ZIMSEC O Level Mathematics Paper 2 June 2007

Question 101

[3 marks]Factorisation, H.C.F and L.C.M

Find the value of

4 - 2\frac{2}{3} \times 1\frac{1}{5}

, giving your answer as a fraction in its lowest terms.

Answer this when you sit the paper.

Question 102

[2 marks]Factorisation, H.C.F and L.C.M

P, Q and R are subsets of

\xi

with

n(\mathrm{P} \cap \mathrm{Q}) = 0

\mathrm{R} \subset \mathrm{Q}

and

\mathrm{P} \cap \mathrm{R} = \phi

. Which arrangement of loops inside the rectangle

\xi

fits all three conditions?

ATwo overlapping loops Q and R, with a smaller loop P drawn wholly inside Q too.
BTwo separate loops P and Q, with a smaller loop R drawn wholly inside Q.
CTwo overlapping loops P and Q, with a smaller loop R drawn inside the overlap.
DTwo separate loops P and R, with a smaller loop Q drawn wholly inside R instead.

Question 103

[2 marks]Factorisation, H.C.F and L.C.M

Factorise completely

3df - d^2 - 3ef + de

Answer this when you sit the paper.

Question 104

[2 marks]Factorisation, H.C.F and L.C.M

Factorise completely

3m^3 - 27m

Answer this when you sit the paper.

Question 201

[2 marks]Matrices

Matrix

A = \begin{pmatrix} 3 & -2 \\ 4 & -5 \end{pmatrix}

and matrix

C = \begin{pmatrix} -1 & 5 \\ 3 & 0 \end{pmatrix}

. Evaluate

2A + C

Answer this when you sit the paper.

Question 202

[2 marks]Matrices

Matrix

A = \begin{pmatrix} 3 & -2 \\ 4 & -5 \end{pmatrix}

and matrix

B = \begin{pmatrix} 2 \\ 3 \end{pmatrix}

. Evaluate

AB

Answer this when you sit the paper.

Question 203

[2 marks]Matrices

Matrix

A = \begin{pmatrix} 3 & -2 \\ 4 & -5 \end{pmatrix}

. Which matrix is

A^{-1}

A $\begin{pmatrix} \frac{3}{7} & -\frac{2}{7} \\ \frac{4}{7} & -\frac{5}{7} \end{pmatrix}$
B $\begin{pmatrix} \frac{5}{7} & -\frac{2}{7} \\ \frac{4}{7} & -\frac{3}{7} \end{pmatrix}$
C $\begin{pmatrix} -\frac{5}{7} & -\frac{2}{7} \\ -\frac{4}{7} & -\frac{3}{7} \end{pmatrix}$
D $\begin{pmatrix} -\frac{5}{7} & \frac{2}{7} \\ -\frac{4}{7} & \frac{3}{7} \end{pmatrix}$

Question 204

[2 marks]Matrices

Given that

V = \sqrt{\dfrac{l + b}{l - b}}

, which expression gives

l

in terms of

b

and

V

A $l = b\left(\dfrac{V + 1}{V - 1}\right)$ , reached by cross multiplying without squaring first.
B $l = \dfrac{b(V^2 - 1)}{V^2 + 1}$ , reached by squaring both sides and gathering the $l$ terms.
C $l = \dfrac{b(V^2 + 1)}{V^2 - 1}$ , reached by squaring both sides and gathering the $l$ terms.
D $l = \dfrac{V^2 + 1}{b(V^2 - 1)}$ , reached by squaring both sides and dividing through by $b$ .

Question 205

[3 marks]Matrices

Given that

V = \sqrt{\dfrac{l + b}{l - b}}

, find

l

when

b = 16

and

V = \dfrac{1}{3}

Answer this when you sit the paper.

Question 301

[3 marks]Inequalities

Solve the inequalities

y - 4 < 3y + 2 \le 6

, leaving the answer in the form

a < y \le b

A $-3 \le y < 1\frac{1}{3}$ , from $y \ge -3$ on the left and $y < \frac{4}{3}$ on the right.
B $3 < y \le 1\frac{1}{3}$ , from $y > 3$ on the left part and $y \le \frac{4}{3}$ on the right.
C $-3 < y \le 2\frac{2}{3}$ , from $y > -3$ on the left and $y \le \frac{8}{3}$ on the right.
D $-3 < y \le 1\frac{1}{3}$ , from $y > -3$ on the left and $y \le \frac{4}{3}$ on the right.

Question 302

[3 marks]Inequalities

Solve the equation

\dfrac{4}{x - 3} = \dfrac{2}{x} + 1

, giving both roots.

Answer this when you sit the paper.

Question 303

[2 marks]Inequalities

A tourist exchanged 4 200 South African Rands for 157 500 Zimbabwean dollars. Calculate the equivalent of R1 in Zimbabwean dollars.

Answer this when you sit the paper.

Question 304

[1 marks]Inequalities

At a rate of 37,50 Zimbabwean dollars to one South African Rand, how many Zimbabwean dollars would a tourist get for R250?

Answer this when you sit the paper.

Question 305

[2 marks]Inequalities

At a rate of 37,50 Zimbabwean dollars to one South African Rand, find the equivalent cost, in Rands, of a gift marked 46 200 Zimbabwean dollars.

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Question 401

[2 marks]Circle Geometry

OACB is a sector of a circle of radius 15 cm with centre O. D is the point on OB for which

\mathrm{A}\hat{\mathrm{D}}\mathrm{O} = 90^\circ

, and OD = 12 cm. Calculate the size of

\mathrm{A}\hat{\mathrm{O}}\mathrm{B}

, to the nearest degree.

Answer this when you sit the paper.

Question 402

[3 marks]Circle Geometry

Sectors each of angle

37^\circ

are drawn inside a full circle without overlapping. Find, in degrees, the angle of the sector left over when as many as possible have been drawn.

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Question 403

[2 marks]Circle Geometry

A, B, C and D lie in that order on a circle with centre O. The tangent at A meets the tangent at C at a point T, and

\mathrm{B}\hat{\mathrm{A}}\mathrm{T} = 30^\circ

. Find

\mathrm{O}\hat{\mathrm{A}}\mathrm{B}

, in degrees.

Answer this when you sit the paper.

Question 404

[3 marks]Circle Geometry

A, B, C and D lie in that order on a circle with centre O. The tangent at A meets the tangent at C at a point T.

\mathrm{B}\hat{\mathrm{A}}\mathrm{T} = 30^\circ

, and the angle

\mathrm{A}\hat{\mathrm{O}}\mathrm{C}

on the same side of the centre as B is

100^\circ

. Find

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

, in degrees.

Answer this when you sit the paper.

Question 501

[2 marks]Statistics and Probability

Planet B is

5,97 \times 10^7

km from the sun and planet D is

2,87 \times 10^9

km from the sun. Find the ratio of D's distance to B's distance in the form

k : 1

, giving

k

to the nearest whole number.

Answer this when you sit the paper.

Question 502

[3 marks]Statistics and Probability

Planet C is

7,78 \times 10^8

km from the sun and light travels at

3 \times 10^8

metres per second. Calculate the time, correct to the nearest minute, that light takes to travel from the sun to planet C.

Answer this when you sit the paper.

Question 503

[1 marks]Statistics and Probability

A packet contains 60 identical sweets, of which 40 are red and the rest are yellow. One sweet is picked at random. Find the probability that it is red.

Answer this when you sit the paper.

Question 504

[2 marks]Statistics and Probability

A packet holds 60 sweets, 40 red and 20 yellow. One sweet is picked and not replaced, and it turns out to be red. What is the probability that the second sweet picked is yellow?

A $\frac{19}{59}$ , because one yellow sweet is treated as having already left the packet.
B $\frac{39}{59}$ , because 39 of the 59 sweets that remain in the packet are red ones.
C $\frac{20}{59}$ , because 59 sweets are left and all 20 yellow ones are still among them.
D $\frac{20}{60}$ , because both the packet size and the yellow count are treated as unchanged.

Question 505

[3 marks]Statistics and Probability

Two sweets are picked without replacement from a packet of 60, of which 40 are red and 20 are yellow. Find the probability that the two sweets are of the same colour.

Answer this when you sit the paper.

Question 601

[2 marks]Constructions and Loci

AB and AD are two straight lines drawn from the point A. What is the locus of points equidistant from AB and AD?

AThe line through A at right angles to AB, drawn on the same side of AB as D.
BThe bisector of the angle DAB, the line from A splitting that angle into two equal parts.
CThe perpendicular bisector of DB, the line that cuts the segment DB in half at right angles.
DA circle with centre A whose radius is half of the shorter of the two lines.

Question 602

[1 marks]Constructions and Loci

Name the locus of all the points that are 5 cm from a fixed point A.

Answer this when you sit the paper.

Question 603

[2 marks]Constructions and Loci

AD is a straight line and B is a point on one side of it. What is the locus of points 3 cm from AD and on the same side of AD as B?

AA pair of straight lines parallel to AD, one drawn 3 cm from it on each side.
BA circle of radius 3 cm whose centre is the midpoint of the line AD.
CA straight line parallel to AD, drawn 3 cm from it on the same side as B.
DAn arc of radius 3 cm centred on A, drawn on the same side of AD as B is.

Question 604

[2 marks]Constructions and Loci

In quadrilateral ABCD, AB = 10 cm, AD = 6 cm, BC = 13 cm,

\mathrm{A}\hat{\mathrm{B}}\mathrm{C} = 30^\circ

and

\mathrm{D}\hat{\mathrm{A}}\mathrm{B} = 120^\circ

. Find the size of

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

, to the nearest degree.

Answer this when you sit the paper.

Question 701

[2 marks]Consumer Arithmetic

The cash price of an electric stove is 3 400 000 dollars. On hire purchase the buyer pays a deposit of 35 % of the cash price. Calculate the deposit, in dollars.

Answer this when you sit the paper.

Question 702

[2 marks]Consumer Arithmetic

A stove is bought on hire purchase with a deposit of 1 190 000 dollars and 12 equal monthly instalments of 300 000 dollars each. Calculate the total amount paid, in dollars.

Answer this when you sit the paper.

Question 703

[2 marks]Consumer Arithmetic

A stove has a cash price of 3 400 000 dollars and a hire purchase price of 4 790 000 dollars. Calculate the difference between the two prices, in dollars.

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Question 704

[2 marks]Consumer Arithmetic

Convert a speed of 120 km/h to a speed in m/s, giving your answer correct to 3 significant figures.

Answer this when you sit the paper.

Question 705

[2 marks]Consumer Arithmetic

A bus travelling at

\frac{100}{3}

m/s slows uniformly to rest in 30 seconds. Calculate the deceleration, in m/s

^2

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 706

[2 marks]Consumer Arithmetic

A bus travelling at

\frac{100}{3}

m/s slows uniformly to rest in 30 seconds. Calculate the distance, in metres, that it covers while slowing down.

Answer this when you sit the paper.

Question 801

[2 marks]Vector Geometry

Points P and Q have position vectors

\mathbf{p}

and

\mathbf{q}

relative to the origin O. Given that

\mathbf{p} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}

and

\overrightarrow{\mathrm{PQ}} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}

, find

\mathbf{q}

Answer this when you sit the paper.

Question 802

[2 marks]Vector Geometry

Given that

\overrightarrow{\mathrm{PQ}} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}

, find

\left|\overrightarrow{\mathrm{PQ}}\right|

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 803

[2 marks]Vector Geometry

Given that

\mathbf{p} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}

\mathbf{r} = \begin{pmatrix} -3 \\ 2 \end{pmatrix}

\mathbf{t} = \begin{pmatrix} 1 \\ 6 \end{pmatrix}

and

m\mathbf{p} + n\mathbf{r} = \mathbf{t}

, which pair of equations follows?

A $2m - 3n = 1$ and $5m + 2n = 6$ , taking one equation from each row of the vectors.
B $2m - 3n = 6$ and $5m + 2n = 1$ , taking one equation from each row of the vectors.
C $2m + 3n = 1$ and $5m - 2n = 6$ , taking one equation from each row of the vectors.
D $2m + 5n = 1$ and $-3m + 2n = 6$ , taking one equation from each column of the vectors.

Question 804

[3 marks]Vector Geometry

Solve the simultaneous equations

2m - 3n = 1

and

5m + 2n = 6

, and write down the value of

m

as a fraction.

Answer this when you sit the paper.

Question 805

[3 marks]Vector Geometry

Solve the equation

2x^2 + 4x - 3 = 0

, giving both answers correct to 2 decimal places.

Answer this when you sit the paper.

Question 901

[1 marks]Functional Graphs

For the curve

y = \dfrac{8}{x^2}

, a row of values gives

y = p

when

x = 4

. Find the value of

p

Answer this when you sit the paper.

Question 902

[2 marks]Functional Graphs

Find the gradient of the curve

y = \dfrac{8}{x^2}

at the point where

x = 2

Answer this when you sit the paper.

Question 903

[2 marks]Functional Graphs

Find the equation of the straight line passing through

(2; 3)

with gradient

-\dfrac{5}{4}

A $y = 5,5 - 1,25x$ , since $y - 3 = -\frac{5}{4}(x - 2)$ rearranges to give that form.
B $y = 1,25x + 0,5$ , since a gradient of $-\frac{5}{4}$ is read as a rise of 1,25 units.
C $y = 3 - 1,25x$ , since the line must cut the $y$ -axis at the given $y$ -value of 3.
D $y = 0,5 - 1,25x$ , since the $x$ -value 2 is taken away from the $y$ -value 3 here.

Question 904

[3 marks]Functional Graphs

For

1 \le x \le 6

, the curve

y = \dfrac{8}{x^2}

meets the line

y = 5,5 - 1,25x

at two points. Write down the

x

-coordinate of each point, correct to 1 decimal place.

Answer this when you sit the paper.

Question 905

[2 marks]Functional Graphs

The curve

y = \dfrac{8}{x^2}

and the line

y = 5,5 - 1,25x

cross where

x = 4

. Find the

y

-coordinate of that crossing point.

Answer this when you sit the paper.

Question 1001

[2 marks]Geometrical Transformation

A kite K has a vertex at

(3; 3)

. K is reflected in the line

x = -1

. Write down the coordinates of the image of that vertex.

Answer this when you sit the paper.

Question 1002

[3 marks]Geometrical Transformation

Kite K has vertices

(1; 3)

(2; 1)

(3; 3)

and

(2; 4)

. Kite K

_2

has vertices

(-1; 2)

(-3; 1)

(-4; 2)

and

(-3; 3)

. Describe completely the single transformation which maps K onto K

_2

AA rotation through $90^\circ$ clockwise about the origin $(0; 0)$ , sending $(x; y)$ to $(y; -x)$ .
BA rotation through $180^\circ$ about the origin, which sends $(x; y)$ to $(-x; -y)$ .
CA reflection in the line $y = x$ , which sends the point $(x; y)$ to the point $(y; x)$ .
DA rotation through $90^\circ$ anticlockwise about the origin, sending $(x; y)$ to $(-y; x)$ .

Question 1003

[2 marks]Geometrical Transformation

The point

(2; 4)

is mapped by an enlargement of scale factor

-\dfrac{1}{2}

with the origin as centre. Write down the coordinates of its image.

Answer this when you sit the paper.

Question 1004

[2 marks]Geometrical Transformation

The point

(2; 1)

is mapped by a translation of vector

\begin{pmatrix} 1 \\ -4 \end{pmatrix}

. Write down the coordinates of its image.

Answer this when you sit the paper.

Question 1005

[2 marks]Geometrical Transformation

A kite K is enlarged by scale factor

-\dfrac{1}{2}

about the origin to give K

_3

. The area of K

_3

is what fraction of the area of K?

Answer this when you sit the paper.

Question 1101

[1 marks]Measures and Mensuration

A fuel tank 5 metres long lies horizontally on its side. The flat surface of the fuel inside it is a rectangle ABCD of length 5 m and width AB. Safety standards require this surface to be at least 2 square metres. Calculate the minimum value of AB, in metres.

Answer this when you sit the paper.

Question 1102

[2 marks]Measures and Mensuration

The circular cross-section of a fuel tank has centre O and diameter 2 metres. The fuel in it is 1,8 metres deep, and AB is the horizontal chord at the fuel surface. Calculate the length of AB, in metres.

Answer this when you sit the paper.

Question 1103

[2 marks]Measures and Mensuration

In a circle of centre O and radius 1 m, the chord AB is 1,2 m long and its perpendicular distance from O is 0,8 m. Calculate the area of triangle AOB, in square metres.

Answer this when you sit the paper.

Question 1104

[2 marks]Measures and Mensuration

In a circle of centre O and radius 1 m, the chord AB is 1,2 m long. Calculate the acute angle

\mathrm{A}\hat{\mathrm{O}}\mathrm{B}

, in degrees, correct to 1 decimal place.

Answer this when you sit the paper.

Question 1105

[2 marks]Measures and Mensuration

A minor sector AOB of a circle of radius 1 m has an angle of

73,7^\circ

at the centre. Taking

\pi

to be 3,142, calculate the area of the sector, in square metres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1106

[3 marks]Measures and Mensuration

A tank 5 m long has a circular cross-section of radius 1 m. The fuel surface is the chord AB, the minor sector AOB above it has area 0,643 m

^2

and triangle AOB has area 0,48 m

^2

. Taking

\pi

to be 3,142, calculate the volume of fuel, in cubic metres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1201

[1 marks]Prime Numbers, Sequences & Types of Numbers

In a number pattern each row starts and ends with a number one more than the row before it, and every other entry is the sum of the two entries above it. The 3rd row is 7, 12, 7 and the 4th row is 8,

p

, 19, 8. Find the value of

p

Answer this when you sit the paper.

Question 1202

[1 marks]Prime Numbers, Sequences & Types of Numbers

In the same pattern, where each row starts and ends one higher than the row before and every other entry is the sum of the two above it, the 5th row is 9, 27, 38, 27, 9 and the 6th row is 10, 36, 65, 65,

q

, 10. Find the value of

q

Answer this when you sit the paper.

Question 1203

[2 marks]Prime Numbers, Sequences & Types of Numbers

The 6th row of a number pattern is 10, 36, 65, 65, 36, 10. Find the row total

r

Answer this when you sit the paper.

Question 1204

[3 marks]Prime Numbers, Sequences & Types of Numbers

The row totals of a number pattern are 5; 12; 26; 54; 110; 222;

s

;

t

; ... where each total comes from the one before it by the same rule. Find the value of

t

Answer this when you sit the paper.

Question 1205

[3 marks]Prime Numbers, Sequences & Types of Numbers

The row total for the

n

th row of a pattern is given by

2^{n-1}(a + 2) - 2

, where

a

is the number in the 1st row. For a pattern whose 1st row is 5, find the total for the 9th row.

Answer this when you sit the paper.

Question 1206

[2 marks]Prime Numbers, Sequences & Types of Numbers

A pattern is built the same way, each row starting and ending one higher than the row before and every other entry the sum of the two above it, but starting with 3 in the 1st row. What is its 4th row?

A6, 13, 13, 6, whose entries add to a row total of 38 altogether.
B5, 13, 13, 5, whose entries add to a row total of 36 altogether.
C6, 14, 14, 6, whose entries add to a row total of 40 altogether.
D6, 12, 12, 6, whose entries add to a row total of 36 altogether.

Mathematics Paper 2 June 2007

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The questions

Question 101

Question 102

Question 103

Question 104

Question 201

Question 202

Question 203

Question 204

Question 205

Question 301

Question 302

Question 303

Question 304

Question 305

Question 401

Question 402

Question 403

Question 404

Question 501

Question 502

Question 503

Question 504

Question 505

Question 601

Question 602

Question 603

Question 604

Question 701

Question 702

Question 703

Question 704

Question 705

Question 706

Question 801

Question 802

Question 803

Question 804

Question 805

Question 901

Question 902

Question 903

Question 904

Question 905

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1106

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

Question 1206

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