MATHEMATICS PAPER 1 Marking Scheme - 2019 KCSE Prediction Answers Set 1

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SECTION I (50 marks)

Answer all the questions in this section

  1. Use logarithms to evaluate (4 marks)
    maths p1 q1
    p1 ans 1
  2. Solve for x in the equation given below (3 marks)
    92x+1 = 30 − 344x
    p1 ans 2
  3. The lengths of a triangle are in the ratio 5:6:9. if its area is 250√2, calculate its perimeter (4 marks)
    p1 ans 3
  4. Simplify completely without using mathematical tables or calculator (3 marks)
    maths p1 q4
    p1 ans 4
  5. Four metal rods of length 15x3h2y, 3x2yh5 and 9x4y2h are to be cut into smaller pieces of the same length such that there is no metal left. What is the maximum length of one of the pieces (1 mark)
    p1 ans 5
  6. Given that the log 2 = 0.30103 and log 7 = 0.84510, find without using mathematical tables or calculator, log 9.8 (3 marks)
    p1 ans 6
  7. Given that 3x − y = 17, find the value of y2 + 9x2 − 6xy − 9 (2 marks)
    p1 ans 7
  8. The center of a circle is C (6,7). A tangent to the circle passes through the point P (8,3) lying on the circle. Find the eqaution of the tangent (4 marks)
    p1 ans 8
  9. Coffee grade 1 at sh 30 per 50 g is blended with coffe grade 2 at sh 20 per 25g and the mixture is sold at sh 96 per 100g at a profit of 22%. Calculate the ratio in which the two brands of coffee were mixed. (4 marks)
    p1 ans 9
  10. In the figure below, O is the centre of the circle which passes through thepoints C, T and D. CT is parallel to OD and line ATB is tangent to the circle at T. If the angle BTC is 44o, find the size of angle TOD. (3 marks)
    p1 ans 10
  11. Two similar solids have masses of 1000g and 1728g. if it costs Ksh 1080 to paint the outside of the larger solid, how much will it cost to paint the outside of the smaller solid (3 marks)
    p1 ans 11
  12. The figure below shows part of a circle. AB = 12 cm and CD= 6 cm. AC = CB. Calculate the shaded area. (4 marks)
    p1 q12
    p1 ans 12i
    p1 ans 12ii
  13. Three interior angles of a polygon are 155o, 153o and 160o. Each of the other interior angles is 148o. How many sides does the polygon have (3 marks)
    p1 ans 13
  14. Write down the inequalities that describe the set of points in the unshaded region P. (3 marks)
    p1 ans 14i
    p1 ans 14ii
  15. The position vectors position vectors p1 AsAvdrespectively. Given that the length of AB is 7√2, find the value of b. (3 marks)
    p1 ans 15
  16. A line segment AB is shown below. Construct a triangle CBA = 30o. Hence use the constructed line AC to find a point T such that B divides At in the ratio 5: −2 (3 marks)
    p1 ans 16

SECTION II (50 Marks)

Answer Five questions only from this section

  1. A cylindrical tank is to be constructed. A model of the tank is made such that it is similar to the actual tank. The curved surface area of the model is 2160 cmand that of the proposed tank is 135m2
    1. Given that the length of the model is 6cm, calculate the height of the tank in metres. (3 marks)
      p1 ans 17a
    2. Caculate the volume of the model given that the diameter of the actual tank is 14 m. (3 marks)
      p1 ans 17b
    3. Determine the volume of the actual tank in m3 (2 marks)
      p1 ans 17c
    4. The actual tank is used to store some liquids whose density is 0.82 g/cm3. If the tank is half full, determine the mass of the liquid in kg. (2 marks)
      p1 ans 17d
    1. A bus travelling at 99km/hr passes a check point at 10.00 a.m and a matatu travelling at 132 km/h in the same direction passes through the check point at 10:15 a.m. If the bus and the matatu continue at their uniform speeds, find the time the matatu will overtake the bus. (6 marks)
      p1 ans 18 a
    2. Two passenger trains A and B which are 240 m apart and travelling in opposite directiosn at 164 km/h and 88km/h respectively approach one another on a straight railway line. Train A is 150 metres long and train B is 100 metres long. Determine time in seconds that elapses before the two trains completely pass each other. (4 marks)
      p1 ans 18b
    1. On the grid provided, draw the garph of the function y = x2 + x + 9 for −3 ≤ x ≤3
       X  −3 −2  −1  0  3
       Y  15  11  9  9  11 15   21
      p1 ans 19a
    2. Calculate the mid-ordinate for 5 strips between x = −2 and x = 3 and hence use the mid-ordinate rule to aapproximate the area under the curve between x = −2, x = 3 and the x - axis
      p1 ans 19 b
       X  −1.5  −0.5   0.5  1.5  2.5
       Y  9.75  8.75  9.75  12.75  17.5
    3. Assuming that the area determined by inetgration to be the actual area, caluate the percentage error in using the mid-ordinate rule. (2 marks)
      p1 ans 19 c
  4. On the graph paper provided plots the points P (2,2) Q (2,5) and R(4,4).
    1. Join them to form a triangle PQR (1 mark)
      p1 ans 20a
    2. Reflect the triangle PQR in the line X = 0 and label the image P'Q'R'. (2 marks)
    3. Triangle PQR is given a translation by vector T(22) to P"Q"R". Plot the triangle P"Q"R". (3 marks)
    4. Rotate triangle P"Q"R" about the origin through −90o. State the coordinates of P'''Q'''R'''. (3 marks)
    5. Identify two pair of triangles that are direct congruence (1 mark)
  5. Three warshisps P,Q,R are at sea such that ship Q is 400 km on a bearing of 030o from ship P. Ship R is 750 km from ship Q and on a bearing of S60oE from ship Q. Ship Q is 100km and to the north of an emeny warship S.
    1. Taking a scale of 1 cm to represent 100 km, locate the position of ships P,Q,R and S. (4 marks)
    2. Find the compass bearing of:
      1. Ship P from ship S (1 mark)
        • N15oW
      2. Ship S from ship R (1 mark)
        • S45oW
    3. Use the scale drawing to determine
      1. The distance of S from P (1 mark)
        • 6.8 cm × 100 = 680 km ± 10 km
      2. The distance of R from S (1 mark)
        • 8.8 cm × 100 = 880 km ± 10 km
    4. Find the bearing of 
      1. Q from R (1 mark)
        • N60oW or 300o
      2. P from R (1 mark)
        • 270o
  6. A certain number of people agreed to contribute equally to buy books worths shs. 1200 for a school library. Five people pulled out and so the others agreed to contribute an extra sh. 40 each. Their contribution enabled them to raise the sh. 1200 expected
    1. If the original number of people was x, write an expression of how much each was originally going to contribute (1 mark)
      • 1200/x
    2. Write down the expression of how much each contributed after the five people pulled out. (1 mark)
      • 1200/x−5
    3. Calculate how many people made the contributions (5 marks)
      p1 ans 22 c
    4. If the prices of books before buying went up in the ratio 5:4, how much extra did each contributor give. (3 marks)
      p1 ans 22d
  7. PQRS is a trapezium where PQ is parallel to SR. PR and SQ intersect at X, so that SX = KSQ and PX = hPR wher k and h are constants. Vectors PQ = 3q and PS = s,SR = q
    1. Show this information on a diagram (1 mark)
      p1 ans 23a
    2. Express vector SQ in terms of s and q (1 mark)
      • SQ = SP + PQ
        = −S + 3Q
        = 3Q − S
    3. Express SX in terms of k, q and s (1 mark)
      • SX = KSQ
        =K(3Q − S )
    4. Express SX in terms of h, q and s (1 mark)
      • S = spp + px
        = −s + hpr
        = −s + h(s+q)
    5. Obtain h and k (4 marks)
      p1 ans 23 e
    6. In what ratio does X divide SQ? (2 marks)
      p1 ans 23f
  8. A solid cylinder has a radius of 21cm and a height of 18 cm. A conical hole of radius r is drilled in the cylinder on one of the end faces. The conical hole is 12 cm deep. If the material removed from the hole is 22/3% of the volume of the cylinder, find: (Use π = 22/7)
    1. The surface area of the hole(5 marks)
      p1 ans 24a
    2. The radius of a spherical ball made out of the material (3 marks)
      volume = 4/3πr3
    3. The surface area of the spherical ball (2 marks)
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