P1: FXS/ABE P2: FXS CHAPTER 7 - Cambridge University · PDF fileP1: FXS/ABE P2: FXS 9780521740494c07.xml CUAU033-EVANS September 9, 2008 7:8 CHAPTER Revision 7 Revision of chapters

P1: FXS/ABE P2: FXS

9780521740494c07.xml CUAU033-EVANS September 9, 2008 7:8

C H A P T E R

Revision7

Revision of chapters 1–67.1 Multiple-choice questions

1 If P2 = 4I, then P−1 equals

A1

4P B

1

2P C

1

2I D 2P E 4P

2 If R = [5 3 1] and S =

0

−1

2

, then RS is

A not defined B [−1] C

0 0 0

−5 −3 −1

10 6 2

D [0 −3 2] E

0

−3

2

3 If A =[

9 8

−11 5

], then det (A) equals

A −43 B − 1

43C

1

133D 17 E 133

4 If A =

1

2

5

and B = [−2 6 4], then BA has dimensions

A 1 × 1 B 3 × 1 C 1 × 3 D 3 × 3 E 3 × 2

5 Given that A =[

5 2

2 1

], B =

[2 −1

6 7

]and C =

[5 4

8 9

], then if AX + B = C,

X equals

A1

20

[−2 19

−2 6

]B

[−1 1

4 0

]C

[−2 19

−2 6

]

D

[3 −10

−4 10

]E

1

20

[1 3

4 5

]

177

SAMPLE

Cambridge University Press • Uncorrected Sample Pages • 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard

P1: FXS/ABE P2: FXS


Rev

isio

n178 Essential Advanced General Mathematics

6 Let P =[

2 −1

3 2

], Q =

[4 2

6 5

], R =

[2 1

−3 2

]and X = PQR. The number of zero

elements in X is

A 0 B 1 C 2 D 3 E 4

7 If X =[

3 5

−1 −2

], then X−1 is

A

[2 5

−1 −3

]B

[−2 −5

1 3

]C

1

3

1

5

−1 −1

2

D

[−3 −1

5 2

]E

[3 −1

−5 −2

]

8 The determinant of the matrix

[4 6

2 4

]is

A 16 B 4 C −16 D1

4E −4

9 If S =[

5 7

2 2

], then S−1 is

A −[

5 7

2 2

]B

[5 −7

−2 5

]C −1

4

[−2 7

2 −5

]

D1

4

[−2 7

2 −5

]E

1

4

[−2 −7

−2 −5

]

10 In algebraic form, five is seven less than three times one more than x can be written as

A 5 = 7 − 3(x + 1) B 3x + 1 = 5 − 7 C (x + 1) − 7 = 5

D 5 = 7 − 3x + 1 E 5 = 3x − 4

113

x − 3− 2

x + 3is equal to

A 1 Bx + 15

x2 − 9C

15

x − 9D

x − 3

x2 − 9E −1

6

12 p varies directly as x and inversely as the square of y. If x is decreased by 30% and y is

decreased by 20%, the percentage change in p is best approximated by

A increase by 10% B decrease by 10% C increase by 9.4%

D decrease by 9.4% E no change

13 The sum of the odd numbers from 1 to n inclusive is 100. The value of n is

A 13 B 15 C 17 D 19 E 21

14 If the sum of the first n terms of a geometric sequence is 2n+1 − 2 , the nth term of the

geometric sequence is

A 2n−1 B 2n C 2n − 1 D 2n−1 + 1 E 2n + 1

SAMPLE


P1: FXS/ABE P2: FXS


Revisio

nChapter 7 — Revision of chapters 1–6 179

15 If m ∝ n and m = 9 when n = 4, then k, the constant of variation, equals

A9

4B 13 C 36 D

4

9E 5

16 If A = {1, 2, 3, 4}, B = {2, 3, 4, 5, 6} and C = {3, 4, 5, 6, 7} then A ∩ (B ∪ C) is equal to

A {1, 2, 3, 4, 5, 6, 7} B {1, 2, 3, 4, 5, 6} C {2, 3, 4}D {3, 4} E {2, 3, 4, 5, 6, 7}

17 The price of painting the outside of a cylindrical tank (the bottom and top are not painted)

of radius r and height h varies directly as the total surface area. If r = 5 and h = 4, the

price is $60. The price when r = 4 and h = 6 is

A $45 B $57.60 C $53.50 D $62.80 E $72

18 If x ∝ y and x = 8 when y = 2, the value of x when y = 7 is

A 20 B 13 C 11 D 28 E 1.75

19 The recurring decimal 0.7̇2̇ is equal to

A72

101B

72

100C

72

99D

72

90E

73

90

20 If x varies directly as y2 and inversely as z, the percentage increase of x when y is

increased by 25% and z is decreased by 20% is best approximated by

A 5% B 50% C 85% D 95% E 100%

21−4

x − 1− 3

1 − x+ x

x − 1is equal to

A 1 B −1 C7x

x − 1D

1

1 − xE none of these

22x + 2

3− 5

6is equal to

Ax − 3

6B

2x + 4

6C

2x − 1

6D

2x − 5

6E

x − 3

3

23 If a = 1 + 1

1 + b, then b equals

A 1 − 1

a − 1B 1 + 1

a − 1C

1

a − 1− 1 D

1

a + 1+ 1 E

1

a + 1− 1

24 When the repeating decimal 0.3̇6̇ is written in simplest fractional form, then the sum of

the numerator and denominator is

A 15 B 45 C 114 D 135 E 150

25 If2x − y

2x + y= 3

4, then

x

yequals

A2

7B

7

2C

3

4D

4

3E Not possible unless the values of x and y are known

26 The sum to infinity of the series1

2− 1

4+ 1

8− 1

16+ · · · is

A 2 B 1 C1

2D

1

3E

2

3

SAMPLE


P1: FXS/ABE P2: FXS


Rev

isio


27 If x varies directly as y and inversely as the square of z and x = 10 when y = 4 and

z = 14, then when y = 16 and z = 7, x equals

A 180 B 160 C 154 D 140 E 120

28 If3

3 + y= 4, then y equals

A1

4B −9

4C

9

4D 0 E −4

9

29 The coordinates of the point where the lines with equations 3x + y = −7 and

2x + 5y = 4 intersect are

A (3, −16) B (−3, 2) C (3, −2) D (−2, 3) E no solution

30 Ifm + 2

4− 2 − m

4= 1

2then m is equal to

A 1 B −1 C1

2D 0 E −1

2

31 46 200 can be written as

A 2 × 3 × 5 × 7 × 11 B 22 × 32 × 52 × 7 × 11

C 2 × 32 × 5 × 72 × 11 D 23 × 3 × 52 × 7 × 11

E 22 × 3 × 53 × 7 × 11

32 Three numbers, y, y − 1 and 2y − 1, are consecutive numbers of an arithmetic sequence.

y equals

A −1 B 1 C 0 D 2 E −2

33 If the integers n + 1, n − 1, n − 6, n − 5, n + 4 are arranged in increasing order of

magnitude then the middle number is

A n + 1 B n − 1 C n − 6 D n − 5 E n + 4

34 If x ∝ 1

y, and y is multiplied by 5, then x will be

A decreased by 5 B increased by 5 C multiplied by 5

D divided by 5 E none of these

35 An arithmetic sequence has 3 as its first term and 9 as its fourth term. The eleventh term is

A 23 B 11 C 63 D 21 E none of these

36 The expression4

n + 1+ 3

n − 1is equal to

A7n − 1

1 − n2B

1 − 7n

1 − n2C

7n − 1

n2 + 1D

7

n2 − 1E

7

n

37 If the second number is twice the first number and a third number is half the first number

and the three numbers sum to 28, then the numbers are

A (8, 16, 4) B (2, 3, 12) C (7, 9, 11) D (6, 8, 16) E (12, 14, 2)

38 (√

7 + 3)(√

7 − 3) is equal to

A −2 B 10 C√

14 − 19 D 2√

7 − 9 E 45

SAMPLE


P1: FXS/ABE P2: FXS


Revisio


39 If13x − 10

2x2 − 9x + 4= P

x − 4+ Q

2x − 1then the values of P and Q are

A P = 1 and Q = 1 B P = −1 and Q = 1 C P = 6 and Q = 1

D P = −6 and Q = 1 E P = 1 and Q = −6

40 The first term of a geometric sequence is a and the infinite sum of the geometric sequence

is 4a. The common ratio of the geometric sequence is

A 3 B 4 C3

4D −3

4E −4

3

41 If5x

(x + 2)(x − 3)= P

x + 2+ Q

x − 3, then

A P = 2, and Q = 3 B P = 2, and Q = −3 C P = −2, and Q = 3

D P = −2, and Q = −3 E P = 1, and Q = 1

42 If n is a perfect square then the next largest perfect square greater than n is

A n + 1 B n2 + 1 C n2 + 2n + 1 D n2 + n E n + 2√

n + 1

43 The area of triangle varies directly as the base length provided the altitude is constant. If

the area equals 14 when the base is 2.4, then the base length (correct to three decimal

places) when the area is 18 will equal

A 3.086 B 5.000 C 6.400 D 9.600 E 0.324

44 Which of the following is not a rational number?

A 0.4 B3

8C

√5 D

√16 E 4.125

45 If1

x= a

band

1

y= a − b, then x + y equals

A2

aB

a2 − b2

aC

ba − b2 + a

a(a − b)D

2a

a2 − b2E

−2b

a2 − b2

46 9x2 − 4mx + 4 is a perfect square when m equals

A 5 B ±12 C 2 D ±1 E ±3

47 If x = (n + 1)(n + 2)(n + 3) where n is a positive integer, then x is not always divisible

by

A 1 B 2 C 3 D 5 E 6

48 The numbers −4, a, b, c, d, e, f, 10 are consecutive terms of an arithmetic sequence. The

sum a + b + c + d + e + f is equal to

A 6 B 10 C 18 D 24 E 48

49 If n and p are both odd numbers, which one of the following numbers must be an even

number?

A n + p B np C np + 2 D n + p + 1 E 2n + p

SAMPLE


P1: FXS/ABE P2: FXS


Rev

isio


7.2 Extended-response questions1 The diagram represents a glass containing milk. When the height

of the milk in the glass is h cm, the diameter, d cm, of the surface

of the milk is given by the formula

d = h

5+ 6

a Find d when h = 10 b Find d when h = 8.5

c What is the diameter of the bottom of the glass?

d The diameter of the top of the glass is 9 cm. What is the

height of the glass?

d cm

h cm

2 The formula A = 180 − 360

ngives the size of each interior angle, A◦, of a regular polygon

with n sides.

a Find the value of A when n equals

i 180 ii 360 iii 720 iv 7200

b As n becomes very large

i what value does A approach? ii what shape does the polygon approach?

c Find the value of n when A = 162. d Make n the subject of the formula.

e Three regular polygons, two of which are octagons, meet at a point so that they fit

together without any gaps. Describe the third polygon.

3 The figure shows a solid consisting of three parts, a cone, a cylinder

and a hemisphere, all of the same base radius.

a Find in terms of w, s, t and � the volume of each part.

b i If the volume of each of the three parts is the same,

find the ratio w : s : t .

ii If also w + s + t = 11, find the total volume in terms of �.

w

s

t

4 The cost, $C, of manufacturing each jacket of a particular type is

given by the formula

C = an + b for 0 < n ≤ 300

where a and b are constants and n is the size of the production run of this type of jacket.

For making 100 jackets, the cost is $108 each.

For making 120 jackets, the cost is $100 each.

a Find the values of a and b.

b Sketch the graph of C against n for 0 < n ≤ 300.

c Find the cost of manufacturing each jacket if 200 jackets are made.

d If the cost of manufacturing each jacket is $48.80, find the size of the production run.

SAMPLE


P1: FXS/ABE P2: FXS


Revisio


5 a In the diagram, OPQ is a sector of radius R.

A circle, centre C1 and radius r1 is inscribed

in this sector.

RQO

P

60°

r3

r2

r1

r1r2r3

C3

C2

C1

i Express OC1 in terms of R and r1.

ii Show thatr1

OC1= 1

2and hence express r1 in terms of R.

b Another circle, centre C2, is inscribed in the sector as shown.

i Express OC2 in terms of r2 and R.

ii Express r2 in terms of R.

c Circles C3, C4, . . . are constructed in a similar way. Their radii are r3, r4, . . .

respectively. It is known that r1, r2, r3, . . . is a geometric sequence.

i Find the common ratio. ii Find rn .

iii Find the sum to infinity of the sequence, and interpret the result geometrically.

iv Find in terms of R and �, the sum to infinity of the areas of the circles with radii r1,

r2, r3, . . . .

6 At the beginning of 1997, Andrew and John bought a small catering business. The profit,

$P, in a particular year is given by

P = an + b

where n is the number of years of operation and a and b are constants.

a Given the table, find the values of a and b.

Year 1997 2001

Number of years of operation (n) 1 5

Profit −9000 15 000

b Find the profit when n = 12. c In which year was the profit $45 000?

7 Two companies produce the same chemical. For Company A the number of tonnes

produced increases by 80 tonnes per month. For Company B production increases by 4%

per month. Each company produced 1000 tonnes in January 2003. (Let n be the number of

months of production. Use n = 1 for January 2003.)

a Find, to the nearest tonne where appropriate,

i the production of Company A in the nth month

ii the production of each company in December 2004 (i.e. for n = 24)

(cont’d)

SAMPLE


P1: FXS/ABE P2: FXS


Rev

isio


iii the total production of Company A over n months (starting with n = 1 for January

2003)

iv the total production of each company for the period January 2003 to December

2004 inclusive.

b Find in which month of which year the total production of Company A passed 100 000

tonnes.

8 The square shown has each side of length one.

a The perimeter of the square is denoted by P1.

What is the value of P1 ?1

1

1

1

b A new figure is formed by joining two squares of

side length1

2to this square, as shown. The

perimeter is denoted by P2. What is the value of P2?1

1

1

1

21

2

1

21

21

2

1

2c What is the perimeter, P3, of this figure? 1

1

11

2

1

2

1

2

1

2

1

2

1

4

1

4

1

4 4

1

4

1

4

1

4

1

4

1

4

d It is known that P1, P2, P3, . . . are the terms of an arithmetic sequence with first term

P1. What is the common difference ?

e i Find P4. ii Find Pn in terms of Pn−1. iii Find Pn in terms of n.

iv Draw the diagram of the figure corresponding to P4.

9 A piece of wire 28 cm long is cut into

two parts, one to make a rectangle three

times as long as it is wide and the other

to make a square.

3x cm

x cm

a What is the perimeter of the rectangle in terms of x?

b What is the perimeter of the square in terms of x?

c What is the length of each side of the square in terms of x?

Let A be the sum of the areas of the two figures.

d Show that A = 7(x2 − 4x + 7)

e Use a graphics calculator to help sketch the graph of

A = 7(x2 − 4x + 7) for 0 < x < 5

f Find the minimum value that A can take and the corresponding value of x.

SAMPLE


P1: FXS/ABE P2: FXS


Revisio


10 A particular plastic plate manufactured at a factory sells at $1.50. The cost of production

consists of an initial cost of $3500 and then $0.50 per plate. Let x be the number of plates

produced.

a Let $C be the cost of production of x plates. Write an expression for C in terms of x.

b Let $I be the income from selling x plates. Write an expression for I in terms of x.

c On the one set of axes, sketch the graphs of I against x and C against x.

d How many plates must be sold for the income to equal the cost of production?

e How many plates must be sold for a profit of $2000 to be made?

f Let P = I − C . Sketch the graph of P against x. What does P represent?

11 n is a natural number less than 50 such that n + 25 is a perfect square.

a Show that there exists an integer a such that

n = a(a + 10)

b Any natural number less than 100 can be written in the form 10p + q where p and q are

digits. For this representation of n show that q = p2.

c Give all possible values of n.

12 a i For the equation√

7x − 5 − √2x = √

15 − 7x square both sides to show that this

equation implies

8x − 10 =√

14x2 − 10x

ii Square both sides of this equation and simplify to form the equation

x2 − 3x + 2 = 0 1

iii The solutions to the equation 1 are x = 1 or x = 2.

Test these solutions for the equation√

7x − 5 −√

2x = √15 − 7x

and hence show that x = 2 is the only solution for the original equation.

b Use the techniques of a to solve the equations

i√

x + 2 − 2√

x = √x + 1 ii 2

√x + 1 + √

x − 1 = 3√

x

13 A geometric series is defined by

x + 1

x2− 1

x+ 1

x + 1− · · ·

a Let r be the common ratio. Find r in terms of x.

b i Find the infinite sum if x = 1.

ii Find the infinite sum if x = −1

4.

iii Find the infinite sum if x = 2.

c Find the possible values of x for which the infinite sum is defined.

SAMPLE


P1: FXS/ABE P2: FXS


Rev

isio


14 a The area, A, of the shaded region varies directly

as the cube of a.

i If A = 4

3when a = 2, find an expression

for A in terms of a.

ii Find A when a = 3.

iii If A = 4500, find a. 0x

y = x(x – a)

y

b The area, A1, of the shaded region varies directly

as the cube of a

i If A1 = 1152 when a = 24, find an

expression for A1 in terms of a.

ii Find A1 when a = 18.

iii Find a when A1 = 3888.x

y

y = x(x – a)

–a2

4a2

,

0

c The area, A2, of the shaded region varies partly as

the reciprocal of a and partly as the reciprocal of b.

x

y

y =1

x2

a b

i Find A2 in terms of a and b if,

when a = 1 and b = 2, A2 = 1

2and

when a = 3 and b = 4, A2 = 1

12ii Find A2 when a = 1 and b = 6.

iii Find A2 when a = 1

4and b = 3.

iv Find A2 when a = 1

100and b = 100.

v Find A2 when a = 1

1000and b = 1000.

15 In a vegetable garden carrots are planted in rows parallel to the fence.

Fence

row 1row 2

row 3row 4

rabbit burrow

0.5 m 1.5 m 1.5 m 1.5 m

a Calculate the distance between the fence and the 10th row of carrots.

b If tn represents the distance between the fence and the nth row, find a formula for tn in

terms of n.

SAMPLE


P1: FXS/ABE P2: FXS


Revisio


c Given that the last row of carrots is less than 80 m from the fence, what is the largest

number of rows possible in this vegetable garden?

d A systematic rabbit has its burrow under

the fence as shown in the diagram. It runs

to the first row, takes a carrot and returns

it to its burrow. It then runs to the

second row, collects a carrot and

returns it to its burrow.

It continues in this way until it has

15 carrots. Calculate the shortest distance the rabbit has to run to accomplish this.

Fence

rabbit burrow Trip 1

Trip 2

row 1row 2

16 The potential energy, P joules, of a body varies jointly as the mass, m kg, of the body and

the height, h m, of the body above the ground.

a For a body of mass 5 kg

i find P in terms of h if P = 980 when h = 20

ii sketch the graph of P against h iii find P if h = 23.2.

b i Find P in terms of h and m if P = 980 when h = 20 and m = 5.

ii Find the percentage change in potential energy if the height (h m) is doubled and the

mass remains constant.

iii Find the percentage change in potential energy if a body has a quarter of the

original height (h m) and double the original mass (m kg).

c If a body is dropped from a height, h m, above ground level its speed, V m/s, when it

reaches the ground is given by V = √19.6h.

i Find V when h = 10. ii Find V when h = 90.

d In order to double the speed a given body has when it hits the ground, by what factor

must the height from which it is dropped be increased?

17 In its first month of operation a soft drink manufacturer produces 50 000 litres of a type of

soft drink. In each successive month the production rises 5000 litres a month.a i The quantity of soft drink, tn , produced in the nth month can be determined from a

rule of the form

tn = a + (n − 1)d

Find the values of a and d.

ii In which month will the factory double its original production?

iii How many litres in total will be produced in the first 36 months of operation?

b Another soft drink manufacturer sets up a factory at the same time as the first. In the

first month the production is 12 000 litres. The production of this factory increases by

10% every month.

i Find a rule for qn , the quantity of soft drink produced in the nth month.

ii Find the total amount of soft drink produced in the first 12 months.

c If the two factories start production in the same month, in which month will the

production of the second factory exceed the production of the first factory?

SAMPLE


P1: FXS/ABE P2: FXS


Rev

isio


18 In a certain country grain production and population statistics are produced.In December 1986 the population of the country was 12.5 million.

In 1986 the grain production was 10 million tonnes.

It was found that since then the population has grown by 5% each year and grain

production has increased by 0.9 million tonnes each year.

Let P1 denote the population in December 1986.

Let p2 denote the population in December 1987.

Pn denotes the population n − 1 years after December 1986.

Let t1 denote the grain production in 1986.

Let t2 denote the grain production in 1987.

tn denotes the grain production in the (n − 1)th year after 1986.

a Find, in millions of tonnes, the grain production in

i 1992 ii 1999

b Find an expression for tn .

c Find the total grain production for the 20 years starting 1986.

d How many years does it take for the grain production to double?

e Find an expression for Pn .

f How many years does it take for the population to double?

19 The diagram shows a straight road OD where

OD = 6 km. A hiker is at A, 2 km from O.

2 km

x kmO

A

X D

6 km

The hiker can walk at 3 km/h when off-road

but at 8 km/h along the road.

a Calculate the time taken, in hours and

minutes, correct to the nearest minute,

if he hikes directly to X then along the

road to D where OX = 3 km.

b Calculate OX, correct to one decimal place (in km) if the total time taken was

11

2hours.

20 Seventy-six photographers submitted work for a photographic exhibition in which they

were permitted to enter not more than one photograph in each of the three classes, black

and white (B), colour prints (C), transparencies (T ). Eighteen entrants had all their work

rejected while 30 B, 30 T and 20 C were accepted.

From the exhibitors, as many showed T only as showed T and C.

There were three times as many exhibitors showing B only as showing C only.

Four exhibitors showed B and T but not C.

a Write the last three sentences in symbolic form.

b Draw a Venn diagram representing the information.

c Find

i n(B ∩ C ∩ T ) ii n(B ∩ C ∩ T ′)

SAMPLE


P1: FXS/ABE P2: FXS


Revisio


21 Let A =[

a b

c d

]with b �= 0 and c �= 0

a Find

i A2 ii 3A

b If A2 = 3A − I, show that

i a + d = 3 ii det(A) = 1

c If A has the properties� a + d = 3� det(A) = 1

show that A2 = 3A − I.

22 The trace of square matrix A is defined to be the sum of the leading diagonal of A, and it is

denoted by Tr(A).

For example, if A =[

6 −3

2 2

], Tr(A) = 8

a Prove each of the following for any 2 × 2 matrices X and Y.

i Tr(X + Y) = Tr(X) + Tr(Y)

ii Tr(−X) = −Tr(X) iii Tr(XY) = Tr(YX)

b Use the results of a to show that there do not exist 2 × 2 matrices X and Y such that

XY − YX = I.

SAMPLE


Documents

P1: FXS/ABE P2: FXS CHAPTER 7 - Cambridge University · PDF fileP1: FXS/ABE P2: FXS 9780521740494c07.xml CUAU033-EVANS September 9, 2008 7:8 CHAPTER Revision 7 Revision of chapters