ZIMSEC O Level Mathematics Paper 2 November 2007

Question 101

[2 marks]Algebra

Simplify

5,2 - 8,3 \times 0,2

Answer this when you sit the paper.

Question 102

[2 marks]Algebra

Evaluate

\left(4 \times 10^{2}\right) + \left(6 \times 10^{3}\right) + \left(1 \times 10^{5}\right)

, giving the answer in standard form.

A $1,64 \times 10^{5}$
B $1,064 \times 10^{5}$
C $1,064 \times 10^{6}$
D $10,64 \times 10^{4}$

Question 103

[3 marks]Algebra

Solve the equation

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

Answer this when you sit the paper.

Question 104

[3 marks]Algebra

Given that

3 = \sqrt{b^{2} + c^{2}}

, make

c

the subject of the formula.

A $c = \sqrt{b^{2} - 9}$
B $c = 9 - b^{2}$
C $c = 3 - b$
D $c = \sqrt{9 - b^{2}}$

Question 201

[3 marks]Matrices

\begin{pmatrix} 1 & -3 \\ 2 & -4 \end{pmatrix} - \begin{pmatrix} e & 0 \\ 5 & -10 \end{pmatrix} = \begin{pmatrix} -1 & -3 \\ -3 & 2f \end{pmatrix}

, find the value of

e

and the value of

f

A $e = 0$ and $f = 3$
B $e = 2$ and $f = 3$
C $e = 2$ and $f = 6$
D $e = -1$ and $f = -7$

Question 202

[3 marks]Matrices

Given that

\begin{pmatrix} 7 & -5 \\ 3 & -1 \end{pmatrix}\mathbf{M} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

, find the matrix

\mathbf{M}

A $\begin{pmatrix} -1 & 5 \\ -3 & 7 \end{pmatrix}$
B $\begin{pmatrix} \frac{1}{8} & -\frac{5}{8} \\ \frac{3}{8} & -\frac{7}{8} \end{pmatrix}$
C $\begin{pmatrix} -\frac{1}{8} & \frac{3}{8} \\ \frac{5}{8} & -\frac{7}{8} \end{pmatrix}$
D $\begin{pmatrix} -\frac{1}{8} & \frac{5}{8} \\ -\frac{3}{8} & \frac{7}{8} \end{pmatrix}$

Question 203

[3 marks]Matrices

One root of

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

is a whole number. Write down the other root.

Answer this when you sit the paper.

Question 301

[2 marks]Circle Geometry

In the diagram, PQRSU is a circle centre O, UQ is a diameter and PT is a tangent at P, with

T\hat{P}Q = 53^\circ

and

U\hat{Q}S = 40^\circ

. Find

P\hat{S}U

in degrees.

Answer this when you sit the paper.

Question 302

[1 marks]Circle Geometry

In the diagram, PQRSU is a circle centre O with

U\hat{Q}S = 40^\circ

. Find

Q\hat{P}S

in degrees.

Answer this when you sit the paper.

Question 303

[1 marks]Circle Geometry

In the diagram, PQRSU is a circle centre O with

U\hat{Q}S = 40^\circ

and R on the arc between Q and S. Find

Q\hat{R}S

in degrees.

Answer this when you sit the paper.

Question 304

[3 marks]Circle Geometry

\xi = \{x : 2 \le x \le 30

, and

x

is a multiple of

4\}

A = \{x : x

is exactly divisible by

8\}

and

B = \{4;\, 8;\, 28\}

. Which elements of

\xi

lie outside both A and B?

A12, 20 and 28
B4, 12, 20 and 28
C8, 16 and 24
D12 and 20

Question 305

[3 marks]Circle Geometry

A polygon has

n

sides. The sum of its interior angles is three times the sum of its exterior angles. Find the value of

n

Answer this when you sit the paper.

Question 401

[3 marks]Variation

Given that

23_{x} = 21_{10}

, where

x

is a base, find the value of

x

Answer this when you sit the paper.

Question 402

[3 marks]Variation

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.

Answer this when you sit the paper.

Question 403

[2 marks]Variation

y

varies directly as

v

and inversely as

(x + 2)

. Express

y

in terms of

v

x

and a constant

k

A $y = \frac{v}{k(x + 2)}$
B $y = \frac{kv}{x + 2}$
C $y = \frac{k(x + 2)}{v}$
D $y = kv(x + 2)$

Question 404

[2 marks]Variation

Given that

y = \frac{kv}{x + 2}

, and that

y = \frac{3}{2}

when

x = 8

and

v = 5

, find the value of

k

Answer this when you sit the paper.

Question 405

[2 marks]Variation

Given that

y = \frac{3v}{x + 2}

, find

y

when

x = -11

and

v = 2

Answer this when you sit the paper.

Question 501

[3 marks]Measures & Mensuration

Factorise completely

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

A $(x - 2)(y - 1)(y + 1)$
B $(x - 2)(y^{2} + 1)$
C $(x + 2)(y - 1)(y + 1)$
D $(x - 2)(y^{2} - 1)$

Question 502

[2 marks]Measures & Mensuration

In the diagram, ABCD is a rectangle inscribed in a circle centre O, with

AD = 3,6

cm and

AB = 1,5

cm. Calculate OC, the radius of the circle, in cm.

Answer this when you sit the paper.

Question 503

[3 marks]Measures & Mensuration

In the diagram, ABCD is a rectangle inscribed in a circle centre O, with

BC = 3,6

cm and the radius

OC = 1,95

cm. Calculate

B\hat{O}C

in degrees, correct to 1 decimal place.

Answer this when you sit the paper.

Question 504

[3 marks]Measures & Mensuration

In the diagram, the shaded region is the minor segment cut off by the chord BC in a circle centre O of radius 1,95 cm, where

B\hat{O}C = 134,8^\circ

BC = 3,6

cm and O stands 0,75 cm from BC. Taking

\pi

to be

\frac{22}{7}

, calculate the area of the shaded segment in cm

^{2}

Answer this when you sit the paper.

Question 601

[3 marks]Constructions & Loci

A triangle PQR is constructed with

PQ = 7

cm,

QR = 8,2

cm and

P\hat{Q}R = 120^\circ

. Calculate the length of PR in cm, correct to 3 significant figures.

Answer this when you sit the paper.

Question 602

[2 marks]Constructions & Loci

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

AThe locus of points equidistant from the lines PQ and QR
BThe locus of points that are the same distance from the line PQ as from R
CThe locus of points equidistant from P and from Q
DThe locus of points 3,5 cm from the midpoint of the line PQ

Question 603

[2 marks]Constructions & Loci

On a construction, what is the locus of points that are 4,5 cm from the point R?

AA circle of radius 4,5 cm with centre R
BA straight line parallel to PQ and 4,5 cm away from R
CA pair of parallel lines drawn 4,5 cm on either side of R
DAn arc of a circle of radius 9 cm drawn with its centre at R

Question 604

[2 marks]Constructions & Loci

On a construction of triangle PQR, what is the locus of points that are 3,6 cm from the line PQ and on the same side of PQ as R?

AA straight line parallel to PQ, 3,6 cm from it, on the same side as R
BA circle of radius 3,6 cm drawn with the midpoint of PQ as its centre
CA pair of straight lines parallel to PQ, one 3,6 cm on each side of it
DAn arc of radius 3,6 cm swept about P and about Q, on R's side of PQ

Question 701

[3 marks]Trigonometry, Bearing & Distances

In the diagram, H, E, F and G are points on level ground with

HE = 35

EF = 20

m and

H\hat{E}F = 102^\circ

. Calculate HF in metres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 702

[1 marks]Trigonometry, Bearing & Distances

Convert 0,08034 ha to square metres.

Answer this when you sit the paper.

Question 703

[3 marks]Trigonometry, Bearing & Distances

In the diagram, triangle HFG has

HG = 50

FG = 38

m and an area of 803,4 m

^{2}

. Calculate the acute angle

H\hat{G}F

in degrees, correct to 1 decimal place.

Answer this when you sit the paper.

Question 704

[2 marks]Trigonometry, Bearing & Distances

A vertical pole 8 m high stands at G. H is a point on level ground 50 m from G. Calculate the angle of elevation of the top of the pole from H, in degrees, correct to 1 decimal place.

Answer this when you sit the paper.

Question 801

[2 marks]Functional Graphs

A table of values for

y = 4 + x - x^{2}

gives

y = p

when

x = -2

and

y = q

when

x = 2

. Find

p

and

q

A $p = -2$ and $q = 2$
B $p = 2$ and $q = -2$
C $p = -2$ and $q = -2$
D $p = 6$ and $q = 2$

Question 802

[2 marks]Functional Graphs

Estimate the roots of the equation

4 + x - x^{2} = -4

, correct to 1 decimal place.

A $x = -1,4$ and $x = 2,4$
B $x = -3,4$ and $x = 2,4$
C $x = -2,4$ and $x = 3,4$
D $x = -2$ and $x = 4$

Question 803

[2 marks]Functional Graphs

Find the gradient of the curve

y = 4 + x - x^{2}

when

x = -\frac{1}{2}

Answer this when you sit the paper.

Question 804

[1 marks]Functional Graphs

For the curve

y = 4 + x - x^{2}

, write down the value of

x

at which

y

is a maximum.

Answer this when you sit the paper.

Question 805

[2 marks]Functional Graphs

The curve

y = 4 + x - x^{2}

passes through

(3;\, -2)

and

(4;\, -8)

and lies below the

x

-axis between them. Using one trapezium, estimate the area, in square units, of the region bounded by the curve, the

x

-axis and the lines

x = 3

and

x = 4

Answer this when you sit the paper.

Question 901

[2 marks]Vector Geometry

In the diagram, OABC is a trapezium with

\mathbf{OC} = 3\mathbf{a} - \mathbf{b}

and

\mathbf{OA} = 2\mathbf{a} + 6\mathbf{b}

. Express

\mathbf{AC}

in terms of

\mathbf{a}

and

\mathbf{b}

Answer this when you sit the paper.

Question 902

[2 marks]Vector Geometry

In the diagram, OABC is a trapezium with OA parallel to CB,

\mathbf{OC} = 3\mathbf{a} - \mathbf{b}

\mathbf{OA} = 2\mathbf{a} + 6\mathbf{b}

and

CB = \frac{1}{2}OA

. Express

\mathbf{AB}

in terms of

\mathbf{a}

and

\mathbf{b}

Answer this when you sit the paper.

Question 903

[2 marks]Vector Geometry

In the diagram, G lies on OB with

OG : GB = 2 : 1

, and

\mathbf{OB} = 4\mathbf{a} + 2\mathbf{b}

. Express

\mathbf{OG}

in terms of

\mathbf{a}

and

\mathbf{b}

A $3\mathbf{a} + \mathbf{b}$
B $\frac{4}{3}\mathbf{a} + \frac{8}{3}\mathbf{b}$
C $\frac{8}{3}\mathbf{a} + \frac{4}{3}\mathbf{b}$
D $\frac{2}{3}\mathbf{a} - \frac{14}{3}\mathbf{b}$

Question 904

[1 marks]Vector Geometry

In the diagram,

\mathbf{AC} = \mathbf{a} - 7\mathbf{b}

and

\mathbf{AG} = k\mathbf{AC}

, where

k

is a scalar. Express

\mathbf{AG}

in terms of

\mathbf{a}

\mathbf{b}

and

k

A $k\mathbf{a} + 7k\mathbf{b}$
B $2k\mathbf{a} - 4k\mathbf{b}$
C $3k\mathbf{a} - k\mathbf{b}$
D $k\mathbf{a} - 7k\mathbf{b}$

Question 905

[3 marks]Vector Geometry

Given that

\mathbf{OG} = \frac{8}{3}\mathbf{a} + \frac{4}{3}\mathbf{b}

and also

\mathbf{OG} = (2 + k)\mathbf{a} + (6 - 7k)\mathbf{b}

, find the value of

k

Answer this when you sit the paper.

Question 1001

[1 marks]Geometrical Transformation

On the grid, triangle A is mapped onto triangle B by a translation. Write down the column vector for this translation.

A $\begin{pmatrix} 2 \\ 8 \end{pmatrix}$
B $\begin{pmatrix} 2 \\ -8 \end{pmatrix}$
C $\begin{pmatrix} -2 \\ 8 \end{pmatrix}$
D $\begin{pmatrix} 8 \\ -2 \end{pmatrix}$

Question 1002

[3 marks]Geometrical Transformation

On the grid, describe fully the single transformation which maps triangle A onto triangle C.

AA rotation through $90^\circ$ clockwise about $(2;\, 8)$
BA rotation through $90^\circ$ anticlockwise about $(2;\, 8)$
CA rotation through $90^\circ$ anticlockwise about $(4;\, 8)$
DA rotation through $180^\circ$ about the point $(3;\, 7)$

Question 1003

[2 marks]Geometrical Transformation

On the grid, a reflection maps triangle A onto triangle D. Write down the equation of the mirror line.

Answer this when you sit the paper.

Question 1004

[2 marks]Geometrical Transformation

On the grid, triangle D is mapped onto triangle F by a single transformation. Write down the matrix which represents it.

A $\begin{pmatrix} 0 & -\frac{1}{2} \\ -\frac{1}{2} & 0 \end{pmatrix}$
B $\begin{pmatrix} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{pmatrix}$
C $\begin{pmatrix} -2 & 0 \\ 0 & -2 \end{pmatrix}$
D $\begin{pmatrix} -\frac{1}{2} & 0 \\ 0 & -\frac{1}{2} \end{pmatrix}$

Question 1005

[2 marks]Geometrical Transformation

On the grid, triangle A is mapped onto triangle E by a one-way stretch. Write down the stretch factor and the equation of the invariant line.

AStretch factor $-\frac{3}{2}$ , invariant line $x = 0$
BStretch factor $\frac{3}{2}$ , invariant line $y = 0$
CStretch factor $-\frac{2}{3}$ , invariant line $x = 0$
DStretch factor $-\frac{3}{2}$ , invariant line $y = 0$

Question 1006

[2 marks]Geometrical Transformation

A transformation represented by

\begin{pmatrix} 3 & 1 \\ -1 & 0 \end{pmatrix}

maps the point R onto the point

(0;\, 2)

. Write down the coordinates of R.

Answer this when you sit the paper.

Question 1101

[2 marks]Consumer Arithmetic

In a pattern, Column 1 runs 0, 1, 2, 3, 4, 5 and against it Column 2 runs 0, 0, 1, 3,

t

, 10 while Column 3 runs 1, 2, 4, 7, 11,

u

. Find

t

and

u

A $t = 7$ and $u = 17$
B $t = 6$ and $u = 15$
C $t = 6$ and $u = 16$
D $t = 5$ and $u = 16$

Question 1102

[1 marks]Consumer Arithmetic

In a pattern, the entry in Column 2 against the value

n

in Column 1 is

\frac{n(n - 1)}{2}

, and against 11 it is 55. Find the value

v

in Column 2 against 10 in Column 1.

Answer this when you sit the paper.

Question 1103

[1 marks]Consumer Arithmetic

In a pattern, a row reads

x

in Column 1,

y

in Column 2 and

z

in Column 3, where Column 2 is

\frac{x(x - 1)}{2}

and Column 3 is

1 + \frac{x(x + 1)}{2}

. Write an expression for

z

in terms of

x

and

y

A $z = x + y + 1$
B $z = xy + 1$
C $z = 2y + x$
D $z = x + y - 1$

Question 1104

[1 marks]Consumer Arithmetic

Mrs Chido imported a television set from South Africa for R1 600. The exchange rate was Z

800 to R1. Convert R1 600 to Z

Answer this when you sit the paper.

Question 1105

[2 marks]Consumer Arithmetic

A television set is valued at Z$1 280 000. Mrs Chido paid 65% of that value as import duty. Calculate the import duty paid in Z$.

Answer this when you sit the paper.

Question 1106

[2 marks]Consumer Arithmetic

A television set was sold for Z$4 600 000, and this amount included 15% Value Added Tax. Calculate the Value Added Tax paid, in Z$.

Answer this when you sit the paper.

Question 1107

[3 marks]Consumer Arithmetic

Mrs Chido bought a television set for Z$1 280 000 and paid Z$832 000 import duty on it. She sold it for Z$4 600 000, of which Z$600 000 was Value Added Tax collected for the state. Calculate her profit in Z$.

Answer this when you sit the paper.

Question 1201

[2 marks]Statistics & Probability

For seven age classes the frequencies are 60, 140, 450, 510,

r

, 150 and 50, and the cumulative frequencies are 60,

p

q

, 1 160, 1 550, 1 700 and 1 750. Find

p

q

and

r

A $p = 140$ , $q = 650$ and $r = 390$
B $p = 200$ , $q = 590$ and $r = 390$
C $p = 200$ , $q = 650$ and $r = 390$
D $p = 200$ , $q = 650$ and $r = 350$

Question 1202

[3 marks]Statistics & Probability

The cumulative frequencies of the ages of 1 750 patients are 60, 200, 650, 1 160, 1 550, 1 700 and 1 750 at the upper boundaries 10, 20, 30, 40, 50, 60 and 70 years. Estimate the median age in years, correct to 1 decimal place.

Answer this when you sit the paper.

Question 1203

[3 marks]Statistics & Probability

The cumulative frequencies of the ages of 1 750 patients are 60, 200, 650, 1 160, 1 550, 1 700 and 1 750 at the upper boundaries 10, 20, 30, 40, 50, 60 and 70 years. Estimate the number of patients whose age

x

satisfies

22 < x \le 45

Answer this when you sit the paper.

Question 1204

[3 marks]Statistics & Probability

Of 1 750 patients, 200 are at most 20 years old and 50 are more than 60 years old. Two patients are chosen at random. Find the probability that one is at most 20 years old and the other is more than 60 years old.

Answer this when you sit the paper.

Mathematics Paper 2 November 2007

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The questions

Question 101

Question 102

Question 103

Question 104

Question 201

Question 202

Question 203

Question 301

Question 302

Question 303

Question 304

Question 305

Question 401

Question 402

Question 403

Question 404

Question 405

Question 501

Question 502

Question 503

Question 504

Question 601

Question 602

Question 603

Question 604

Question 701

Question 702

Question 703

Question 704

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 1006

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1106

Question 1107

Question 1201

Question 1202

Question 1203

Question 1204

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