KCSE 2017 Mathematics Alt A Paper 1 with Marking Scheme

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-prpogrammable 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. Evaluate    correct to four decimal places, hence write the answer in standard form  (3 marks)
2. Two numbers p and q are such that p³ ×q = 189. Find p and q.   (3 marks)
3. A trader bought maize for Ksh 20 per kilogram and beans for Ksh 60 per kilogram. She mixed the maize and beans and sold the mixture at Ksh 48 per kilogram. If she made a 60% profit, determine the ratio maize:beans per kilogram in the mixture.  (4 marks)
4. In the triangle ABC below, BC -12 cm, ABC 80° and Z ACB = 30°.
   
   Calculate, correct to 4 significant figures, the area of triangle ABC.   (3 marks)
5. Given that the exterior angle of a regular hexagon is x, find the size of each interior angle of the hexagon.  (2 marks)
6. Use logarithms to evaluate     (4 marks)
   
7. A line L is perpendicular to the line ^2x/₃ + ^5y/₇=1. Given that L passes through (4,11), find:
   1. gradient of L;   (1 mark)
   2. equation of L in the form y = mx + c, where m and care constants.   (2 marks)
8. Convert 2π^c into degrees.   (2 marks)
                9
9. A garden is in the shape of a right angled triangle. The length of the shortest side is 17m and the area of the garden is 346.8m². Calculate the length of the longest side of the garden.(3 marks)
10. The diagram below shows a region R bounded by three lines L₁, L₂ and L₃  
    
    Form the three inequalities that satisfy the given region R.   (3 marks)
11. Murimi and Naliaka had each 840 tree seedlings. Murimi planted equal number of seedlings per row in x rows while Naliaka planted equal number of seedlings in (x + 1) rows. The number of tree seedlings planted by Murimi in each row were 4 more than those planted by Naliaka in each row. Calculate the number of seedlings Murimi planted in each row.  (4 marks)
12. A Kenyan bank buys and sells foreign currencies as shown below:
                                       Buying  (Ksh)   Selling (Ksh)
    1 Sterling pound (£)      130.10               130.54
     1 South African Rand      9.52                    9.58
    A businessman on a trip to Kenya had £50 000 which he converted to Kenya shillings. While in Kenya, he spent 80% of the money and changed the balance to South African Rand. Calculate, to the nearest Rand, the amount he obtained.
    (3 marks)
13. Use the mid ordinate rule with six strips to find the area bounded by the curve y= x² + 1. the lines x = −4, x = 8 and the x-axis.    (3 marks)
14. Given that a = , c =  and 3a − 2b + 4c = , find b ( 3 marks)
15. A triangle T with vertices A(2,4), B (6,2) and C(4,8) is mapped onto trinagle T' with vertices A'(10,0), B'(8,−4) and C'(14, −2) by a rotation.
    1. On the grid provided draw triangle T and its image.  (2 marks)
    2. Determine the centre and angle of rotation that maps T onto T'.  (2 marks)
16. A construction company employs technicians and artisans. On a certain day 3 technicians and 2 artisans were hired and paid a total of Ksh 9000. On another day the firm hired 4 technicians and 1 artisan and paid a total of Ksh 9500. Calculate the cost of hiring 2 technicicnas and 5 artisans in a day.  (3 marks)

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

17. Two lines L₁: 2y − 3x − 6 = 0 and L₂: 3y + x − 20 = 0 intersect at a point A.
    1. Find the coordinates of A.   (3 marks)
    2. A third line L₃ is perpendicular to L₂ at point A. Find the equation of L₃ in the form y = mx + c, where m and c are constants.  (3 marks)
    3. Another line L₄ is parallel to L₁ and passes through (−1,3). Find the x and y intercepts of L₄ (4 marks)
18. The masses, in kilograms, of patients who attended a clinic on a certain day were recorde follows.
    38   52   46   48   60   59   62   73   49   54
    49   41   57   58   69   72   60   58   42   41
    79   62   58    67   54   60   65   61  48   47
    69   59   70    52   63   58   59   49   51   44
    67   49   51    58   54   59   39   59   54   52
    1. Starting with the class 35 - 39, make a frequency distribution table for the data.  (2 marks)
    2. Calculate:
       1. the mean mass;   (3 marks)
       2. the median mass  (3 marks)
    3. On the grid provided draw a histogram to represent the data (2 marks)
19. A solid S is made up of a cylindrical part and a conical part. The height of the solid is 4.5 m. The common radius of the cylindrical part and the conical part is 0.9 m. The height of the conical part is 1.5 m.
    1. Calculate the volume, correct to 1 decimal place, of solid S.   (4 marks)
    2. Calculate the total surface area of solid S.   (4 marks)
    3. A square base pillar of side 1.6 m has the same volume as solid S. Determine the height of the pillar, correct to 1 decimal place.   (2 marks)
20. In the figure below (not drawn to scale), AB = 11 cm, BC -8cm, AD = 3 cm, AC = 5 cm and ∠DAC is a right angle.
    
    Calculate, correct to one decimal place:
    1. the length DC;   (2 marks)
    2. the size of  ∠ADC;   (2 marks)
    3. the size of  ∠ACB;  (3 marks)
    4. the area of the quadrilateral ABCD.   (3 marks)
21. The comer points A, B, C and D of a ranch are such that B is 8 km directly East of A and C is 6 km from B on a bearing of 30°. Dis 7 km from C on a bearing of 300°.
    1. Using a scale of 1cm to represent 1 km, draw a diagram to show the positions of A, B, C and D.   (4 marks)
    2. Use the scale drawing to determine: 
       1. the bearing of A from D;   (1 mark)
       2. the distance BD in kilometres.   (2 marks)
       3. the perimeter of the ranch in kilometres    (3 marks)
22.  
    1. Given that A =  B = , find values of x for which AB is a singular matrix.  (4 marks)
    2. Mambo bought 3 exercise books and 5 pens for a total of Ksh 165. If Mambo had bought 2 exercise books and 4 pens, he would have spent Ksh 45 less. Taking x to represent the price of an exercise book and y to represent the price of a pen:
       1. Form two equations to represent the above information.  (1 mark)
       2. Use matrix method to find the price of an exercise book and that of a pen.  (3 marks)
       3. A teacher of a class of 36 students bought 2 exercise books and 1 pen for each student. Calculate the total amount of money the teacher paid for the books and pens  (2 marks)
23. An institution intended to buy a certain number of chairs for Ksh 16 200. The supplier agreed to offer a discount of Ksh 60 per chair which enabled the institution to get 3 more chairs. Taking x as the originally intended number of chairs,
    1. Write an expressions in terms of x for:
       1. original price per chair,  (1 mark)
       2. price per chair after discount.  (1 mark)
    2. Determine:
       1. the number of chairs the institution originally intended to buy  (4 marks)
       2. price per chair after discount;  (2 marks)
       3. the amount of money the institution would have saved per chair if it bought the intended number of chairs at a discount of 15%.  (2 marks)
24. The equation of a curve is given as y = 2x³ − ⁹/₂x² − 15x + 3
    1. Find:
       1. the value of r when x=2;     (2 marks)
       2. the equation of the tangent to the curve at x=2    (4 marks)
    2. Determine the turning points of the curve.    (4 marks)

MARKING SCHEME

1.  
    = √0.2581
                          = 0.5080
   Standard form = 5.080 x 10⁻¹
2. 189 = 3 × 3 ×  3 ×  7
         = 3³ × 7
   ∴ p³ × q = 3³ × 7
   p=3, q = 7
3. Let the number of kg of maize be m and number of kg of beans be b
   Buying price = 20m + 60b
   Selling price = 48(m+b)
   60   = 48(m+b) − (20m+60b)
   100             20m+60b
   0.6  = 28m − 12b
             20m+60b
   ⇒ 12m + 36b = 28m − 12b
    = 16m − 48b
   ^m/_b = ³/₁
   ∴ Ratio m:b = 3:1
4. ∠BAC = 180° − (80° + 30°) = 70
     AC     =    12    
   sin 80°     sin 70°
              =12.58cm
   Area of ΔABC = ½×12×12 sin 30°
   = 6 × 12.58 × 0.5
   = 37.74cm²
5. No. of sides of a hexagon = 6
   Each exterior anglex, x = 360
                                          6
                                     = 60°
   Size of each exteriror angle 
   =180° − 60°
   =120°
6.  
   
   = 0.4745
7.  
   1. ^2x/₃ + ^5y/₇=1
      14x + 15y = 21
      y = ⁻¹⁴/₁₅ + ²¹/₁₅
      gradient of L = ¹⁵/₁₄
      
   2. Equation of L
      y − 11 = 15
       x − 4     14
      y = ¹⁵/₁₄x + ⁴⁷/₇
8. π^c  = 180°
   
   =40°
9. Area = ½ × b × h
   Let h be the other shorter side
   346.8 = ½ × 17 × h
   h = 40.8
   Longest side = √(17² + 40.8²)
   = √1953.64
   = 44.2m
10. L₁ : y − x ≤ 1
    L₂ : x < 4
    L₃ : x + 2y ≥ 6
11. 840 − 840 = 4
      x      x+1
    4x² + 4x − 840 = 0
    x² + x − 210 =0
    (x+15)(x−14)=0
    x = 14
    No. of seedling planted by
    Murimi per row = ⁸⁴⁰/₁₄
                           = 60
12. £500000 to Ksh = 50,000 × 130.10
    =Ksh 6,505,000
    Balance after expenditure
    =²⁰/₁₀₀ × 6,505,000
    =Ksh 1,301,000/
    Amount in Rands
    = 1,301,000
           9.58
    =R 153,804
13. Mid ordiantes are
    

    x	−3	−1	1	3	5	7
    y	10	2	2	10	26	50

    Area = 2(10+2+2+10+26+50)
           = 200
14.  
    
    4 − 2x = 10
      − 2x = 6
    x = −3
    −11−2y = −19
           −2y = −8
    y=4
    
15.  
    1.  
       
       
       
    2. Centre of rotation (4, −2)
       Angle of rotation −90°
16. 3t + 2a = 9000
    4t + a = 9500
    a = 9500 − 4t
    3t +2(9500 − 4t) = 9000
    −5t = −10000
      t = 2000
    ⇒ a = 9500 − 8000 = 1500
    Cost of hiring 2 technicians 5 artisans
    = 2 × 2000 + 5 × 1500 = ksh 11500
17.  
    1. 2y − 3x = 6
       3y + x = 20
       2y − 3x = 6
       9y + 3x = 60 
       11y = 60
       y = 6
       x = 20 − 18 
          = 2
       Coordinates of A are (2,6)
    2. L₂: 3y = −x + 20
       y = −¹/₃x + 20
       Gradient of perpendicular = 3
       y − 6 = 3
       x − 2
       y = 3x − 6 + 6
       y = 3x
    3. Gradient of L₄ = gradient of L₁
       ^{= 3}/₂
       y − 3 = ³/₂
       x + 1
       2y − 6 = 3x + 3
       2y − 3x = 9
       When x = 0  y = 4.5
       When y = 0  x = −3
18.  

    1. Mass kg	35 - 39	40-44	45-49	50-54	55-59	60-64	65-69	70-74	75-79
       Freq. (f)	2	4	8	9	11	7	5	3	1

    2.  
       1. Mean=2 × 37 + 4 × 42 + 8 × 47 + 9 × 52 + 11 × 57 + 7 × 62 + 5 × 67 + 3 × 72 + 1 × 77 
                                                                               50
              = 2775
                   50
             = 55.5kg
       2. C.f's 2, 6, 14, 23, 34, 41, 46, 49, 50
          Median = 54.5 + ²/₁₁ × 5
                    = 55.4kg
    3.  
       
19.  
    1. Volume of solid S
       Volume of conical part
       = ¹/₃π × (0.9)² × 1.5 
       = 1.3m²
       Volume of cylindrical part= π × (0.9)² × 3
       = 7.6m³
       Volume of pillar = 1.3 + 7.6
                              = 8.9m³
    2. S.A. of Solid S
       Slant length of conical part
       =√(1.5)² + (0.9)² = 1.7
       S.A. of conical part
       =π × (0.9) × 1.7
       = 4.8m²
       S.A. of cylindrical part
       =2π × 0.9 × 3 + π × (0.9)²
       =19.5m²
       S.A. of Solid s = 19.5m² + 4.8m²
                            = 24.3m²
    3. (1.6)² × L = 8.9m³
                    h = 3.5m
20.  
    1. Length DC = √(3² + 5²)
                       = 5.8cm
    2. tan^−1 5/₃ = 59.0°
    3. Size of angle ACB
       11² = 5² + 8² − 2 × 5 × 8 Cos C
       Cos C = 
             = −0.4
       ∠ACB = Cos⁻¹ (−0.4)
       ∠ACB = 113.6°
    4. Area of ABCD = Area of ACD + Area of ABC
       = ½ × 3 × 5 + ½ × 5 × 8 Sin 113.6
       = 25.8cm²
21.  
    
    
    1. Location of B 
       Location of C
       Location of D
       Complete quadrilateral ABCD
    2.  
       1. Bearing of A from D = 180 + 30
                                       = 210°
       2. Distance BD = 9.2km × 1km 
                             = 902km ± 0.1
       3. Perimeter: 
                    AD = 10.0 ± 0.1km
          Perimeter = 10 + 8 + 6 + 7
                         = 31km
22.  
    1.  
       
       
       ⇒ (3 + 3x) (2x + 2) − 6(x + 7) =0
       6x + 6x² + 6x − 6x − 36 = 0
       6x² + 6x − 36 = 0
       x² + x − 6 = 0
       (x + 3)(x − 2) = 0
       x = 2 or = −3
    2.  
       1. 3x + 5y = 165
          2x + 4y = 120
       2.  
          
          
          Cost of an exercise book =  Ksh. 30
          Cost of a pen = Ksh.15
       3. 2 × 36 × 30 + 36 × 15
          = Ksh 2700
23.  
    1.  
       1. Original price = 16200
                                     x
       2. Price after discount = 16200
                                            x+3
    2.  
       1. 16200 − 60 = 16200
              x                 x+3
          16200 − 60x =    16200
                   x                 x+3
          (16200 − 60x)(x+3) = 16200x
          60x² + 180x − 48600 = 0
          x² + 3x − 810 = 0
          (x + 30)(x − 27) = 0
          x = 27
       2. 16200
           27+3
          =Ksh 540
       3. 16200 × 15 
              27     100
          =Ksh 90
24.  
    1.  
       1. When x = 2
          y = 2(2)³ − ⁹/₂ (2)² − 15(2) + 3
          = −29
       2. ^dy/_dx = 6x² − 9x − 15
          at x = 2
          ^dy/_dx = −9
          Equation of tangent;
          y + 29 = −9
           x−2
          y = −9x + 18 − 29
          y = −9x −11
    2. ^dy/_dx = 6x² − 9x − 15
       At turning point
       6x² − 9x − 15 = 0
       6x² + 6x − 15x − 15 = 0
       (6x − 15) (x + 1) = 0
       x = −1 or 2.5
       at x = −1; y = 11.5
       turning point = (−1, 11.5)
       at x = 2.5, y = −31³/₈
       turning point = (2.5,  −31³/₈)