ZIMSEC O-Level Mathematics Paper 2 June 2015 Question Paper

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL <br> General Certificate of Education Ordinary Level

**MATHEMATICS <br> PAPER 2**

4028/2 JUNE 2015 SESSION 2 hours 30 minutes

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

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 number 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, J2015.

**Section A [64 marks]** Answer all the questions in this section.

Question 1

(b)

Simplify

#### (i)

\(27^{\frac{2}{3}}-\left(\frac{1}{8}\right)^{\frac{1}{3}}\),

#### (ii)

\(3 \frac{4}{5}-\left(1 \frac{2}{3}+\frac{7}{15}\right)\), giving the answer in its lowest terms.

(c)

Given that \(p=0,045\) and \(r=2,513 \times 10^{-4}\),

#### (i)

express \(p\) in standard form,

#### (ii)

evaluate \(pr\) giving the answer in standard form.

Question 2

(a)

It is given that \(\xi=\{x: 1 \leq x \leq 10\), where \(x\) is an integer \(\}\)

$$ \begin{aligned} & A=\{x: x \text { is a prime number }\} \text { and } \\ & B=\{x: x \text { is a factor of } 20\} . \end{aligned} $$

#### (i)

List all the elements of A .

#### (ii)

List all the elements of \((\mathrm{A} \cup \mathrm{B})^{\prime}\).

#### (iii)

Find \(n(A \cap B)\).

#### (iv)

Draw a clearly labelled Venn diagram to show the sets and their elements.

(b)

Solve the equation \(\frac{3 x+1}{3}-\frac{x-4}{5}=\frac{1}{2}\).

Question 3

(a)

It is given that 300 cattle are to be shared in the ratio \(12: 10: 8\).

#### (i)

Express the ratio in its simplest form.

#### (ii)

Calculate the difference between the largest and smallest shares.

(b)

An advert in a shop read, "VALENTINE SPECIAL 25 % OFF ALL RED SHIRTS". The red shirts were originally marked at \(\$ 25,00\) each.

#### (i)

Joko bought a red shirt.

Calculate the amount that Joko paid for the shirt.

#### (ii)

Tindo bought 10 such shirts and sold them at \(\$ 23,00\) each.

Calculate the total profit Tindo made.

Question 4

(a)

The diagram shows points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D on the circumference of a circle centre O . EAF is a tangent to the circle at A .

Given that \(\mathrm{BAF}=40^{\circ}\) and \(\mathrm{CAD}=30^{\circ}\), calculate

#### (i)

\(\mathrm{A} \hat{\mathrm{D}} \mathrm{B}\),

#### (ii)

\(\mathrm{D} \hat{\mathrm{A}} \mathrm{E}\),

#### (iii)

\(\mathrm{A} \hat{\mathrm{B}} \mathrm{F}\),

#### (iv)

\(\mathrm{C} \hat{\mathrm{D}} \mathrm{B}\)

(b)

#### (i)

Convert \(65_{10}\) to a number in base 3 .

#### (ii)

Simplify \(3102_{4}+11101_{2}\), giving the answer in base 4 .

Question 5

(a)

Given that \(\mathrm{f}(x)=3 x^{2}-7 x+1\),

#### (i)

evaluate \(\mathrm{f}(-1)\),

#### (ii)

find the values of \(x\) when \(\mathrm{f}(x)=-1\).

(b)

Given that \(P=\frac{n}{2}\{2 a+(n-1) d\}\),

#### (i)

express \(a\) in terms of \(d, n\), and \(P\),

#### (ii)

find the value of \(a\) when \(n=10, d=4\) and \(P=20\).

Question 6

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 done on a single diagram.

From a point, M , on level ground, the angle of elevation of a bird on top of a vertical pole is \(30^{\circ}\). From another point, N, 10 metres closer to the pole such that M and N are on the same side of the pole, the angle of elevation of the same bird is \(45^{\circ}\).

(a)

Using a scale of 1 cm to represent 2 metres, construct a diagram to show the positions of \(M, N\) and the vertical pole.

(b)

Use the diagram to find the

#### (i)

height of the pole,

#### (ii)

distance of M from the bottom of the pole.

**Section B [36 marks]**

Answer any three questions in this section

Question 7

(a)

\(M\) is directly proportional to \((d-1)^{2}\). Given that \(M=12\) when \(d=4\), calculate \(M\) when \(d=7\).

(b)

The diagram ABCDE is a cross-section of a tobacco shed that is 20 m long. \(\mathrm{AB}=\mathrm{BC}=\mathrm{CD}=\mathrm{DE}=5 \mathrm{~m}\) and \(\mathrm{AE}=8 \mathrm{~m}\).

Calculate the

#### (i)

perpendicular height of C above the side BD ,

#### (ii)

area of the cross-section ABCDE ,

#### (iii)

volume of the shed,

#### (iv)

number of bales of tobacco that can be stored in the shed up to BD , given that each bale has a volume of \(4 \mathrm{~m}^{3}\).

Question 8

Answer the whole of this question on a sheet of graph paper. A girl is given \(\$ 6,00\) to buy fireworks for her birthday party. She buys \(x\) rockets at 60 c each and \(y\) crackers at 30 c each.

(a)

Write down an inequality in \(x\) and \(y\) and show that it reduces to \(2 x+y \leq 20\).

(b)

She wants to buy at least 4 rockets and the number of crackers should be more than or equal to twice the number of rockets.

Write down two inequalities that satisfy these conditions.

(c)

Using a scale of 2 cm to 2 units on both axes, show by shading the UNWANTED regions, the region in which \((x ; y)\) must lie.

(d)

Use your graph to find

#### (i)

the combination that uses the maximum amount of money available,

#### (ii)

1. the combination that uses the minimum amount of money, 2. the change she would get in (ii) 1 .

Question 9

Answer the whole of this question on a sheet of graph paper. Quadrilateral ABCD has vertices at \(\mathrm{A}(1 ; 0), \mathrm{B}(2 ; 0) \mathrm{C}(2 ; 2)\) and \(\mathrm{D}(1 ; 2)\) Using a scale of 2 cm to represent 1 unit on each axes, draw the \(x\) and \(y\) axes for \(-4 \leq x \leq 6\) and \(-5 \leq y \leq 5\).

(a)

#### (i)

Draw and label \(A B C D\).

#### (ii)

State the special name given to quadrilateral ABCD .

(b)

Quadrilateral \(\mathrm{ABC}_{1} \mathrm{D}_{1}\) has coordinates at \(\mathrm{A}(1 ; 0), \mathrm{B}(2 ; 0), \mathrm{C}_{1}(6 ; 2)\) and \(\mathrm{D}_{1}(5 ; 2)\).

#### (i)

Draw and label quadrilateral \(A B C_{1} D_{1}\).

#### (ii)

Describe fully the single transformation that maps ABCD onto \(\mathrm{ABC}_{1} \mathrm{D}_{1}\).

(c)

Quadrilateral ABCD is mapped onto quadrilateral \(\mathrm{A}_{2} \mathrm{~B}_{2} \mathrm{C}_{2} \mathrm{D}_{2}\) by a reflection in the line \(y=x+2\).

#### (i)

Draw and label line \(y=x+2\).

#### (ii)

Draw and label quadrilateral \(\mathrm{A}_{2} \mathrm{~B}_{2} \mathrm{C}_{2} \mathrm{D}_{2}\).

(d)

\(\mathrm{A}_{3} \mathrm{~B}_{3} \mathrm{C}_{3} \mathrm{D}_{3}\) is the image of ABCD under an enlargement of scale factor -1 with \((-1 ;-1)\) as centre.

Draw and label quadrilateral \(\mathrm{A}_{3} \mathrm{~B}_{3} \mathrm{C}_{3} \mathrm{D}_{3}\).

Question 10

(a)

The point, M , has coordinates \((7 ;-3)\) and \(\overrightarrow{\mathrm{RM}}=\binom{6}{4}\).

Calculate

#### (i)

the coordinates of R ,

#### (ii)

\(\quad \overrightarrow{\mathrm{MR}}\).

(b)

The diagram is a quadrilateral ORST in which \(\overrightarrow{\mathrm{OR}}=u\), \(\overrightarrow{\mathrm{OT}}=2 v\) and \(\overrightarrow{\mathrm{TS}}=2 u+v\). Diagonals OS and RT intersect at P .

#### (i)

Express in terms of \(u\) and/or \(v\).

1. \(\quad \overrightarrow{\mathrm{RT}}\), 2. \(\quad \overrightarrow{\mathrm{OS}}\).

#### (ii)

Given that \(\overrightarrow{\mathrm{OP}}=k \overrightarrow{\mathrm{OS}}\), express in terms of \(k, u\) and/or \(v\)

3. \(\overrightarrow{\mathrm{OP}}\), 4. \(\quad \overrightarrow{\mathrm{RP}}\) and show that it reduces to \((2 k-1) u+3 k v\).

#### (iii)

Given also that \(\overrightarrow{\mathrm{RP}}=h \overrightarrow{\mathrm{RT}}\), express \(\overrightarrow{\mathrm{RP}}\) in terms of \(h, u\) and/or \(\boldsymbol{v}\).

#### (iv)

Using the results in (ii) 2 and (iii), calculate the value of \(h\) and the value of \(k\).

Question 11

(a)

Solve the simultaneous equations:

$$ \begin{aligned} & 3 x-2 y=8 \\ & 5 x-4 y=12 \end{aligned} $$

(b)

The diagram shows three points, \(\mathrm{P}, \mathrm{Q}\) and R such that P is 4 km North of \(Q\) and \(R\) is 6 km from \(P\) on a bearing of \(073^{\circ}\).

Calculate