ¡x

218 Find

Zx

1 +px + 2

dx.

219 Finddy

dxif sin(xy) + y2 = x.

220 The lines r =

Ã3¡22

!+ ¸

Ãa¡12

!and

x¡ 4

2= 1 ¡ y =

z + 2

3intersect at point P.

a Find the value of a and hence find the coordinates of P.

b Find the acute angle between the two lines.

c Find the equation of the plane which contains the two lines.

221 For what values of a does the graph of y = ax+2 cut the graph of y = 3x2¡2x+5 in

two distinct points?

222 The height of a cone is always twice the radius of its base. The volume of the cone is

increasing at a constant rate of 5 cm3 s¡1. Find the rate of change in the radius when

the height is 20 cm.

223 Find the value of a if the line passing through the points A(0, 5, 6) and B(4, 1, ¡2)

and the line r =

Ãa32

!+ s

Ã2¡11

!are coplanar.


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224 Let f(x) = x tanp

1 ¡ x2, ¡1 6 x 6 1.

a Sketch the graph of y = f(x).

b Write down an expression for V , the volume of the solid formed by rotating the

region bounded by y = f(x), x = 0 and x = 1 through 360o about the x-axis.

225 Let f(x) = xe1¡2x2

.

a Find f 0(x) and f 00(x).

b Find the exact coordinates of the stationary points of the function and determine

their nature.

c Find the exact values of the x-cordinates of the points of inflexion of the function.

d Discuss the behaviour of the function as x ! §1:

e Sketch the graph of the function.

f Find the exact value of k if k > 0 and the region bounded by the y = f (x), the

y-axis, and the line x = k has area equal to 14(e¡ 1) units2.

226 a Find a and k if the line l1 given by r =

Ã1¡12

!+¸

Ã1a¡1

!lies on the plane P1

with equation 3x¡ ky + z = 3.

b Show that the plane P2 with equation 2x¡ y ¡ 4z = 9 is perpendicular to P1 .

c Find the equation of l2, the line of intersection of P1 and P2 .

d Find the point of intersection of l1 and l2 .

e Find the angle between the lines l1 and l2 .

227 Construct a quartic polynomial f(x) with integer coefficients such that f(x) < 0 for

all x 2 R . Write your polynomial in expanded form.

228 P (z) = z3 + az2 + bz + c where a, b and c 2 R .

Two of the roots of P (z) are ¡2 and ¡3 + 2i.

Find a, b and c and also find possible values of z when P (z) > 0.

229 f(x) = 2 tan(3(x¡ 1)) + 4 for x 2 [¡1, 1 ]. Find:

a the period of y = f(x)

b the equations of any asymptotes

c the transformations that transform y = tanx into y = f(x).

d the domain and range of y = f(x).


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30. miscellaneous questions

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