2011_maths_olevel_mj_paper2

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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Ordinary Level

MATHEMATICS 4028/2
PAPER 2

JUNE 2011 SESSION 2 hours 30 minutes

Additional materials:
Answer paper
Geometrical instruments
Graph paper (3 sheets)
Mathematical tables
Plain paper (1 sheet)

TIME 2 hours 30 minutes
INSTRUCTIONS TO CANDIDATES

Write your name, Centre number and candidate number in the spaces provided on the
answer paper/answer booklet.

Answer all questions in Section A and any three questions from Section B.

Write your answers on the separate answer paper provided.
If you use more than one sheet of paper, fasten the sheets together.

All working must be clearly shown. It should be done on the same sheet as the rest of the answer,
Omission of essential working will result in loss of marks.

If the degree of accuracy is not specified in the question and if the answer is not exact, the

answer should be given to three significant figures. Answers in degrees should be given

to one decimal place. .

INFORMATION FOR CANDIDATES

The mimber of marks is given in brackets [ ] at the end of each question or part question| .
Mathematical tables or electronic calculators may be used to evaluate explicit
numerical expressions.

This question paper consists of 11 printed pages and 1 blank page.

Copyright: Zimbabwe School Examinations Council, J2011.
OZIMSEC J2011 (Turn over

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(b)

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(d)

(a)

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Section A [64 marks]

Answer all the questions in this section.

1 ,3 2
implify |3—-i—|+i-.
Simp a > 2)« 3

Find the exact value of
(i) 10,03 x 0,17;

(i) 7,2 +0,018.

Three farmers share 120 hectares of land in the ratio 3:4: 5.

Calculate the area of the largest piece of land.
Find the Highest Common Factor (H.C.F) of

27x" yz,
T2xy*2" and
108.xyz°.

It is given that r = 2g — 5 and q = 3p +2.

(13) Express r in terms of p.

Gi) Given also that p = —3, find the numerical value of r.

@ Express 5+ *

(ii) Hence or otherwise solve the equation zt

Express 150 g as a percentage of 3 kg.

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x-4

as a single fraction in its simplest form.

= 4.

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[2]

[2]

[2]

[3]

[4]

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(a)

The diagram shows points A, B, C and D on the circumference of a circle
such that BC = CD. AC and BD intersect at F and CD is produced to E.
ACD = 33° and ADE = 87°. Calculate

@) ABD,
(ii) CBD,
(iii) ~~ BAC,
(iv) ADB,
(vy) CED.

16]

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(b)

Take x to be 3,142.

The diagram shows a sector of a circle centre 0. POQ = 37°, PO= 8 cm
and PT is perpendicular to QO. Calculate

(i) = PT,
(i area of the sector,
(iii) area of triangle PQO,

A (iv) the area of the shaded segment.

[7]

(a) _ Factorise completely
@ 3np — 6ng + ap - 2aq,
(i) 12- 4g-¢7. 14
(b) Inasurvey of 72 girls, it was found that every girl watched at least one
of the following TV programmes, Teen Scene or Fashion Show. Fifty
girls watched Teen Scene and 62 girls watched Fashion Show. Find the
number of girls who watched

@ both programmes,

(ii) Fashion Show only. [4]

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5 fa) i) Convert 1123 to a number in base 5. (ii) Evaluate 11012 + 10112, giving your answer in base 2. [3]

(b) =} = Show that 2log,(3x+2)-log,2 =1 reduces to 3x7+4x-2=0.

(ii) Solve the equation 3x7 + 4x-2 = 0, giving your answers correct
to two decimal places. (9]

Answer the whole of this question on a sheet of plain paper.

Use ruler and compasses only and show clearly all construction lines and arcs.
All constructions should be in a single diagram.

4
A secondary school, Baterekwa; is to be built to service three primary schools
Tugwi, Shinga and Hlangu, Shinga is 16 km from Tugwi on a bearing of 150°.
Hlangu is 15 km North-East of Shinga.

(a) Using a scale of 1 cm to represent 2 km, construct a diagram to show
the positions of the three primary schools. [7]

(b) _ Boterekwa is 8 km from Shinga and equidistant from Tugwi and Hlangu.

{i) Construct the locus of points 8 km from Shinga and the locus of
points equidistant from Tugwi and Hlangu

(ii) Mark and label B, and By the possible positions of Boterekwa. [4]

(©) Use your diagram to find the shorter distance between Boterekwa and

Tugwi. [2]

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6 Section B [36 marks] Answer any three questions in this section.

Each question carries 12 marks,

3 -4
7 (a) Given that A= ( }
1 -2
a) find the inverse of matrix A,

(ii) __ hence or otherwise solve the equations

‘3x-4y =-3,
ma (5]
x2y =-2,
(b)
The diagram shows a trapezium OABC where OC is parallel to AB,
with OA = x and OC = = y. Diagonals OB and AC intersect at D such
that AD : DC =3: 2,
Express, in terms of x and/or y,
@® @ AC,
(b) AD.
(ii) Given that AB = KOC, express OB in terms of &, x and y.
(iii) = Given also that OB = AOD, express OB in terms of A, x and ye
(iv) Using resuits from (ii) and (iii) above, find the numerical value
of # and the numerical value of k.
[7]

A

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7

Answer the whole of this question on a sheet of graph paper.

Triangle ABC has vertices A(2; 1), B(4; 1) and C(4; 4). Using a scale of 1 cm to

represent | unit on both axes, draw the x and y-axes for 4< x =14 and
-10sy<12.

(a) Draw and label clearly triangle ABC.

(b) Triangle AyByC; is a reflection of triangle ABC in the line y = -2.
Draw and label clearly triangle AiBiC).

(ce) Triangle AgBoC, has vertices A2(6; 4) Bo(10; 4) and C,(10; 10).

i Draw and label clearly triangle A2B2Co.

(i) Describe fully the single transformation that maps triangle ABC

onto triangle A2B2C>.

(d) Triangle ABC is rotated through 90° anticlockwise about (0; 0) onto

triangle A3B3C3.
(i) Draw and label clearly triangle A3B3C3.

(ii) = Write down the matrix that represents this transformation.

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[4]

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8 9 Answer the whole of this question on a sheet of graph paper.

The table below shows the grades obtained by candidates in a Mathematics
examination.

Gea Gan GG

(a) Calculate the number of candidates who wrote the examination. {2]

(b) State the modal grade. eh]

(c) Using a scale of 2 cm to represent 5 candidates, draw a bar graph to show
the information. [4]

(d) Given that the passing grades are A, B and C, find the probability that
two candidates chosen at random passed the examination. [3]

(e) Ifa pie-chart is drawn for this information, calculate the angle of the
sector that represents grade C. [2]

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{a)

(b)

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The length, /, of a rectangle of constant area varies inversely as 5,
the width of the rectangle.

@ State the relationship between /, 5 and a constant k.

(ii) Given that /= 8,5 cm when b = 6 cm, find b when /= 10,2 cm.

Answer this part of the question on a sheet of graph paper.

Using a scale of 1 cm to represent 1 unit on the x-axis and 2 cm to
represent I unit on the y-axis draw the x and y axes for 0< x <14
and-3 sy<5,

(i) Show by shading the unwanted regions, the region which is

defined by x 2 3,
ye ~2,
and x+2y <8.

(ii) From the region defined, find the coordinates of a point that
gives a maximum value of 3x-2y.

(iii) State the maximum value of 3x-2y.

(4]

[8]

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(2)

(b)

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The pattern below refers to the number of elements in a set and the
number of subsets of that set. Study the pattern and answer the questions
that follow.

Number of elements in a set Number of subsets

(i) Find the value of p and the value of g.
(ii) Express r in terms of n. (4]
Mbudzi Investments borrowed $6 000 from a bank to start a project.

The bank charged them interest and expected the company to pay
$120 per month to service the loan. The following is an incomplete

Loan Account Statement for Mbudzi Investments.
Debit Credit Balance
$ $ $

6 000
Interest 6 080
Repayment 5 960
Interest 6 039,47
Repayment 3 919,47
Interest 5 998,40
Repayment 5 878,40
Interest x

Repayment z

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@ Find the rate of simple interest per annum.

(ii) Calculate the values of

(@) ™,
(b) ox,
© »,
@ z

[8]

2 Answer the whole of the question on a sheet of graph paper.
The velocity of a particle moving along a straight line is given by v= 15+ 7¢-2f.

The table below shows corresponding values of v and t.

ror PPP

(a) _ Find the value of p and the value of q. [2}

(b) ~—_— Using a scale of 2 cm to represent 5 m/s on the y-axis and 2 cm to

represent 1 second on the axis, draw a graph of

v= 15+ 7¢~2f from t=0 to f= 6. [4]
(ce) Use your graph to estimate

@ the maximum value of y,

(ii) _ the acceleration of the particle when t = 3 seconds,

(ii) _ the distance travelled by the particle from ¢=0 to 1=5. [6]

ae

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