Mathematics Paper 1 Questions and Answers - KCSE 2021 Westlands Mock Exams

Questions

Instructions to candidates

• The paper contains TWO Sections: Section I and Section II.
• Answer ALL the questions in Section I and Only five questions from Section II.
• Show all the steps in your calculations, giving your answers at each stage in the spaces below each question.
• Marks may be given for correct working even if the answer is wrong.
• Non-programmable silent electronic calculators and KNEC Mathematical tables may be used except where stated otherwise.
• Answer all the questions in English.

SECTION I (50 marks)
Answer all questions in this section in the spaces provided

1. All odd numbers from are arranged in descending order to form a number.
   1.        
      1. Write the number (1 mark)
      2. Write the total value of the second digit of the number formed in (a) (i) (1 mark)
      3. Express the value of the number in (a) (ii) as a product of its prime factors in power form.(2 marks)
2. A shopkeeper bought a bag of sugar. He intends to repack the sugar in 40g,250g and 750g . Determine the least mass in grams of sugar that was in the bag. (3 marks)
3. Given that Log₁₀2= 0.3010 and Log₁₀2= 0.4771 without using tables or calculator find correct to 4 significant figures. (3 marks)
4. Evaluate (3 marks)
   ½ of ³/₂+1½(2½ - ²/₃)
     ¾ of 2½ ÷½
5. Using the grid provided below, solve the simultaneous equation (3 marks)
   
6. Given that a chord of length subtends an angle of at the circumference of the circle. Calculate the radius of the circle. (3 marks)
7. When a shopkeeper sells articles at , he makes a profit on the cost price. During a sale, he reduced the price of each article to . Calculate the percentage profit on an article sold at the sale price. (3 marks)
8. The size of one interior angle of an irregular polygon is 80°. Each of the other interior angles is 128°. Find the number of sides of the polygon. (3 marks)
9. Simplify 81^¾ - (¹/₅)^-1 - 27⁰(2 marks)
10. Given the inequalities x-6 < -3x + 2 < -2x + 9
    1. Solve the inequality (3 marks)
    2. Represent on a number line (1 mark)
11. The diagram below represents a right rectangular based pyramid of 5 cm by 4 cm. The slant edge of the pyramid is 6 cm. Draw and label the net of the pyramid. (3 marks)
    
12. Vectors OA = 4i + 3j,OB= -2i -j and OC=-5i-3j . Show that points A, B and C are collinear.(3 marks)
13. Find the period , amplitude and phase angle of the function 2y=3sin (½x - 60°)(3 marks)
14. Simplify (3 marks)
    20 - 11x - 3x²
      16x - 12x²
15.  Write the following ratios in ascending order 2:3,15:16 ,7:6 13:15, (3 marks)
16. Under an enlargement, the image of the points A(3,1) and B(1,2) are A'(3,7) and B'(7,5) . Find the centre and scale factor of enlargement. (4 marks)

SECTION II (50 marks)
Answer only five questions in this section in spaces provided

17.  A straight line passes through P(-1,1) and Q(3,4) .
    
    1. Find the length of line PQ(2 marks)
    2. Find the equation of the perpendicular bisector of line PQ , leaving the equation in the form y=mx+c(4 marks)
    3. Determine the equation of line parallel to line and passes through point (2,3), leaving your answer in double intercept form. Hence state the y intercept. (4 marks)
18. The marks scored by 30 students in test were recorded as follows
    41    43   34   28   19   22
    32    38   22   18   25   33
    30    41   36   31   28   37
    35    34   19   22   29   23
    29    44   26   27   29   36
    
    1. Starting with the class 18-22 , make a frequency distribution table for the data. (2 marks)
    2. Using the frequency distribution in (a) above calculate :
       1. the mean (2 marks)
       2. the median (3 marks)
    3. Draw a frequency polygon to represent the data. (3 marks)
19. The solid below is made up of hemispherical part and a frustum of cone. The top and bottom radius of the frustum are and respectively. The vertical height of the frustum is .
    
    
    1. Determine the vertical height of the cone from which the frustum was cut. (2 marks)
    2. Calculate
       1. The volume of the solid correct to 2 decimal places (3 marks)
       2. The surface area of the solid correct to 2 decimal places (5 marks)
20.               
    1.        
       1. Draw the graph of the function for y= 2x² - 3x - 5 for -2< x < 3(5 marks)
       2. Use the graph to solve the equation 2x² - 3x - 5=0(1 mark)
    2. Use the graph to solve the simultaneous equation y=2x² - 3x - 5 and y=-2x-2 (3 marks)
    3. Write down the quadratic equation which the line y=-2x-2 is solving. (1 marks)
21. The diagram below shows the speed time graph for a bus travelling between two stations, the bus starts from rest and accelerates uniformly for 75 seconds. It then travels at constant speed for 150 seconds and finally decelerates uniformly for 100 seconds.
    
    
    1. Given that the distance between the two stations is . Calculate
       1. maximum speed in attained by the bus. (3 marks)
       2. the acceleration of the bus (2 marks)
    2. A van left Nairobi at 8.30 a.m and travelled towards Mombasa at an average speed of At 8.30 am a car left Nairobi and travelled along the same road at an average speed of .
       1. Calculate the distance covered by the car to catch up with the van. (4 marks)
       2. Find the time of the day when the car caught up with van. (1 mark)
22. On the Cartesian plane below, triangle PQR has vertices P(2, 3), Q(1, 2) and R(4, 1) while triangle P’’Q’’R’’ has vertices P’’(-2, 3), Q’’(-1, 2) and R’’(-4, 1).
     
    
    1. Describe fully the transformation which maps triangle PQR onto triangle P’’Q’’R’’.(1 mark)
    2. On the same plane, draw triangle P’Q’R’, the image of triangle PQR under a reflection in the line (2 marks)
    3. Describe fully a single transformation which maps triangle P’Q’R’ onto triangle P’’Q’’R’’ (2 marks)
    4. Draw triangle P’’’Q’’’R’’’ such that it can be mapped onto triangle PQR by a positive quarter turn about (0, 0) (3 marks)
    5. State a pair of triangles that is
       1. oppositely congruent (1 mark)
       2. directly congruent (1 mark)
23. The equation of the curve is y=x³ - 2x² -1
    1. Determine
       1. the stationary points (4 marks)
       2. the nature of the stationary points in (a) (i) above (2 marks)
    2. Determine
       1. the equation of the tangent to the curve at x=1 (2 marks)
       2. the equation of the normal to the curve at x=1 (2 marks)
24. The boundaries of ranch AB, BC , CD and DA are straight lines such that B is 075° from A and a distance of 50km. C is due east of B and a bearing of N80°E from A. D is due south of C and a distance of 70km.
    1. Using a scale of 1 cm to represent 10 km. show the relative positions of ABCD.(3 marks)
    2. From the scale drawing, determine
       1. the distance in kilometres between and (2 marks)
       2. the bearing of from (2 marks)
       3. the shortest distance from to border (1 mark)
    3. Calculate the area of the ranch in square kilometer. (2 marks)

Marking Scheme

1.         
   1. 97531
   2. 7000
   3. 2³ x 5³ x 7
      
      
2. 40= 2³ x 5
   250= 2 x 5³
   750 = 2 x 3 x 5³
   LCM= 2³ x 3 x 5³
   = 3000
   
   
3. =log(2²x 3²)
        10³
   2log2 + 2log 3 - 3log10
   2(0.3010) + 2(0.4771) - 3
   = -1.4438
   = -1.444
   
   
4. Num ³/₄ x 1¹/₂ x ¹¹/₆ = ³/₄ + ¹¹/₄ = ¹⁴/₄
   Den ³/₄ x ⁵/₂ ÷ ¹/₂ = ¹⁵/₄
   = ¹⁴/₄ x ⁴/₁₅ = ¹⁴/₁₅
   
   
5.    
   
   
   
6. Angle at the centre = 1.2° x 2 = 2.4°
   2.4 x 180/π = 137.51°
   137.51° ÷ 2 = 68.75°
   sin 68.75°= ⁵/_r
   r= 5.365cm
   
   
7. Cost price = 100/130 x 24.05 = 18.5
   =22.95 - 18.5 = 4.45
   =   4.45   x 100%
      18.5
   = 24.05%
   
   
8. 80 + 128(n-1)= (n-2)180
   80 + 128n - 128 = 180n - 360
   -48 + 360 = 180n - 128n
   312= 52n
   n=6
   
   
9. = (3⁴)^¾ - 5 - 1
   =27 - 5 - 1 =21
   
   
10.       
    1. x - 6 < -3x + 2
       4x < 8
       x< 2
       -3x + 2 < -2x + 9
       -7 < x
       -7 < x < 2
       
       
    2.   
       
       
       
11.      
    
    
    
12.          
    
    AB = ²/₃AC
    ^AB/_AC
    A, shared point
    hence point A, B and C are colinear
    
    
13. Amplitude = 1.5
    period 360 ÷ 0.5 = 720°
    phase angle - 60°
    
    
14. Num
    3x² + 11x - 20
    3x² + 15x - 4x - 20
    3x(x+5)- 4(x + 5)
    (3x - 4)(x + 5)
    Den 4x(4-3x)
    = (3x-4)(x+5)
          4x(4-3x)
    = -x-5
        4x
15. 2 x 80 = 160
    3 x 80     240
    
    15 x 15 = 225
     16 x 15   240
    
    7 x 40 = 280
    6 x 40    240
    
    13 x 16 = 208
    15 x 16    240
    
    7:6, 15:16 , 13:15, 2:3
    
    
16.    
    
    -2(3 - x)= 3 - x
    -6 + 2x= 3 - x
    3x = 9
    x=3
    -2(1-y)= 7 -y
    -2 + 2y = 7-y
    3y=9
    y=3
    (3,3)   
    
    
17.      
    1.    
       
       = √(3² + 4²)
       = 5
    2. Mid point
       (-1 + 3 , 1 + 4)= (1, 2.5)
             2          2
       Gradient of PQ
       =   4-1    = ³/₄
           3 + 1 
       m₂ = ^-4/₃
       y - 2.5 = ^-4/₃
         x -1
       3(y-2.5)= -4(x -1)  
       3y = -4x + 11.5
       y = ^-4/₃x + 3⁵/₆
       
       
    3.   y - 3  = ³/₄
         x -2      
       4y - 12 = 3x -6
       ^4y/₆ - ^3x/₆ = ⁶/₆
       ^-x/₂ + ^y/_1.5 = 1
       y intercept = 1.5
18.           
    1.    
       

       Marks	Frequency	midpoint(x)	fx	cf
       18 - 22 23 - 27 28 - 32 33 - 37 38 - 42 43 - 47	7 3 8 6 4 2	20 25 30 35 40 45	140 75 240 210 160 90	7 10 18 24 28 30

    2.         
       1. Σfx = 915
          Σf       30
          = 30.5
          
          
       2. = 28.5 + (^{15 - 10}/₈)5
          = 31.625
    3.    
       
19.        
    1.    x     = 5/15
       x+24
       15x = 5x + 120
       10x = 120
       x =12cm
    2.        
       1. volume = (¹/₃ x ²²/₇ x 15² x 36 - ¹/₃ x ²²/₇ x 5² x 12) + ²/₃ x ²²/₇ x 15³
          (8,845.71 - 314.29) + 7071.43
          = 15242.86
          
          
       2. √(12² + 5²) = 13
          √(36² + 15²) = 39
          S.A = (²²/₇ x 15 x 39 - ²²/₇ x 5 x13)+ 2 x ²²/₇ x 15² + ²²/₇ x 5²
          = 3048.57cm2
          
          
20.        
    1.        
       1.      
          
       2. x = -1 and x = 2.5
          
          
    2. x = -1, y=0
       x = 1.5, y=-5
       
       
    3. 2x² - 3x - 5 = -2x -2
       2x² - x -3 = 0   
21.          
    1.     
       1. 1/2(150 + 325)h = 5225
          h= 22 m/s
          
          
       2. 22 - 0 = 0.2933 m/s
            75
    2.        
       1. time difference 8:30 am - 8:00am = 0.5hr
          Distance covered by van 0.5 x 80 = 40km/hr
          R.S = 120 - 80 = 40km/hr
          time taken by van to catch up = 40/40 = 1 hr
          ditance covered by car 1 x 120 = 120 km/h
          
          
       2. time of day = 8:30 am + 1hr = 9:30 am
22.                
      
    1. reflection in line x = 0
    2. Rotation through -90° about (0,0)
    3. P'Q'R' and PQR , P"Q"R" and P'" Q'" R"',
       PQR and P" Q" R", P'Q'R' and P'"Q'"R'"
23.           
    1.       
       1. ^dy/_dx = 3x² - 4x
          3x² - 4x = 0
          x(3x - 4)= 0
          x=0
          x = 11/3
          y= (0)³ - 2(0)² - 1=-1
          (0, -1)
          y = (⁴/₃)³ - 2(⁴/₃)² - 1= -2⁵/₂₇
          (1¹/₃, -2⁵/₂₇)
       2. (0, -1) maximum
          (1¹/₃, -2⁵/₂₇) minimum
    2.          
       1. when x = 1, gradient = 1
          when x =1, y=-2
           y+2 = -1
           x -1
          y + 2= -x + 1
          Y= -x-1
       2. gradeint = 1
          y + 2 = 1
          x -1
          y + 2 = x -1
          y = x -3
24.             
    1.      
            
    2.        
       1. 2.7 ± 0.1 x 10 = 27km ± 1
          
          
       2. 306° ± 1°
          9.6 ± 0.1 X 10= 96km ± 1
          
          
       3. 7.7 ± 0.1 x 10 = 77km ± 1
          
          
    3. (¹/₂ x 56 x 77)+ 1/2(77 + 27)14 = 2884km²