﻿ Mathematics Report

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### Mathematics Report

5.0 MATHEMATICS ALT A (121)

The 2010 KCSE Mathematics Alternative A was tested in two papers. Paper 1 (121/1) and Paper 2 (121/2). The papers are equally weighted with each having two sections; Section (50 marks) short answer questions of not more than four marks each and Section 11(50 marks), a choice of eight questions of 10 marks each where candidates answer any five.

Paper 1(121/1) tests mainly Forms I and 2 work while Paper 2(121/2) tests mainly forms 3 and 4 work.
It is hoped that this report will be helpful to teachers in the teaching/learning process as well as in preparing candidates for future examinations.

5.1 CANDIDATES GENERAL PERFORMANCE

The table below shows the performance of both papers in the last four years.
Table 10: Candidates Performance in Mathematics for the last four yea rs

 Year Paper Candidature Maximum Score Mean Score Standard Deviation 2007 1 2 Overall 273504 100 100 200 19.55 19.91 39.46 19.09 20.74 39.83 2008 1 2 Overall 304908 100 100 200 22.76 19.82 42.59 22.76 19.56 41.53 2009 1 2 Overall 335615 100 100 200 22.37 19.89 42.26 19.71 18.78 37.65 2010 1 2 Overall 356072 100 100 200 26.21 19.92 46.07 20.63 20.35 40.02

From the table the following observations can be made:

5.1.1 The overall performance in Mathematics Alt A shown a slightly improvement compared to the previous years.
5.1.2 There is a notable improvement in the performance of Paper 1(121/1) from a mean of 22.27 in the year
2009 to a mean of 26.21 in the year 2010.

5.1.3 Paper 2 (121/2) shown a slight improvement from a mean of 19.89 in the year 2009 to a mean of 19.92 in the year 2010

5.1.4 There has been a significant increase in the candidature over the years.

5.2 INDIVIDUAL QUESTION ANALYSIS

The following is a discussion of the questions in which the candidates performed poorly.

5.2.1 PAPER! (121/1)

Question 4
A bus left a petrol station at 9.20 am. and traveled at an average speed of 75 km/h to a town N. At 9.40 a.m. a taxi, travelling at an average speed of 95 km/h, left the same petrol station and followed the route of the bus.
Determine the distance, from the petrol station, covered by the taxi at the time it caught up with the bus.
(3 marks)

The question tested on relative speed in the topic of linear motion.

Weaknesses

Calculation of the distance covered.

Expected response

Let the distance be d km
d d
or 75 95
dd2O
75 9560
d=118.75km

Give more practical examples in relative motion.

Question 10

Using a ruler and a pair of compasses only, construct a rhombus QRST in which angle
TQR 60� and QS 10cm. (3 marks)

The question tested on basic construction of a rhombus. The candidates were required to have knowledge of the properties of a rhombus.

Weaknesses

The location of point S. Candidates who did not score in this question took the length of the diagonal as equal to the length of one of the sides.

Expected response

Emphasis on construction of basic plane figures is essential. Give guidance on the correct labeling of plane
figures.

Question 15
The figure below shows two sectors in which CD and EF are arcs of concentric circles, centre 0. AngIe COD
radians and CE DF = 5 cm.

If the perimeter of the shape CDFE is 24cm, calculate the length of 0G.
(3 marks)

The question tested on arc length of a circle. Use of thc relationship s = rO where s is the arc length andO is
the angle at the centre measured in radians was required.

Weaknesses

Use of the radian measure in calculating the arc length. i.e. s = rO ,where 6 is in radians

Expected response

Emphasis on the radian measure and on conversion from degrees to radians and vice versa

Question 16

The histogram shown below represents the distribution of heights of seedlings of a
certain plant.
Height (cm) of seedlings
The shaded area in the histogram represents 20 seedlings. Calculate the percentage number of seedlings with
heights of at least 23 cm but less than 27 can
(3 marks)

The question is on representation of data with unequal width using a histogram. The students were required to
calculate the frequency density of each class in order to answer the question.

Weaknesses

Most candidates could not interpret the histogram properly and thus unable to answer question.

Expected response

This is an area which been performed poorly whenever it's tested. Teachers are advised to teach this area
thoroughly and give more practice in the area for the concept to be understood clearly.

Question 22
In the figure below, ABCD is a square. Points P, Q, R and S are the midpoints of AB, BC, CD and DA respectively.

(a) Describe fully:
(i) a reflection that maps triangle QCE onto triangle SDE; (1 mark)

(ii) an enlargement that maps triangle QCE onto triangle SAE; (2 marks)

(iii) a rotation that maps triangle QCE onto triangle SED. (3 marks)

(b) The triangle ERC is reflected on the line BD. The image of ERC under the reflection is rotated clockwise through an angle of 90o about P.
Determine the images of R and C:
(i) tinder the reflection; (2 marks)
(ii) after the two successive transformations. (2 marks)

The question tested on transformations. Candidates were required to know the general properties of transformations, i.e. reflection, rotation and enlargement.

Weaknesses

This question was unpopular with most of the candidates. Some of those who attempted the question had weaknesses in the description of the transformation.
Expected responses
(a) (i) Reflection in the line PR or ER
(ii) Enlargement centre E Scale factor = -l
(iii) Rotation about point R through 990 clockwise
p.S
C
(ii) R
C

The question was unpopular to most of the candidates. Thus there is need for more emphasis on transformations and use of more practical situations other than the ones in the text books only.

5.2.2 PAPER 2 (121/2)

Question 10

The points 0, A and B have the coordinates (0, 0), (4, 0) and (3, 2) respectively. Under a shear represented by
Emphasis on transformation is important and also use of different approaches to teach the topic.

Question 16
The circle shown below cuts the x-axis at (2, 0) and (4, 0). It also cuts y-axis at (0, 2) and
(0,-4).

Determine the:
(a) (i) coordinates of the centre;
(1 mark) (1 mark)

(b) equation of the circle in the form x2 -by2 + ax -4- by = c where a, b and care constants.
(2 marks)

the matrix , triangle OAB maps onto triangle OAR'.
(a) Determine in terms of k, the x coordinate of point B' . (2 marks)

(b) If OAB' is a right angled triangle in which angle OB'A is acute, find two possible values of k. (2 marks)

The question was on matrix transformation. Knowledge of the shear and stretch was important in answering this question

Weaknesses

The question was unfamiliar to both students and teachers especially in part (b). There was wrong interpretation of x andy coordinates with the students. Correct understanding of the shear was also a problem.

Expected responses

11 k(3 (3�2k
(a) II 1=1
0 1}2) 2
x ordinate 3+2k
(b) 3�2k=4
3�2k=0 k=-
(4 marks)

The question tested on equation of a circle. The candidates were required to use knowledge of chords in answering the question. Point of intersection of the perpendicular bisector of the chords gives the center on the circle

Weaknesses

Use of the chords to find the coordinates of the centre of the circle was a problem due to failure to relate the perpendicular bisectors of the chords and the centre of the circle.

Expected responses

(a) Coordinates of centre (1, -1)
(b) Equation
(x 1)2 + (y +1 f =10
x2 2x+1+y22y+1=10
x +y22x�2y=8
(4 marks)

Revise on chord of a circle and their perpendicular bisectors

Question 18

In the figure below OJKL is a parallelogram in which OJ 3p and OL 2r.

(a) If A is a point on LK such that LA = AK and a point B divides the line JK externally in the ratio 3:1, express OB and AJ in terms of p and r. (2 marks)

(b) Line OB intersects AJ at X such that OX mOB and AX nAJ.
(i) Express OX in terms of p, rand m. (1 mark)

(ii) Express OX in terms of p, rand n. (I mark)

The question tested on vectors and ration theorem.

Weaknesses

interpretation of a ratio for external division.

Expected responses

(a) OB3p�3r
AJ = 2p � 2r
(h) OX tn(08) in(3p -4- 3r) OX 2r�p+n(2p2r)

(iii) m(3p+3r) = 2r-2nr-I-p-}-2np
3mp+3mr r(2-2n)+p(1+2n)
3mp (1+2n)p
3m = lfln (i)
3mr r(2-2n)
3m 2-2ri (ii)
1-2n 2-2n
1
4n = 1 n
4
Subst. for n in (i)
4
1
3m1+2x
4
1 3 1
3m1=' i;2=
4 2x3 2
The ratio in which x divides AJ
AX=nAJ �AJ
Ratio 1:3
(10 marks)

Emphasize on different situations in external division.

Question 22
The first term of an Arithmetic Progression (A.P.) with six terms isp and its common difference is c. Another
A.P. with five terms has also its first term asp and a common difference of d. The last terms of the two
Arithmetic Progressions are equal.

(a) Express din terms of c. (3 marks)

(b) Given that the 4th term of the second A.P. exceeds the 4th term of the first one by1,find the values of c
and d. (3 marks)

(c) Calculate the value ofp if the sum of the terms of the first A.P. is 10 more than the sum of the terms of the
second A.P. (4 marks)

The question tested on Arithmetic progression (A.P). Candidates were required to calculate the common differences of the two APs and the first term.

Weaknesses

Relating the terms in the nvo progressions.

Expected responses

Give more practical examples on the topic of sequence and series.

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