ZIMSEC O Level Mathematics Paper 2 November 2006

Question 101

[3 marks]Fractions, algebraic expressions and functions

Evaluate

5\frac{1}{4}-1\frac{2}{3}\times 2\frac{1}{10}

, giving the answer as a mixed number in its lowest terms.

Answer this when you sit the paper.

Question 102

[2 marks]Fractions, algebraic expressions and functions

Noma has

(4x-3y)

dollars and Rudo has

(5x-y)

dollars. Find, in its simplest form, the amount of money they have altogether.

Answer this when you sit the paper.

Question 103

[1 marks]Fractions, algebraic expressions and functions

Noma has

(4x-3y)

dollars and Rudo has

(5x-y)

dollars. Find, in its simplest form, the amount of money Rudo has more than Noma.

Answer this when you sit the paper.

Question 104

[2 marks]Fractions, algebraic expressions and functions

f(x)=x^3-3x^2+kx-4

, find

k

given that

f(3)=11

Answer this when you sit the paper.

Question 201

[2 marks]Factorisation and changing the subject of a formula

Factorise completely

3mp+np-6mq-2nq

Answer this when you sit the paper.

Question 202

[2 marks]Factorisation and changing the subject of a formula

Factorise completely

16-9r^2

Answer this when you sit the paper.

Question 203

[2 marks]Factorisation and changing the subject of a formula

It is given that

A=\pi r\left(h^2-r^2\right)

. Which expression correctly makes

h

the subject of this formula?

A $h=\sqrt{r^2+\dfrac{A}{\pi r}}$ , dividing by $\pi r$ and then taking the square root
B $h=\sqrt{\dfrac{A}{\pi r}}-r$ , taking the square root of each of the two terms separately
C $h=\sqrt{r^2-\dfrac{A}{\pi r}}$ , keeping the minus sign that stands in the original bracket
D $h=\dfrac{A}{\pi r}+r^2$ , dividing through by $\pi r$ but leaving the square untouched

Question 204

[3 marks]Factorisation and changing the subject of a formula

Given that

A=\pi r\left(h^2-r^2\right)

, find the value of

h

when

A=330

r=7

and

\pi=\frac{22}{7}

Answer this when you sit the paper.

Question 301

[3 marks]Venn diagrams and algebraic fractions

In a class of 40, every pupil studies at least one of Mathematics, Geography and Accounts. 4 pupils study Mathematics and Geography, 5 study Mathematics and Accounts, 7 study Geography and Accounts, 15 study Mathematics only, 13 study Geography only and 4 study Accounts only. Find the number of pupils who study all three subjects.

Answer this when you sit the paper.

Question 302

[3 marks]Venn diagrams and algebraic fractions

Express

\dfrac{b}{a^2-ab}+\dfrac{a}{b^2-ab}

as a single fraction in its lowest terms.

A $\dfrac{a+b}{ab}$ , from adding the two numerators over the product $ab$
B $\dfrac{a^2+b^2}{ab(a-b)}$ , from keeping the factor $a-b$ in the denominator
C $\dfrac{a-b}{ab}$ , from subtracting one numerator from the other over $ab$
D $-\dfrac{a+b}{ab}$ , the sum of the letters over their product, negated

Question 303

[3 marks]Venn diagrams and algebraic fractions

Solve the equation

\dfrac{1}{2-m}-\dfrac{3}{m-4}=0

Answer this when you sit the paper.

Question 401

[2 marks]Regular polygons, area and arc length

A regular octagon has centre O. A and D are two adjacent vertices of the octagon. Determine the size of

A\hat{O}D

, in degrees.

Answer this when you sit the paper.

Question 402

[2 marks]Regular polygons, area and arc length

A triangle AOD has

OA=OD=5

cm and the included angle

A\hat{O}D=45^\circ

. Calculate its area, in cm

^2

, correct to 2 decimal places.

Answer this when you sit the paper.

Question 403

[3 marks]Regular polygons, area and arc length

Two regular concentric octagons have the same centre O. Each vertex of the inner octagon is 5 cm from O and each vertex of the outer octagon is 6 cm from O. Calculate the area between the two octagons, in cm

^2

, correct to 1 decimal place.

Answer this when you sit the paper.

Question 404

[2 marks]Regular polygons, area and arc length

Calculate the angle, in degrees, through which the minute hand of a clock turns in 18 minutes.

Answer this when you sit the paper.

Question 405

[2 marks]Regular polygons, area and arc length

The minute hand of a clock is 4 cm long. Taking

\pi

to be 3,142, calculate the distance, in cm, that the tip of the hand moves in 18 minutes.

Answer this when you sit the paper.

Question 501

[2 marks]Matrices and vectors

Given that

\mathbf{M}=\begin{pmatrix}4&-9\\-2&5\end{pmatrix}

and

\mathbf{N}=\begin{pmatrix}1&3\\0&-1\end{pmatrix}

, find

\mathbf{M}+2\mathbf{N}

Answer this when you sit the paper.

Question 502

[3 marks]Matrices and vectors

Given that

\mathbf{M}=\begin{pmatrix}4&-9\\-2&5\end{pmatrix}

and

\mathbf{N}=\begin{pmatrix}1&3\\0&-1\end{pmatrix}

, find

\mathbf{MN}

Answer this when you sit the paper.

Question 503

[2 marks]Matrices and vectors

The matrix

\mathbf{L}=\begin{pmatrix}2d&4\\1&3\end{pmatrix}

is singular. Find the value of

d

Answer this when you sit the paper.

Question 504

[3 marks]Matrices and vectors

Points O, P, Q, R and T are such that

\overrightarrow{OP}=2\mathbf{p}

\overrightarrow{OQ}=3\mathbf{q}

and

\overrightarrow{PR}=3\mathbf{p}-\mathbf{q}

. Express

\overrightarrow{RQ}

in terms of

\mathbf{p}

and

\mathbf{q}

Answer this when you sit the paper.

Question 601

[3 marks]Constructions and loci

A quadrilateral ABCD is constructed with AB = 4 cm, BC = 6 cm, CD = 5 cm,

A\hat{B}C=135^\circ

and

B\hat{C}D=120^\circ

, with A and D on the same side of BC. Find the length of AD, in cm, correct to 1 decimal place.

Answer this when you sit the paper.

Question 602

[3 marks]Constructions and loci

A quadrilateral ABCD is constructed with AB = 4 cm, BC = 6 cm, CD = 5 cm,

A\hat{B}C=135^\circ

and

B\hat{C}D=120^\circ

, with A and D on the same side of BC. Find

B\hat{A}D

, in degrees, correct to the nearest degree.

Answer this when you sit the paper.

Question 603

[2 marks]Constructions and loci

A quadrilateral ABCD is constructed with AB = 4 cm, BC = 6 cm and

A\hat{B}C=135^\circ

. What is the locus of the points that are equidistant from AB and from BC?

AThe circle whose centre is the vertex B and whose radius is half the length of BC
BThe bisector of angle ABC, drawn from B through the crossing of two equal arcs
CThe perpendicular bisector of the line AC, drawn from equal arcs centred on A and on C
DA straight line parallel to BC and lying exactly halfway between BC and the point A

Question 604

[2 marks]Constructions and loci

A quadrilateral ABCD is constructed with AB = 4 cm and BC = 6 cm. What is the locus of the points that are 4 cm from the vertex B?

AA single straight line drawn parallel to BC at a perpendicular distance of 4 cm from it
BA pair of straight lines parallel to AB, each of them lying 4 cm away from AB
CA circle of radius 4 cm with its centre at B, drawn in one sweep of the compasses
DThe part of the bisector of angle ABC that lies more than 4 cm away from vertex B

Question 701

[3 marks]Quadratic equations and similar solids

Solve the equation

2x^2+6x+1=0

, giving your answers to 2 decimal places, and write down the larger of the two roots.

Answer this when you sit the paper.

Question 702

[3 marks]Quadratic equations and similar solids

A model globe is a sphere of diameter 8 cm. Taking

\pi

to be 3,142 and using surface area

=4\pi r^2

, calculate the surface area of the model, in cm

^2

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 703

[3 marks]Quadratic equations and similar solids

A geographical globe has a diameter of 48 cm and a model of it has a diameter of 8 cm. On the globe, Zimbabwe covers an area of 23,04 cm

^2

. Calculate the corresponding area on the model, in cm

^2

Answer this when you sit the paper.

Question 704

[3 marks]Quadratic equations and similar solids

A solid model globe is a sphere of diameter 8 cm. Taking

\pi

to be 3,142 and using volume

=\frac{4}{3}\pi r^3

, calculate the volume of the model, in cm

^3

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 801

[3 marks]Linear programming

R is the region bounded by the lines

m

n

l

and p. One of the inequalities that defines R is

y\le-\frac{2}{5}x+32

, which comes from line

l

. Which inequality comes from line

m

A $5x+8y\ge 320$ , which places R on the far side of line $m$ from the origin
B $5x+8y\le 320$ , the line joining $(0;40)$ to the point $(64;0)$
C $8x+5y\le 320$ , the line joining $(0;64)$ to the point $(40;0)$
D $3x+5y\le 200$ , the line joining $(0;40)$ to the point $(66\frac{2}{3};0)$

Question 802

[2 marks]Linear programming

For the region R, state the maximum value of

y

Answer this when you sit the paper.

Question 803

[3 marks]Linear programming

Given that

(x;y)

is a point of the region R with

x

and

y

both integers, find the greatest value of

x+y

Answer this when you sit the paper.

Question 804

[3 marks]Linear programming

Find the maximum value of

40x+20y

for points

(x;y)

in the region R.

Answer this when you sit the paper.

Question 901

[2 marks]Graph of a reciprocal function

A value

p

in a table for the function

y=\dfrac{12}{x}-1

is the value of

y

when

x=5

. Calculate

p

Answer this when you sit the paper.

Question 902

[3 marks]Graph of a reciprocal function

The curve

y=\dfrac{12}{x}-1

is drawn for

1\le x\le 8

. Estimate the gradient of the curve at

x=2

Answer this when you sit the paper.

Question 903

[3 marks]Graph of a reciprocal function

Use the trapezium rule with three strips each of width 1 unit to estimate the area of the region between the curve

y=\dfrac{12}{x}-1

, the

x

-axis and the lines

x=3

and

x=6

. Give the answer in square units.

Answer this when you sit the paper.

Question 904

[3 marks]Graph of a reciprocal function

Solve the equation

\dfrac{12}{x}-1=x+4

for the root that lies between 1 and 8, correct to 1 decimal place.

Answer this when you sit the paper.

Question 1001

[3 marks]Circle theorems and trigonometry

A, B, C and D lie in that order on the circumference of a circle with centre O, and

A\hat{B}C=71^\circ

. Calculate the reflex angle

A\hat{O}C

, in degrees.

Answer this when you sit the paper.

Question 1002

[2 marks]Circle theorems and trigonometry

O is the centre of a circle, C is a point on the circle and CT is the tangent to the circle at C. B is another point on the circle with

B\hat{C}O=47^\circ

, and B and T lie on the same side of OC. Calculate

B\hat{C}T

, in degrees.

Answer this when you sit the paper.

Question 1003

[3 marks]Circle theorems and trigonometry

A, B, C and D lie in that order on a circle with centre O, with

A\hat{B}C=71^\circ

and

B\hat{C}O=47^\circ

. Calculate

B\hat{A}O

, in degrees.

Answer this when you sit the paper.

Question 1004

[2 marks]Circle theorems and trigonometry

PQXS is a parallelogram in which

PQ=PS=6

cm and

P\hat{Q}X=41^\circ 37'

. Calculate its area, in cm

^2

, correct to 1 decimal place.

Answer this when you sit the paper.

Question 1005

[2 marks]Circle theorems and trigonometry

In a trapezium PQRS, X lies on SR so that PQXS is a parallelogram with

PQ=PS=6

cm and

P\hat{Q}X=41^\circ 37'

. Given that

QR=7,6

cm, calculate

Q\hat{R}X

, in degrees, correct to 1 decimal place.

Answer this when you sit the paper.

Question 1101

[3 marks]Household bills, percentage and rates

An electricity bill charges 269 units of energy at a single rate and the energy charge comes to \$1 894,57. Calculate that rate, in cents per unit, correct to 2 decimal places.

Answer this when you sit the paper.

Question 1102

[2 marks]Household bills, percentage and rates

An electricity meter read 31 834 units at the start of January and 32 331 units at the end of that month. Calculate the consumption for January, in units.

Answer this when you sit the paper.

Question 1103

[3 marks]Household bills, percentage and rates

A January electricity bill carries a balance brought forward of \$2 788,81, a payment of \$5 000,00 credited, an energy charge of \$9 496,84, a fixed monthly charge of \$1 067,69 and VAT of \$1 253,00. Calculate the amount due.

Answer this when you sit the paper.

Question 1104

[2 marks]Household bills, percentage and rates

On a December electricity bill the energy charge was \$1 894,57, the fixed monthly charge was \$530,49 and the Value Added Tax charged on those two items came to \$363,76. Calculate the rate at which VAT was charged, as a percentage.

Answer this when you sit the paper.

Question 1105

[2 marks]Household bills, percentage and rates

The fixed monthly charge on an electricity bill rose from \$530,49 in December to \$1 067,69 in January. Calculate the percentage increase, correct to the nearest whole number.

Answer this when you sit the paper.

Question 1201

[2 marks]Grouped data, histograms and probability

The marks of 50 pupils are grouped as

20<x\le 30

with 2 pupils,

30<x\le 45

with 5,

45<x\le 50

with 4,

50<x\le 60

with 16,

60<x\le 70

with 14,

70<x\le 80

with 6 and

80<x\le 100

with 3. State the modal class.

A $50<x\le 60$ , which has both the largest frequency and the largest frequency density
B $60<x\le 70$ , which holds the second largest number of pupils in the whole set
C $80<x\le 100$ , the widest class of the seven and so the one that covers the most marks
D $30<x\le 45$ , the class holding the second smallest number of pupils recorded

Question 1202

[3 marks]Grouped data, histograms and probability

The marks of 50 pupils are grouped as

20<x\le 30

with 2 pupils,

30<x\le 45

with 5,

45<x\le 50

with 4,

50<x\le 60

with 16,

60<x\le 70

with 14,

70<x\le 80

with 6 and

80<x\le 100

with 3. Calculate an estimate of the mean mark.

Answer this when you sit the paper.

Question 1203

[2 marks]Grouped data, histograms and probability

In a grouped frequency distribution the class

45<x\le 50

contains 4 items. Calculate the frequency density for that class.

Answer this when you sit the paper.

Question 1204

[2 marks]Grouped data, histograms and probability

In a grouped frequency distribution the class

80<x\le 100

contains 3 items. Calculate the frequency density for that class.

Answer this when you sit the paper.

Question 1205

[3 marks]Grouped data, histograms and probability

Of 50 pupils who sat a test, 39 scored above 50 marks and so passed. Two of the 50 pupils are chosen at random, without replacement. Calculate the probability that both of them passed, correct to 3 decimal places.

Answer this when you sit the paper.

Mathematics Paper 2 November 2006

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 104

Question 201

Question 202

Question 203

Question 204

Question 301

Question 302

Question 303

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 901

Question 902

Question 903

Question 904

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

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The answers, and why they are the answers

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