Mathematics Paper 1 Questions and answers - Sukellemo Joint Pre Mock Exams 2022

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QUESTIONS
SECTION 1 (50 MARKS)

Answer all the questions in the space provided below each question

  1. Find the equation of a straight line passing through the points A (1,-3) and B (-2, 5).Express your answer in the form ax + by = c where a, b and c are integers. (3marks)
  2. Evaluate without using mathematical tables or calculator -10÷2+6×4-8×5 (3marks)
                                                                                                 -5+(-12)÷3×2
  3. Solve for x in the equation Cos(2x-30)°= tan45° (3marks)
                                               Sin(3x+10)°
  4. Two taps P and Q together can fill a water tank in 6 minutes. Tap P alone takes 5 minutes longer than tap Q. How many minutes does it take tap P alone to fill the tank? (3marks)
  5. Given that, 275x-2y=243 and 812x-y=3, Calculate the values of x and y. (3marks)
  6. A point P is mapped onto P’ by a negative quarter turn about the origin. P’ is mapped onto P’’ by a translation represented by the vector (-23) . If P’’ has coordinates (11,-5) determine the coordinates of p. (3marks)

  7. A metallic pipe which is 21 meters long has an internal radius of 13 cm and an external radius of 15 cm. if the density of the metal is 8620 kg/ m3, find its mass. (3marks)
  8. Using logarithms evaluate ∛(82.73×0.29432)(3marks)
                                                       613.5
  9. A proper fraction is such that the denominator exceeds the numerator by 3. If 2 is subtracted from both the numerator and denominator, the fraction formed is 1/8 less. Determine the original fraction. (3 marks)
  10. Given that OM = 2i +3j -6k and ON = -3i + 5j +k.Find the magnitude of MN to 2 decimal places. (3marks)
  11. Find the range of the integral values of x in the inequality 10<3(x+2)<35 , giving your answer in the form a≤x≤b (3marks)
  12. Simplify completely    2-2x    ÷  x-1  (3marks)
                                   6x2-x-12    2x-3
  13. The marked price of a recliner sofa set in a furniture store was ksh 400,000.A customer bought the recliner at 10% discount. The dealer still made a profit of 20%, Calculate the amount of money the dealer paid for the recliner. (3marks)
  14. Draw a line AB of length 9 cm. On one side of line AB construct the locus of a point P such that the area of triangle ABC is 13.5 cm2.On this locus locate two positions of a point P1 and P2 such that
  15. Given that the area of an image is four times the area of the object under a transformation whose matrix is , find the possible value of x .(3 marks)
  16. Construct a triangle ABC in which AB = 5cm and AC = 8cm and ∠ABC=105°. Using line AC, locate point x on AB produced such that AX: XB =3: -2. (4marks)

SECTION II (50 MARKS)
Answer only five questions in this section

  1. The table below shows the weekly salary (k£) paid to workers in a school.
    Salary (k£) 50≤x≤100 100≤x≤150 150≤x≤250 250≤x≤350 350≤x≤500
    No. of Workers 25 27 30 26 24
    1. Calculate the differences between the mean and the median. (6 marks)
    2. Draw a frequency polygon to illustrate the above information. (4marks)
      1
  2.      
    1. Complete the table of values for the equation, y=-2x^(2 )+x+8. (2marks)
      x -3 -2 -1 0 1 2 3 4
      y                
    2. Use the values above to draw the graph of y=-2x2+x+8 . (3marks)
      1
    3. Using the graph drawn above Solve the equations:-
      1. 2x2=x+8 (2marks)
      2. -2x2+4x+12=0 (3marks)
  3. Three towns P, Q and R are such that Q is 16 km north of P and the distance of R is 12 km from P and on a bearing of 60º from Q.
    1. Using a scale of 1cm to represent 4km, Make a scale drawing showing the relative positions of the three towns. (3marks)
    2. Using the scale drawing above, find the
      1. Distance of R from Q. (1mark)
      2. Bearing of P from Q. (1mark)
      3. How far town R is east of Q (1mark)
    3. A Passenger in an aero plane after take-off from town R spotted town P at an angle of depression of 48º, by means of a scale drawing determine the vertical height of the plane at town R. (3marks)
  4.      
    1. The equation of a straight line L1 is of the form 3y+2x=5.L1 is perpendicular to L2 and meets it at the point where X=-2, determine the equation of L2 in the form y = mx+c where m and c are constants. (5marks)
    2. L3 is parallel to the line L2 and passes through the point (-3,2).,find the equation of L3, leaving your answer in its double intercept form. (3marks)
    3. Determine the angle of inclination of L2 to the Y-axis. (2marks)
  5. The points P, Q, R and S, have position vectors 2p, 3p, r and 3r respectively, relative to an origin O. A point T divides PS internally in the ratio 1:6.
    1. Find, in its simplest form OT, QT and TR in terms of p and r. (6 marks)
    2. Show that the points Q, T and R, are collinear. (3marks)
    3. Determine the ratio in which T divides QR. (1mark)
  6. In the figure below, O1 and O2 are the centers of the circles whose radii are 5 cm and 8 cm respectively. The circles intersect at A and B and angle AO1O2 = 64˚.
    2
    Calculate the area of the:-
    1. Sector
      1. AO1B (2marks)
      2. AO2B (3 marks)
    2. Intersecting region. (3marks)
    3. The shaded region. (2marks)
  7.      
    1. Find the x –intercept of the curve y = (x+2) (x-1)2. (1mark).
    2. Find the gradient function of the curve y = (x+2) (x-1)2 (2marks)
    3. Find the co-ordinates of the turning point. Hence sketch the curve y= (x+2) (x-1)2. (4 marks)
    4. Calculate the exact area enclosed by the curve and the x - axis (3marks)
  8. P and Q are two points on latitude 40°N.Their longitudes are 30°E and 150°W respectively. Find to one decimal place :( Take the radius of the earth = 6370km andπ=22/7)
    1. The distance in km between P and Q along the parallel of latitudes. (2marks)
    2. The shortest distance along the earth’s surface between P and Q in km. (3marks)
    3. A weather forecaster reports that the center of a cyclone at (40°N, 60°W) is moving due north at 24 knots. How long will it take to reach a point (45°N, 60°W). (2marks)
    4. A plane leaves P at 2.15 pm at a speed of 350 knots to town R (40°N, 65°E). Determine the time at R when the plane arrived. (3marks)


MARKING SCHEME

  1. Gradient = 5/-2 = 3/-1 = -8/3
    y - 5-8/3
    x + 2
    3(y - 5) = -8(x + 2)
    3y - 15 = -8 - 16
    3y  + 8x = -1
  2. Numerator = -5 + 24 - 40
    -21
    den. = -5 - 8 
    = -13
    -21/-13
    = 18/13
  3. cos(2x - 30)º = sin(3x + 10)º
    cos(2x - 30)º = cos 90 -(3x + 10)
    2x - 30 = 90 - (3x + 10)
    2x - 30 + (3x + 10) = 90
    5x - 20 = 90
    5x = 110
    x = 22
  4. 1/x+5 + 1/x = 1/6
    6(x + x + 5) = x(x + 5)
    6x + 6x + 30 = x2 + 5x
    -7x - 30 = 0
    - 10x + 3x - 30 = 0
    x(x - 10) + 3(x - 10) = 0
    (x + 3)(x - 10) = 0
    x = -3 or 10
    Top P = 5 + 10
    =15mins
  5. 33(5x - 2y) = 35
    3(5x - 2y)= 5
    15x - 6y = 5
    34(2x - y) = 31
    4(2x - y) = 1
    (15x - 6y = 5)4
    (8x - 4y = 1)6
    60x - 24y = 20
    48x - 24y = 6
    12x = 14
    x = 11/6
    8x7/6 - 4y = 1
    56/6 - 4y = 1 
    81/3 = 4y
    21/12 = y

  6. 3
    y - 2 = 11
    y = 13
    x + 3 = -5
    x = -8
    P(-8, 13)
  7. External volume - Internal volume
    22/7 x 152 x 2100 - 22/7 x 132 x 2100
    22/7 x 2100 (225 - 169)
    22/7 x 2100 x 56
    369,600cm3
    0.3696m3
    Mass = 8620 x 0.3696
    = 8620 x 3696
                   1000
    3185.952
    = 3186kg

  8. No Log
    0.29432

    82.73

    613.5



    0.2269
    T.4688 x 2
    2.9376
    1.9177 + 
    0.8553
    2.7878-
    2.0675
    2.0675 = 3/3 + 1.0675 = T + 0.3558
    33
    T.3558
  9. x/y 
    y - x = 3
    y = x = 3
    y = 3 + x
    x - 2x/y - 1/8
    y - 2 
      x - 2  =  x   1/8
     3+x-2    3+x
    (x - 2) (x + 3 )8 = x (x + 1)8-((x+1)(x + 3))
    (x2 + x - 6)8 = 8x2 + 8x - (x2 + 4x + 3)
    8x2 + 8x - 48 = 8x2 + 8x - x- 4x - 3
    0 = -y-4x + 45
    x+ 4x - 45 = 0
    x+ 9x - 5x - 45 - 0
    x(x + 9) -5(x + 9) = 0
    (x - 5)(x + 9) = 0
    x = 5 or -9 
  10. MN = √(-3-2)2 + (5-3)2 + (1-6)2
    √25+ 4 + 49
    √78
    8.832
    8.83
  11. 10<33(x + 2)
    10<3x + 6
    4<3x
    11/3 < x
    3x + 6 < 35
    3x < 29
    x < 92/3

    2≤x≤9
  12.         2(1-x)         
    6x2 - 9x + 8x - 12
          -2(x - 1)        
    3x(2x-3)+4(2x-3)
         -2(x-1)    
    (3x+4)(2x-3)
    =  -2   
      3x+4
  13. 100% = 400000
    90% = 90 x 400000
                     100
    = 360000
    120% = 360000
    100% = 100 x 360000
                       120
    = 300000

  14. 4
    ½ x 9 x h = 13.5
    h = 3cm
  15. x2 - ((x-4)(x+8)) = 4
    x- (+4x - 32) = 4
    x2- x2 - 4x + 32 = 4
    - 4x = -28
    x = 7
  16.     
    5
  17.    

    1. 0-50 f x fx cf fd
      50-100 25 75 1875 25 0.5
      100-150 27 125 3375 52 0.54
      150-250 30 200 6000 82 0.3
      250-350 26 300 7800 108 0.26
      350-500 24 425 10200 132 0.16
      500-650 Σf=132   Σfx= 29250    
      Mean = 29250
      132 
      = 221.60
      Median = 66th position
      150 + 82
      150 + (66 - 52) x 100
                   200
      221.60 - 157
      = 64.60
    2.    
      6.png
  18.    
    1.    
      x -3 -2 -1 0 1 2 3 4
      y -13 -2 5 8 7 2 -7 -20
    2.     
      7
    3.      
      1. 2x2 + x + 8 = 0
        -2x2 + x + 8 = y
        0 = y
        x = -1.78 or x =2.2
      2. -2x2 + x + 8 = y
        -3x - 4 = y
        x = -1.55
        x = 3.67
  19.      
    1.     
      8
    2.    
      1. 5.3cm = 5.3 x 4 = 21.2km 
        21 ± 0.4
      2. 215º
        214 ± 1 
      3. 4.7cm = 4.7 x 4 = 18.8km
        18.1 ± 0.4
    3.     
      9
  20.      
    1. 3y = -2x + 5
      y = -2/3x + 5/3
      Gr = 3/2
      at x = -2
      y = 4/3 + 5/39/3 = 3
      (-2, 3)
      y - 33/2
      x + 2 
      2y - 6 = 3x + 6 
      2y = 3x + 12
      y = 3/2x + 6
    2.  y- 2 3/2
      x + 3
      3(x + 3) = 2(y - 2)
      3x + 9 = 2y - 4
      3x - 2y = -13
      3x/-13 + 2y/13 = 1
         x   +  y   = 1
      -13/   13/
        x    +  y  = 1
      -41/3   6½

    3. 10
      tanθ = 3/2
      θ = 56.31º
      90 - 56.31º = 33.69º
      angle 33.69º or 146 .31º
  21.      

    1. 11
      OT = 3/7r + 12/7p
      QT = QO + OT
      = -3p +3/7r + 12/7p
      = 3/7r - 9/7p
      TR = TO + OR
      = -3/7r - 12/7p + r
      = 4/7r -12/7p
    2. QT = 3/7(r - 3p)
      TR = 4/7(r - 3p)
      4/7QT = 3/7TR
      4QT = 3TR
      QT//TR
    3. QT/TR = ¾
      QT:TR = 3:4

  22. 12     
    1.      
      1. 128/360 x 22/7 x 25
        27.94
      2. 8/sin64 = 5/sinx
        sin x = 5 x sin64
                         8
        = 0.5617
        = 34.18º
        2x = 68.36º
        68.3622/7 x 64
         360
        = 38.19
    2. (27.94 - x 25 x sin128) + 38.19 - x 64 x sin68..36
      27.94 - 9.850                        38.19 - 29.74
      18.09 + 8.45
      26.54
    3. 9.850 + 29.74 - 26.54
      39.59 - 26.54
      13.05
  23.        
    1. (x+2)(x-1) = 0
      x = 1 or -2
    2. y = (x+2)(x2 - 2x + 1)
      y = x3 - 2x2 + 2x2 + x - 4x
      y = x3 - 3x
      dy/dx = 3x2 - 3 
    3. 3(x2 -1) = 0
      3(x-1)(x+1) = 0
      3(x-1)(x-1) = 0
      x = 1 or -1
      when x = 1, y = 0
      (1,0)
      d2y = 6x at x=1
      dx2
      = 6
      (1,0)minimal
      when x = -1
      y = (-1+2)(-1-1)2
      y = 4
      (-1,4)
      d2y = 6x at x = -1
      dx2
      22y = -6
      dx
      13

    4. 18
      -1.25 - (-2)
      0.75
  24.      

    1. 14
      dif in longitude = 180
      18022/7 x 2 6370 x cos40
      360
      = 15336.2

    2. 15
      100
      22/7 x 2 x 6370
      360
      = 8272.7

    3. 16
      differences in latitude = 45-40
      = 50
      = 5 x 60 = 300NM
      Time = 300/24
      = 12.5hrs

    4. 17
      differences in longitude = 65-0
      = 35º
      24hrs = 360º
      35 x 24 
        360
      2hrs 20mins
      Distance = 35 x 60 x c0s40
      = 1,608.7NM
      Time = 1608.7
                    350
      = 4.596hrs
      4hrs, 36 mins 
      = 8.11p.m

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