JUNE 1988

JUNE 1988PAPER 2SECTION 1

Answer ALL the questions in this section

1. You must not use slide rules, tables or calculators to work your answers to this question. All steps and all calculations must be clearly shown to earn credit for your solution.

a. Calculate, correct to 2 decimal places,

(i) 0.03 x 1.3

(ii) 6(3 1.47)

(3 marks)

(b) Calculate the exact value of 3.12

0.3 x 0.13

(3 marks)

(c) A woman bought a stove for $2800. After using it for 2

years she decided to trade in the stove. The company estimated

a depreciation of 15% for the first year of its use and a further

15% on its reduced value, for the second year.

(i) Calculate the value of the stove after the two years.

(ii) Express the value of the stove after two years as a

Percentage of the original value.

(6 marks)

2. Given that:

U is the set of persons in Tobago

C is the set of persons in Tobago who like calypso

S is the set of persons in Tobago who like steelband

P is the set of persons in Tobago who like pop-music

All persons who like calypso also like steelband but do

not like pop-music.

Some persons who like steelband also like pop-music.

(a) Draw a carefully labeled Venn diagram to illustrate this data.

(b) Given that n(U) = 40000, n(S) = 25000, n(C) = 10000 and n(P) = 8000.

Determine n(S ( C) and n(C ( P).

( 2 marks)

3. The co-ordinate of A and B are (3,5) and (7,1) respectively. X

is the mid-point of AB

(a) Calculate

(i) the length of AB

(ii) the gradient of AB

(iii) the co-ordinates of X.

(4 marks)

(b) Determine the equation of the perpendicular bisector of

AB and state the co-ordinates of the point at which the

Perpendicular bisector meets the y-axis.

(6 marks)

4. (a) Simplify

(i) 2(5x - y) 3(3x y)

(ii) 811/2 x 271/3

(6 marks)

b. Calculate correct to one decimal place the values of x for which

2x2 + 2x 8 = 3x 6

(6 marks)

5. (a) Given that

A = 1 0

B = p q

0 3 0 r

and that AB = A + 2B,

calculate the values of p, q and r.

(7 marks)

(b)

In the diagram above, not drawn to scale, PQ and LM

Represent parallel edges of an east-west river bank.

Angle LPR = 55 and angle RQM = 25.

Given that PQ = 10 km, calculate

(i) the acute angle PRQ

(ii) the distance RQ

(iii) the width of the river.

(7 marks)

6. The scores obtained by 100 applicants on an aptitude test for

selection into a programme are shown in the frequency table

below.

Score

FrequencyCumulative Frequency

0 9

10 - 19

20 29

30 39

40 49

50 59

60 69

70 798

13

17

20

19

14

6

3

(a) Copy in your answer booklet the table above and

complete the cumulative frequency column.

(c) Draw the cumulative frequency curve using 2 cm to

represent each class interval and 2 cm to represent 10

applicants.

(d) Using your curve, answer the following:

(i) A score of 45 was considered as acceptable for the programme. estimate the number of applicants who qualified for entry.

(ii) Assuming there were places for only 15

applicants, estimate the lowest score that

would be used to select them.

(iii)Calculate the probability that an applicant

chosen at random obtained a score of at least

45.

(12 marks)

7.Given f:x = 2x - 3 and g: x = x + 2

x - 1

(a)evaluate f(-2) and gf(-2)

(b)determine f-1(x)

(c)calculate the value of x, if f(x) = 12

(d)calculate the value of x, for which

(i) f(x) = 0

(ii) f(x) is undefined

(12 marks)

8. (a)Use rulers and compasses only for this question. All

construction lines and areas must be clearly shown and

be of sufficient length and clarity to permit assessment.

(i) Construct a triangle ABC in which AB = 8

cm, A = 45and B = 60.

(ii) Construct also the perpendicular bisector of

AC to meet AB at X.

(iii) Measure accurately and state length of AC

and AX.

(5 marks)

(b)A rectangular steel pyramid of height 6 cm and base

dimensions 11 cm by 16 cm, is melted down and rolled

into a cylinder of height 7 cm.

Calculate (i) the radius of the cylinder in cm

(ii)the mass of the cylinder in kg, if

the density of the steel is 5g/cm3.

Note:Volume of pyramid = 1 Ah

3

Volume of cylinder = r2h

Take to be 22

7

(7 marks)

SECTION II

Answer TWO questions in this section.

RELATIONS AND FUNCTIONS

9. (a) Copy and complete the tables below for the functions

x = cos x and x = tan x, where x is measured in

degrees.

x01020304050

Cos x1.000.9850.9400.866

Tan x00.1760.364

(b) Draw the graph of x = cos x for 0 x 50,

using a horizontal scale of 2 cm to represent

10 and a vertical scale of 10 cm to represent

1 unit.

(c) Using the same scale and on the same axes, draw the

graph of x = tan x, for 0 x 50.

(d) Using your graphs,

(i)solve the equation tan x = cos x in the range

0 x 50.

(ii)derive the value for which cos x = tan x in

the range 90 x 180.

(iii) deduce the corresponding value of tan x, for

each value of x, in the range 0 x 180.

(15 marks)

10. (a) Find the range of values of x for which

(i) x(x + 1) > 0

(ii) x(x + 1) 6

(8 marks)

(b)

The diagram above shows a shaded region ABC bounded by the line AB represented by the equation 2x+5y=10 and the broken line BC drawn parallel to

AO.

(i) State the THREE in equalities in x and y to describe the shaded region ABC.

(ii) Given that P = 3x + 2y, find the maximum value of P for any point in the region ABC

(7 marks)

TRIGONOMETRY

(a ) Given that tan = 2

K

Calculate the value of(i) cos

(ii) sin (90 - )

(iii) cos (90 - )

(5 marks)

(b) P and Q are two points on level ground. P is 10 meters

due south of a vertical tower and the angle of elevation of the top of the tower from P is 75. Q lies 30 meters on a bearing of 60 from P.

(i) Illustrate this information by means of a

sketch, taking care to indicate clearly the

appropriate distances and angles.

(ii) Calculate, in meters,

(i) the height of the tower

(ii) the distance of Q from the foot of the tower.

(10 marks)

12. (a) A model airplane, flying on a course due north with

an air speed of 10 kilometers per hour, encounters a

wind blowing from the west which causes the

airplane to go 30 off course.

Use a scale of 1 cm to represent 1 kmh-1, determine by

drawing,

(i) the wind speed

(ii) the ground speed of the airplane.

(6 marks)

c. A helicopter is situated at a point p(25.8N, 2W) on the earths surface. It has a range of 600 km. Calculate the latitude and longitude of the furthest places which the

Helicopter can reach by flying

(i) due east

(ii) due north

[Note:Consider that the earth is a sphere with circumference

40 000 km.]

VECTORS AND MATRICES

13.

(a)In the figure above, not drawn to scale, the

transformation R is an anti-clockwise

rotation about the origin O through an angle

of 60.

The image of the points L, and N under R

are respectively L, M and N. L is the

Point (2, 0) and M is the point (0, 6).

N is the point (3, -1), not shown.

(i)Calculate the co-ordinate of L and M.

(ii)Hence, determine the image of

the vector 0 and 1

1 0

under the transformation R and

write the matrix for R.

(b)The transformation T is represented by the

matrix 0 -1

-1 0 .

Describe fully the transformation T.

(c)Calculate the co-ordinate of the image N

under the combination of transformation

denoted by TR.

(d) Determine the 2 x 2 matrix which represents

the combination of transformation denoted

by RT.

(7 marks)

14.(a)(i) Calculate the determinant of the matrix

4 3

-1 1/2 .

(ii) Find the inverse of the matrix 4 3

-1

(iii) Hence, use a matrix method to solve

simultaneously the pair of equations

8x + 6y = 28

-2x + y = 8

(8 marks)

(b) Given that a and b are two vectors in the

same plane.

OP = 3a + 2b

OQ = 5a 3b

OR = a + 7b

(i) express the vector PQ in terms of a and b

(ii) show that the points P, Q and R lie on a straight line and indicate, on a diagram, their relative positions.(7 marks)

END OF EXAM

JUNE 1989 PAPER 2SECTION 1

Answer ALL the questions in this section

1.(a)Find the exact value of

4 4 ( 2 2 - 1 1 2

5 3 3

(5 marks)

(b) Write your answer to part (a) as a decimal

correct to 2 significant figures.

(2 marks)

(c) A sum of money is to be divided among A,

B, and C in the ratio 2 : 3 : 5. The largest

share amounts to $1200.

Calculate

(i) The total sum of money to be shared

(ii) As share

(iii) the percentage of the total amount that

B receives.

(5 marks)

2.(a)Given that x = y 2 , express y in terms of x

y 3

(3 marks)

(b)Factorize completely x2 y2 4x + 4y

(3 marks)

(c)Solve 3(x 2) - x 3 = 4.

2 4

(4 marks)

3.(a)Solve 6x2 + 17x 14 = 0(4 marks)

(b)A survey on a sample of persons who read at

least one of the magazines P, Q, and R

yielded the following data:

72 persons read P

53 persons read Q

29 persons read R

14 persons read only P and R

9 persons read only P and Q

2 persons read only Q and R

44 persons read P only.

(i) Use x to represent the number of persons who

read all three magazines. Draw a carefully

labeled Venn Diagram to represent the data.

(ii) Determine the value of x.

(iii)Calculate the number of persons in the sample.

(6 marks)

(a)The figure above, not drawn to scale,

represents a fish tank in the shape of a

cuboid of height 30 cm.

(i) Calculate the volume of the tank.

(ii) If there are 40 litres of water in the tank, calculate the height of water in the tank.

(5 marks)

(b) Construct a triangle DAB such that AB = 8

cm and angle DAB 70. Through D, construct DC parallel to AB. Construct also the line BC perpendicular to AB. Measure and state the length of DC. (Show all construction lines clearly.)

(7 marks)

5.(a)State THREE properties which define a

rhombus, with respect to its sides, angles

and diagonals.

(b)

ABCD is a rhombus (not drawn to scale)

with AO = 4.8 cm and BO = 3.6 cm.

Calculate

(i) the length of AB

(ii) the size of the angle BAD to the nearest degree

(iii) the area of ABCD.

(8 marks)

Use the answer sheet provided for this section (page 14 )

6.(a)Given that R is an anticlockwise rotation of

90 about the origin and MN is reflection on

the x-axis, draw accurate diagrams to show

(i) the image ABC of ABC under R

(ii) the image ABC of ABC under M.

(b)(i)If the transformation Q = MR,

obtain a single matrix to

represent Q.

(ii)Determine clearly Q as a single

geometrical transformation.

(9 marks)

7.

Given that f(x) = 5x and g(x) = x 2

(i) calculate f(2) and gf(2)

(ii) determine x when fg(x) = 0

(iii) prove (gf)-1 (23) 5

(b)

The figure above represents the distance-time graph of the two persons, Andy and Bob, journeying between two offices, M and N. Andy leaves M at the same time that Bob leaves N. Use the graph to

(i) determine the distance MN

(ii) determine how much time Andy spent in walking between M and N

(iii) determine how far from M Andy and Bob met

(iv) calculate Andys average, average speed for the whole distance

(v) explain fully the segment PQ on Bobs graph

(7 marks)

8. The cumulative frequency distribution of the daily

wages, in dollars, for a group of workers is shown in

the table below:

Wages ($)Cumulative

Frequency

41-45

46-50

51-55

56-60

61-655

17

45

56

60

(a)Using 2 cm to represent $5 and 1 cm to

represent 5 workers, construct on graph

paper, a cumulative frequency curve for the

data.

(b) Using your graph estimate

(i)the semi-interquartile range for

the distribution

(ii)the number of workers who

received more than $47.50 in

daily wages.

(iii)the probability that a worker

selected at random receives a

wage which is more than $45.50

but less than $55.00.

(12 marks)

SECTION B

Answer two questions in this section

Trigonometry

9.Two places, P and Q, are on the same parallel of

latitude 48 N but on longitude 57 E and 123 W

respectively.

(a)Calculate, to the nearest hundred kilometers,

the distance PQ, travelling

(i) due east along their parallel on latitude.

(ii) along the great circle over the North Pole

(b)Two aircrafts, X and Y, traveling at the

same ground speed left from P for Q at the

same time. If X used the great circle route

over the north pole while Y traveled due east

calculate the longitude of the point at which

aircraft Y was, at the time when aircraft X

reached Q.

[Take the radius of the earth to be 6400 km

and to be 3.14]

(15 marks)

10.(a)Prove that 1 + tan A = 1___

tan A cos A sin A

(4 marks)

(b)Three stations P, Q and R, are on level

ground such that P is due south of Q and

R is on a bearing of 071 from Q. The

bearing of R from P is 036. A vertical

tower TQ is situated at Q. Given that TQ =

26m and QR is 84m, calculate

(i)the distance PR to the nearest

metre

(ii)the angle of elevation of T from

P.

(11 marks)

11.

RELATIONS AND FUNCTIONS

TypeFees per pupil

Full-time pupil

Part-time pupil$200

$100

The table above shows the cost of tuition per week for a private school. The conditions under which the school

operates are as follows:

1.The maximum number of places

at the school is 75.

2.There must be a minimum of 20

full-time pupils.

3.The number of part-time pupils

must be at least half the number

of full-time pupils.

4.The minimum weekly income must be $6000.

Let x represent the number of full-time pupils and y

represent the number of part-time pupils.

(a)Write FOUR inequalities not including x (

0, y ( 0, to represent the conditions stated

above.

(b)Using a scale of 1 cm to 5 units on both axes

draw the graphs of the inequalities.

(c)It the profit on each full-time pupil is $40

and that on each part-time pupils $30,

determine from your graphs

(i)the number of full-time and part-

time pupils for the maximum

profit to be obtained.

(ii)the sum representing the profit.

(15 marks)

12.(a)Copy and complete the table below for

y = 3 sin 2x

x0

843

8

25

83

47

8

Y02.12-2.12

(b)Using the scale of 2cm to represent radians

on the x-axis, and 2cm to 1 unit on the

y-axis, draw the graph of the function above

0 x .

(c) Using your graph

(i) state the maximum value of the fraction

(ii) state the value of x within the domain 0 x , for which the fraction is negative.

(iii) determine the solution of 3 sin 2x = 1.5.

(15 marks)

VECTORS AND MATRICES

13.(a)Given x = p q and y = 3 0

r s 0 4

(i) evaluate XY and X + Y

(ii) if XY = X + Y, determine the value of p, q and r.

(b)Given that A and B are two points whose position vectors with respect to the origin are a and b respectively.

(i)draw on a diagram the point C

whose position vector is

2a + b

3

(ii)prove that ABC is a straight line,

using a vector method

(iii) determine the ratio AB:BC

(9 marks)

14.

(a) Using the graph above

(i)express each of the position

vectors OA and OB in the form

x

y

(ii)if OA + OB = c, show that

c = 34

(4 marks)

(b)With 0 as origin, (OMN is transformed

under a translation (T) to (PQR such that

N (3, 2) moves to R (1, 1).

(i)Determine the matrix of the

transformation.

The point R is mapped to the point S (-2, -2)

under an enlargement (E), centre 0, scale

factor e.

(ii)State the value of e.

(iii)Hence determine the co-ordinates

of the mapping of the origin 0

under the combined

transformation represented by ET

(11 marks)

Answer sheet for Question 6- June 1989 (page 8)

JUNE 1990PAPER 2

SECTION I

Answer ALL question in this section

1.Calculators, slide rules and mathematical tables must be NOT be used to answer this question. Show ALL steps clearly.

(a)Calculate the exact value of

4 1/3 - 1 5/6

1 3/7 x 2 2/3

(3 marks)

(b) Calculate the value of

0.023

0.351

giving your answer correct to two significant figures.

(3 marks)

(c)(i)Some years ago, US$1.00 (one

United States dollar) was

equivalent to J$3.50 (three

dollars and fifty cents, Jamaican

currency). Calculate the amount

of US currency that was

equivalent to $8400.

(ii)After devaluation J$1.00 was worth 70% of its original value.

Calculate the new rate of exchange for US$1.00 and hence calculate the amount of Jamaican dollars that would be equivalent to US$2400.

(7 marks)

2.Consider the folowing three statements:

1.Some students play basketball

2. Tall students are over 2 metres in height

3. All basketball players are tall students.

(a)Reprsent the statements in a suitable Venn

diagram and stating an appropriate universal

set.

(b)Show on your Venn Diagram that

(i) Nina is 1.5 m tall

(ii) Robert, who is 2.2 m tall, does not play basketball.

(5 marks)

3.Solve the following equations for x:

(a)3(x + 2)2 = 7(x + 2)

(5 marks)

(b)92x = 1

(4 marks)

27

(c)3x = y, given that sin y = 3 where 0 < y < 180.

2

(4 marks)

4.(a)(i)Using a scale of 1 cm to 1 unit on

each axis, draw on graph paper

triangle ABC whose vertices are

A(2, 2), B(5, 2) and C(2, 4).

(ii)( ABC is the image of (

ABC under an enlargement,

centre (0, 0) and scale factor k =2

Draw ( ABC, and state the

co-ordinate of the vertices.

(iii) (ABC is the image of (ABC under a transformation T. The vertices are A(-2, -4),

B(-5, -4) and C(-2, -2). Give a full geometrical description of

the transformation T.

(9 marks)

5.(a)Given that f(x) = 2 sin x/3, copy and

complete the following table

x0

6

4

3

22

33

4

f(x)0.350.520.681.29

(b)Using a scale of 2 cm to represent /12 unit on the x-axis and 10 cm to represent 1 unit on the f(x) axis, draw the graph of the function f(x) for the domain 0 x .

(c) By drawing a suitable tangent, determine the gradient of the curve at x =

2.(10 marks)

6.(a)A = 3 5 and B = 6 4

1 0

-1 -2

(i) Calculate the matrix product AB.

(ii) If C = x -8 and 2A + C =AB

4 y

calculate the values of x and y.

(8 marks)

(b)The ratio of the prices of two different

sheets of glass is 2 : 5.

The total bill for 20 sheets of the cheaper glass and 10 sheets of the more expensive one is $1080. If d dollars represent the cost one sheet of the cheaper glass, determine

(i)an expression in d for the cost of

ONE sheet of the more expensive glass.

(ii) the value of d.

(iii)the cost of ONE sheet of the

more expensive glass.

(6 marks)

7.(a)The frequency distribution of the length of

100 rods measured to the nearest mm is

given in the following table.

Using an assumed mean of 222 mm, calculate the mean length of all the rods.

Length (mm)Frequency

200 204

205 209

210 214

215 219

220 224

225 229

230 234

235 - 2393

7

11

14

25

23

11

6

(7 marks)

(b)A two-digit number in base ten in formed by choosing the ten digit from the set {4, 5, 6, 7, 8} and the units digit from the set {1, 2, 3, 9}.

(i)Determine the total number of

different two-digit numbers that

may be formed.

(ii)Calculate the probability that the resulting number is even.

(iii) Hence, deduce the probability that the resulting number is odd.

(5 marks)

8.(a)LMNOP is a hexagon (not drawn to scale)

with (Q = 130, (P = 110 and (O = 90

(L = (M = (N

(i) Calculate the value of(L.

(ii) Given that PQ =4 cm and the area of (NOP is 12 cm2, calculate the length of PN in cm, giving your answer correct to one decimal place.

(7 marks)

(b) (Take as 22)

7

The diagram below, not drawn to scale, represents a plot of land ABCDE in the shape of a square of slides 21 m with a semicircle at one end.

(i) Calculate in metres, the perimeter of the plot of land.

(ii)WXYZ is a rectangular flower

bed of length 15 m and width 12m. Calculate in square metres, the area of the shaded section.

(iii)The soil in the flower bed is

replaced to depth of 30 cm.

Calculate, in cubic centimetres,

the volume of the soil replaced,

writing your answer in standard

form.

SHAPE \* MERGEFORMAT

(7 marks)

SECTION II

Answer TWO questions in this section

RELATIONS AND FUNCTIONS

9.(a)Copy and complete the following table for

the function

f : x = 1 for real values of x in the domain

x2

0.5 x 3..5.

x0.51.02.251.52.02.53.03.5

f(x)1.00.640.160.08

(b)Using a scale of 4 cm to represent 1 unit on

the x-axis and 2 cm to represent 0.1 unit on

the f(x) axis, draw the graph of f : x = 1 for

x2

real values of x in the domain 1 x < 4.

(c)(i)Using the same scale and axes,

draw the graph of the function

g : x = x for the same domain.

10

(ii)Use your graph to solve the equation 1 = x for real values

x2 10

of x such that 1 x < 4.

(iii)Hence, deduce one solution of the equation, 64x3 = 10 for real values of x.

(15 marks)

10.At a factory, two types of motor cycles, M1 and M2 are assembled. The factory can produce a maximum of 80 motor cycles daily. For each M2 motor cylcle made, there must be at least two M1 motor cycles produced. It takes one workman to produce each M1 motor cycle but three workmen to produce each M2 motor cycle. Factory policy requires that at least 90 workmen be employed.

Let x represent the number of M1 motor cycles and y the number of M2 motor cycles produced daily.

(a)Write down three inequalities, not including

x ( 0, y ( 0, which represent the above

conditions.

(b)Using a scale of 2 cm to 10 motor cycles on EACH axis, shade, on graph paper, the region which satisfies the inequalities.

(c) Given that a profit of $250 is made on each M1 motor cycle and $200 on each M2 motor cycle, use your graph to determine the values of x and y which give the maximum profit.

Calculate the maximum profit. (15 marks)

TRIGONOMETRY

11.The foot, F, of a hill and the base B, of vertical tower TB, 21 metres tall, are on the same horizontal level. From the top, T, of the tower, the angle depression of F is 36.9. P is a point on the hill 24. 5 m away from F along the line of greatest slope. T, B, F, and P all lie in the same vertical plane. The angle of depression of P from T is 19.4.

(a)Draw a sketch to represent the information

given above.

(b)Show that

(i) TF is 3. 5 m approximately

(ii) sin TPF = 3/7 approximately.

(c)Calculate

(i) the gradient of the hill

(ii)the height, in metres, of P above

F, giving your answer correct to

one decimal place.

(15 marks)

12.(a)An helicopter is set on a course of 095. A

wind is blowing from a direction of 50 east

of north at a speed of 40 kmh-1.

Using a scale of 1 cm to 20 kmh-1, determine by drawing,

(i)the track and ground speed of the helicopter when it has an air speed of 100 kmh-1

(ii) the course which must be set if the helicopter is to fly on a track of 095.

(8 marks)

(b)In an experiment, the distance d of a moving

object from its starting point is given by the

equation

d = 3 2 sin

where the angle is measured in radians.

(i)For -2 2, determine all

the values of for which d = 0.

(ii)State the values of d when = 4

3

giving your answer in surd form.

(iii)state the largest possible value of

d, giving your answer in surd

form.

(7 marks)

VECTORS AND MATRICES

13.

The diagram, not drawn to scale, shows a regular 8-sided polygon ABCDEFGH, centre 0. OE = .

(a)Show that COE = 90

(b)Calculate

(i)the angle CDE

(ii)the length of CE in terms of

(c)Given that P is the midpoint of CE, show

that OP = PE

(d)Given that r represents the unit vector in the

direction OD and s is the unit vector in the

direction OF, express in terms of r and s

(i) OE

(ii) OG

(15 marks)

14(a)Four points on a plane have the following

co-ordinates:

O(0, 0); P(0, 12); Q(6, 0); R(0, -3)

The triangle QOR is mapped onto triangle QOP by a combination of two transformation a reflection and a stretch.

(i)describe fully the transformation

and determine the matrix for

(a) the reflection

(b) the stretch

(c) the combination of the 2 transformation

(ii)Determine the matrix which

maps triangle QOP onto triangle

QOR.

(8 marks)

(b) The vertices of triangle ABC have

co-ordinates A(2, 30; B(4, 6); C(6, 3) and the triangle ABC has verices A(3, 3); B(6. 5, 6); C (7, 3). ABC is the image of ABC under a shear by a positive scale factor

Determine, for the shear,

(i) the equation of the invariant line

(ii) its scale factor.

(7 marks)

JUNE 1991PAPER 2

SECTION I

Answer ALL the question in this section.

1.All steps in your calculation must be clearly shown.

(a)Calculate the exact value of:

(2 1/3 1 5/6) ( 1 1/3

(4 marks)

(b)The simple interest on $15000 for 4 years

is $8100. Calculate the rate percent per

annum.

(3 marks)

(c)The sum of $2500 is divided among Peter,

Queen and Raymond. Raymond received

half, Peter received $312.50 and Queen

received the remainder.

Calculate(i) Raymonds share

(ii) Queens share

(iii) the ratio in which the $2500

was divided among the three

persons

(iv) the percentage of the total

that Peter received.

(5 marks)

2.(a)Factorize completely:

(i) 1 (a + b)2(ii) (2x2 + xy y2) + 2x y

(3 marks)

(b) Solve:

x - 5 = x + 4

5 15

(4 marks)

(c) Solve the simultaneous equations:

x + y = 5

xy = 6

(5 marks)

3.(a)The scores obtained by a class of ten

students in a test were:

2, 3, 4, 4, 5, 6, 6, 7, 11, 12

Calculate

(i) the median mark

(ii) the mean mark

(iii) the standard deviation of the marks; showing clearly all steps in your calculation.

(8 marks)

(b)Determine the probability that a pupil

selected at random from the class obtained

(i) exactly 6 marks

(ii) at least 6 marks

(4 marks)

4.(a)

SHAPE \* MERGEFORMAT

The figure ABCDEF above, not drawn to

scale, represents a wedge with measurement

as shown. BC is perpendicular to the plane

FEDC.

Calculate

(i) the height, in cm, of BD

(ii) the surface area in cm2, of the wedge

(iii) the volume, in cm3, of th wedge

(iv) the size of angle BDC.

(7 marks)

(b)Using ruler and compasses only, construct a

parallelogram ABCD, such that AB = 6. 5

cm, AD = 5. 7 cm and the angle DAB =60.

Measure and state the length of BD in

centimetres.

[Note: All construction line must be clearly

shown]

(4 marks)

5.There are 50 students in Form VI

All students study Mathematics.

17 study Biology

18 study Chemistry

24 study Physics

5 study Physics, Chemistry and Mathematics

7 study Physics, Biology and Mathematics

6 study Chemistry, Biology and Mathematic

2 study all four subjects.

(a)Draw a carefully labelled Venn Diagram to

represent the data, using the universal set as

a the set of students who study Mathematics.

(3 marks)

(b)Determine the number of students who study

at LEAST TWO subjects.

(2 marks)

(c) Calculate the number of students who study

Mathematics only. (1 marks)

6.(a)The function f and g are defined by;

f : x = 5 + x

g : x = x3

Determine expressions for the function

(i) fg

(ii) g-1(b)The distance S, in metres, moved by a

particle from its starting point, at t seconds,

is given by

S = 3t + t2

(i)Copy and complete the table

below for S = 3t + t2t012345678

S108405470

(ii)Using a scale of 1 cm to represent 1 second and 1 cm to represent 10 m, draw a graph of the function

S = 3t + t2 for the range 0 t 8

(iii)Using the graph, estimate the

distance moved in 4.5 seconds.

(iv)Draw a tangent to the curve at t = 6. Estimate the value of the tangent at this point. Give an interpretation of this value.

(8 marks)

7.

(a)In the graph above, A and B are points such

that OA = a and OB = b. The point P (not

shown) is such that

OP = a + b,

(i) Write OP in the form x

y

(ii) Determine the length of OP

(5 marks)

(b)The rectanhle WXYZ, with co-ordinates

(2, 2), (5, 2), (5, 4) and (2, 4) respectively, is

mapped onto parallelogram WXYZ with

co-ordinates (5, 2), (8, 2), (10, 4) and (7, 4)

respectively under a shear (S).

(i) Plot the rectangle WXYZ and its

image WXYZ.

(ii)Determine the equation of the

invariant line.

(iii) Calculate the area of WXYZ.

(iv)State the co-ordinates of the

image of the point (2, 3) under

the shear S.

(7 marks)

8. (a)

In the diagram above, not drawn to scale, the

points X, Q and R are on a straight line in

the same horizontal plane. The angle of

depression of a point Q from the top of a

tower PX, 10 m high, is 70. The angle of

depression of R from the top of the tower is

40.

Calculate the length QR to one decimal

place.

(5 marks)

(b)

SHAPE \* MERGEFORMAT

In the diagram above, not drawn to scale,

PQ, XY and SR are parallel lines. QY = 10

cm, YR = 5 cm and XY = 3 cm.

(i)Prove that triangle PQX and

RSX are similar.

(ii)Calculate the lengths PQ and RS.

(iii)Calculate the ratio of the areas of

triangles PQX and RSX.

(8 marks)

SECTION II

Answer ANY TWO questions in this section

TRIGONOMETRY

9.(a)Prove that 1 + 1 = 1.

sin2 cos2 sin2 cos2

(2 marks)

(b)A ship sails 4 n autical miles from port P on

a course 069 to a point Q, changes course to

295 and sails a further 5 nautical miles to a

port R.

(i) Draw a carefully labelled diagram of the entire route taken. Show the north direction.

(ii) Calculate the distance PR to 2 significant figures.

(iii) Determine the bearing of P from R.

(13 marks)

10.(a)If sin A = 3/5, cos B = 15/17 and A and B

are acute angles, determine the exact value

of tan A and B.

(3 marks)

(b)[Take the radius of the earth to be 6400 km

and to be 3.142]

(i)The distance between Town

P(80E, 72N) is 800 km. If

Town R is east of Town P,

determine xE, the longitude of

R. Give your answer to the

nearest degree.

(ii)The shortest distance between

Town P(80E, 72N) and Town

Q(100W, yN) is 10054.4 km.

Determine yN, the latitude of Q.

(12 marks)

RELATIONS AND FUNCTIONS

11.(a)(i)Use the method of completing

the square to determine the value

of x for which the expression

4x2 + 4x + 3 is minimum.

(ii)Hence deduce the minimum

value of 4x2 + 4x + 3.

(6 marks)

(b)A motorist starting a car from rest,

accelerates uniformly to a speed of 40

kmh-1 in 2 minutes. He maintains this

speed for another 3 minutes. He then

applies the brakes and decelerates

uniformly to rest in 1 minute.

(i)Draw a diagram of the velocity-

time graph to show the different

segments of the journey.

Determine

(ii)the acceleration, in kmh-2,

during the first two minutes.

(iii)the retardation, in kmh-2,

during the last minute

(iv)the total distance of the journey

(v)the average speed for the whole

journey in kmh-1

(9 marks)

12.(a)Copy and complete the table below for y =

5/x

x123456

y5.001.000.83

(b)Using a scale of 2 cm to represent 1 unit on

both x and y axes, draw the graph of the

function above for 1 x 6.

(c)Using the same axes and scale given above,

draw the graph of the function

x + y = 6 for 0 x 6

(d)(i)Shading on your graph the region

which satisfies the inequalities

y 5/x, x + y 6, x ( 0 and y( 0

(ii)Indicate a point Z in the region

which satisfies the inequalities

y 5/x and x + y > 6.

(iii)Identify, by writing the letter T,

the region bounded by the curve

and the straight line.

(iv)State the inequalities which

describe the region bounded by

thhe curve and the straight line.

(15 marks)

VECTORS AND MATRICES

13.(a)If A = 3 2

2 -3 ,

(i) evaluate the determinant of A]

(ii) determine A-1

(iii) using A-1 solve the simultaneous equations:

3x + 2y =1

2x - 3y =5

(7 marks)

(b)S is the transformation represented by

-1 0 x 3

0 1 y + 0

(i) Perform the transformation S

on a square with vertices at

A(0, 0), B(0, 1), C(1, 1), D(1, 0)

and write in co-ordinate form, the

images of A, B, C and D.

(ii)Describe, in words, the single

transformation represented by

transformation S.

(8 marks)

14.(a)(i)T is a matrix

p q

r s

Determine the elements of T

which map A(2, 1) onto

A(4, 3) and B(3, 2) onto

B(6, 6).

(ii)Derive the matix S, such that TS

represents the transformation

matrix for a reflection in the x-

axis.

(10 marks)

(b)

The figure ABC is a triangle with X and Y

the mid-points of AB and AC respectively.

Using a vector method prove that]

(i) BC is parallel to XY

(ii) BC = 2XY.

(5 marks)

JUNE 1992PAPER 2

SECTION I

Answer ALL the question in this section.

1.All working must be clearly shown.

(a)Calculate the exact value of

5 2/7 + 3 5/7

4 - 2 4/5 (4 marks)

(b)A piece of string 64 cm long, is divided in

three pieces in the ratio 1:2:5. Calculate the

length of the longest piece.

(3 maks)

(c)A merchant sold a pen for $5.35, therby

making a profit of 7% on the cost to him.

Calculate

(i)the cost price of the pen to the

merchant.

(ii)the selling price the merchant

should request in order to make

a 15% profit.

(5 marks)

2.(a)Solve 2x 1 - x + 5 = 2

2 3

(4 marks)

(b)

The Venn Diagram above illustrates some of

the information given below.

There are 100 members in a foreign

language club.

48 members speak Spanish

45 members speak French

52 members speak German

15 members speak Spanish and French

18 members speak Spanish and German

21 members speak German and French

Each member speak AT LEAST ONE of the

three languages.

Let the number of members who speak all

three languages be x.

(i) Write an algebraic expression to represent the number of members in the shaded region

(ii) Describe the region shaded

(iii)Write an equation to show the total number

of members in the club

(iv)Hence, determine the number of members

who speak all languages.

(7 marks)

3.(a)Solve the simultaneous equations

4x - 4y = 2

7x + 2y = 17 (4 marks)

(b) Factorise completely:

(i) 1 - 9x2(ii) 3x2 7x 6

(3 marks)

(c)The vector p translates the point (2, 3) to the

point (4, 6).

The vector q translates the point (2, 7) to the

point (-4, 3).

(i) Write p and q as column vectors.

(ii) Hence, determine the vector

(p + q)

(5 marks)

4.(a)(i)Using ruler and compasses only,

construct a quadrilateral ABCD

in which AB = AD = 6 cm,

BC = 4 cm , angle BAD = 60

and angle ABC = 90.

(ii)Measure and state

the length of DC

the size of angle of ADC

(6marks)

(b)The images of L(1, 1) and N(2, 3) under a

single transformation Q are L(-1, 1) and

N(-3, 2) respectively.

(i)Describe geometrically the

transformation Q.

(ii)Determine the equation of the

line LN.

(6 marks)

5.(a)Copy in your answer booklet the table for

f(x) = 2x2 x 3 for the domain -2 x 3,

and calculate the missing values.

x-2-10123

y7-2312

(b)Using a scale of 2 cm to represent 1 unit on

the x-axis and 1 cm to represent 1 unit on

the f(x) axis, draw the graph of f(x) for

-2 x 3.

(c) From your graph, determine,

(i) the value of x for which f(x) = 0

(ii) the minimum value of f(x)

(11 marks)

Note:You should attach your graph carefully to

the page on which you did the other working

for this question.

6.The frequency distribution of the Mathematics marks

obtained by 100 candidates is given below.

MarksNumber of CandidatesCumulative

Frequency

0-10

11-20

21-30

31-40

41-50

51-60

61-70

71-80

81-90

91-1004

7

9

12

18

13

12

11

9

54

11

(a) Copy the table above in your answer booklet

and complete the cumulative frequency

column.

(b)Using a scale of 1 cm to represent 10 marks

on the x-axis and 1 cm to represent 5

candidates on the y-axis, draw the

cumulative frequency curve for the data.

(c)From your cumulative frequency curve,

estimate

(i)the number of candidates who

scored at LEAST 45 marks

(ii)the probability that a candidate

chosen at random scored less

than 45 marks.

(11 marks)

7.In this question, take = 3.14

In the figure shown here, not drawn to scale, the

chord HK subtends angle HOK at O, the centre of the

circle. Angle HOK = 120 and OH = 12 cm.

Calculate to three significant figures

(a)the area of the circle

(b)the area of the minor sector OHK

(c)the area of triangle HOK

(d) the length of the minor are HK.

(11 marks)

8.

In the figure above, not drawn to scale, ABCDE is a

pentegon inscribed in a circle of centrem 0. The

diameter AB is produced to F. Angle CDF = 136 and

angle BAD = 72.

(a)Calculate, giving reasons for your answer,

the magnitude of angles

(i) CDA

(ii) BCD

(iii) AED

(6 marks)

(b)Given that OA =15 cm and angle EAD = 35

calculate the length of AE.

(4 marks)

SECTION II

Answer TWO of the questions in this section.

RELATIONS AND FUNCTIONS

9.A manufacturer produces two types of ball-point pens:

Type L and Type M. There are at least 50 of Type L

and at least 25 of Type M pens.

The manufacture, however, does not produces more

than 80 Type L or more than 60 of Type M or more

than 120 of both Type L and Type M taken together.

(a)Using x to represent the number of Type L

pens and y the number of Type M pens

produced, write THREE inequalities (not

including x ( 0 and y ( 0) which represent

the above conditions.

(b)Using scale of 1 cm to represent 10 pens on

each axis, draw the graph of the inequalities

Identify the region which satisfies the

inequalities.

(c)The manufacturer makes a profit of $1.50

on each Type L pen and $1.10 on each Type

M pen.

(i)Write an expression to represent

his total profit.

(ii)Use the graph to determine the

values of x and y which give a

maximum profit, and hence

determine the maximum profit.

(15 marks)

10.(a)Solve the equation 3x2 +5x = 6, giving your

answer correct to two decimal places.

(5 marks)

(b)

The graph above records, the journeys of

two cyclists travelling between towns A and

B. The cyclists begin their journey at the

same time.

Calculate

(i)the distance between the two

towns

(ii)the time the cyclists from B takes

for the journey

(iii)the average speed of the cyclists

from B, in metres per second

(iv)the distance from town B where

the cyclist met

(v)the average speed, in metres per

second, at which the cyclist from

A would need to travel after he

met the cyclist from B, in order

to complete the journey in the

same time as the cyclist from B.

(10 marks)

TRIGONOMETRY

11. (a)Prove that 2 cos2 1 = 2 sin2

(2 marks)

(b)The angles of depression from the top of the

tower T to R and S are 32 and 22

repectiely. The points R and S and the foot

of the tower are on the same horizontal

plane. The height of the tower TX is 52 m.

The bearings of R and S from X are 270

and 220 respectively.

(i)Draw a sketch to represent the

information given above.

(ii)Hence or otherwise, calculate

- the distance RS to one decimal

place

- the bearing of S from R.

(13 marks)

12.In this question take the radius of the earth to be 6400 km and to be 3.142.

(a)The co-ordinates of the points P and Q on

the earths surface are (26 S, 25 W) ans(60 N, 25 W) respectively.

Calculate

(i) the shortest distance from P to Q

(ii) the circumference of the circle of

latitude 60 N.

(b)Two tracking stations X and Y are both

situated on latitude 60 N. Station X is

situated at (60 N, 10 E) and station Y is

situated west of X. The distance between X

and Y along latitude 60 N is 1800 km.

Calculate the position of the tracking station

Y.

(8 marks)

VECTORS AND MATRICES

13.(a)Points U, V and W have position

-1 2 and 5 respectively.

-2 6

If U, V and W are collinear, determine

the value of .

(6 marks)

(b) The matrix 5 12 represents a

3 5

transformation T.

(i)Determine the co-ordinates of the

image of the point (1, -1) under

the transformation T.

(ii)Derive the equation of the line

onto which the line x + y = 0 is

mapped by the transformation T.

(9 marks)

14.(a)The position vectors of the points A, B, C

and D are a, b, 3a b and a + b respectively

(i) Prove that CD is parallel to AB

(ii) Determine the ratio AB : CD.

(b)The poin t E(3, 2), under a shear S, where

the invariant line is the x-axis, has image

E(7, 2).

Determine

(i) the scale factor of the shear S.

(ii) the (2 x 2) matrix which represents S.

(iii) the co-ordinates of the image F of the point F(2, 1) under the transformation S.

JUNE 1993 PAPER 2

SECTION I

Answer ALL the question in this section

1.All working must be clearly shown.

(a)Calculate the exact value

(3 2/7 + 1 2/3) ( 1 1/7

(4 marks)

(b)Evaluate 0.0004 x 10-6, giving your

answer in standard form.

(3 marks)

(c)A tourist exchanged US$200.00 for

Jamaican currency at the rate US$1.00 =

J$18.81. She had to pay a government tax

of 2 % of the amount exchanged.

Calculate in Jamaican currency

(i) the tax paid

(ii) the amount the tourist received.

(4 marks)

2. (a)There are 68 students in Form V.

15 students study Mathematics only.

12 students study Physics only.

8 students study Physics and Chemistry only.

2 students study Physics and Mathematics only.

3 students study Mathematics, Physics and Chemistry.

4 students do not study any of these subjects.

(i)Draw a carefully labelled Venn diagram to

represent the information given above.

(ii)Determine the number of students who study

Physics.

(iii)Given that x students study Mathematics and

Chemistry only, and twice as many study

Chemistry only, write an algebraic equation

to represents the information given and

hence, calculate the value of x.

(6 marks)

(b)

Given that a and b are unit vector as shown in the

diagram above,

(i)write the position vectors OP and OQ in

terms of a and b.

(ii)determine the length of OP.

(5 marks)

3.(a)Solve 2P + 5 = 3

5 P(4 marks)

(b)Given that y varies inversely as x2 and that

y = 3 when x = 2, calculae the value of y

when x = 3.

(3 marks)

(c)The cost of four chairs and a small table is

$684. The cost of six chairs and a large

table is $1196. The cost of the large table

is TWICE the cost of the small table.

Given that a is the cost, in dollars, of a chair

and b is the cost, in dollars, of a small table.

(i)write a pair of simultaneous

equations to represent the

information given

(ii)calculate the cost of the large

table.

(5 marks)

4.(a)(i)Using rulers and compasses only,

construct a triangle ABC with

AB = 9.5 cm, AC = 7. 5 cm and

angle BAC = 60.

(ii)Locate the point D such that DB

is perpendicular to AB and CD is

parellel to AB.

Measure and state the length of

BD in centimetres.

(6 marks)

(b)Triangle PQR with P(1, 1), Q(11, 2) and

R(1, 9) is mapped onto triangle PQR with

P(3, 3), Q(8, 3.5), and R(3, 3).

(i)Using a scale of 2 cm to 1 unit on

both axes, draw on graph paper

triangles PQR and PQR.

Note: Draw the x-axis on the longer side of the graph paper.(ii)Hence, describe fully the

transformation which maps

triangle PQR onto triangle

PQR.

(6 marks)

5.(a)An aeroplane travel distance of 3700 km in

7 hours. Calculate its average speed.

(2 marks)

(b)An aeroplane left Kingston at 21:00 hours

local time and travelled for 9 hours

arriving at Los Angeles airport at 01:30 hrs

local time on the following day. Calculate

the difference in time between Kingston and

Los Angeles.

(4 marks)

(c)(i)The angle of elevation from a

point P on the ground to the top

of a tower 20 m tall, is 65.

Calculate the distance of P from

the foot of the tower.

(ii)A point Y is 3.2 m nearer than P

to the foot of the tower.

Calculate the angle of depression

of Y from the top of the tower.

(6 marks)

6.A survey was taken among 100 customers to find out

the time spent waiting in lines for service at the bank.

The following table shows the result of the survey.

Waiting time

(in minutes)No. of

Customers

1-5

6-10

11-15

16-20

21-25

26-30

31-355

12

15

19

21

25

3

(a)Construct a cumulative frequency table to

represent the data above.

(2 marks)

(b)Using a scale of 2 cmto 5 minutes on the

horizontal axis and 2 cm to 10 customers

on the verical axis, draw a cumulative

frequency curve to illustrate the information

(4 marks)

(c)Estimate the proportion of customers who

waited more than 16 minutes.

(2 marks)

(d)Calculate the probability that a customer

chosen at random would have waited for

more than 27 minutes.

(3 marks)

7.(a)Copy and complete the table below for the

function

y = 5 + x 2x2

x-3-2-101/2123

y-1624-1-10

(b)Using a scale of 2 cm to 1 unit on the x-axis

and 1 cm to 1 unit on the y-axis, draw the

graph

y = 5 + x 2x2 for -3 x 3.

(4 marks)

(c)Using your graph or otherwise, determine

the range of values of x for which

x 2x2 > -3

(5 marks)

8.Note for this question: Take = 22/7

Volume of cone 1/3 r2h

The diagram above, not drawn to scale, shows the

MAJOR sector, AOB, of a circle of radius 6 cm. It

represents the net of a cone.

(a)show by calculation that the circumference

of the base of the cone is 22 cm.

(2 marks)

(b)Calculate

(i) the radius of the base of the cone

(ii) the height of the cone, giving your answer correct to one decimal place.

(5 marks)

(c)Calculate to two significant figures the

volume of liquid, in litres, that the cone

holds when filled.

(3 marks)

SECTION II

Answer any TWO questions in this section.

RELATIONS AND FUNCTIONS

9.(a)Given that f(x) = x + 3 2x2

(i)derive f(x) in the form

f(x) = c + a(x + b)2, where a, b,

and c are constants

(ii)determine the value of x at which

the maximum value f(x) occurs

(iii) state the maximum value f(x)

(6 marks)

(b)A composite function k is defined as

k(x) = (2x 1)2.

(i)Express k(x) as g(x), where f(x)

and g(x) are two simple functions

(ii)Show that k-1(x) = f-1g-1(x)

(9 marks)

10.A boy has $280. He wants to buy x records at $35 each and y tapes at $40 each. He must buy more than one

but not more than four tapes. He must also but at least

three records.

(a)Write THREE inequalities in x and y to

represent the above information.

(b)(i)Using a scale of 2 cm to

represent 1 unit on EACH axes,

draw the graphs of the inequalitie

(ii)SHADE the region that satisfies

the THREE inequalities.

(7 marks)

(c)Determine the maximum amount spent and

state the (x, y) value that gives this amount.

(4 marks)

11.(a)In the figure shown here, not drawn to scale,

the quadrilateral PQRS is inscribed in the

circle, centre O. PR passes through O. The

tangents TP and TS are drawn to the circle

from T. Angle RSV = 20.

Calculate, giving reasons,

(i)angle PQR

(ii)angle SPR

(iii)angle PST

(iv) angle PTS

(7 marks)

(b)In the triangle, ABC shown here, not drawn

to scale, AB =4 cm, AC = 7 cm and angle

BAC is .

Given that sin2 = 0.64, determine

(i) the exact value of cos2

(ii) the value of , if 90 < < 180

(iii) the length of BC, correct to one decimal place.

(8 marks)

12.A point K is on a bearing of 025 from another point M. The boat leaves K to got to M. The engine speed of the boat is 55 kmh-1. A wind is blowing from the East at a

speed of 25 kmh-1.

(a)Using a scale of 1 cm to represent 5 kmh-1, find by accurate drawing

(i) the course of the boat

(ii) the resultant speed of the boat.

(12 marks)

(b)If K is 12 km away from M, and the boat

leaves K at 07:00 hrs, calculate the time at

which the boat reaches M.

(3 marks)

VECTORS AND MATRICES

13.(a)Using a scale of 2 cm to represent 1 unit on

EACH axis, draw ( PQR with P(1, 4),

Q(3, 1) and R(4, 2).

(3 marks)

(b)( PQR is transformed by the matrix

1 2

0 1

(i)Determine the co-ordinates of the

image , ( PQR.

(ii)Draw triangle PQR and

describe the transformation fully.

(7 marks)

(c)Determine the 2 x 2 matrix that will

transform ( PQR onto ( PQR.

(5 marks)

14.(a)P is the point (6, 4) and Q is the point (8, 2).

M and N are the mid-points of OP and OQ

respectively, where O is the origin.

(i) Determine the vector PQ

(ii) Determine the vector MN

(iii) State the relationships between MN and PQ.

(8 marks)

(b)Given the equations

x y = -5

3x + 2y = -5

(i)write the equations in matrix

form

(ii)determine the inverse of the

matrix

(iii)hence, solve the equations.

(7 marks)

JUNE 1994 PAPER 2

SECTION I

Answer ALL the questions in this sectionAll working must be clearly shown.

1.(a)Given that a = 4, b = -2, and c = 3

calculate the value of a2 - bc

b + c

(4 marks)

(b)Factorise (i) 9a2 b2

(ii) 3x 8y -4xy +6

(4 marks)

(c)Given that 2x + 4 =1,

3 y

express y in terms of x.

(3 marks)

2.(a)Evaluate 3.7 x 102 + 2.4 x 103, giving your

answer in standard form.

(3 marks)

(b)Janets gross salary is $2400 per month.

Her tax-free allowances are shown in Table

A below.

TABLE A: Tax-free Allowances

National Insurance5% of gross salary

Personal Allowance$3000 per year

Calculate

(i) her gross yearly salary

(ii) her total tax-free allowance for the year

(iii) her taxable yearly income

(iv) A 10% tax is changed on the first $20000 of taxable income.

A 20% tax is changed on the portion of taxable income above $20000

Calculate the amount of income tax Janet pays for a year.

(8 marks)

3.(a)Given the following information

U = {3, 6, 9, 12,., 27}

E = {even numbers}

G = {numbers greater than 15}

E and G are subsets of U.

(i) List the members of E and of G.

(ii) Draw a Venn diagram to represent the above data

(iii) State n(E ( G).

(6 marks)

(b)(i)The width of a rectangular field

is w metres. The length is 6

metres more than twice the width

Write, in terms of w, algebraic

expressions for

the length of the field

the area of the field

(ii)The area of the field 360 m2 Write an algebraic equation for the area of the field

Determine the value of w.

(6 marks)

4. (a)(i)The scale used for a map is

1:250 000. The distance MN

on the map 4.4 cm.

Calculate, in kilometres, the

actual distance of M from N.

(ii)A car leaves M at 09: 50 hrs and

arrives at N at 10: 04 hrs the

same day.

Calculate

the time, in minutes, taken for the journey

the average speed of the car in kilometres per hour, giving your answer to the nearest whole number.

(6 marks)

(b)

SHAPE \* MERGEFORMAT

In the figure above, not drawn to scale, TF is

perpendicular to FY, FX = 40 m, XY = 20 m

and angles TXF = 42.

Calculate, correct to 2 significant figures,

(i) the height TF

(ii) angle XTY.

(6 marks)

5.(a)Using ruler and protractor, construct a

quadrilateral VWXY in which YX = 8 cm,

angle XYV = 80, VY = 6 cm, XW = 7 cm

and angle XYW = 35.

Measure and state the length of VW correct

to one decimal place.

(5 marks)

(b)

SHAPE \* MERGEFORMAT

In the figure above, not drawn to scale, C is the centre

of the circle of radius 5 cm. AB is a chord of length 6

cm. J is a point on the circumference so that JA = JB.

(i)Calculate the perpendicular distance of C

from AB

(ii)Show that JA = 9. 5 cm approx.

(6 marks)

6. (a)The points P(2, 1), Q(4, 2) and R(3, -5) are

the vertices of (PQR. (PQR is mapped

onto (PQR by a translation -1 .

2

(i)On graph paper, using 1 cm to

represent 1 unit on each axis, plot

the points P, Q, and R.

(ii)Write the position vectors of P in

the form a

b .

(iii)Calculate the vector OP, using

the sum of two vectors.

(iv)Calculate the co-ordinates of P,

Q and P.

(6 marks)

(b)(PQR is mapped by a reflection onto (PQR where P is (-4, 1) and Q(-6,2)

(i) Plot the points P and Q

(ii) Draw on the same axis the mirror line .

(iii) Write the equation of

(iv) Find the co-ordinates of R.

(5 marks)

7.(a)Given that 2 -3 x = -2 + 3

1 2 -1 y -1

calculate the values of x and y.

(5 marks)

(b)A straight line HK cuts the y-axis at H(0,-1).

The gradient of HK is 2/3

(i)Show that the equation of the line

HK is 2x 3y = 3.

(ii)On graph paper, using 2 cm to

represent 1 unit on each axis,

draw on the same axes the graph

of 2x 3y = 3 and x + y ( 4.

(iii) Shade only the region to represent both inequalities:

2x 3y ( 3 and x + y ( 4.

(iv)State the x values which satisfy

both inequalities.

(6 marks)

8.(a)The results of a test for defective bottles are

shown in the table below, where x is the

number of defectie bottles per carton, and n

is the number of cartons.

x01234

n1916573

Illustrate the information given above on a bar chart.

(3 marks)

(b)Each carton contains a total of 48 bottles.

Calculate, for the sample

(i) the total number of bottles

(ii) the total number of defective bottles.

(4 marks)

(c)The company produces 250 cartons each day

but only ships cartons which contain no

defective bottles. Using the results of the

above sample.

(i) calculate the probability of randomly choosing a carton which contain no defective bottles

(ii) estimate the number of cartons that is likely to be shipped in a five-day week. (4 marks)

SECTION II

Answer TWO questions in this section.

RELATIONS, FUNCTIONS AND GRAPHS

9.Use the answer sheet provided (page 34) to answer part

(a) of this question.

(a)(i)Indicate, on your answer sheet,

the value of x for which 2x =12

Stae this x-value.

(ii)Indicate also, the value of y for

which y = 121.4.

State this y-value.

(iii)Determine the gradient of the

curve y = 2x at x = 2, giving

your answer correct to 2

significant figures.

(8 marks)

(b)

The following above shows the velocity-time graph

BRSE of a moving body.

Calculate, clearly stating the units,

(i)the acceleration of the body during the first

two minutes.

(ii)the acceleration of the body during the last

minute

(iii)the total distance moved during the first ten

minutes.

(7 marks)

10.(a)Given that f(x) = 2x2 + x 3, copy and

complete the table below.

x-3-2-1012

F(x)1237

(3 marks)

(b)Using 2 cm to represent 1 unit on the x-axis,

1 cm to represent 1 unit on the y-axis, draw

the graph of

f(x) = 2x2 + x -3 for -3 x 2.

(4 marks)

(c)Using the graph,

(i)determine the value of x for the

minimum value of f(x)

(ii)determine the minimum value of

f(x)

(iii) state the values of x for f(x) > 0.

(4 marks)

(d)Using the same scale and the same axes as

in part (b)

(i) draw the graph of g(x) = 3x 1

(ii) shade the area which repersents f(x) g(x).

(4 marks)

TRIGONOMETRY AND GEOMETRY

11.(a)

In the diagram above, not drawn to scale, O

is the centre of the circle and AOB is a

diametre. D is a point on the circumference

and F is the midpoint of the chord BE.

Angle ABE = 70.

(i) Calculate angle BDE

(ii) Show that (OFB and (AEF are similar.

(iii) Calculate angle AOF.

(Note: State reasons and show necessary working.)

(9 marks)

(b)VMNPQ is pyramid on a square base

MNPQ of side 40 cm.

(i)Draw a diagram to represent the

pyramid. Clearly label the

vertices.

(ii)Draw a plan of the pyramid,

viewed from above.

State the scale used.

(iii)The height of the pyramid is 20

cm. Show that the length of the

sloping edge VM is 20 3 cm.

(6 marks)

12.(a)

(In this question take the radius of the earth to be 6400 km and take to be 3.142)(i)A ship X travelled due South from P(15W,

30 N to Q.

- Calculate the circmference of the circle that passes through P and

Q

-Given that the distance PQ is

3114 km, calculate the position

of Q.

(ii)Another ship Y travelled due East from

P(15 W, 30 N) to T(30 E, 30 N).

-Show that the radius of the circle

that passes through P and T is

6400 cos 30.

-Calculate the distance PT

(7 marks)

(b)In triangle VXR, VX = 5 cm, VR = 4 cm

and angle XVR = 55. Calculate to 3

significant figures

(i)the area of triangle VXR

(ii)the perpendicular distane of X

from VR

(iii)the magnitude of angle XRV

(8 marks)

VECTORS AND MATRICES

13.OFGH is a parallelogram where O(0, 0) is the origin. The position vectors of F is 3 and the position vectors

1

of g is 2 . M is the midpoint of OG.

4

(a)Sketch a diagram to show the position

vectors of F and G.

(2 marks)

(b)Determine the vector FG.

(3 marks)

(c)Show that the position vector

(i) of H is -1

3

(ii) of M is 1

2 .

(2 marks)

(d)Use vectors to show that

(i) the diagonals of the paralleogram of OFGH bisect one another

(ii) OFGH is a square.

(8 marks)

14.(a)Show, with the aid of diagrams, that

(i)the transformation represented by

the matrix 0 1 is not a

1 0 rotation.

(ii)the transformation represented by

the matrix 2 0 is not a

0 1 rotation.

(4 marks)

(b)Determine the matrix which represents a

reflection in the line y = x.

(5 marks)

(c) The matrix A= -1 0 and the matrix K= -1 0

0 -1

0 1

(i) Calculate AK

(ii) Identify the co-ordinates of the image of the point (5, 3) under the combined transformation represented by AK

(iii) Describe completely the combination of the two transformation represented by AK.

(6 marks)

Answer Sheet for Question 9 (a)

(Page 32)

June 1995 PAPER 2

SECTION I

Answer ALL the questions in this section.

1.All working must be clearly shown.

(a)Calculate 0.05181 ( 3.14 and write your

answer

(i)correct to 2 decimal places

(ii)correct to 3 significant figures

(iii)in standard form

(4 marks)

(b)Calculate the exact value of

(3 3/5 x 1 2/3) - 2 2/7

(3 marks)

(c)The simple interest on a sum of money

invested at 3% per annum for 2 years was

$39.75. Calculate the sum of money

invested.

(3 marks)

2.(a)Factorise completely:

(i) 9 25m2(ii) 2x2 x 15

(iii) x + y ax ay

(6 marks)

(b)Solve P 1 - P 2 = 1

2 3

(3 marks)

(c)If a * b = ab, where the positive root is

taken, calculate

(2 * 18) * 24.

(3 marks)

3.(a)Given that -1 0 and b = 5

0 1 4

(i) sketch vector a + b(ii) express a + b as a column vector

(iii) determine the magnitude of a + b

(4 marks)

(b)A survey of 156 visitors to the Caribbean

found that:

118 persons visited Barbados

98 persons visited Antigua

110 persons visited Tobago

25 persons visited Barbados & Antigua only

35 persons visited Barbados & Tobago only.

30 persons visited Tobago & Antigua only

x visitors had visited all three countries.

(i)Draw a carefully Labelled Venn

diagram to represent the

information above.

(ii)Write an algebraic expression in

x to represent the number of

travellers who visited Barbados

only

(iii)Write an equation in x to show

the total number of visitors in the

survey.

(iv)Calculate the number of

travellers who visited all three

countries.

(7 marks)

4.(a)A straight line is drawn through the points

A(-5, 3) and B(1, 2).

(i) Determine the gradient of AB

(ii) Write the equation of the line AB

(4 marks)

(b)(i)Using rulers and compasses only,

construct

- the triangle CAB with angle

CAB = 60, with AB = 8cm

and AC = 9 cm.

the perpendicular bisector

of AB to meet AC at X and

AB at Y.

(ii)Measure and state the length of

XY

(iii)Measure and state the size on

angle ABC.

Note:Credit will be given for construction lines

clearly shown.

(7 marks)

5.(a)In the trapezium DEFG shown here, not

drawn to scale, DE = 10 cm, DG = 13 cm

and GX = 5 cm. Angle EFX and DXF are

right angles.

SHAPE \* MERGEFORMAT

Calculate

(i) the length of DX.

(ii) the area of trapezium DEFG.

(5 marks)

(b)In this problem, take to be 22/7.

A piece of wire, formed into a circle,

encloses an area of 1386 cm2.

(i) Calculate the radius of the circle.

(ii) Calculate the length of the wire used to form the circle.

(iii) The wire is then bent to form a square. Calculate, in cm2, the area of the square.

(7 marks)

6.(a)In the diagram shown here, not drawn to

scale, PR = 7 cm, QO = 10 cm and angles at

R and S are right angles.

SHAPE \* MERGEFORMAT

Calculate

(i) the size of angle POR

(ii) the length of RS

(7 marks)

(b)

In the diagram above, the transformation M

followed by another transformation N, maps

triangle XUV on to triangle TUW.

Describe fully transformation M and N.

(5 marks)

7The table below shows the heights of a sample of

seedlings measured to the nearest centimetre.

Height (cm)1 - 34 - 67 910 1213 - 15

Frequency4142093

(a)Calculate

(i)the number of seedlings in the

sample.

(ii)the mean height of the seedlings

in the sample.

(4 marks0

(b)Draw a histogram to display the data

(4 marks)

(c)Calculate the probability that a seedling

chosen at random will measure 10 cm or

more in height.

(3 marks)

8.(a)Given that f(x) = x and g(x) = x 2

Calculate

(i) g(-2)

(ii) fg(4)

(iii) f-1(4)

(6 marks)

(b)Copy on graph paper the diagram here, and

show by shading ONLY, the region which

satisfies both the inequalities.

x > 5

x y

(5 marks)

SECTION II

Answer any TWO questions in this setion.

RELATIONS AND FUNCTIONS

9.(a)Given radians = 180, express

(i) /6 radians in degrees

(ii) 210 in radians

(2 marks)

(b)A function f is defined on the set of real

numbers by f(x) = sin x.

(i)Complete the table for given

values of x where x is stated in

radians and the value of f(x) is

given to two places of decimals.

(3 marks)

x0/62/63/64/65/66/67/6

F(x)00.500. 80. 5-0. 50

(ii)Using a scale of 2 cm to

represent a unit of /6 radians on

the x-axis and 5 cm to represent

1 unit on the y-axis, draw the

graph of f(x) for

0 x 7/6.

(6 marks)

(c)By drawing an appropriate straight line on

the graph of f(x), estimate the range of

values of x (in degrees) for which

sin x ( 0. 8.

(4 marks)

10. (a)Solve the following equations:

x2 + 9y2 = 37

x - 2y = -3

(8 marks)

(b)If x = 1 is one root of the equation

(x c)2 = 4(x + c + 2), calculate to 2

decimal places, the possible values of the

constant c.

(7 marks)

TRIGONOMETRY

11.In this question, take the radius of the earth to be 6400 km and use = 3. 14.

(a)Using the diagram here, not drawn to scale,

state the co-ordinates of:

(i) Point A

(ii) Point R

(iii) Point T

(3 marks)

(b)Calculate

(i)the latitude of Q, if Q lies 1600

km due north of A.

(ii)the distance in km between Q

and M, if M is due east of Q an

on longitude 20 E.

(iii)the longitude of L, if lies 1200

km due west of R.

(12 marks)

12.(a)Prove the identity

sin2 = (1 + cos) (1 cos )(2 marks)

(b)

In the triangle above, not drawn to scale, the

right angle and the angle are indicated.

Given that cos = 3/5, calculate the value of

sin + tan

(5 marks)

(c)

In the diagram above, not drawn to scale, O

is the centre of the circle LMNT and PTQ a tangent to the circle at T. Given that (LTQ = 65, calculate, stating your reasons, the size of:

(i)