ZIMSEC O-Level Mathematics Paper 2 November 2007 Question Paper

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Ordinary Level **MATHEMATICS** 4008/2

**PAPER 2** **NOVEMBER 2007 SESSION**

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. Electronic calculators must not be used. 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 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, N2007. ©ZIMSEC N2007 [Turn over

**Section A [64 marks]**

Answer all the questions in this section.

Question 1

(a)

Simplify \(5,2-8,3 \times 0,2\).

(b)

Evaluate \(\left(4 \times 10^{2}\right)+\left(6 \times 10^{3}\right)+\left(1 \times 10^{5}\right)\) giving your answer in standard form.

(c)

Solve the equation

$$ \frac{2}{3}(x-1)-\frac{1}{4}(3 x-5)=1 $$

(d)

Given that \(3=\sqrt{b^{2}+c^{2}}\), make \(c\) the subject of the formula.

Question 2

(a)

#### (i)

If \(\left(\begin{array}{cc}1 & -3 \\ 2 & -4\end{array}\right)-\left(\begin{array}{cc}e & 0 \\ 5 & -10\end{array}\right)=\left(\begin{array}{cc}-1 & -3 \\ -3 & 2 f\end{array}\right)\), find the value of \(e\) and the value of \(f\).

#### (ii)

Given that \(\left(\begin{array}{ll}7 & -5 \\ 3 & -1\end{array}\right) \mathbf{M}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\), find the matrix \(\mathbf{M}\).

(b)

Solve the equation

$$ 5 x^{2}-3 x-2=0 $$

Question 3

(a)

In the diagram, PQRSU is a circle centre O. UQ is a diameter and PT is a tangent to the circle at P. \(\mathrm{TPQ}=53^{\circ}\) and \(\mathrm{U} \hat{\mathrm{Q} S}=40^{\circ}\).

Find

#### (i)

\(\mathrm{P} \hat{\mathrm{S} U}\),

#### (ii)

\(\mathrm{QPS}\),

#### (iii)

\(\mathrm{Q} \hat{\mathrm{R} S}\).

(b)

\(\xi=\{x: 2 \leq x \leq 30, \quad x\) is a multiple of 4\(\}\), \(\mathrm{A}=\{x: x\) is exactly divisible by 8\(\}\), \(B=\{4 ; 8 ; 28\}\).

Draw a clearly labelled Venn diagram to show the elements in each region.

(c)

A polygon has \(n\) sides. If the sum of the interior angles of the polygon is three times the sum of its exterior angles, find the value of \(n\).

Question 4

(a)

Given that \(23_{x}=21_{10}\), where \(x\) is a base, find the value of \(x\).

(b)

The mean of eight numbers is 13. When four more numbers are added, the mean of the twelve numbers is 11.

Find the mean of the four numbers which were added.

(c)

\(y\) varies directly as \(v\) and inversely as \((x+2)\).

#### (i)

Express \(y\) in terms of \(v, x\) and a constant \(k\).

#### (ii)

Given that when \(y=\frac{3}{2}, x=8\) and \(v=5\), find the value of \(k\),

#### (iii)

Find \(y\) when \(x=-11\) and \(v=2\).

Question 5

(a)

Factorise completely

$$ y^{2}(x-2)-x+2 $$

(b)

In this question take \(\pi\) to be \(\frac{22}{7}\).

In the diagram, ABCD is a rectangle inscribed in a circle centre O. \(AD=3.6 \mathrm{~cm}\) and \(AB=1.5 \mathrm{~cm}\).

Calculate

#### (i)

OC, the radius of the circle,

#### (ii)

\(\mathrm{B} \hat{\mathrm{O} C}\),

#### (iii)

the area of the shaded segment.

Question 6

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

Use ruler and compasses only for all constructions and show clearly all construction arcs and lines.

(a)

Construct, on a single diagram,

#### (i)

the triangle PQR with \(\mathrm{PQ}=7 \mathrm{~cm}, \mathrm{QR}=8.2 \mathrm{~cm}\) and \(\mathrm{PQR}=120^{\circ}\),

#### (ii)

the locus of points that are \(4.5 \mathrm{~cm}\) from R,

#### (iii)

the locus of points that are \(3.6 \mathrm{~cm}\) from PQ and on the same side of PQ as R,

#### (iv)

the perpendicular bisector of PQ.

(b)

Measure and write down the length of PR.

(c)

Describe fully the locus represented by the perpendicular bisector of PQ.

**Section B [36 marks]**

Answer any three questions in this section.

Question 7

In the diagram, H, E, F and G are points on level ground. \(HE=35 \mathrm{~m}, EF=20 \mathrm{~m}\), \(\mathrm{FG}=38 \mathrm{~m}, \mathrm{HG}=50 \mathrm{~m}\) and \(\mathrm{HEF}=102^{\circ}\).

(a)

Calculate HF.

(b)

If the area of triangle HFG is \(0.08034\) ha,

#### (i)

convert \(0.08034\) ha to \(\mathrm{m}^{2}\),

#### (ii)

calculate \(\mathrm{H} \hat{\mathrm{G} F}\).

(c)

At G, there is a vertical pole 8 m high (not shown on diagram). Calculate the angle of elevation of the top of the pole from H.

Question 8

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

Below is a table of values for the function \(y=4+x-x^{2}\).

| \(x\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | \(y\) | -8 | \(p\) | 2 | 4 | 4 | \(q\) | -2 | -8 |

(a)

Calculate the value of \(p\) and the value of \(q\).

(b)

Using a scale of 2 cm to represent 1 unit on the \(x\)-axis and 2 cm to represent 2 units on the \(y\)-axis, draw the graph of \(y=4+x-x^{2}\) for \(-3 \leq x \leq 4\).

(c)

Use the graph to estimate

#### (i)

the roots of the equation \(4+x-x^{2}=-4\),

#### (ii)

the gradient of the curve when \(x=-\frac{1}{2}\) by drawing a suitable tangent,

#### (iii)

the value of \(x\) when \(y\) is maximum,

#### (iv)

the area of the region bounded by the graph, the \(x\)-axis and the lines \(x=3\) and \(x=4\).

Question 9

In the diagram, OABC is a trapezium with OA parallel to CB. OB and AC intersect at G. \(\mathrm{OG}: \mathrm{GB}=2: 1\) and \(\mathrm{CB}=\frac{1}{2} \mathrm{OA}\). \(\mathrm{OC}=3 a-b\) and \(\mathrm{OA}=2 a+6 b\).

(a)

Express in terms of \(a\) and/or \(b\).

#### (i)

AC,

#### (ii)

AB,

#### (iii)

OG.

(b)

Given that \(\mathrm{AG}=k \mathrm{AC}\),

#### (i)

express AG in terms of \(a, b\), and scalar \(k\),

#### (ii)

show that \(\mathrm{OG}=(2+k) a+(6-7 k) b\).

(c)

Using the two expressions of OG, find the value of \(k\).

Question 10

(a)

[Do not re-draw the graph]

Use the graph above to answer the questions which follow.

#### (i)

\(\triangle \mathrm{A}\) is mapped onto \(\triangle \mathrm{B}\) by a translation. Write down the column vector for this translation.

#### (ii)

Describe fully the single transformation which maps \(\triangle \mathrm{A}\) onto \(\triangle \mathrm{C}\).

#### (iii)

A reflection maps \(\triangle \mathrm{A}\) onto \(\triangle \mathrm{D}\). Write down the equation of the mirror line.

#### (iv)

\(\triangle \mathrm{D}\) is mapped onto \(\triangle \mathrm{F}\) by a single transformation. Write down the matrix which represents this transformation.

#### (v)

\(\triangle \mathrm{A}\) is mapped onto \(\triangle \mathrm{E}\) by a one-way stretch.

Write down

(a) the stretch factor,

(b) the equation of the invariant line.

(b)

A transformation represented by \(\left(\begin{array}{cc}3 & 1 \\ -1 & 0\end{array}\right)\) maps point R onto point \((0 ; 2)\).

Write down the coordinates of R.

Question 11

(a)

Study the patterns below.

| Column 1 | Column 2 | Column 3 | | :--- | :--- | :--- | | 0 | 0 | 1 | | 1 | 0 | 2 | | 2 | 1 | 4 | | 3 | 3 | 7 | | 4 | \(t\) | 11 | | 5 | 10 | \(u\) | | | | | | 10 | \(v\) | | | 11 | 55 | | | ⋮ | ⋮ | ⋮ | | \(x\) | \(y\) | \(z\) | | ⋮ | ⋮ | |

Use the patterns to write down

#### (i)

the value of \(t\),

#### (ii)

the value of \(u\),

#### (iii)

the value of \(v\),

#### (iv)