ZIMSEC O Level Mathematics Paper 2 June 2008

Question 101

[3 marks]Algebraic Fractions

Solve the equation

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

Answer this when you sit the paper.

Question 102

[2 marks]Algebraic Fractions

Factorise completely

6y^2 - y - 12

Answer this when you sit the paper.

Question 103

[2 marks]Algebraic Fractions

Express

\frac{2}{2x - 1} - \frac{3}{x}

as a single fraction in its lowest terms.

A $\frac{4x - 3}{2x^2 - x}$
B $\frac{6x - 3}{2x^2 - x}$
C $\frac{3 - 4x}{2x^2 - x}$
D $\frac{2x - 3}{2x^2 - x}$

Question 104

[2 marks]Algebraic Fractions

Given that

z = r\sqrt{n - 1}

, find

z

when

r = 0,3

and

n = 50

Answer this when you sit the paper.

Question 105

[2 marks]Algebraic Fractions

Given that

z = r\sqrt{n - 1}

, express

n

in terms of

z

and

r

A $n = \left(\dfrac{z}{r} + 1\right)^2$
B $n = \dfrac{z^2}{r^2} + 1$
C $n = \dfrac{z^2}{r^2} - 1$
D $n = \dfrac{z}{r^2} + 1$

Question 201

[2 marks]Circle Geometry

TR and TS are tangents from a point T to a circle with centre O. Q lies on the major arc SR, SR is parallel to OQ and

S\hat{O}R = 116^\circ

. Calculate

S\hat{Q}R

, in degrees.

Answer this when you sit the paper.

Question 202

[2 marks]Circle Geometry

TR and TS are tangents from a point T to a circle with centre O. Q lies on the major arc SR, SR is parallel to OQ and

S\hat{O}R = 116^\circ

. Calculate

R\hat{S}Q

, in degrees.

Answer this when you sit the paper.

Question 203

[2 marks]Circle Geometry

TR and TS are tangents from a point T to a circle with centre O, and

S\hat{O}R = 116^\circ

. Calculate

R\hat{T}S

, in degrees.

Answer this when you sit the paper.

Question 204

[2 marks]Circle Geometry

ABCD is a rectangle. A square BCEF is drawn inside it with F on AB and E on DC. Given that

DA = x

cm,

AF = y

cm,

x + y = 15

and

x - y = 7

, calculate the area of ABCD, in cm

^2

Answer this when you sit the paper.

Question 205

[3 marks]Circle Geometry

Six angles of an octagon are

140^\circ

each and the remaining angles are equal. Find the size of each of the remaining angles.

A $135^\circ$ , since every octagon has equal interior angles
B $150^\circ$ , since the exterior angles must add up to $360^\circ$
C $120^\circ$ , since the two remaining angles share $240^\circ$
D $105^\circ$ , since the eight angles average $135^\circ$ each

Question 301

[2 marks]Matrices

Given that

\begin{pmatrix} x & 2 \end{pmatrix}\begin{pmatrix} 3 & 1 \\ 0 & y \end{pmatrix} = \begin{pmatrix} 15 & -7 \end{pmatrix}

, find the value of

y

Answer this when you sit the paper.

Question 302

[2 marks]Matrices

\xi

= {all triangles}, A = {all equilateral triangles}, B = {all isosceles triangles} and C = {all right angled triangles}. Which statement describes how the three sets sit inside

\xi

AB lies wholly inside A, and C meets A but keeps clear of B
BA lies wholly inside B, and C meets B but keeps clear of A
CA, B and C overlap one another in pairs, with a common region
DA and B overlap partly, and C lies wholly outside both of them

Question 303

[2 marks]Matrices

In a number pattern, Column 2 runs 1, 3, 6, 10, ... down the rows, and each Column 3 entry is the square of the Column 2 entry beside it. Column 1 numbers the rows. Find the Column 3 entry in the row numbered 5.

Answer this when you sit the paper.

Question 304

[2 marks]Matrices

In a number pattern, Column 1 numbers the rows and Column 2 runs 1, 3, 6, 10, ... down the rows. Find the Column 1 entry in the row whose Column 2 entry is 78.

Answer this when you sit the paper.

Question 305

[2 marks]Matrices

In a number pattern, each Column 3 entry

v

is the square of the Column 2 entry

w

beside it, the first four rows being 1 and 1, 3 and 9, 6 and 36, 10 and 100. Express

v

in terms of

w

Answer this when you sit the paper.

Question 401

[2 marks]Measures and Mensuration

Take

\pi

to be

\frac{22}{7}

. A drinking trough is made by bisecting a closed cylindrical drum lengthwise, so its cross-section is a semicircle. The trough has a diameter of 56 cm. Calculate the area of the cross-section, in cm

^2

Answer this when you sit the paper.

Question 402

[3 marks]Measures and Mensuration

A trough has a semicircular cross-section of area 1 232 cm

^2

and a capacity of 110 litres. Calculate its length, in cm, correct to 3 significant figures.

Answer this when you sit the paper.

Question 403

[3 marks]Measures and Mensuration

A drum was bought for

\$25\,500

and this represents a 70% increase on its price in the previous year. Calculate the price of such a drum in the previous year, in dollars.

Answer this when you sit the paper.

Question 404

[3 marks]Measures and Mensuration

Solve the equation

4x^2 - 2x - 3 = 0

, giving your answers correct to 2 significant figures.

A $x = 2,3$ or $x = -1,3$
B $x = 0,65$ or $x = -1,2$
C $x = 1,5$ or $x = -0,50$
D $x = 1,2$ or $x = -0,65$

Question 501

[2 marks]Vector Geometry

In a triangle PQR,

\overrightarrow{PQ} = \mathbf{p}

and

\overrightarrow{PR} = \mathbf{q}

. Express

\overrightarrow{QR}

in terms of

\mathbf{p}

and

\mathbf{q}

A $\mathbf{p} - \mathbf{q}$
B $\mathbf{p} + \mathbf{q}$
C $\tfrac{1}{2}(\mathbf{p} + \mathbf{q})$
D $\mathbf{q} - \mathbf{p}$

Question 502

[2 marks]Vector Geometry

In a triangle PQR,

\overrightarrow{PQ} = \mathbf{p}

\overrightarrow{PR} = \mathbf{q}

and L lies on QR with QL : LR = 2 : 1. Express

\overrightarrow{LR}

in terms of

\mathbf{p}

and

\mathbf{q}

A $\tfrac{1}{3}(\mathbf{q} - \mathbf{p})$
B $\tfrac{2}{3}(\mathbf{q} - \mathbf{p})$
C $\tfrac{1}{3}(\mathbf{p} - \mathbf{q})$
D $\tfrac{1}{2}(\mathbf{q} - \mathbf{p})$

Question 503

[3 marks]Vector Geometry

In a triangle PQR,

\overrightarrow{PQ} = \mathbf{p}

and

\overrightarrow{PR} = \mathbf{q}

. M is the midpoint of PR, L lies on QR with QL : LR = 2 : 1, N lies on PL and MN is parallel to RQ. Given that

\overrightarrow{NM} = k\,\overrightarrow{QR}

, find the scalar

k

Answer this when you sit the paper.

Question 504

[2 marks]Vector Geometry

Take

\pi

to be

\frac{22}{7}

. A hemispherical bowl has an internal diameter of 14 cm. Calculate the capacity of the bowl in litres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 505

[3 marks]Vector Geometry

Take

\pi

to be

\frac{22}{7}

. A hemispherical bowl is made of wood 2 cm thick and has an internal diameter of 14 cm. Calculate the mass of the bowl, in grams, given that the density of the wood is 0,8 g/cm

^3

Answer this when you sit the paper.

Question 601

[2 marks]Similarity and Congruency

ABCD is a quadrilateral with AB parallel to DC, and the diagonals AC and BD meet at E. Name, in the correct order, the triangle that is similar to

\triangle

ABE.

A $\triangle$ DCE
B $\triangle$ DEC
C $\triangle$ CDE
D $\triangle$ CED

Question 602

[2 marks]Similarity and Congruency

ABCD is a quadrilateral with AB parallel to DC, and the diagonals AC and BD meet at E. Given AB = 6 cm, BE = 3 cm and DC = 15 cm, calculate DE, in cm.

Answer this when you sit the paper.

Question 603

[2 marks]Similarity and Congruency

ABCD is a quadrilateral with AB parallel to DC, and the diagonals AC and BD meet at E, with BE = 3 cm and DE = 7,5 cm. Given that the area of

\triangle

BEC is 22,5 cm

^2

, calculate the area of

\triangle

DEC, in cm

^2

Answer this when you sit the paper.

Question 604

[3 marks]Similarity and Congruency

ABCD is a quadrilateral with AB parallel to DC, and the diagonals AC and BD meet at E. AB = 6 cm, DC = 15 cm and the area of

\triangle

DEC is 56,25 cm

^2

. Find the ratio of the area of

\triangle

ABE to the area of

\triangle

ADC, in its simplest form.

Answer this when you sit the paper.

Question 701

[3 marks]Geometrical Transformation

Triangle A has vertices

(6; 4)

(8; 6)

and

(4; 6)

. A reflection in the line

y = x + 2

maps triangle A onto triangle B. State the vertices of triangle B.

A $(8; 2)$ , $(10; 4)$ and $(6; 4)$
B $(2; 8)$ , $(4; 10)$ and $(4; 6)$
C $(4; 6)$ , $(6; 8)$ and $(6; 4)$
D $(6; 4)$ , $(8; 6)$ and $(8; 2)$

Question 702

[2 marks]Geometrical Transformation

The transformation with matrix

\begin{pmatrix} 1 & 0 \\ -1\frac{1}{2} & 1 \end{pmatrix}

maps the point

(8; 6)

onto the point

(8; k)

. Find

k

Answer this when you sit the paper.

Question 703

[2 marks]Geometrical Transformation

State the name of the transformation represented by the matrix

\begin{pmatrix} 1 & 0 \\ -1\frac{1}{2} & 1 \end{pmatrix}

AA shear with the $y$ -axis invariant and factor $-1\frac{1}{2}$
BA shear with the $x$ -axis invariant and factor $-1\frac{1}{2}$
CA one-way stretch with the $y$ -axis invariant and factor $-1\frac{1}{2}$
DAn enlargement with centre the origin and scale factor $-1\frac{1}{2}$

Question 704

[3 marks]Geometrical Transformation

Triangle A has vertices

(6; 4)

(8; 6)

and

(4; 6)

. A clockwise rotation of

90^\circ

about the centre

(0; 10)

maps triangle A onto triangle E. State the vertices of triangle E.

A $(6; -4)$ , $(4; -2)$ and $(4; -6)$
B $(-6; 4)$ , $(-4; 2)$ and $(-4; 6)$
C $(6; 16)$ , $(4; 18)$ and $(4; 14)$
D $(-6; -4)$ , $(-4; -2)$ and $(-4; -6)$

Question 705

[2 marks]Geometrical Transformation

Triangle A has vertices

(6; 4)

(8; 6)

and

(4; 6)

. Find its area, in square units.

Answer this when you sit the paper.

Question 801

[3 marks]Constructions and Loci

Three schools P, Q and R are such that the bearing of Q from P is

045^\circ

and the bearing of R from P is

300^\circ

. The distance PR is 18 km and Q is due east of R. Calculate the distance PQ, in km, correct to 3 significant figures.

Answer this when you sit the paper.

Question 802

[2 marks]Constructions and Loci

The bearing of Q from P is

045^\circ

and the bearing of R from P is

300^\circ

. Calculate the size of angle

Q\hat{P}R

A $75^\circ$
B $135^\circ$
C $255^\circ$
D $105^\circ$

Question 803

[3 marks]Constructions and Loci

Three schools P, Q and R are such that the bearing of Q from P is

045^\circ

and the bearing of R from P is

300^\circ

. The distance PR is 18 km, PQ is 12,7 km and Q is due east of R. Calculate the area of the triangular region PQR, in km

^2

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 804

[2 marks]Constructions and Loci

A plan is drawn to a scale of 1 cm to represent 2 km. Two schools P and R are 18 km apart. Find the length, in cm, of the line joining them on the plan.

Answer this when you sit the paper.

Question 805

[2 marks]Constructions and Loci

The bearing of R from P is

300^\circ

. Find the bearing of P from R.

A $120^\circ$
B $060^\circ$
C $240^\circ$
D $300^\circ$

Question 901

[2 marks]Functional Graphs

For the curve

y = x^2 + \frac{1}{x}

, calculate the value of

y

when

x = 1

Answer this when you sit the paper.

Question 902

[2 marks]Functional Graphs

For the curve

y = x^2 + \frac{1}{x}

, calculate the value of

y

when

x = 2,5

, correct to 1 decimal place.

Answer this when you sit the paper.

Question 903

[3 marks]Functional Graphs

The curve

y = x^2 + \frac{1}{x}

meets the line

2y = 5x + 2

. Which equation do the

x

-coordinates of the meeting points satisfy?

A $x^3 - 2,5x^2 + x - 1 = 0$
B $x^3 - 2,5x^2 - x + 1 = 0$
C $x^3 - 2,5x^2 - x - 1 = 0$
D $x^3 + 2,5x^2 - x + 1 = 0$

Question 904

[3 marks]Functional Graphs

The curve

y = x^2 + \frac{1}{x}

meets the line

2y = 5x + 2

at two points in the interval

0,25 \le x \le 3

. Write down the smaller of the two

x

-coordinates.

Answer this when you sit the paper.

Question 905

[2 marks]Functional Graphs

Estimate the area, in square units, enclosed between the curve

y = x^2 + \frac{1}{x}

, the line

2y = 5x + 2

, and the lines

x = 1

and

x = 2

Answer this when you sit the paper.

Question 1001

[2 marks]Trigonometry, Bearing & Distances

A vertical aerial mast is 20,5 m high. Calculate, to the nearest degree, the angle of elevation of the top of the mast from a point on horizontal ground 32,6 m from the foot of the mast.

Answer this when you sit the paper.

Question 1002

[2 marks]Trigonometry, Bearing & Distances

\triangle

LMN, LM = 6 cm, LN = 7 cm and MN = 12 cm. Using the cosine rule,

\cos(M\hat{L}N)

equals

A $-\frac{59}{84}$
B $\frac{59}{84}$
C $-\frac{59}{144}$
D $\frac{85}{84}$

Question 1003

[3 marks]Trigonometry, Bearing & Distances

\triangle

LMN, LM = 6 cm, LN = 7 cm and MN = 12 cm. Calculate

M\hat{L}N

, in degrees, correct to 1 decimal place.

Answer this when you sit the paper.

Question 1004

[2 marks]Trigonometry, Bearing & Distances

A co-operative deposited

\$60

million into a bank for

1\frac{1}{2}

years at the rate of 20% per annum simple interest. Calculate the interest the co-operative made, in millions of dollars.

Answer this when you sit the paper.

Question 1005

[3 marks]Trigonometry, Bearing & Distances

A co-operative earned

\$18

million in interest, was then charged 15% tax on that interest and

\$4,32

million in bank charges. Calculate its net profit, in millions of dollars.

Answer this when you sit the paper.

Question 1101

[2 marks]Statistics and Probability

In a grouped frequency distribution, 12 students walked a distance

x

km in the class

20 < x \le 25

. Find the frequency density for that class.

Answer this when you sit the paper.

Question 1102

[2 marks]Statistics and Probability

In a grouped frequency distribution, the class

25 < x \le 40

has a frequency density of 1. Find the number of students in that class.

Answer this when you sit the paper.

Question 1103

[2 marks]Statistics and Probability

70 students walked distances

x

km grouped as

10 < x \le 20

with 30 students,

20 < x \le 25

with 12,

25 < x \le 40

with 15, and

40 < x \le 50

with 13. State the modal class.

A $40 < x \le 50$ , because it reaches the largest distance walked
B $20 < x \le 25$ , because it is the narrowest of the four classes
C $25 < x \le 40$ , because it holds the widest spread of distances
D $10 < x \le 20$ , because it has the greatest frequency density

Question 1104

[3 marks]Statistics and Probability

In a sponsored walk, 15 students covered a distance

x

km in the class

25 < x \le 40

and 13 covered

40 < x \le 50

. A sponsor paid

\$10\,000

per km. Calculate an estimate of the total amount paid, in dollars, to those who walked more than 25 km.

Answer this when you sit the paper.

Question 1105

[3 marks]Statistics and Probability

Of 70 students, 30 walked at most 20 km and 12 walked more than 20 km but at most 25 km. Two of the 70 are chosen at random. Calculate the probability that one of them walked at most 20 km and the other walked more than 20 km but at most 25 km.

Answer this when you sit the paper.

Question 1201

[3 marks]Inequalities and Linear Programming

Mr Hove makes

x

tables and

y

chairs. A table needs 5 m of softwood and 3 m of hardwood, a chair needs 3 m of softwood and 4 m of hardwood, and he holds 45 m of softwood and 48 m of hardwood. Which pair of inequalities states the limits on his wood?

A $5x + 3y \ge 45$ and $3x + 4y \ge 48$
B $5x + 4y \le 45$ and $3x + 3y \le 48$
C $5x + 3y \le 45$ and $3x + 4y \le 48$
D $3x + 5y \le 45$ and $4x + 3y \le 48$

Question 1202

[2 marks]Inequalities and Linear Programming

Mr Hove makes

x

tables and

y

chairs. To make a profit he should manufacture more than 2 tables and at least 4 chairs. Which pair of inequalities states these conditions?

A $x \ge 2$ and $y \ge 4$
B $x > 2$ and $y \ge 4$
C $x \ge 2$ and $y > 4$
D $x > 2$ and $y > 4$

Question 1203

[2 marks]Inequalities and Linear Programming

Mr Hove makes

x

tables and

y

chairs subject to

5x + 3y \le 45

3x + 4y \le 48

x > 2

and

y \ge 4

. Find the greatest total number of items, tables and chairs together, that he can make.

Answer this when you sit the paper.

Question 1204

[3 marks]Inequalities and Linear Programming

Mr Hove makes

x

tables and

y

chairs subject to

5x + 3y \le 45

3x + 4y \le 48

x > 2

and

y \ge 4

. Which combinations give the maximum number of tables and chairs manufactured?

A3 tables with 9 chairs, and 4 tables with 8 chairs
B3 tables with 9 chairs, and 5 tables with 7 chairs
C5 tables with 7 chairs, and 6 tables with 6 chairs
D4 tables with 8 chairs, and 6 tables with 6 chairs

Question 1205

[2 marks]Inequalities and Linear Programming

A table needs 3 m of hardwood and a chair needs 4 m. Mr Hove holds 48 m of hardwood and makes 4 tables and 8 chairs. Find the length of hardwood, in metres, that he has left.

Answer this when you sit the paper.

Mathematics Paper 2 June 2008

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 104

Question 105

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 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 1201

Question 1202

Question 1203

Question 1204

Question 1205

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