KCSE 2017 Mathematics Alt B Paper 1 with Marking Scheme

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INSTRUCTIONS TO CANDIDATES

  • This paper consists of two sections: Section I and Section II
  • Answer all questions in Section I and only five questions from Section II.
  • Show all the steps in your calculations, giving your answer at each stage 
  • Marks may be given for correct working even if the answer is wrong
  • Non-programmable silent electronic calculator and KNEC Mathematical tables may be used, except where stated otherwise
  • Candidates should answer the questions in English

SECTION I (50 marks)

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

  1.  
    1. Express 4732 in terms of its prime factors.  (1 mark)
    2. Find the smallest positive number that must be multiplied by 4732 to make it a perfect square.   (1 mark)
  2. Three people Juma, Weru and Njeri went round a circular racing track, 3.12 km long. They all started from the same point and moved in the same direction. Juma walked at 48 m per minute, Weru ran at 120 m per minute while Njeri cycled at 156 m per minute.
    If they started travelling at 0700 h, find the time when they were first together again.    (3 marks)
  3. Evaluate 
    KCSE 2017 Maths Alt A PP2 Q3   (3 marks)
  4. Without using a calculator, evaluate 
    KCSE 2017 Maths Alt A PP2 QS4      (3 marks)
  5. Use logarithms to evaluate:
    KCSE 2017 Maths Alt A PP2 QS5correct to 4 significant figures.      (4 marks)
  6. The diagram below represents a cube of side 10 cm from which a cuboid measuring xem by xcm by 10cm is removed as shown.
    KCSE 2017 Maths Alt A PP2 QS6
    Write an expression in terms of x, for the surface area of the remaining solid.  (3 marks)
  7. A cylindrical tank 1.4 m in diameter contains 3234 litres of water. Find the depth, in metres, of the water. (Take π = 22/7  ).   (3 marks)
  8. The figure below represents a quadrilateral ABCD in which angle DAB = 60°, angle BCD = 30° and BC = DC = 40cm. Side AB = AD.
    KCSE 2017 Maths Alt B PP1 QS8
    Calculate the area of the quadrilateral ABCD correct to 4 significant figures.  (4 marks)
  9. The area of a sector of a circle is 36.96 cm2. The sector subtends an angle of 135° at the centre of the circle. Find the radius of the circle. (Take π = 22/7 )     (3 marks)
  10. Evaluate the expression KCSE 2017 Maths Alt B PP1 QS10  given that t=5 and r = 27.   (2 marks)
  11. Two employees Njoka and Okoth contributed ¼ and 1/6 of their salaries respectively to start a project The contribution amounted to Ksh 16 000. If Njoka contributed 4/9 and Okoth 1/3 of their salaries, the contribution would have been ksh 30 000. Calculate each person's salary.   (3 marks)
  12. Solve x − 8 ≤ − x ≥ 4 −3x and represent the integral values of x on a number line.   (4 marks)
  13. Figure ABCDEF is a regular hexagon. Line AE and BF intersect at G.
    KCSE 2017 Maths Alt B PP1 QS13
    Find the size of angle FGE.   (3 marks)
  14. Using a ruler and a pair of compasses only, construct triangle PQR in which PQ = 8 cm, ∠RPQ = 60° and ∠PRQ = 75°. Measure PR.   (4 marks)
  15. The marked price of a TV set is Ksh 36000. A dealer sold the set and allowed a 12% discount on the marked price and still made a 25% profit on the cost price. Find the cost price of the set.    (3 marks)
  16. Figure A'B'C'D' is the image of ABCD under a rotation. By construction, determine the centre P and the angle of rotation.   (3 marks)
    KCSE 2017 Maths Alt B PP1 QS16

SECTION II (50 marks)

Answer any five questions in this section in the spaces provided

  1. A saleslady earns a monthly salary of Ksh 60 000. She gets a commission of 4% on the value of goods she sells above Ksh 250 000 but less than Ksh 400 000. On goods sold above Ksh 400 000, she gets a commission of 7.5%.
    1. In a certain month, she sold goods worth Ksh 525 000. Calculate her total earnings that month.    (4 marks)
    2. In another month, she earned a total of Ksh 94 500. Find the value of goods that she sold that month.
      (6 marks)
  2. Lines y + 2x = 4 and 3x − y = 1 intersect at point T.
    1. Find the equation of line L, which passes through point T and (3,−2).   (5 marks)
    2. A line L2 passes through (5,4) and is parallel to L1 Find the equation of L2 in the form y=mx+c where m and care constants.  (2 marks)
    3. Another line is L3 perpendicular to L1 at T. Find the equation of L3 in the form ar + by = c where a, b and care integers.   (3 marks)
  3. A car travelled from town A to town B. The car started from rest at A and moved with a constant acceleration for 2 minutes and attained a speed of 1.2 km/minute. It then maintained this speed for a further 10 minutes before decelerating at a constant rate for another four minutes. The car finally came to rest at B.
    1. On the grid provided, draw a speed-time graph for the car.   (4 marks)
    2. Use the graph to calculate:
      1. the distance, in metres, the car travelled during its deceleration;   (2 marks)
      2. the distance, in kilometres, covered by the car in the whole journey;   (2 marks)
      3. the average speed, in km/h, for the whole journey.    (2 marks)
  4. The figure below is a square of side x cm. The square is divided into four regions A, B, C and D. Regions A and Care squares. Square C is of side yem. Regions B and D are rectangles.
    KCSE 2017 Maths Alt B PP1 QS20
    1. Find the total area of the following regions in terms of x and y in factorised form:
      1. A and C;     (1 mark)
      2. B and D;     (2 marks)
      3. A, B, C and D.    (2 marks)
    2. Find the total area of B and D in terms of x given that y = 2 cm.    (2 marks)
    3. Factorise 25c2 − 16. Evaluate without using mathematical tables:
      1. 50242 − 49762   (1 mark)
      2. 8.962 − 1.042     (1 mark)
  5. The figure below represents a right pyramid VEFGH mounted on a cuboid ABCDEFGH. Line AB = 6 cm, DA = 8 cm and AF = BG = CH = DE = 3 cm. Line VE = VF = VG = VH = 13 cm.
    KCSE 2017 Maths Alt B PP1 QS21
    Calculate, correct to 2 decimal places:
    1. the surface area of the rectangular faces;   (3 marks)
    2. the surface area of the triangular faces.    (5 marks)
    3. the total surface area of the solid.    (2 marks)
  6. The figure below is a solid which consists of a frustum of a cone, a cylinder and a hemispherical top. The internal radii of the frustum are 42 cm and 21 cm. The vertical height of the original cone was 40 cm and the height of the cylinder is 30 cm.
    KCSE 2017 Maths Alt B PP1 QS22
    Calculate:
    1. the volume of the frustum part;     (5 marks)
    2. the volume of the cylindrical part;   (2 marks)
    3. the total volume of the solid.       (3 marks)
  7. Four posts A, B, C and D stand on a level horizontal ground. Post B is 240 m on a bearing of 060° from A, C is 210m to the south of B and D is 150 m on a bearing of 140° from A.
    1. Using a scale of 1 cm to represent 30m, show the relative positions of the posts.   (4 marks)
    2. Use the scale drawing to:   
      1. find the distance and the bearing of C from D;    (2 marks)
      2. determine how far A is to the west of B.    (2 marks
    3. The height of post D is 18m. Calculate, correct to 2 decimal places, the angle of elevation of the top of post D from the foot of post A.    (2 marks)
  8. The vertices of a triangle are A(-2, 2), B(2, 2) and C(2,8).
    1. On the grid provided, draw triangle ABC and its image A'B'C' under a rotation of −90° about R(1,1).
      (3 marks)
    2. The vertices of triangle A"B"C"are A"(-1,-5), B"(-1, -3) and C"(-4,-3).
      1. Draw triangle A"B"C".  (1 mark)
      2. Describe fully the transformation X that maps Δ A'B'C' onto Δ A"B"C".    (3 marks)
    3. Triangle A'''B'''C''' is the image of triangle A"B"C" under a reflection in the line x = 0. On the same grid, draw triangle A"B""C"".   (1 mark)
    4. State the type of congruence between:      (1 mark)
      1. ΔABC and ΔA'B'C'.   (1 mark)
      2. Δ A''B''C'' and Δ A''' B''' C '''    (1 mark)


MARKING SCHEME

  1.  
    1. 4732 = 22 × 7 × 132
    2. 22 × 7 × 132 × 7 = 22 × 72 × 132 is a perfect square.
      ∴ smallest factror is 7
  2. Time taken : Juma 3120 = 65 min
                                 48
    Weru 3120 = 26 min, Njeri = 3120 = 20min
              120                            156
    LCM of 65, 26, 20
    KCSE 2017 Maths Alt B PP1 Ans 3
    LCM = 260min or 4h 20min
    Time together is 1120h
  3.  
    KCSE 2017 Maths Alt A PP2 Q3 
    −10
        −5
    = 2
  4.  
    KCSE 2017 Maths Alt B PP1 Ans4
                        = 33/20 × 5/11 
                        = 3/4   
  5.  
    KCSE 2017 Maths Alt B PP1 Ans5
  6.  4 × 10 × 10 + 2 × (10 × 10 − x2)
    = 400 + 200 − 2x2
    = 600 − 2x2
  7. 22/7 × 0.7 × 0.7 × h = 3.234
    1.54h = 3.234
    h = 3.234
           1.54
       = 2.1m

  8. KCSE 2017 Maths Alt B PP1 Ans8
    Area of ΔABC
    Area of ΔBDC 
    BM = sin 15° ⇒ BM = 40 sin 15°
                           BC = 2 × 40 sin 15°
                                = 20.71cm
    Area of the qusdrilateral
    =½ × (20.71)2 × sin 60 + ½ × 402 × sin 30
    =185.7 + 400
    =585.7cm2
  9. 135/360 × 22/7 × r2 = 36.96
    r2 = 36.96 × 360 × 7
                 135 × 22
     r = √31.36
       = 5.6cm
  10.  
    KCSE 2017 Maths Alt B PP1 Ans10
                   =549
  11. Let Njoka's salary be x, and Okoth's salary be y 
    ¼x + 1/6y = 16000
    4/9 x + 1/3y = 30000
    3x + 2y = 192000
    4x + 3y = 270000
    9x + 6y = 576000
    8x + 6y = 540000
    x           = 36000
    y           =42000
    Njoka's sh.36000 and Okoth's sh 42000
  12. x − 8 ≤ − x                −x ≥ 4 −3x
    2x ≤ 8                        2x ≥ 4
    x ≤ 4                            x ≥ 2
    2 ≤ x ≤ 4 
    KCSE 2017 Maths Alt B PP1 Ans12
  13. ∠BAF = 120° interior angle of a regular hexagon
    ∠AEF = ∠FAE = 180 − 120 = 30°
                                 2
    In ΔEFG, ∠EFG = 120 − 30 = 90°
    ∴ ∠FGE = 180 − (90+30) = 60°
  14.   
    KCSE 2017 Maths Alt B PP1 Ans14
    PR = 5.9 ± 0.1cm
  15. Sellinig price was Ksh 36000 × 88/100
         = Ksh 31680
    Cost price was ksh31680 × 100/125
         = Ksh 25344
  16.  
    KCSE 2017 Maths Alt B PP1 Ans16
    Centre P 
    Angle of rotation, −71°
  17.   
    1. 4/100 × (400000 − 250000) + 7.5/100 × (525000 − 400000) + 60000
      =6000 + 9375 + 60000
      =Ksh 75375
    2. 94500 = 60000 + 6000 + x
      x = 28500
      Value of goods for commission of ksh 28500
      = 28500 × 100
            7.5
      = Ksh 380000
      Total sale = 250000 + 150000 + 380000
      = ksh 780000
  18.  
    1. y + 2x = 4
      −y =3x = 1
      5x = 5
      x = 1
      y +  2 = 4 ⇒ y = 2
      T(1,2)
      Grad 
      −2−2−4 = −2
       3−1       2
      y − 2 = −2
      x − 1
      y = − 2x + 4
    2. Grad = −2
      y − 4 = −2
      x − 5
      ⇒ y − 4 = −2x + 10
         y = −2x +14
    3. Grad = ½
      y − 2 = ½
      x − 1
      y − 2  = ½(x − 1)
      2y − 4 = x − 1
      −x + 2y = 3
  19.  
    1.  
      KCSE 2017 math alt B Q19
      Scale
      Acceleration part
      Constant acceleration
      Deceleration
    2.  
      1. ½ × 4 × 1.2 × 1000m
        =2400
      2. ½ × 2 × 1.2 + 1.2 × 10 + ½ × 4 × 1.2km
        =15.6km
      3. Average speed
        KCSE 2017 Maths Alt B PP1 Ans19b
        =58.5km/h
  20.  
    1.  
      1. Area of A + C = (x − y) (x − y) + y2
      2. Area of B + D = y(x − y) + y(x − y)
                            =2y(x − y)
      3. (x − y)2 + y2 + 2y(x − y)
        =(x − y)(x − y) + y2 + 2yx − 2y2
        =x2 − 2yx + y2 + y2 + 2yx − 2y2
        =x2
    2. 2(x−2) +2(x−2)
      =4x − 8
    3. 25c2 − 16 = (5c)2 − 42
      = (5c+4)(5c−4)
    4.  
      1. 50242 − 49762 = (5024 + 4976)(5024 − 4976)
        =10000 × 48
        =480000
      2. 8.962 −1.042 = (8.96 + 1.04)(8.96 − 1.04)
        = 10 × 7.92
        =79.2
  21.  
    1. Area of the base of the cuboid
      = 8 × 6cm2 = 48cm2
      Area of 4 faces of the side of th cuboid
      =(2×8×3+2×6×3)cm2
      =48+36cm = 84cm2
      Total 48+84 = 132cm2
    2. Consider faces VEF and VHG use Hero's formula
      S = ½(13+13 + 8) = 17cm
      Area of VEF and VHG = 2√(17(17−13)(17−13)(17−8))cm
      = 2√(17x4x4x9) = 98.96cm
      Consider faces VFG and VEH
      S =½ (13+13+6)=16cm
      Area of VFG
      = 2√(16(16−13)(16–13)(16 −6))
      = 2√(16 x 3 x 3 x 10)
      =75.90cm2
      Area of triangular faces
      98.96+75.90
      = 174.86cm2
    3. Surface area of the solid
      =132+174.86
      =306.86cm2
  22.  
    1. Vertical height of small cone:
      AE = 21 ⇒ AE = 20cm 
      Volume of frustum
      1/3 × 22/7 × 422 x 40 – 1/3 × 22/7 × 212 × 20
      = 73920 − 9240
      = 64680cm3
    2. Volume of cylindrical part
      = 22/7 x 212 x 30
      = 41580 cm2
    3. Volume of hemispherical part
      ½ × 4/3 × 22/7 × 213 
      =19404cm2
      Total volume
      = 64680+41580+19404
      =125 664 cm2
  23.  
    1.  
      KCSE 2017 Maths Alt B PP1 Ans23
    2.  
      1. Distance CD = 3.7 × 30
        =111km
        Bearing of C from D = 076°
      2. Distance of A to the west of B 
        =6.9 × 30 
        =207km
    3. tan θ = 18/150
               =0.12
      θ = tan−1 0.12
      =6.84°
  24.  
    1.  
      KCSE 2017 Q24 a Alt B
      Δ ABC correctly drawn 
      A'B'C' correctly plotted
      Δ A'B'C' drawn
    2.  
      1. Δ A"B"C" drawn enlargement
      2. scale factor, −½ centre (0, −2)
    3. Δ A'''B'''C''' drawn
    4.  
      1. Directly congruent
      2. Opposite congruent

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