Math PP May 2007





Paper 02—General Proficiency

2 hours 40 minutes

24 May 2007 (am)




  1.  Answer ALL questions in Section I and ANY TWO in Section II.
  2. Write your answers in the booklet provided.
  3. All working must be shown clearly.
  4. A list of formulae is provided on page 2 of this booklet.

Required Examination Materials

Electronic Calculator

Geometry Set

Graph Paper (Provided)




Volume of a prisim

V= Ah where A is the area of a cross-section and h is the perpendicular length.

Volume of cylinder

V=∏r2h where r is the radius of the base and h is the perpendicular height.


C=2∏r where r is the radius of the circle.

Area of a circle

A=∏r2 where r is the radius of the circle.

Area of a trapezium

A=1/2 (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 ax2 + bx + c = 0; then x = -b + √b2 – 4ac/ 2a

Trigonometric Ratios

Sin Ѳ = Opposite side/Hypoteneuse

Cos Ѳ = Adjacent side/Hypoteneuse

Tan Ѳ = Opposite side/Adjacent Side


 Area of triangle

Area of triangle= 1/2 bh where b is the length of the base and h is the perpendicular height.

Area of triangle ABC= 1/2 ab sinC

Area of triangle ABC=√s(s-a) (s-b) (s-c) where s = (a+b+c)/2

area of triangle

Sine rule





Answer ALL the questions in this section.

All working must be clearly shown.

1. (a) Using a calculator, or otherwise, determine the exact value of May 2007 S1 1

(3 marks)

(b) A total of 1,200 students attended Top View High School.

The ratio of teachers to students is 1:30.

(i) How many teachers are there at the school?

(2 marks)

Two-fifths of the students own personal computers.

(ii) How many students do NOT own personal computers?

(2 marks)

Thirty percent of the students who own personal computers also own play stations.

(iii) What fraction of the students in the school own play stations?

Express your answer in its lowest terms.

(4 marks)

Total 11 marks

2. (a) Given that May 2007 S1 2


(i) 4*8

(ii) 2*(4*8)

(4 marks)

(b) Simplify, expressing your answer in its simplest form.

May 2007 S1 2b

(2 marks)

(c) A stadium has two sections A and B.

Tickets for section A cost $a each.

Tickets for section B cost $b each.

Johanna paid $105 for 5 Section A tickets and 3 Section B tickets.

Raiyah paid $63 for 4 Section A tickets and and 1 Section B ticket.

(i) Write two equations in a and b to represent the information above.

(ii) Calculate the values of a and b.

(5 marks)

Total 11 marks

3. The Venn Diagram below represents information on the type of games played by members of a youth club. All members of the club play at least one game.

May 2007 S1 3

S represents the set of members who play squash.

T represents the set of members who play tennis.

H represents the set of members who play hockey.

Leo, Mia and Neil are three members of the youth club.

(i) State what game(s) is/are played by

(a) Leo

(b) Mia

(c) Neil

(ii) Describe in words the members of the set May 2007 S1 3a

(5 marks)

(b) (i) Using a pencil, a ruler and a pair of compasses only,

  (a) Construct a triangle PQR in which QR = 8.5 cm, PQ = 6 cm   and PR = 7.5 cm.

  (b) Construct a line PT such that PT is perpendicular to QR       and meets QR at T.

(ii) (a) Measure and state the size of angle PQR.

     (b) Measure and state the length of PT.

(7 marks)

Total 12 marks

4. (a) The diagram below shows the map of a golf course drawn on a grid of 1 cm squares. The scale of the map is 1:4000.

May 2007 S1 4

Using the map of the golf course, find:

(i) the distance, to the nearest m, from South Gate to East Gate.

(ii) the distance, to the nearest m, from North Gate to South Gate.

(iii) the area on the ground represented by 1 cmon the map.

(iv) the actual area of the golf course, giving the answer in square metres.

(6 marks)

(b) The diagram below, not drawn to scale, shows a prism of volume 960 cm3. The cross section ABCD is a square. The length of the prism is 15 cm.

May 2007 S1 4b


(i) the length of the edge AB, in cm.

(ii) the total surface area of the prism in cm2.

(5 marks)

Total 11 marks

5. Two variables, x and y are related such that ‘y varies inversely as the square of x’.

(a) Write an equation in x, y and k to describe the inverse variation, where k is the constant of variation.

(2 marks)

(b)              May 2007 S1 5b

Using the information in the table above, calculate the value of

(i) k, the constant of variation

(ii) r

(iii) f

(6 marks)

(c) Determine the equation of the line which is parallel to the line y = 2x + 3 and passes through the coordinate (4,7).

(4 marks)

Total 12 marks

6. (a) Answer sheet is provided for this question. 

L’M’N’ is the image of LMN under an enlargement.

(i) Write on your answer sheet

(a) the scale factor for the enlargement

(b) the coordinates of the centre of the enlargement.

L”M”N” is the image of LMN under a reflection in the line y=-x.

(ii) Draw and label the triangle L”M”N” on your answer sheet.

(5 marks)

(b)                May 2007 S1 6b


Three towns P, Q and R are such that the bearing of P from Q is 070°. R is 10 km due east of Q and PQ = 5 km.

(i) Calculate, correct to one decimal place, the distance PR.

(ii) Given that angle QPR is 142°, state the bearing of R from P.

(6 marks)

Total 11 marks

7. A class of 32 students participated in running a 400 m race in preparation for their sports day. The time, in seconds, taken by each student is recorded below.

May 2007 S1 7

(a) Copy and complete the frequency table to represent this data.

May 2007 S1 7a

(2 marks)

(b) Using the raw scores, determine the range for the data.

(2 marks)

(c) Using a scale of 2 cm to represent 5 seconds on the horizontal axis and a scale of 1 cm to represent 1 student on the vertical axis, draw a frequency polygon to represent the data.

NOTE: An empty interval must be shown at each end of the distribution and the polygon closed.

(6 marks)

(d) To qualify for the finals, a student must complete the race in less than 60 secs. What is the probability that a student from this class will qualify for the finals?

(2 marks)

Total 12 marks

8. Rectangle WXYX below represents one whole unit which has been divided into seven smaller parts.

May 2007 S1 8

(a) Copy and complete the following table, stating what fraction of the rectangle each part represents.

May 2007 S1 8a

(5 marks)

(b) Write the parts in order of the size of their perimeters.

(2 marks)

(c) The area of G is 2 square units. E, F and G are rearranged to form a trapezium.

(i) What is the area of the trapezium in square units?

(ii) Sketch the trapezium clearly showing the outline of each of the 3 parts.

(3 marks)

Total 10 marks


Answer TWO questions in this section.


9. (a) Given that g(x) = 2x+1/5 and f(x) = x+4

(i) Calculate the value of g(-2).

(ii) Write an expression for gf(x) in its simplest form.

(iii) Find the inverse function g-1(x).

(7 marks)

(b) The length of the rectangle below is (2x – 1) cm and its width is (x + 3) cm.

May 2007 S1 9b

(i) Write an expression in the form ax2 + bx + c for the area of the rectangle.

(ii) Given that the area of the rectangle is 294 cm2 , determine the value of x.

(iii) Hence, state the dimensions of the rectangle in centimetres.

(8 marks)

Total 15 marks

10. A company manufactures gold and silver stars to be used as party decorations. The stars are placed in packets so that each packet contains x gold stars and y silver stars.

The conditions for packaging are given in the table below.

May 2007 S1 10

(a) Write down the inequalities to represent conditions (2) and (3).

(2 marks)

(b) Describe, in words, the condition presented by the inequality x < 2y.

(2 marks)

(c) Using a scale of 2 cm to represent 10 units on both axes, draw the graphs of ALL FOUR inequalities represented in the table above.

(7 marks)

(d) Three packets of stars were selected for inspection. Their contents are shown below.

May 2007 S1 10d

Plot the points A, B and C on your graph. Hence determine which of the three packets satisfy ALL the conditions.

(4 marks)

Total 15 marks


11.Given that May 2007 S1 11

(i) Express in fractional or surd form.

    (a) cos Ѳ

    (b) tan Ѳ

(ii) Hence, determine the exact value of May 2007 S1 11a

(7 marks)

(b) For this question take ∏=3.14 and R=6370 km, where R is the radius of the earth. The diagram below, not drawn to scale, shows a sketch of the earth with the North and South poles labelled N and S respectively. The circle of latitude 37°N is shown.

May 2007 S1 11b

(i) Calculate to the nearest kilometer, the length of the circle of latitude 37°N.

(ii) Two towns, A and B, have co-ordinates (37°N, 50°W) and (37°N, x°E) respectively. The distance from A to B measured along their common circle of latitude is 5390 km, calculate the value of x.

(8 marks)

Total 15 marks

12. (a) The figure below, not drawn to scale, is a rectangular octagon with centre X, and XY=6cm.

May 2007 S1 12


(i) the size of YXZ

(ii) the area of the triangle YXZ, expressing your answer correct to one decimal place.

(iii) area of the octagon

(6 marks)

(b) In the diagram below, not drawn to scale, LM is a tangent to the circle at the point, T. O is the centre of the circle and angle MTS=23°.

May 2007 S1 12b

Calculate the size of each of the following angles, giving reasons for your answer

a) angle TPQ

b) angle MTQ

c) angle TQS

d) angle SRQ

(9 marks)

Total 15 marks


13.                  May 2007 S1 13

OK and OM are the position vectors such that May 2007 S1 13a

(a) Sketch the diagram above. Show the approximate positions of points R and S such that

R is the mid-point of OK

S is a point on OM such that May 2007 S1 13ai

(2 marks)

(b) Write down in terms of k and m the vectors,

May 2007 S1 13b

(8 marks)

(c) L is the midpoint of RM. Using a vector method, prove that RS is parallel to KL.

(5 marks)

Total 15 marks

14. (a) A, B and C are three 2×2 matrices such that May 2007 S1 14


(i) 3A

(ii) B-1

(iii) 3AB-1

(iv) the value of a, b, c and d given that 3A +B-1 =C.

(7 marks)

(b) The diagram below shows a parallelogram EFGH and its images after undergoing two successive transformations.

May 2007 S1 14c

(i) Describe in words the geometric transformations

    a) J which maps EFGH onto E’F’G’H

    b) K which maps E’F’G’H onto E”F”G”H”

(ii) Write the matrix which represents the transformation described above as

    a) J

    b) K

(iii) the point P (6,2) is mapped onto P’ by the transformation J’. State the co-ordinates of P’.

(iv) The point Q (5,-4) is mapped onto Q’ by the transformation K. State the coordinates of Q’.

(8 marks)

Total 15 marks


May 2007 S1 14d

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