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)