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MATHEMATICS PAPER 1 Marking Scheme- Kenya Certificate of Secondary Education (KCSE 2016)

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SECTION I (50 marks)

Answer all the questions from this section in the spaces provided. 

  1. Without using a calculator evaluate, (3marks)  
    evaluate
    answerp1q1
  2. Simplify (4 marks)
    simplify
    answerp1q2
  3. The external length, width and height of an open rectangular container are 41 cm, 21cm and 15.5 cm respectively. The thickness of the materials making the container is 5 mm. If the container has 8 litres of water, calculate the internal height above the water level. (4 marks)
    answerp1q3
  4. The figure below shows a net of a solid (measurements are in centimetres).
    Net of a solidBelow is a part of the sketch of the solid whose net is shown above. Complete the sketch of the solid, showing the hidden edges with broken lines. (3 marks)
    part of sketch
    answerp1q4
  5. Given that OA = 2i + 3j and OB = 3i — 2j, find the magnitude of AB to one decimal place. (3 marks)
    answerp1q5
  6. A bus travelling at an average speed of 63 km/h left a station at 8:15 a.m. A car later left the same station at 9:00 a.m. and caught up with the bus at 10:45 a.m. Find the average speed of the car. (3 marks)
    answerp1q6
  7. Given that x is an acute angle and cos x° = 2/5√5 find, without using mathematical tables or a calculator, tan (90 — x)°. (2 marks)
    answerp1q7
  8. Without using mathematical tables or a calculator, evaluate (3 marks)
    evaluate without using tables
    answerp1q8
  9. A minor arc of a circle subtends an angle of 105° at the centre of the circle. If the radius of the circle is 8.4 cm, find the length of the major arc. (Take π it = 22/7). (3 marks)
    answerp1qu9
  10. The gradient of the tangent to the curve y= — ax3 + bx at the point (1,1) is —5. Calculate the values of a and b. (4 marks)
    answerp1q10

  11. A line with gradient of —3 passes through the points (3, k) and (k, 8). Find the value of k and hence express the equation of theline in the form ax + by — c, where a, b and c are constants. (3 marks)

    answerp1q11
  12. Points L and M are equidistant from another point K. The bearing of L from K is 330°. The bearing of M from K is 220°. Calculate the bearing of M from L. (3 marks)
    answerp1q12
  13. In this question, mathematical tables should not be used.
    A Kenyan bank buys and sells foreign currencies as shown below:
    buy and sell foreign currenciesA tourist arrived in Kenya with 105 000 Hong Kong Dollars and changed the whole amount to Kenya Shillings. While in Kenya, she spent Sh 403 879 and changed the balance to South African Rands before leaving for South Africa. Calculate the amount in South African Rand, that she received. (3 marks)
    answerp1q13
  1. A small cone of height 8 cm is cut off from a bigger cone to leave a frustum of height 16 cm. If the volume of the smaller cone is 160 cm3, find the volume of the frustum. (3 marks)
    answerp1q14
  2. The production of milk, in litres, of 14 cows on a certain day was recorded as follows: 
    22, 26, 15, 19, 20, 16, 27, 15, 19, 22, 21, 20, 22 and 28.
    Determine:
    1. the mode: (1 mark)
    2. the median. (2 marks)
      answerp1q15
  3. Given that Log 4 = 0.6021 and Log 6 = 0.7782, without using mathematical tables or a calculator, evaluate log 0.096. (3 marks)
    answerp1q16


SECTION II (50 marks)

Answer any five questions from this section in the spaces provided. 

  1.  
    1. Solve the equation    (4 marks)
      solve
      Answer
      X = 5 or - 6
    2. The length of a floor of a rectangular hall is 9 m more than its width. The area of the floor is 136 m2.
      1. Calculate the perimeter of the floor. (4 marks)

        Answer
        =50 m
      2. A rectangular carpet is placed on the floor of the wall leaving an area of 64 m2. If the length of the carpet is twice its width, determine the width of the carpet (2 marks)

        Answer
        =
         6 m

  2. Three business partners: Asha, Nangila and Cherop contributed Ksh 6 000, Ksh 85 000 and Ksh 105 000 respectively. They agreed to put 25% of the profit back into business each year. They also agreed to put aside 40% of the remaining profit to cater for taxes and insurance. The rest of the profit would then be shared among the partners in the ratio of their contributions. At the end of the first year, the business realised a gross profit of Ksh 225 000.
    1. Calculate the amount of money Cherop received more than Asha at the end of the first year. (5 marks)
      answerp1q18
    2. Nangila further invested Ksh 25 000 into the business at the beginning of the second year. Given that the gross profit at the end of the second year increased in the ratio 10:9, calculate Nangila’s share of the profit at the end of the second year. (5 marks)
      answerp1q18b



  3. The frequency table below shows the daily wages paid to casual workers by a certain company.
    Wages in shillings  100-150  150-200  200-300  300-400  400-600 
    No. of workers 160 120 380 240  100
    1. In the grid provided, draw a histogram to represent the above information. (5 marks)
      answerp1q19a
    2.  
      1. State the class in which the median wage lies. (1 mark)
        Answer
        median class: 200-300
      2. Draw a vertical line, in the histogram, showing where the median wage lies. (1 mark)
        Answer
        Shown by dotted line(257.895)

    3. Using the histogram, determine the number of workers who earn Sh 450 or less per day. (3 marks)
      Answer
      Number of workers who earn Sh. 450 or less per day= 900+50×0.5=925

  4. In the diagram below, the coordinates of points A and B are (1, 6) and (15, 6) respectively. Point N is on OB and that 30N = 20B. Line OA is produced to L such that OL = 30A
    coordinates
    1. Find vector LN. (3 marks) 
      answerp1q20a
    2. Given that a point M is on LN such that LM:MN = 3:4, find the coordinate of M. (2 marks)
      answerp1q20b
    3. If line OM is produced to T such that OM:MT = 6:1 
      1. Find the position of vector T (1 mark
        answerp1q20ci
      2. Show that points L, T and B are collinear. (4 marks)
        answerp1q20cii
  5.  
    1. The ratio of Juma’s and Akinyi’s earnings was 5:3. Juma’s earnings rose to Ksh 8 400 after an increase of 12%.
      Calculate the percentage increase in Akinyi’s earnings given that the sum of their new earnings was Ksh 14 100. (6 marks)
      answerp1q21a
    2. Juma and Akinyi contributed all the new earnings to buy maize at Ksh 1175 per bag. The maize was then sold at Ksh 1762.50 per bag. The two shared all the money from the sales of the maize in the ratio of their contributions. Calculate the amount that Akinyi got. (4 marks)
      answerp1q21b
  6. Using a pair of compasses and ruler only, construct:
    1.  
      1. Triangle ABC in which AB = 5 cm, <BAC = 30° and <ABC = 105°. (3 marks)
      2. A circle that passes through the vertices of the triangle ABC. Measure the radius. (3 marks)
      3. the height of triangle ABC with AB as the base. Measure the height. (2 marks)
    2. Determine the area of circle that lies outside the triangle correct to 2 decimal places. (2 marks)
  7. The equation of a curve is y = 2x3 + 3x2.
    1. Find:
      1. the x-intercept of the curve: (2 marks)
        answerp1q23ai
      2. the y-intercept of the curve. (1 mark)
        answerp1q23aii
    2.  
      1. Determine the stationary points of the curve. (3 marks)
        answerp1q23bi
      2. For each point in (b) (i) above, determine whether it is a maximum or a minimum. (2 marks)
        answerp1q23bii
    1. Sketch the curve. (2 marks)
      answerp1q23c
  8.  
    1. On the grid provided, draw a graph of the function (3 marks)
      function
      answerp1q24a
    2. Calculate the mid-ordinates for five strips between x = 1 and x = 6, and hence use the mid-ordinate rule to approximate the area under the curve between x = 1, x = 6 and the x - axis. (3 marks)
    3. Assuming that the area determined by integration to be the actual area, calculate the percentage error in using the mid-ordinate rule. (4 marks)
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