ZIMSEC O Level Mathematics Paper 2 November 2000

Question 101

[2 marks]Algebra, factorising and rates

Simplify

4rt^2(3r-t^3)

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Question 102

[2 marks]Algebra, factorising and rates

Solve the equation

4(3x+5)-7(6-x)=16

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Question 103

[2 marks]Algebra, factorising and rates

Factorise completely

x^3+7x^2+12x

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Question 104

[2 marks]Algebra, factorising and rates

Factorise completely

ab-ad-bc+cd

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Question 105

[3 marks]Algebra, factorising and rates

Express as a single fraction in its simplest form

\frac{y}{4y-1}+\frac{3}{5}

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Question 106

[3 marks]Algebra, factorising and rates

The time taken to cook w kg of meat is

(48w+23)

minutes. Calculate how long it takes to cook 3kg of meat, giving the answer in hours and minutes.

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Question 201

[2 marks]Pythagoras and rhombus geometry

In a diagram,

B\hat{A}D=B\hat{C}D=90°

AB=AD

BC=7cm

and

CD=1cm

. Using Pythagoras in triangle BCD (right-angled at C), calculate

BD^2

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Question 202

[2 marks]Pythagoras and rhombus geometry

In a diagram,

B\hat{A}D=90°

AB=AD

and

BD^2=50

(from triangle BCD). Using Pythagoras in triangle ABD, calculate the length of AB.

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Question 203

[3 marks]Pythagoras and rhombus geometry

A rhombus has sides of length 8cm and one of its angles is 70°. Calculate the length of the longer diagonal of the rhombus.

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Question 204

[3 marks]Pythagoras and rhombus geometry

A rhombus has sides of length 8cm and one of its angles is 70°. Calculate the area of the rhombus.

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Question 301

[2 marks]Matrices and sets

Given

U=\begin{pmatrix}4&-1\\0&2\end{pmatrix}

, calculate the determinant of U.

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Question 302

[2 marks]Matrices and sets

Given

U=\begin{pmatrix}4&-1\\0&2\end{pmatrix}

with

\det U=8

, what is

U^{-1}

A(1/8) x [[4,1],[0,2]]
B(1/8) x [[2,1],[0,4]]
C(1/8) x [[2,-1],[0,4]]
D[[2,1],[0,4]] (no scalar factor)

Question 303

[2 marks]Matrices and sets

Given

U=\begin{pmatrix}4&-1\\0&2\end{pmatrix}

V=\begin{pmatrix}2&3\\0&t\end{pmatrix}

, and

UV=VU

, calculate the value of t.

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Question 304

[2 marks]Matrices and sets

Let

\xi=\{x: 1\le x<10, x \text{ an integer}\}

and

A=\{x: x \text{ is a perfect square}\}

. List the elements of A.

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Question 305

[1 marks]Matrices and sets

Let

\xi=\{x: 1\le x<10, x \text{ an integer}\}

A=\{1,4,9\}

(perfect squares) and

B=\{3,6,9\}

(multiples of 3). Find

n(A\cap B)

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Question 306

[2 marks]Matrices and sets

Let

\xi=\{x: 1\le x<10, x \text{ an integer}\}=\{1,...,9\}

A=\{1,4,9\}

and

B=\{3,6,9\}

. Write down

(A\cup B)'

by listing its elements.

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Question 401

[2 marks]Circle geometry and congruent triangles

KL is a diameter of circle KLMN. NM is parallel to KL and

M\hat{K}L=22°

. Calculate

K\hat{N}M

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Question 402

[2 marks]Circle geometry and congruent triangles

KL is a diameter of circle KLMN. NM is parallel to KL,

M\hat{K}L=22°

and

K\hat{N}M=112°

. Calculate

M\hat{K}N

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Question 403

[2 marks]Circle geometry and congruent triangles

In a diagram,

BP=BQ

P\hat{B}A=C\hat{B}Q

and

B\hat{A}Q=B\hat{C}P

. Triangle ABQ is congruent to which triangle, and by which case?

ATriangle CBP, case AAS
BTriangle PBC, case SSS
CTriangle CBP, case SAS
DTriangle BCP, case AAS

Question 404

[3 marks]Circle geometry and congruent triangles

In a diagram,

BP=BQ

P\hat{B}A=C\hat{B}Q

B\hat{A}Q=B\hat{C}P

, and triangle ABQ is congruent to triangle CBP. Write down the pairs of equal elements, not already given, in the congruent triangles.

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Question 501

[3 marks]Number, bills and percentage loss

Find the average of the three numbers

1\frac{3}{4}

-2\frac{1}{2}

3\frac{5}{8}

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Question 502

[3 marks]Number, bills and percentage loss

A householder used 960 units of electricity, charged at 35,2 cents per unit for the first 100 units, 30,5 cents per unit for the next 200 units, and 26 cents per unit for the remainder. Calculate the total cost.

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Question 503

[2 marks]Number, bills and percentage loss

A greengrocer bought 30kg of bananas at \$18,75 per kg. He sold 40% of them (12kg) at \$20,50 per kg, and

66\frac{2}{3}\%

of the remaining 18kg (12kg) at \$19,50 per kg; the last 6kg went unsold. Calculate the total revenue from the bananas he sold.

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Question 504

[3 marks]Number, bills and percentage loss

A greengrocer bought 30kg of bananas at \$18,75 per kg (cost price \$562,50 total) and sold them for a total revenue of \$480. Find the percentage loss he made, giving the answer correct to 2 significant figures.

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Question 601

[1 marks]Constructions & Loci

Triangle ABC has

BC=10cm

CA=7cm

AB=8,5cm

. Q lies on AC with

AQ=4cm

. By accurate construction and measurement, find the length of BQ.

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Question 701

[2 marks]Equations and variation

Calculate the discriminant of

3x^2-5x-4=0

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Question 702

[3 marks]Equations and variation

Solve

3x^2-5x-4=0

, giving both answers correct to 2 decimal places.

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Question 703

[2 marks]Equations and variation

Given that

\frac{1}{g}=\frac{1}{e}+\frac{1}{f}

, express

\frac{1}{e}

in terms of f and g.

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Question 704

[2 marks]Equations and variation

Given that

\frac{1}{g}=\frac{1}{e}+\frac{1}{f}

and

\frac{1}{e}=\frac{f-g}{fg}

, express e in terms of f and g.

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Question 705

[3 marks]Equations and variation

y is inversely proportional to

(x-3)

, and

y=4

when

x=2

. Express y in terms of x.

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Question 801

[2 marks]Graphs and quadratic curves

Using the graph of

y=x^2-6x+8

together with the line

y=6

(since

x^2-6x+2=0 \Leftrightarrow x^2-6x+8=6

), estimate the solutions of

x^2-6x+2=0

correct to 1 decimal place.

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Question 802

[2 marks]Graphs and quadratic curves

In the diagram, line l intersects the curve

y=x^2-6x+8

at P and Q. Using the diagram, write down the coordinates of P and Q.

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Question 803

[1 marks]Graphs and quadratic curves

A line l passes through P(1,3) and Q(6,8). Find the gradient of l.

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Question 804

[2 marks]Graphs and quadratic curves

A line l passes through P(1,3) and Q(6,8) and has gradient 1. Write down the equation of l.

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Question 805

[3 marks]Graphs and quadratic curves

A line l intersects the curve

y=x^2-6x+8

at P(1,3) and Q(6,8). Write down the equation in x, in the form

ax^2+bx+c=0

with a, b, c integers, whose solutions are the x-coordinates of P and Q.

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Question 806

[2 marks]Graphs and quadratic curves

Using the diagram, estimate the area of the shaded region, which is bounded above by the line

l

, at the sides by

x=2

and

x=5

, and below by the

x

-axis and the curve

y=x^2-6x+8

Answer this when you sit the paper.

Question 901

[2 marks]Statistics & Probability

The masses of Form 1 pupils fall into 5kg classes with frequencies 5, 30, 23, 25, 12, 5 (from 35kg up to 65kg). The cumulative frequency at 40kg is 5 and at 50kg is 58. Find the cumulative frequencies at 45kg, 55kg and 60kg, in that order.

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Question 902

[2 marks]Statistics & Probability

The cumulative frequency distribution of Form 1 pupils' masses passes through (45,35), (50,58) and (55,83), out of 100 pupils in total. Estimate the median mass.

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Question 903

[2 marks]Statistics & Probability

Out of 100 Form 1 pupils, 23 have mass in

45<m\le50

and 25 have mass in

50<m\le55

. One pupil is picked at random. Find, as a fraction in its lowest terms, the probability that the pupil's mass is more than 45kg but at most 55kg.

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Question 904

[2 marks]Statistics & Probability

Out of 100 Form 1 pupils, the probability that a randomly picked pupil's mass is more than m kg is

\frac{1}{10}

. Given the cumulative frequency curve reaches 90 at m=57,5kg, find the value of m.

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Question 1001

[3 marks]Bearings and trigonometry

E, F, G, H are points on a sportsfield. E is due north of F, H is due east of F (so

E\hat{F}H=90°

F\hat{E}H=65°

and

FH=15,2m

. Calculate the distance EF.

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Question 1002

[2 marks]Bearings and trigonometry

In triangle FGH,

HG=12,5m

FH=15,2m

and

F\hat{G}H=75°20'

. Using the sine rule, calculate

\sin(G\hat{F}H)

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Question 1003

[2 marks]Bearings and trigonometry

In triangle FGH,

\sin(G\hat{F}H)\approx0,795

. Find angle

G\hat{F}H

correct to 1 decimal place.

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Question 1004

[2 marks]Bearings and trigonometry

H is due east of F (bearing 090° from F), and G lies a further

52,7°

round from FH. Calculate the bearing of G from F.

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Question 1005

[3 marks]Bearings and trigonometry

A vertical pole of height 5,6m stands at H, and

FH=15,2m

. Calculate the angle of elevation of the top of the pole from F.

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Question 1101

[3 marks]Geometrical Transformation

Triangle X has vertices (-4,4), (-2,2) and (0,4). A single transformation Z maps triangle X onto triangle Z(X) with vertices (6,-6), (3,-3) and (0,-6). Describe fully the transformation Z.

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Question 1102

[2 marks]Geometrical Transformation

Triangle X has a vertex at (-4,4). A translation T maps this vertex onto (8,2). Write down the translation vector T.

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Question 1103

[3 marks]Geometrical Transformation

Triangle X has a vertex at (-4,4). The transformation R is a clockwise rotation of 90° about (4,2). Find the coordinates of the image of (-4,4) under R.

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Question 1201

[3 marks]Inequalities and vectors

List the integer values of x that satisfy the inequality

14-x \le 5x < 27

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Question 1202

[3 marks]Inequalities and vectors

Ratidzo has \$200 pocket money to buy c crayons at \$12,50 each and b books at \$25 each. She wants at least 5 crayons and at most 4 books. Write down three inequalities, other than

c\ge0

and

b\ge0

, that describe these conditions.

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Question 1203

[1 marks]Inequalities and vectors

In a diagram, OPA and OBR are straight lines,

\overrightarrow{OA}=\mathbf{a}

and

\overrightarrow{OB}=\mathbf{b}

. Express

\overrightarrow{AB}

in terms of a and/or b.

Answer this when you sit the paper.

Question 1204

[2 marks]Inequalities and vectors

In a diagram, OPA and OBR are straight lines,

\overrightarrow{OA}=\mathbf{a}

\overrightarrow{OB}=\mathbf{b}

\overrightarrow{OP}=4\overrightarrow{PA}

(so

\overrightarrow{OP}=\frac{4}{5}\mathbf{a}

) and

\overrightarrow{OR}=3\overrightarrow{OB}

. Express

\overrightarrow{PR}

in terms of a and/or b.

Answer this when you sit the paper.

Question 1205

[3 marks]Inequalities and vectors

In a diagram,

\overrightarrow{OP}=\frac{4}{5}\mathbf{a}

\overrightarrow{PR}=-\frac{4}{5}\mathbf{a}+3\mathbf{b}

, and Q is on PR with

\overrightarrow{PQ}=h\overrightarrow{PR}

. Express

\overrightarrow{PQ}

in terms of h, a and b.

Answer this when you sit the paper.

Mathematics Paper 2 November 2000

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The questions

Question 101

Question 102

Question 103

Question 104

Question 105

Question 106

Question 201

Question 202

Question 203

Question 204

Question 301

Question 302

Question 303

Question 304

Question 305

Question 306

Question 401

Question 402

Question 403

Question 404

Question 501

Question 502

Question 503

Question 504

Question 601

Question 701

Question 702

Question 703

Question 704

Question 705

Question 801

Question 802

Question 803

Question 804

Question 805

Question 806

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 1201

Question 1202

Question 1203

Question 1204

Question 1205

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