CXC MATH PAST PAPER JAN 2021 Paper 2
LIST OF FORMULAE
Volume of a prism V = Ah where A is the area of a cross-section and h is the perpendicular
length.
Volume of a cylinder V = πr
2
h where r is the radius of the base and h is the perpendicular height.
Volume of a right pyramid V = — Ah where A is the area of the base and h is the perpendicular height.
Circumference C = 2Ï€r where r is the radius of the circle.
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Arc length S = —— × 2Ï€r where θ is the angle subtended by the arc, measured in
degrees.
Area of a circle A = πr
Area of a sector A = —— × Ï€r
2
where r is the radius of the circle.
2
where θ is the angle of the sector, measured in degrees.
Area of a trapezium A = — (a + b) h where a and b are the lengths of the parallel sides and h
is the perpendicular distance between the parallel sides.
Roots of quadratic equations If ax
2
+ bx + c = 0,
then x = ——————
Trigonometric ratios sin θ = —————————
cos θ = —————————
tan θ = —————————
Area of a triangle Area of Δ = — bh where b is the length of the base and h is the perpendicular
height.
Area of Δ ABC = — ab sin C
Area of Δ ABC = √ s (s – a) (s – b) (s – c)
where s = ————
Sine rule —— = —— = ——
Cosine rule a
a
sin A
2
1
3
θ
360
θ
360
1
2
= b
2
–b + √ b
+ c
1
2
2
– 4ac
2a
2
length of opposite side
length of hypotenuse
length of adjacent side
length of hypotenuse
length of opposite side
length of adjacent side
a + b + c
2
b
sin B
1
2
c
sin C
– 2bc cos A
SECTION I
Answer ALL questions.
All working must be clearly shown.
1. (a) (i) Using a calculator, or otherwise, calculate the EXACT value of
1
4
7
—
+
2
3
—
– 1
5
6
—
.
.................................................................................................................................
(2 marks)
(ii) Write the value of
√ 27
3
——
9
2
as a fraction in its LOWEST terms.
.................................................................................................................................
(2 marks)
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(b) The thickness of one sheet of cardboard is given as 485 × 10
mm. A construction worker
uses 75 sheets of the cardboard, stacked together, to insulate a wall.
(i) Show that the exact thickness of the insulation is 363.75 mm.
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.................................................................................................................................
(1 mark)
(ii) Write the thickness of the insulation
a) correct to 2 signiicant igures
........................................................................................................................
(1 mark)
b) correct to 1 decimal place
........................................................................................................................
(1 mark)
c) in standard form.
........................................................................................................................
(1 mark)
(c) Marko is on vacation in the Caribbean. He changes 4500 Mexican pesos (MXN) to
Eastern Caribbean dollars (ECD). He receives 630 ECD.
Complete the statement below about the exchange rate.
1 ECD = ................................. MXN (1 mark)
Total 9 marks
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2. (a) Factorize the following expression completely.
12n
2
– 4mn
..............................................................................................................................................
(1 mark)
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(b) (i) Show that
x
1 – x
——
– 4x =
x(4x – 3)
1 – x
————
.
.................................................................................................................................
(2 marks)
(ii) Hence, solve the equation
x
1 – x
——
– 4x = 0.
.................................................................................................................................
(2 marks)
(c) Make v the subject of the formula p = √ 5 + vt .
..............................................................................................................................................
(2 marks)
(d) The distance needed to stop a car, d, varies directly as the square of the speed, s, at which
it is travelling. A car travelling at a speed of 70 km/h requires a distance of 40 m to make
a stop. What distance is required to stop a car travelling at 80 km/h?
..............................................................................................................................................
(2 marks)
Total 9 marks
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3. (a) The diagram below shows two pentagons, P and Q, drawn on a grid made up of squares.
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(i) Select the correct word from the following list to complete the statement below.
opposite relected congruent translated
Pentagon P is ...................................................................... to Pentagon Q.
(1 mark)
(ii) Give the reason for your choice in (a) (i).
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
(1 mark)
(b) The diagram below, not drawn to scale, shows the pentagon VWXYZ. In the pentagon,
YZ is parallel to XW and YX is parallel to VW. Angle XYZ = 114° while angle VZY = 98°.
Determine the value of
(i) angle WXY
.................................................................................................................................
(1 mark)
(ii) angle ZVW.
.................................................................................................................................
(2 marks)
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(c) The letter ‘A’ and a point C(6, 6) are shown on the grid below.
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On the diagram, draw accurately, EACH of the following transformations.
(i) The enlargement of letter ‘A’ by scale factor 2, about centre, C(6, 6). (2 marks)
(ii) The translation of letter ‘A’ using the vector T =
–3
. (2 marks)
2
Total 9 marks
4. (a) The function f is deined as
f : x → 3 – 2x.
(i) The diagram below shows the mapping diagram of the function, f. Determine the
value of a.
a = ..........................................................................................................................
.................................................................................................................................
(1 mark)
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(ii) Determine, in their simplest form, expressions for
a) the inverse of the function f, f
–1
(x)
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........................................................................................................................
(1 mark)
b) the composite function f
2
(x).
........................................................................................................................
(2 marks)
(iii) State the value of f f
–1
(–2).
.................................................................................................................................
(1 mark)
(b) (i) Using a ruler, draw the lines x =
1
(ii) On the grid, label as R, the region where x >
1
2
—
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, y = x and x + y = 5, on the grid below.
(3 marks)
—
2
, y > x and x + y < 5. (1 mark)
Total 9 marks
5. (a) Sixty students took an algebra test, which comprised 15 multiple choice questions. The
number of correct answers that each student obtained is recorded in the table below.
Number of
Correct Answers
Number of
Students
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Using the table, determine
8 6
9 14
10 2
11 6
12 2
13 11
14 9
15 10
(i) the number of students who had exactly 13 correct answers
.................................................................................................................................
(1 mark)
(ii) the modal number of correct answers
.................................................................................................................................
(1 mark)
(iii) the median number of correct answers
.................................................................................................................................
(1 mark)
(iv) the probability that a student chosen at random had at least 12 correct answers.
..................................................................................................................................
(1 mark)
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(b) A group of students wrote a Physics examination. Each of the students achieved a Grade
I, II, III or IV. The pie chart below shows the results.
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Thirty-nine students achieved a Grade III.
(i) Determine the TOTAL number of students who wrote the examination.
.................................................................................................................................
(2 marks)
(ii) The ratio of the number of students who achieved a Grade I, II or IV is 2:4:3. A
student passed the examination if he/she achieved a Grade I, II or III.
How many students passed the examination?
.................................................................................................................................
(2 marks)
(iii) Determine the value of the angle for the sector representing Grade I in the pie
chart.
.................................................................................................................................
(1 mark)
Total 9 marks
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6. In this question, take π to be
22
7
—
.
The diagram below shows a rectangular tank, with base 50 cm by 40 cm, that is used to store
water. The tank is illed with water to a depth of 15 cm.
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40 cm
(a) Calculate the volume of water in the tank.
50 cm
15 cm
..............................................................................................................................................
(2 marks)
(b) The cylindrical container shown in the diagram below is used to fetch more water to ill
the rectangular tank. The container, which is completely illed with water, has a radius of
20 cm and a height of 21 cm.
All the water in this container is added to the water in the rectangular tank. Calculate the
TOTAL volume of water that is now in the rectangular tank.
..............................................................................................................................................
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(3 marks)
(c) Show that the new depth of water in the rectangular tank is 28.2 cm.
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..............................................................................................................................................
(2 marks)
(d) The vertical height of the rectangular tank is 48 cm. Determine how many more cylindrical
containers of water must be poured into the rectangular tank for it to be completely illed.
.............................................................................................................................................
(2 marks)
Total 9 marks
7. The diagrams below show a sequence of igures made up of circles with dots. Each igure has one
dot at the centre and 4 dots on the circumference of each circle. The radius of the irst circle is
one unit. The radius of each new circle is one unit greater than the radius of the previous circle.
Except for the irst igure, a portion of each of the other igures is shaded.
Figure 1 Figure 2 Figure 3 Figure 4
(a) Complete the rows in the table below for Figure 5 and Figure n.
Figure
Number
Number of
Dots
Area of
Outer
(Largest)
Circle
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Area of
Shaded
Region
1 5 π π 2π
2 9 4Ï€ 3Ï€ 6Ï€
3 13 9Ï€ 5Ï€ 12Ï€
4 17 16Ï€ 7Ï€ 20Ï€
Total Length of
Circumference
of all Circles
5 _________ 25Ï€ _________ _________
(i) (3 marks)
…
…
…
…
…
n
_________ _________ _________ _________
(ii) (4 marks)
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(b) Determine the value of n, when the number of dots in Figure n is 541.
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..............................................................................................................................................
(2 marks)
(c) Write down, in terms of p and π, the area of the LARGEST circle in Figure 3p.
..............................................................................................................................................
(1 mark)
Total 10 marks
SECTION II
Answer ALL questions.
ALGEBRA, RELATIONS, FUNCTIONS AND GRAPHS
8. (a) The straight line graph of x = 5 – 3y intersects the x-axis at P and the y-axis at Q.
(i) Determine the coordinates of P and Q.
P (.........., ..........) Q (.........., ..........) (2 marks)
(ii) Calculate the length of PQ, giving your answer to 2 decimal places.
.................................................................................................................................
(2 marks)
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(iii) R is the midpoint of PQ. Determine the coordinates of R.
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.................................................................................................................................
(1 mark)
(b) The functions f and g are deined as follows
f : x → 5 – x and g : x → x
2
– 2x – 1.
The graphs of f(x) and g(x) meet at points M and N. Determine the coordinates of the
points M and N.
..............................................................................................................................................
(4 marks)
(c) Monty is cycling at 12 metres per second (m/s). After 4.5 seconds he starts to decelerate
and after a further 2.5 seconds he stops. The speed–time graph is shown below.
Calculate
(i) the constant deceleration
.................................................................................................................................
(1 mark)
(ii) Monty’s average speed over the 7 seconds.
.................................................................................................................................
(2 marks)
Total 12 marks
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GEOMETRY AND TRIGONOMETRY
9. (a) In the diagram below, A, B, C and D are points on the circumference of a circle, with centre
O. AOC and BOD are diameters of the circle. AB and DC are parallel.
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(i) State the reason why angle ABC is 90°.
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
.................................................................................................................................
(1 mark)
(ii) Determine the value of EACH of the following angles. Show detailed working
where necessary and give a reason to support your answer.
a) Angle BAC
Reason
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
(2 marks)
b) Angle q
Reason
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
...........................................................................................................................
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(2 marks)
(iii) Calculate the value of angle r.
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(b) From a harbour, H, the bearing of two buoys, S and Q, are 185° and 311° respectively. Q
is 5.4 km from H while S is 3.5 km from H.
(i) On the diagram below, which shows the sketch of this information, insert the value
of the marked angle, QHS. (1 mark)
Q
S
H
N
185°
(1 mark)
(ii) Calculate QS, the distance between the two buoys.
.................................................................................................................................
(2 marks)
(iii) Calculate the bearing of S from Q.
.................................................................................................................................
(3 marks)
Total 12 marks
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10. (a) Given the matrix W =
3 6
VECTORS AND MATRICES
–2 5
, determine
(i) the 2 × 2 matrix, L, such that W + L =
0 0
0 0
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.................................................................................................................................
(1 mark)
(ii) the 2 × 2 matrix, P, such that WP =
1 0
0 1
.
.................................................................................................................................
(2 marks)
(b) A right-angled triangle, M, has vertices X(1, 1), Y(3, 1) and Z(3, 4). When M is transformed
by the matrix N =
0 1
1 0
, the image is M′.
Find the coordinates of the vertices of M′.
..............................................................................................................................................
(2 marks)
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(c) The diagram below shows triangle OPQ in which
OP
= 3u and
OQ
= v. Q is the midpoint
of OR and M is the midpoint of PQ. L is a point on OP such that OL =
2
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3
—
OP.
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(i) Write in terms of u and v, an expression for
a)
LM
...........................................................................................................................
(2 marks)
b)
PR
.
.................................................................................................................................
(1 mark)
(ii) Prove that the points L, M and R are collinear.
..................................................................................................................................
(4 marks)
Total 12 marks
END OF TEST
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