ZIMSEC O Level Mathematics Paper 2 November 2015

Question 101

[3 marks]Fractions, Factorisation and Functional Notation

Find the value of

\left(1\frac{3}{4} + 2\frac{1}{3}\right) \div \frac{5}{6}

Answer this when you sit the paper.

Question 102

[2 marks]Fractions, Factorisation and Functional Notation

Factorise completely

y^2 + 10y - 24

Answer this when you sit the paper.

Question 103

[2 marks]Fractions, Factorisation and Functional Notation

Factorise completely

27 - 3x^2

Answer this when you sit the paper.

Question 104

[1 marks]Fractions, Factorisation and Functional Notation

It is given that

f(x) = 10 + 3x - x^2

. Find

f(2)

Answer this when you sit the paper.

Question 105

[3 marks]Fractions, Factorisation and Functional Notation

It is given that

f(x) = 10 + 3x - x^2

. Find the values of

x

for which

f(x) = 0

Answer this when you sit the paper.

Question 201

[2 marks]Trigonometry, Polygons and Percentages

In quadrilateral ABCD,

BC = 8

cm,

AC = 12

cm and

A\hat{B}C = 90^\circ

. Calculate AB, in centimetres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 202

[2 marks]Trigonometry, Polygons and Percentages

In quadrilateral ABCD,

AC = 12

cm,

C\hat{A}D = 46,5^\circ

and

A\hat{C}D = 90^\circ

. Calculate CD, in centimetres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 203

[3 marks]Trigonometry, Polygons and Percentages

Quadrilateral ABCD has

AB = 8,94

cm,

BC = 8

cm,

AC = 12

cm and

CD = 12,6

cm, with right angles at B and at C. Calculate the area of ABCD, in square centimetres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 204

[2 marks]Trigonometry, Polygons and Percentages

Four interior angles of a nonagon have a sum of

460^\circ

. The remaining interior angles are equal. Find the size of each of the equal angles.

Answer this when you sit the paper.

Question 205

[2 marks]Trigonometry, Polygons and Percentages

A shop sells a refrigerator at \$540. In the previous year the same type of refrigerator cost 8 % less. Calculate the cost price of the refrigerator in the previous year.

A\$532,00, subtracting 8 dollars rather than 8 per cent of the price
B\$496,80, since the previous year's price is 92 % of the \$540 charged now
C\$586,96, taking \$540 as 92 % of the earlier price and dividing by 0,92
D\$583,20, since 8 % of \$540 is added on to give the earlier price

Question 301

[2 marks]Matrices, Algebraic Fractions and Mensuration

Given

M = \begin{pmatrix} 8 & -4 \\ -5 & 3 \end{pmatrix}

and

N = \begin{pmatrix} 1 \\ 3 \end{pmatrix}

, find MN.

Answer this when you sit the paper.

Question 302

[2 marks]Matrices, Algebraic Fractions and Mensuration

Find the inverse of

M = \begin{pmatrix} 8 & -4 \\ -5 & 3 \end{pmatrix}

A $\frac{1}{44}\begin{pmatrix} 3 & 4 \\ 5 & 8 \end{pmatrix}$ , taking the determinant as $24 + 20$ instead of $24 - 20$
B $\frac{1}{4}\begin{pmatrix} 3 & -4 \\ -5 & 8 \end{pmatrix}$ , swapping the leading diagonal but leaving the other two signs alone
C $\frac{1}{4}\begin{pmatrix} 8 & 4 \\ 5 & -3 \end{pmatrix}$ , changing the signs on the leading diagonal instead of swapping them
D $\frac{1}{4}\begin{pmatrix} 3 & 4 \\ 5 & 8 \end{pmatrix}$ , with determinant $24 - 20 = 4$ and both diagonals treated correctly

Question 303

[2 marks]Matrices, Algebraic Fractions and Mensuration

Given

M = \begin{pmatrix} 8 & -4 \\ -5 & 3 \end{pmatrix}

, find

M^2

Answer this when you sit the paper.

Question 304

[3 marks]Matrices, Algebraic Fractions and Mensuration

Express

\frac{2x-1}{4} - \frac{3x-5}{12}

as a single fraction in its simplest form.

Answer this when you sit the paper.

Question 305

[2 marks]Matrices, Algebraic Fractions and Mensuration

The area of a trapezium is 63 cm

^2

and the sum of the lengths of its two parallel sides is 22,5 cm. Calculate the perpendicular distance, in centimetres, between the two parallel sides.

Answer this when you sit the paper.

Question 401

[2 marks]Sets, Probability and Laws of Indices

In a class of 40 music students, 6 enjoyed both the mbira and the guitar, and 3 enjoyed all three of the mbira, the piano and the guitar. Find the value of

w

, the number who enjoyed the mbira and the guitar but not the piano.

Answer this when you sit the paper.

Question 402

[3 marks]Sets, Probability and Laws of Indices

In a class of 40 music students, 20 enjoyed the guitar, 6 enjoyed the mbira and the guitar, 4 enjoyed the guitar and the piano, and 3 enjoyed all three instruments. Find the value of

z

, the number who enjoyed the guitar only.

Answer this when you sit the paper.

Question 403

[2 marks]Sets, Probability and Laws of Indices

In a class of 40 music students, 20 enjoyed playing the guitar. Two students are selected at random from the class. Find the probability that both enjoyed playing the guitar.

A $\frac{19}{78}$ , since the second student comes from the 39 left, 19 of whom play guitar
B $\frac{19}{39}$ , using the second draw by itself and forgetting to count the first one
C $\frac{1}{2}$ , which is the proportion of the whole class that enjoyed the guitar
D $\frac{1}{4}$ , treating the two choices as independent with 20 out of 40 each time

Question 404

[2 marks]Sets, Probability and Laws of Indices

Solve the equation

9^{m-1} = 27

Answer this when you sit the paper.

Question 501

[3 marks]Change of Subject of Formula and Circle Geometry

Make

x

the subject of the formula

y = \frac{p - 2x}{x}

Answer this when you sit the paper.

Question 502

[1 marks]Change of Subject of Formula and Circle Geometry

ABCD is a circle with centre O and CD is a diameter. Given that

A\hat{O}D = 50^\circ

, find

O\hat{C}A

, in degrees.

Answer this when you sit the paper.

Question 503

[2 marks]Change of Subject of Formula and Circle Geometry

ABCD is a circle with centre O and

A\hat{O}D = 50^\circ

. Find

O\hat{A}D

, in degrees.

Answer this when you sit the paper.

Question 504

[2 marks]Change of Subject of Formula and Circle Geometry

ABCD is a cyclic quadrilateral in a circle with centre O, CD is a diameter and

A\hat{D}C = 65^\circ

. Find

A\hat{B}C

A $90^\circ$ , treating CB as a diameter so that the angle at B stands in a semicircle
B $115^\circ$ , since opposite angles of a cyclic quadrilateral add up to $180^\circ$
C $130^\circ$ , doubling the $65^\circ$ angle at D as though it stood at the centre of the circle
D $65^\circ$ , taking $A\hat{B}C$ to equal $A\hat{D}C$ because they look like angles in the same segment

Question 505

[2 marks]Change of Subject of Formula and Circle Geometry

In circle ABCD with centre O, OA is parallel to CB,

O\hat{A}C = 25^\circ

and

A\hat{B}C = 115^\circ

. Find

C\hat{A}B

, in degrees.

Answer this when you sit the paper.

Question 601

[2 marks]Constructions and Loci

A parallelogram ABCD is constructed and the bisector of

A\hat{B}C

is drawn. Describe the locus that this bisector represents.

AThe locus of points equidistant from the points A and C, the two ends of that diagonal
BThe locus of points equidistant from the lines BA and BC, the two arms of the angle
CThe locus of points a fixed distance from the vertex B, which is an arc of a circle
DThe locus of points equidistant from the points B and C, a perpendicular bisector

Question 602

[2 marks]Constructions and Loci

In parallelogram ABCD,

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

. Write down the size of

B\hat{C}D

, in degrees.

Answer this when you sit the paper.

Question 603

[2 marks]Constructions and Loci

State what has to be constructed to show the locus of points equidistant from the two points B and C.

AThe bisector of the angle at B, the line that splits $A\hat{B}C$ into two equal parts
BThe line BC itself, extended equally far beyond B and beyond C in both directions
CThe perpendicular bisector of BC, the line through the midpoint of BC at right angles to it
DA circle with BC as its diameter, drawn so that it passes through both B and C

Question 604

[2 marks]Constructions and Loci

In parallelogram ABCD,

AB = 8

cm and

BC = 10

cm. Calculate the perimeter of the parallelogram, in centimetres.

Answer this when you sit the paper.

Question 701

[1 marks]Functional Graphs

A table of values for

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

gives the value of

y

x = 3

p

. Calculate the value of

p

Answer this when you sit the paper.

Question 702

[2 marks]Functional Graphs

State the maximum value of

y

for the function

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

Answer this when you sit the paper.

Question 703

[2 marks]Functional Graphs

Write down the range of values of

x

for which

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

is positive.

Answer this when you sit the paper.

Question 704

[2 marks]Functional Graphs

State the equation of the straight line that must be drawn together with the curve

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

in order to solve the equation

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

Answer this when you sit the paper.

Question 705

[3 marks]Functional Graphs

Solve the equation

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

, giving the answers correct to 2 decimal places.

A $x = 0$ or $x = 5$ , the two points at which the curve itself crosses the $x$ -axis
B $x = 3$ or $x = -1$ , the roots of $x^2 - 2x - 3 = 0$ found by dropping the fraction
C $x = 3,56$ or $x = -0,56$ , the two roots of $x^2 - 3x - 2 = 0$ to 2 decimal places
D $x = 2$ or $x = -1$ , read off where the line $y = x - 1$ meets the two axes

Question 706

[2 marks]Functional Graphs

For

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

the values of

y

x = 2

x = 3

and

x = 4

are 3, 3 and 2. Using the trapezium rule with these three ordinates, estimate the area bounded by the curve, the

x

-axis and the lines

x = 2

and

x = 4

, in square units.

Answer this when you sit the paper.

Question 801

[2 marks]Quadratic Equations and Linear Programming

The equation

\frac{2}{1-x} - \frac{4}{x} = 3

reduces to

3x^2 + 3x - 4 = 0

. Calculate the value of the discriminant

b^2 - 4ac

of that quadratic.

Answer this when you sit the paper.

Question 802

[3 marks]Quadratic Equations and Linear Programming

Solve the equation

3x^2 + 3x - 4 = 0

, giving the answers correct to 3 significant figures.

Answer this when you sit the paper.

Question 803

[2 marks]Quadratic Equations and Linear Programming

The unshaded region R is defined by three inequalities, one of which is

y \geq 4 - x

. Its other two boundaries are a vertical line through

x = 4

and a line joining

(0; 4)

(8; 0)

, with the shading outside R in each case. Write down the other two inequalities.

A $x \leq 4$ and $x + 2y \leq 8$ , the vertical boundary and the shallower slanting boundary
B $x \geq 4$ and $x + 2y \geq 8$ , taking the shaded side of each boundary as the wanted one
C $x \leq 4$ and $y \leq 4 - 2x$ , reading the shallow line's gradient as $-2$ and not $-\frac{1}{2}$
D $y \leq 4$ and $x + 2y \leq 8$ , using a horizontal boundary in place of the vertical one

Question 804

[3 marks]Quadratic Equations and Linear Programming

The region R has vertices at

(0; 4)

(4; 0)

and

(4; 2)

. Find the maximum value of

3y + x

for points in R.

Answer this when you sit the paper.

Question 901

[2 marks]Vector Geometry

v = \begin{pmatrix} 8 \\ u \end{pmatrix}

and

|v| = 17

, find the two possible values of

u

Answer this when you sit the paper.

Question 902

[1 marks]Vector Geometry

In triangle XYZ,

\overrightarrow{XZ} = p

and

\overrightarrow{XY} = q

. Express

\overrightarrow{YZ}

in terms of

p

and

q

Answer this when you sit the paper.

Question 903

[2 marks]Vector Geometry

In triangle XYZ,

\overrightarrow{XZ} = p

\overrightarrow{XY} = q

and M lies on YZ with

3YM = YZ

. Express

\overrightarrow{XM}

in terms of

p

and

q

Answer this when you sit the paper.

Question 904

[2 marks]Vector Geometry

In triangle XYZ,

\overrightarrow{XM} = \frac{1}{3}p + \frac{2}{3}q

, where

p

and

q

are not parallel. Given also that

\overrightarrow{XM} = hp + kq

, find the value of

h

and the value of

k

A $h = \frac{2}{3}$ and $k = \frac{1}{3}$ , matching the two coefficients the other way round
B $h = \frac{1}{3}$ and $k = \frac{1}{3}$ , taking M as the midpoint of YZ rather than one third along
C $h = \frac{1}{3}$ and $k = \frac{2}{3}$ , matching each coefficient to the vector it multiplies
D $h = 3$ and $k = 3$ , reading $3YM = YZ$ as multiplying by 3 rather than dividing by 3

Question 905

[2 marks]Vector Geometry

N lies on XZ with

\overrightarrow{XN} = \frac{1}{3}\overrightarrow{XZ}

. Write down the numerical value of

\frac{XN}{NZ}

Answer this when you sit the paper.

Question 906

[2 marks]Vector Geometry

N lies on XZ with

XN = \frac{1}{3}XZ

, and Y is the third vertex of triangle XYZ. Write the ratio area of triangle XYN : area of triangle XYZ in its simplest form.

Answer this when you sit the paper.

Question 1001

[2 marks]Geometrical Transformation

Triangle LMN, with L (3; 1), M (2; 2) and N (0; 1), is reflected in the line

y = 1

. Two of the three vertices are unmoved by this reflection. Write down the letters of those two vertices.

Answer this when you sit the paper.

Question 1002

[3 marks]Geometrical Transformation

Triangle LMN, with L (3; 1), M (2; 2) and N (0; 1), is mapped onto triangle

L_2M_2N_2

by a rotation through

180^\circ

about the point

(-1; 0)

. Find the coordinates of

L_2

M_2

and

N_2

A $L_2(-3; -1)$ , $M_2(-2; -2)$ , $N_2(0; -1)$ , rotating about the origin instead of $(-1; 0)$
B $L_2(-5; -1)$ , $M_2(-4; -2)$ , $N_2(-2; -1)$ , using the rule $(x; y) \to (-2 - x; -y)$
C $L_2(-5; 1)$ , $M_2(-4; 2)$ , $N_2(-2; 1)$ , reflecting in the line $x = -1$ rather than rotating
D $L_2(5; 1)$ , $M_2(4; 2)$ , $N_2(2; 1)$ , changing the sign of $x$ only and leaving $y$ untouched

Question 1003

[2 marks]Geometrical Transformation

The point L (3; 1) is mapped by the matrix

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

onto the point

L_3

. Write down the

x

-coordinate of

L_3

Answer this when you sit the paper.

Question 1004

[3 marks]Geometrical Transformation

Describe fully the single transformation P represented by the matrix

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

AAn enlargement centred at the origin with scale factor $-2$ , doubling both coordinates
BA reflection in the $y$ -axis on its own, since the $x$ -coordinates simply change sign
CA one-way stretch parallel to the $y$ -axis, with the $x$ -axis invariant, scale factor $-2$
DA one-way stretch parallel to the $x$ -axis, with the $y$ -axis invariant, scale factor $-2$

Question 1005

[2 marks]Geometrical Transformation

The transformation represented by the matrix

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

leaves one line fixed point by point. Write down the equation of that line.

Answer this when you sit the paper.

Question 1101

[3 marks]Statistics

The marks of 80 students are grouped in classes with mid-values 10, 25, 35, 45, 55, 65, 75, 85 and 95 and frequencies 0, 5, 19, 18, 16, 14, 4, 2 and 2. Estimate the mean mark, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1102

[1 marks]Statistics

A cumulative frequency distribution for 80 marks reads 0, 5, 24, 42, 58, 72, q, 78 and 80 for

m \leq 20

, 30, 40, 50, 60, 70, 80, 90 and 100. The class

70 < m \leq 80

has frequency 4. Find the value of q.

Answer this when you sit the paper.

Question 1103

[3 marks]Statistics

For 80 students the cumulative frequency is 24 at a mark of 40 % and 42 at a mark of 50 %. Estimate the median mark, in per cent, to the nearest whole number.

Answer this when you sit the paper.

Question 1104

[3 marks]Statistics

Reading from a cumulative frequency curve for 80 students, about 33 students scored 45 % or less and about 74 scored 75 % or less. Estimate the number of students whose marks were more than 45 % but less than 75 %.

Answer this when you sit the paper.

Question 1105

[2 marks]Statistics

A grouped distribution has the class

30 < m \leq 40

with frequency 19, and the running total up to

m \leq 40

is 24. State the point that must be plotted for this class when a cumulative frequency curve is drawn.

A $(40; 24)$ , the upper class boundary plotted against the running total up to it
B $(35; 24)$ , the mid-value of the class plotted against the running total up to it
C $(40; 19)$ , the upper class boundary plotted against that one class's frequency
D $(30; 24)$ , the lower class boundary plotted against the total up to the upper one

Question 1201

[2 marks]Measures and Mensuration

A right cone has a base radius of 8 cm and a slant height of 10 cm. Calculate the perpendicular height of the cone, in centimetres.

Answer this when you sit the paper.

Question 1202

[2 marks]Measures and Mensuration

A right cone has a base radius of 8 cm and a slant height of 10 cm. Taking

\pi

to be

\frac{22}{7}

, calculate the curved surface area of the cone, in square centimetres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1203

[2 marks]Measures and Mensuration

A right cone has a base radius of 8 cm and a perpendicular height of 6 cm. Taking

\pi

to be

\frac{22}{7}

, calculate the volume of the cone, in cubic centimetres, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1204

[3 marks]Measures and Mensuration

A cone of base radius 8 cm and slant height 10 cm is cut open to make a sector ABC of a circle centre O, where the shaded sector is the major one. Taking

\pi

to be

\frac{22}{7}

, calculate reflex

A\hat{O}C

, in degrees.

Answer this when you sit the paper.

Question 1205

[3 marks]Measures and Mensuration

A cone PQR is similar to a right cone of base radius 8 cm and slant height 10 cm, and has a slant height of 18 cm. Taking

\pi

to be

\frac{22}{7}

, calculate the base area of the cone PQR.

A $362$ cm $^2$ , scaling the base area by $1,8$ instead of by $1,8$ squared
B $1\,173$ cm $^2$ , scaling the base area by $1,8$ cubed as though it were a volume
C $652$ cm $^2$ , since lengths scale by $1,8$ , so the base radius becomes $14,4$ cm
D $201$ cm $^2$ , the base area of the original cone, left unscaled altogether

Mathematics Paper 2 November 2015

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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 706

Question 801

Question 802

Question 803

Question 804

Question 901

Question 902

Question 903

Question 904

Question 905

Question 906

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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