KCSE 2018 Mathematics Alternative B Paper 2 with Marking Scheme

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KCSE 2018 Mathematics Alternative B Paper 1 with Marking Scheme

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QUESTIONS

SECTION I (50 marks)
Answer all the questions in this section in the spaces provided.

Round off each of the numbers in the expression below correct to 3 significant figures.
2436 X0.2562
0.05117
Hence, without using mathematical tables or a calculator, evaluate the expression. (3 marks)
Makau saved some money in a bank. Every month he deposited twice as much money as the previous month. In the 6th month he deposited Ksh 1600. Calculate the amount of money he deposited in the 10th month.(3 marks)
A matatu owner charged 14 passengers Ksh 2800 for a distance of 80km. If the charges per kilometre were constant, calculate the amount of money charged to 9 passengers for a distance of 150km.(3 marks)
1. Given that y = 4x - x², complete the table below for 0≤x≤4. (2 marks)
  
  x 0 1.5 1 1.5 2 2.5 3 3.5 4
  
  y 0 3 3.75 1.75
2. Use trapezium rule to estimate the area enclosed by the curve y = 4x - x², the lines x=0.5, x = 3.5 and the x-axis. (2 marks)
Charo had x bags of maize. The mass of 1 bag of maize was (x-10) kg. He had a total of 2000kg of maize. Calculate the mass of 35 such bags. (4 marks)
Make x the subject in:
(2 marks)
1. Use a ruler and a pair of compasses only to construct a triangle ABC such that, BC = 3.6 cm, AC = 4.8 cm, AB = 6 cm and ∠BCA = 90°. (1 mark)
2. Construct an inscribed circle for triangle ABC and measure its radius. (3 marks)
Given that the local time at point X(20°N, 15°E) is 3.30a.m. Find the local time at point Y(20°N, 30°W). (3 marks)
In the triangle PQR shown below, PQ = 6 cm, QR = 4 cm and ∠PQR = 40°.

Find, correct to 1 decimal place:
1. PR; (2 marks)
2. the obtuse ∠QRP. (2 marks)
The tangent to the curvey = x² + 3 at x = 1 passes through the point (4,10). Find the instantaneous rate of change of the curve at x = 1. (2 marks)
Kamau deposited some money in a financial institution which offered a simple interest of 8% per annum. At the end of 2 years the interest was Ksh 6 400. Mwau deposited a similar amount of money at a simple interest rate of 6% per annum for 3 years. Calculate the interest earned by Mwau. (3 marks)
The monthly income tax rates for a certain year were as shown in the table below:

monthly income in kenyan shillings (Ksh) tax rate in each shilling

0-11180 10%

11181-21714 15%

21715-32248 20%

32249-42782 25%

above 42782 30%

A monthly tax relief of Ksh 1280 was allowed. Barasa carned a monthly gross income of Ksh 120000. Calculate his net monthly pay. (4 marks)
The mass, in kilograms, of 20 sheep were recorded as follows:
1. Starting with the class 8-12, make a frequency distribution table for the data. (2 marks)
2. State the class size. (1 mark)
The vertices of a triangle LMN are L(2, 2), M(2, 4) and N(4, 5). Triangle LMN is mapped on to triangle L'M'N' whose vertices are L'(-2,2), M'(-4, 2) and N'(-5,4) by a matrix T. Find T. (3 marks)
There were 6 apples and 8 oranges in a basket. Two fruits were picked one at a time without replacement. Determine the probability of picking an orange and an apple. (3 marks)
Given that a = 2i + 5j and b = 5i +2j, find [b-a], correct to 1 decimal place. (2 marks)

SECTION II (50 marks)
Answer any five questions from this section in the spaces provided.

Eight workers can dig a trench in 24 days.
1. Calculate the number of days it would take 10 workers to dig the trench.(2 marks)
2. If the task was to be completed in 16 days, calculate how many more workers would be required to dig the trench.(3 marks)
3. The total cost of labour was Ksh 80 640. The 8 workers worked for 7 hours a day. Calculate the amount of money paid to each worker per hour.(2 marks)
4. After working for 10 days, 3 workers left. Calculate how long it took the remaining workers to complete the task.(3 marks)
In the figure below, line EDC is a tangent to the circle at D. Angle DAB = 45°, AD = AB = 5 cm and DC = 9.2 cm.
1. Find the size of:
  1. ∠BDC: (1 mark)
  2. ∠ADE;(2 marks)
  3. ∠BCD.(1 mark)
2. Find, correct to 1 decimal place, the lengths of:
  1. BC; (4 marks)
  2. DB. (2 marks)
1. Complete the table below for the values of y = cos x and y = - sin x, in the range of 0° ≤ x ≤ 360° (2 marks)
  
  x 0 30 60 90 120 150 180 210 240 270 300 330 360
  
  cos x 1 0.5 0 -0.5 -0.9 0 0.9 1
  
  - sin x 0 -0.9 -0.5 0 0.5 1 0.9 0
2. On the grid provided, draw the graphs of y = cos x and y = -sin x, in the range 0° ≤ x ≤ 360° (5 marks)
3. Use the graph to find:
  1. the values of x when cos x + sin x = 0 (2 marks)
  2. the values of y when cos x + sin x = 0. (1 mark)
Three quantities m, d and v are such that, m varies directly as d and inversely as the cube ofr. When m = 16, d = 8 and r = 3.
1. Find an equation connecting m, d and r. (4 marks)
2. When m = 40.5, d = 6, find r. (3 marks)
3. Calculate, correct to 1 decimal place, the percentage change in m, when r increases by 10%. (3 marks)
1. A one hectare piece of land was bought for Ksh 1 500 000. After 2 years, the land was subdivided into 8 equal plots and each plot was sold for Ksh 350000.
  Calculate, correct to 1 decimal place:
  1. the percentage profit made from the sale of the plots; (2 marks)
  2. the appreciation rate, in percentage per annum, of the original piece of land. (3 marks)
2. The value of a car depreciated at the rate of 3% per annum. After 2 years the value of the car was Ksh 1 129080.
  Calculate:
  1. the original value of the car; (3 marks)
  2. the percentage decrease, correct to 1 decimal place, in value after the car depreciated for the 2 years. (2 marks)
The mass, in kilograms, of patients in a clinic were recorded as follows.

mass(kg) 11-20 21-30 31-40 41-50 51-60 61-70

frequency 3 5 8 18 10 6
1. State the modal class. (1 mark)
2. Estimate the mean mass. (4 marks)
3. 1. On the grid provided, draw an ogive for the data. (4 marks)
  2. From the ogive, estimate the median. (1 mark)
A tin contains 3 black buttons and 4 red buttons. All the buttons are identical except for the colour. Two buttons are picked at random from the tin, one at a time with replacement.
1. Illustrate this information on a tree diagram. (2 marks)
2. Find the probability that:
  1. both buttons are black; (2 marks)
  2. both buttons are of the same colour, (3 marks)
  3. both buttons are of different colours. (3 marks)
1. Given that A = . find A^-1 (3 marks)
2. Ndunda bought 5 spoons and 3 plates for Ksh 110. Tirop bought 7 spoons and 5 plates for Ksh 170. One spoon costs Ksh x and one plate Ksh y.
  1. Form two linear equations to represent the above information. (2 marks)
  2. Use matrix method to solve the equations in part (b) (i) above simultaneously. (3 marks)
3. Kantai bought 30 spoons and 25 plates, calculate the total amount of money he paid. (2 marks)

MARKING SCHEME

2440 x 0.2562 = 2440 x 0.2560 x 10000
0.0512 0.0512 x 10000
=12200
r=2
ar⁵ = 1600
6th month; a x 2⁵ = 1600 = a = 50
10th month; 50 x 2⁹ = 50 x 512
Ksh 25 600
Charges per km = 2800
80x14
=Ksh 2.50
Charges for 150km for 98 = 2.5 x 150 x 9
= Ksh 3375
1. y values to fill: 1.75; 3.75; 4; 3; 0
2. Area = 0.5 {(1.75 + 1.75) + (3+3.75 + 4 + 3.75+3)}
  2
  =0.5(1.75+3+3.75 +4 +3.75 +3)
  = 9.625 sq units
x(x-10) = 2000
x² - 10x - 2000 = 0
(x+40)(x-50)=0
x=-40 or x = 50
The mass of 35 bags = 35 x (50-10)kg
= 1400 kg
x+y=9
x-Y
-8x=-10y
x = 10y
8
x = 5y
4
radius = 1.2cm ± 0.1
Difference in angle = 15+ 30 = 45°
Difference in time = 45 x 4 = 3hrs
60
Time at y = 3.30 – 3hrs
= 00.30
= 12.30a.m.
1. PR² = 36 + 16 - 2x6 x 4 x cos 40
  PR = √15.23
  = 3.9cm
2. sin 40 = sin ∠ORP
  3. 9 6
  sin ∠QRP = 6 x sin 40
  3.9
  =0.9889
  ∠QRP = 81.4°
  = 180° - 81.4°
  = 98.6°
at x=1, y = 4
Inst. Rate of change
= 10-4
4-1
= 6
3
= 2
Principle
= 6400 x 100 x 1
8 2
= 40000
Mwau's Interest = 40000 x 6 x 3
100
=Ksh 7200
Tax = 11180 x 0.1+10534 (0.15 + 0.20 + 0.25)+77218 x 0.30
=Ksh 30 603.80
Monthly Income Tax = 30603.80 - 1280
=Ksh 29 323.80
Monthly Net Salary = 120 000 - 29323.80
=Ksh 90 676.20
1. classes frequency
  
  8-12
  13-17
  18-22
  23-27
  28-32 4
  3
  5
  4
  4
2. Class size = 5
P(OA) or P(A,0) =
8 x 6 + 6 x 8
14 13 14 13
= 48
91
1. 8 x 24 = 19.2 days
  10
2. Workers for 16 days = 8 x 24 = 12 workers
  16
  No. of workers needed = 12-8-4 workers
3. Pay per hour = 80640 = ksh 60
  8 x 24 x 7
4. Fraction left after 10 days = 24 - 10 = 14
  24 24 24
  Time to be taken by 5 workers
  = 8 x 24 x 14
  5 24
  = 22.4 days
1. 1. ∠BDC=45°
  2. ∠ADE = ∠ABD = 180-45
    2
    = 67.5°
  3. ∠BCD=67.5-45
    = 22.5°
2. 1. Let BC = x
    x(x+5)=(9.2)²
    x² + 5x - 84.64 = 0
    
    x = 7.0
  2. Using Δ ABD
    BD² = 5² +5² - 2 x 5 x 5 cos 45
    = 16.64
    BD = 3.8cm
1. x 0 30 60 90 120 150 180 210 240 270 300 330 360
  
  cos x 0.9 -0.9 -1 -0.5 0.5
  
  sin x -0.5 -1 -0.9 0.9 0.5
3. 1. cos x + sin x = 0 x=135° or x = 315°
  2. At cos x =-sin x, y = 0.75 ± 0.01
    or y=-0.75 ± 0.01
1. M=kd
  r³
  16 = 8k;
  27
  k = 16 x 27
  8
  = 54
  M = 54 d
  r³
2. 40.5 = 54 x 6
  r³
  r³= 54 x 6 = 8
  40.5
  r = ³√8
  = 2
3. new M = 54d
  (1.1r)³
  Change in M= 54d - 54d = - 0.249 54d
  1.331r³ r³ r³
  =0.249 x 100%
  = 24.9% decrease
1. 1. 350000 x 8 - 1500000 x 100%
    1500000
    = 86.7%
2. 1. Original value = 1 200 000
  2. % decrease = 1200000 - 1129080
    1200000
    = 70920 x 100%
    1200000
    = 5.91%
1. Modal Class = 41-50
2. classes x f fx c.f
  
  11-20
  21-30
  31-40
  41-50
  51-60
  61-70 15.5
  25.5
  35.5
  45.5
  55.5
  65.5 3
  5
  8
  18
  10
  6 46.5
  127.5
  284
  819
  555
  393 3
  8
  16
  34
  44
  50
3. median = 43 ± 0.5
1. Tree diagram
2. 1. P(B,B) = 3 × 3 = 9
    7 7 49
  2. P(B, B)+P(R, R) = 3 × 3 × 4 × 4
    7 7 7 7
    = 9 + 16 = 25
    49 49 49
  3. P(B, R)+P(R, B)= 3 × 4 × 4 × 3
    7 7 7 7
    = 12 + 12
    49 49
    = 24
    49
1. Det A = 25 - 21 = 4
2. 1. 5x+3y = 110
    70+ 5y=170
  2. x = 10 and y = 20
3. Kantai paid = 30 x 10 + 25 x 20
  = Ksh 800