KCSE 2016 Mathematics Paper 1 Questions with Marking Scheme

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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
  2. Simplify (4 marks)
    simplify
  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)
  4. The figure below shows a net of a solid (measurements are in centimetres).
    Net of a solid
    Below 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
  5. Given that OA = 2i + 3j and OB = 3i — 2j, find the magnitude of AB to one decimal place. (3 marks)
  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)
  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)
  8. Without using mathematical tables or a calculator, evaluate (3 marks)
    evaluate without using tables
  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)
  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)
  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 the line in the form ax + by — c, where a, b and c are constants. (3 marks)
  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)
  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 currencies
    A 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)
  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)
  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)
  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)


SECTION II (50 marks)

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

  1.  
    1. Solve the equation    (4 marks)
      solve
    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)
      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)
  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)
    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)
  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)
      grid
    2.  
      1. State the class in which the median wage lies. (1 mark)
      2. Draw1 a vertical line, in the histogram, showing where the median wage lies. (1 mark)
    3. Using the histogram, determine the number of workers who earn Sh 450 or less per day. (3 marks)
  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) 
    2. Given that a point M is on LN such that LM:MN = 3:4, find the coordinate of M. (2 marks)
    3. If line OM is produced to T such that OM:MT = 6:1 
      1. Find the position of vector T (1 mark
      2. Show that points L, T and B are collinear. (4 marks)
  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)
    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)
  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)
      2. the y-intercept of the curve. (1 mark)
      1. Determine the stationary points of the curve. (3 marks)
      2. For each point in (b) (i) above, determine whether it is a maximum or a minimum. (2 marks)
    2. Sketch the curve. (2 marks)
  8.  
    1. On the grid provided, draw a graph of the function (3 marks)
      function
      grid
    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)

MARKING SCHEME

  1. -2(5+3)-9+3+5 = -14 = -2
     -3x-5+(-2)x4        7
  2.     (p+q)(p+q)        
    p(p2-q2)+q(p2-q2)

       (p+q)(p+q)    
    (p+q)(p+q)(p-q)
    =   1   
        p-q

  3. Internal Dimensions: 40,20, and 15
    Volume unoccupied= 40 x 20 x 15 - 8000
    =4000
    Height above water level= 4000/40x20=5cm

  4. mathsCSE2016p1qa4b
  5.      
    mathsCSE2016p1qa5
  6. Distance covered by bus= 63 x (10.45 - 8.15)
    = 63 x 2.5
    =157.5
    Speed of car
    = 157.5 
         1.75
    = 90km/h

  7. l2 = 52 - (2√52)=5
    l=√5
    ∴ tan(90-x)º= 2√5/√5 or 2

  8. 272/3 x (81/16)1/4 = (33)2 x (34/24)1/4
    =32 x (3/2)-1
    =32 x 2/3
    =6  

  9. Angle of major arc= 360 - 105= 255
    length of arc = 255/360 x 2 x 8.4 x 22/7
    =37.4cm  

  10. dy/dx = 3ax2 + b
    3a+b=-5
    a+b=1
    a=-3
    b=4  

  11. k-8 = -3
    3-k     1

    k-8 = -9 +3k
    2k         1
     k = ½

    y-8 -3
    x-½     1

    y-8 = -3(x -½)
    y-8= -3x + 3/2
    2y-16= 6x+3
    6x+2y=19

  12. <LKM=110º
    <KLM=35º or KML=35º
    Bearing =185º

  13. 105000 x 9.74
    = sh1022700

    1022700 - 403879
           12.11
    = 618821
         12.11
    =51100 rands

  14. L.S.F= 8:24= 1:3
    V.S.F= 1:27
    Volume of frustrum= 160 x 27 - 160
    =4160cm3   

  15.    
    1. mode =22
    2. Median 
      15,15,16, 19,19, 20, 20, 21, 22,22,22, 26, 27, 28
      median = 20 + 21/2
      = 20.5

  16. Log 0.096= Log ( 42 x 6 x 10-3)
    = 2( 0.6021) + ‾3.7782
    =‾2.9824 or
    (-1.0176)

  17.     
    1.  x = 5 or -6
    2.  50m
    3. 6m

  18.    
    1.  % profit for taxes and insurance
      =  40   x  75  
         100     100
      amount shared
      = 100 - (25 + 30) x 225000
               100
      = 45/100 x 225000
      = 101250
      amount cherop received more than Asha: ratio of contribution
      60000: 85000: 105000
           12 :     17 : 21
      21- 12 x 101250
        50 
      = 18225

    2.  profit during 2nd year
      225000 x 10/9
      = 250000
      Nangila's new ratio
      = 110000 = 2
         275000     5
      ∴ Nangila's new share profit
      = 2/5 x 112500
      = 45000   

  19.      
    1.     
      Frequency (f) 160 120 380 240 100
      Class width(c) 50 50 100 100 200
      Freq density(f/c) 3.2 2.4 3.8 2.4 0.5

      mathsCSE2016p1qa19a
    2. Median class= 200-300
    3. Shown by dotted line ( 257.875)
    4. Number of workers who earn sh 450 or less per day = 900 + 50 x 0.5 = 925
  20.         
    1.    
      mathsCSE2016p1qa20a
    2. OM= OL + 3/7 LN    
      mathsCSE2016p1qa20b
      =M(6,12)
    3.     
      1. OT=7/6OM   
        mathsCSE2016p1qa20ci   
      2.    
        mathsCSE2016p1qa20cii
        LB=3LT
        L is the common point
  21.               
    1. Juma's earnings before increase
      112% → 8400
      100% → 8400 x 100/112 = 7500
      Akinyi's earnings before increase:
      3/5 x 7500 = 4500
      Increase in Akinyi's earningws
      14100 - 8400 - 4500 = 1200
      % increase in Akinyi's earnings
      1200/4500 x 100
      = 26 2/3

    2. No. of bags bought
      14100/ 1175
      =12 bags
      profit = (1762.50 - 1175) x 12 = 7050
      Ratio 5700:8400 = 19:28
      Profit for Akinyi= 7050 x 19/47 = 2850
      Total earning for Akinyi
      5700 + 2850= 8550

  22.        
    mathsCSE2016p1qa22
    construction of 135º angle between lines AB=4cm and BC=6cm
    Construction of 60º angle between lines AB=4cm and AD=3cm
    Completion of quadilateral ABCD= <BCD = 31º ± 1º 
  23.              
    1.          
      1. x- intercepts
        when y=0
        x2(2x+3)=0
        x=0 and x = -3/2

      2.  y - intercept
        when x =0, y=0    

    2.           
      1. Stationary points of curve
        dy/dx = 6x2 + 6x
        stationary points when dy/dx = 0
        ie 6x2 + 6x = 0
        6x( x + 1)= 0
        x=0 or x=-1
        ∴ stationary points are
        (0,0) and (-1, 1)

      2.     
        x -2 -1½ -1 0 ½ 1
        dy/dx 12 -1½ 0 12

        minimum poin t(0,0)
        maximum point (-1,1)
    3.        
      mathsCSE2016p1qa23c
      points are at ( -1½, 0), (-1, 1) and (0,0)
      smooth curve
  24.      
    x 0 1 2 3 4 5 6
    y=½x2-x+3 3 7 10½ 15

    mathsCSE2016p1qa24a

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