Mathematics Paper 2 Questions and Answers - Form 3 End Term 1 Exams 2022

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  1. Find all the integral values of X Which satisfy the following inequalities. 2x-7<x + 2 ≤ 2 (x-2) (4MKS)
  2. Using method of completing the square, find the values of x that satisfies the quadratic equation below.x2+ x – 6 = 0 (3mks)
  3. A triangle PQR is such that <PQR =60º length QR = 600cm and length QP=540cm.Find the area of triangle PQR and leave your answer in M2 (3mks)
  4. Use logarithms tables to evaluate ∛239.5 ×0.004832
                                                              sin⁡30º (4mks)
  5. Juma, Ali and Choge are three masons together to complete building a stone wall together in 12 hours.Juma and Ali working together complete the work in 18 hours while Juma and Choge working together complete the work in 24 hours. For how long each would take to complete building of the stone wall while alone. (3mks)
  6. Find the value of X that satisfy the equation
    Log (x + 5) + log (x + 2)=log 4 (3mks)
  7. Simplify   X   + 3x - 2 (2mks)
                  x+2      X2-4
  8. Without using a calculator or mathematical tables, express Cos 30º in surd form and  Tan 45°+√3
    simplify leaving your answer in the form a+b√ c where a, b and c are rational numbers. (3mks)
  9. The figure below shows a quadrilateral ABCD which is cyclic. Solve for x. (3MKS)
    1
  10. Two cyclists moving towards each other take two hours to meet. If one cycles at a speed of 25km/h and the other at Y km/hr,find the value of Y if their distance apart is 92km. (3mks)
  11. Using a ruler and a pair of compass only construct a triangle in which AB is 7cm and BC=9cm and angle BAC =75º measure AC.Construct an inscribed circle in triangle ABC and measure its radius 4mks
  12. Given that cos (x - 20)° = Sin (2x + 32)° and x is an acute angle.Find tan (x - 4)(3mks
  13. Given that    3       3√5   = a + b√5 Find the values of a and b (4mks)
                    3 + √5     3 - √5
  14. In the figure below, 0 is the centre of the circle which passes through the points T,C and D. Line TC is parallel to OD and line ATB is a tangent to the circle at T. Angle DOC = 38°. Calculate the size of angle CTB (3mks)
    2
  15. In the circle below chords PQ and RS interest intersect at X .Given that RX -8cm, XS=3cm and PQ =10cm.Calculate PX (2mks)
    3
  16. The length and width of a rectangular window pane measured to the nearest millimeter are 8.6cm and 5.3 respectively.
    Find to four significant figures. The percentage error in the area of the windowpane. 3mk

DO FIVE QUESTIONS IN SECTION B

  1. Using a ruler and a pair of compasses only, draw a parallelogram ABCD, such that angle DAB = 750. Length AB = 6.0cm and BC = 4.0cm.From point D, drop a perpendicular to meet line AB at N. (7 Marks)
    1. Measure length DN (1 Mark)
    2. Find the area of the parallelogram. (2 Marks)
  2. The table below shows income tax rates in the year 2013.
    A constable earned a monthly salary of ksh. 42500 , house allowance of 10,000. Travelling allowance of 5000.
    Monthly Income in Ksh Tax rate in each shilling
    Up to 9680  10% 
    9681-18800  15% 
    18801 – 27920  20% 
    27921 – 37040  25% 
    Over 37040 30%
    In that year, a monthly personal tax relief of ksh 1056 was allowed. Calculate the monthly
    1. taxable income (2mks) 
    2. Tax due (3mks)
    3. Payable tax (1mk)
    4. Net pay of the constable 4mks
  3. The figure below shows a circle centre O, PQRS is a cyclic quadrilateral and QOS is a straight line.
    4
    Giving reasons for your answers, find the size of
    1. Angle PRS (2 Marks)
    2. Angle POQ (2 Marks)
    3. Angle QSR (3 Marks)
    4. Reflex angle POS (3 Marks)
  4. A group of people planned to contribute equally towards a water project which was to cost Ksh 2000,000 to complete.40 members of the group withdrew before the project started. Hence the remaining members had to contribute Ksh. Using x as the original no. of members,
    1. Form expressions to show the original and the new contribution for each member. 2 mks
    2. Form a quadratic expression to represent the information above (no. 24) (2mks)
    3. find the original number of members 3mks
    4. luckily, 45% of the value of the project was funded by CDF. Calculate amount contributed by each member (3mks)
  5. The figure below shows a frustum of a right pyramid whose top face is a rectangle of side 3 cm by 5 cm and the bottom face is also a rectangle of side 6cm by 10cm. The perpendicular distance between the top and bottom faces (height) is 25cm.
    5
    Find;
    1. the height of the chopped off pyramid (2marks)
    2. the volume of the frustrum (4marks)
    3. The surface area of the frustum. (4marks)
  6. The figure below shows the position of a boat Q which is observed sailing directly towards the pier P at the base of a vertical cliff PT. The angle of elevation of the top of the cliff from Q is 25.4º. After 14 seconds the boat is at point R, and the angle for elevation of T is now 64.7º.
    6
    If the cliff is 50m high, calculate
    1. The distance PQ (2 Marks)
    2. The distance QR (4 Marks)
    3. The speed of the boat in km/h (4 Marks)
  7. A surveyor recorded the measurements of a field book using XY=400m as the base line as shown below.
    To E 200

    To F 250
     
    Y
    320
    210
    170
    50


    150 To D
    150 To C
    225 To B
    100 To A
    1. Use a scale of 1cm to represent 50m to draw the map of the field. (5mks)
    2. Find the area of the field in hectares (5mks)
  8. In the figure below PQ = 2500m, U T= 1000m and TS =2350m. PQR is a straight line. Parallel to UT and angle UPQ =22.5º.
    7
    1. Calculate to the nearest meter
      1. U Q (2marks)
      2. T V (2marks)
      3. V S (2marks)
      4. P U (2marks)
    2. Find the perimeter of the figure. (2marks)


MARKING SCHEME

  1. Find all the integral values of X Which satisfy the following inequalities. 2x-7<x + 2 ≤ 2 (x-2) (4MKS)
    2x - 7 < x + 2
    2x - x <9
    x < 9
    x + 2 ≤ 2(x - 2)
    x + 2 ≤ 2x -4
    x - 2x ≤ -6
    -x ≤ -6
    x ≥ 6 
  2. Using method of completing the square, find the values of x that satisfies the quadratic equation below.x2+ x – 6 = 0 (3mks)
    x2 + x = 6
    x2 + x + (½)2 = 6 + (½)
    (x2 + ½)2 = 6 + 0.25
    (x + ½) = ±√6.25
    Either x = 2.5 - 0.5
    = 2
    or
    (x + 0.5) = -2.5
    x = -2.5
    x = -3
  3. A triangle PQR is such that <PQR =60º length QR = 600cm and length QP=540cm.Find the area of triangle PQR and leave your answer in M2 (3mks)
    8
    A = ½abSinθ
    = ½ x 540 x 600Sin60º
    = 16933.61
    = 1.6933m2
  4. Use logarithms tables to evaluate ∛239.5 ×0.004832
                                                              sin⁡30º (4mks)
    No  Stand Log
    239.5
    2.395 x 10
    4.832 x 10-3
    2.3793
    - 3.6841
    Sin 30
    0.5 


    5x 10-1



    0.0634
    1.6990
    - 0.3644
    3
    1.322    0.1214
  5. Juma, Ali and Choge are three masons together to complete building a stone wall together in 12 hours.Juma and Ali working together complete the work in 18 hours while Juma and Choge working together complete the work in 24 hours. For how long each would take to complete building of the stone wall while alone. (3mks)
    In 1 hour Juma and Ali complete 1/18 of the job hence work done by Choge
    1/121/186 - 4
                        72
    2/72
    Hence Choge takes 36hrs
    Amount of work done by Juma
    1/241/36 =3 - 21/72 
                       72
    Juma takes 72 hrs
  6. Find the value of X that satisfy the equation
    Log (x + 5) + log (x + 2)=log 4 (3mks)
    log(x + 5)(x + 2) = log4
    x2 + 7x + 10 = 4
    x2 + 7x + 6 = 0
    (x2 + x) + (6x + 6)
    x(x + 1) + 6(x + 1)
    (x + 6) (x + 1)
    x = -6  or x = -1
  7. Simplify   X   + 3x - 2 (2mks)
                  x+2    X2- 4
    x (x - 2) + 3x - 2x- 2x + 3x - 2 
        (x - 2) (x + 2)        (x - 2) (x + 2)
         x + x - 2    (x + 2)(x - 1)(x - 1)
    (x - 2) (x + 2)     (x - 2) (x + 2)   (x - 2)
  8. A trader marks his goods 20% above the cost price, He gives 10% discount to a customer and sells goods at Ksh486. Calculate the cost price of the goods(3mks)
    120C  90   = 486
     100      100
    C = 486 x 100
              108
    = Ksh450
  9. The figure below shows a quadrilateral ABCD which is cyclic. Solve for x. (3MKS)
    1
    2x + 60 = 180
    2x = 120
    x = 60
  10. Two cyclists moving towards each other take two hours to meet. If one cycles at a speed of 25km/h and the other at Y km/hr,find the value of Y if their distance apart is 92km. (3mks)
    In one hour both travel (25 + y)km
        92    = 2
    25 + y
    92 = 50 + 2y
    42 = 2y
    y = 21Km/h
  11. Using a ruler and a pair of compass only construct a triangle in which AB is 7cm and BC=9cm and angle BAC =75º measure AC.Construct an inscribed circle in triangle ABC and measure its radius (4mks)
    9
  12. Given that cos (x - 20)° = Sin (2x + 32)° and x is an acute angle.Find tan (x - 4)(3mks)
    (x - 20) + (2x + 32) = 90
    3x + 12 = 90
    3x = 78
    = 26
  13. Given that    3       3√5   = a + b√5 Find the values of a and b (4mks)
                    3 + √5     3 - √5
    Numerator
    9 - 3√5 + 9√5 - 3√5
    3(3√5) + 3√5(3√5)
    9 -3√5 + 9√5 + 3√25
     9 + 6√5 + 15
    24 + 6√5
         4
    Denominator
    (3 + √5) (3 - √5)
    3(3√5) + √5(3 - √5)
    9 - 3√5 + 3√5 - √25
    = 4
    6 + 3/2√5
  14. Using a ruler and a pair of compasses only, draw a parallelogram ABCD, such that angle DAB = 750. Length AB = 6.0cm and BC = 4.0cm.From point D, drop a perpendicular to meet line AB at N. (7 Marks)
    10
    1. Measure length DN (1 Mark)
      3.9 ± 0.1
    2. Find the area of the parallelogram. (2 Marks)
      (3.9 ± x 6) = 23.4cm2
      11
  15. The table below shows income tax rates in the year 2013.
    A constable earned a monthly salary of ksh. 42500 , house allowance of 10,000. Travelling allowance of 5000.
    Monthly Income in Ksh Tax rate in each shilling
    Up to 9680  10% = 968
    9681-18800  15% = 1.368
    18801 – 27920  20% = 1824
    27921 – 37040  25% = 2280
    Over 37040 30% = 6138
    In that year, a monthly personal tax relief of ksh 1056 was allowed. Calculate the monthly
    1. taxable income (2mks) 
      Salary + allowances
      42500 + 10000 + 5000 = Ksh57,500
    2. Tax due (3mks)
      968 + 1368 + 1824 + 2280 + 6138 = Ksh 12578
    3. Payable tax (1mk)
      12578 - 1056 = Ksh11522
    4. Net pay of the constable (4mks)
      Gross Pay - Deductions
      57500 - 1056 = Ksh56444
  16. The figure below shows a circle centre O, PQRS is a cyclic quadrilateral and QOS is a straight line.
    4
    Giving reasons for your answers, find the size of
    1. Angle PRS (2 Marks)
      55º - opp angles in cyclic add up to 180
    2. Angle POQ (2 Marks)
      140º Angle at centre is twice the angle at the circumfrence
    3. Angle QSR (3 Marks)
      55º opp angles in cyclic
      add up to 180º
    4. Reflex angle POS (3 Marks)
      360 - 40 = 320º
      angle of a point add up to 360
  17. A group of people planned to contribute equally towards a water project which was to cost Ksh 2000,000 to complete.40 members of the group withdrew before the project started. Hence the remaining members had to contribute Ksh. Using x as the original no. of members,
    1. Form expressions to show the original and the new contribution for each member. (2 mks)
      New contribution = 2000000
                                        x - 40
      Original Contribution = 2000000
                                                 x
    2. Form a quadratic expression to represent the information above (no. 24) (2mks)
      20000002000000 = 2500
         x - 40           x
      2000000x - 2000000x - 80000 = 0
      x2 - 200x + 160x - 32000 = 0
      x2 - 40x - 3200 = 0
    3. find the original number of members (3mks)
      (x + 160)(x -200) = 0
      x = 200members
    4. luckily, 45% of the value of the project was funded by CDF. Calculate amount contributed by each member (3mks)
       55  x 2000000 x   1   = 6.875shillings
      100                     160
  18. The figure below shows a frustum of a right pyramid whose top face is a rectangle of side 3 cm by 5 cm and the bottom face is also a rectangle of side 6cm by 10cm. The perpendicular distance between the top and bottom faces (height) is 25cm.
    12
    Find;
    1. the height of the chopped off pyramid (2marks)
      25 + x = 10  
           x        5
      125 + 5x = 10x
      125 = 5x
      x = 25
      Height = 25cm
    2. the volume of the frustrum (4marks)
      1/3AH - 1/3ah
      50x1/(6 x 10) - 1/x 3 x 5 x 25
      1000 - 41.67 = 958.33cm2
    3. The surface area of the frustum. (4marks)
  19. The figure below shows the position of a boat Q which is observed sailing directly towards the pier P at the base of a vertical cliff PT. The angle of elevation of the top of the cliff from Q is 25.4º. After 14 seconds the boat is at point R, and the angle for elevation of T is now 64.7º.
    6
    If the cliff is 50m high, calculate
    1. The distance PQ (2 Marks)
      PQ =   50   
             Tan25.4
      PQ = 105.3
    2. The distance QR (4 Marks)
      QR =   50   
              tan64.7
      = 23.63
      QR = 105.3 - 23.63
      = 81.67
    3. The speed of the boat in km/h (4 Marks)
      81.67 ÷ 14 = 5.834 m/sec
      km/h = 5.834 x 1000
                               3600
      = 1.620km/hr
  20. A surveyor recorded the measurements of a field book using XY=400m as the base line as shown below.
    To E 200

    To F 250
     
    Y
    320
    210
    170
    50


    150 To D
    150 To C
    225 To B
    100 To A
    1. Use a scale of 1cm to represent 50m to draw the map of the field. (5mks)
      13
    2. Find the area of the field in hectares (5mks)
      1. 100 + 225 x 50 = 8.125
                 2
      2. 225 + 150  x 120 = 22500
                 2
      3. =40 x 150 = 6000
      4. ½ x 150 x 190 = 14250
      5. ½ x 200 x 80 = 8000
      6. 250 + 200 x 150 = 33750
              2
        ½ x 170 x 250 = 21250
        TOTAL: 113875m2 
                       10000
        11.3875 Hectares
  21. In the figure below PQ = 2500m, U T= 1000m and TS =2350m. PQR is a straight line. Parallel to UT and angle UPQ =22.5º.
    7
    14
    1. Calculate to the nearest meter
      1. U Q (2marks)
        15
        Cos22.5 =   TV  
                          2350
      2. T V (2marks)
        2350 Cos22.5
      3. V S (2marks)
        Sin 22.5 =  VS 
                          TV
        VS = 2171 Cos22.5
        = 899.30
      4. P U (2marks)
        Cos25 = PQ 
                      PU
        PU =   PQ  
                Cos25
        =1035.53
          Cos22.5
        = 2706m
    2. Find the perimeter of the figure. (2marks)
      = 2706 + 1000 + 2350 + 899 + 1036 + 3171 + 2500 
      = 13662m

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