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

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SECTION A: ATTEMPT ALL THE QUESTIONS

  1. Evaluate
    (32÷4 of (-4)×6+28)  (4mks)
             -40÷5×2
  2. The angle of depression of a hen on a horizontal surface on the ground to a crow perched on top of a building is 28 degrees. The building is 8m tall vertically. When the hen sees the crow, the hen moves directly towards base of the building to a point such that the angle of elevation of the cow from B is 320.
    Calculate the distance hen moved. (3mks)
  3. Simplify the expression. (3 marks)
    15a²b - 10ab²
    3a²-5ab+2b²
  4. The position vectors of points P and Q are
    1
    Find the position vector of point X, which divides line PQ internally in ratio, 2:3 (3mks)
  5. Three Villages X,Y and Z are such that y is 12 km south of X and Z is 15km from X..Village. Z is on a bearing of 330º from Y. Calculate the bearing of Z from X. (3mks)
  6. A line AB is formed by coordinates A (1,2) and B (4,6). Another line 2 is the perpendicular bisector to line AB/. Find the equation of line 2. (3mks)
  7. Ann bought four pens and three exercise books for a total of sh.17 while peter bought five pens and two similar pens and two exercise books for a total of sh.16.
    Find the cost of a pen and an exercise book (3mks)
  8. Three villages are situated such that they form 3 sides of a triangle. The distance of villages P to R is 40km while R to Q is 60km.When represented on a paper angle PRQ is 78º.Find the distance of village P to Q. (4mks)
  9. A water tank is cylindrical in shape is filled by a pipe p in 6 hours. This same tank is emptied when filled in 12 hours by a pipe. When someone starts with the tank empty, both pipes are opened at the same time and then pipe Q turned off after 4 hours. Find the time taken to fill the tank. (3mks)
  10. The figure below shows triangle PQR in which PR = 12cm. T is a point on PR such that TR =4cm. Line ST is parallel to QR. If the area of triangle PQR is 336cm². Find the area of the quadrilateral STQR. (4 marks)
    2
  11. Kamau spent 3/5 of his money on buying food,and ½ of what was left on rent.After that,he finds that he has sh.1400 left in his pocket. Determine the amount money spent on buying food. (3mks)
  12. Find the centre of enlargement if P(0,-1) is the image of P(2,3) under enlargement scale factor 3 (3mks)
  13. Use the reciprocal tables to evaluate to determine 3 decimal places, (4mks)
         2     -    3    
    0.4821   24.71.
  14. A bank in Kenya buys and sells foreign currencies as follows.
    Currency                 Buying (Ksh)     Selling (Kshs)
    1 Sterling pound         134.20              134.65
    1 US dollar                  71.40                 71.84
    A tourist arrived in Kenya with $4500. He converted al the dollars to Kenya shillings at the bank, while in Kenya he spend Kshs 215,000 and then converted the remaining amount in Kenya shillings to sterling pound.
    Calculate the amount he received in sterling pound. (3mks)
  15. Three students shared a certain amount of money. The first got sh.120 more than the second thrice what the third student got. If the second got sh.300, how much did each of the other two get? (3mks)
  16. In the figure below, BT is tangent to the circle at B. AYCT and BYD are straight lines. AY=6cm, CT=8cm, BY=4.8cm and YD = 5cm
    3Find length of:
    1. CY
    2. BT

SECTION B: ATTEMPT ANY FIVE QUESTIONS IN THIS SECTION

  1. Three points P, Q and S are pm the vertices of a triangular plain field. P is 400m from Q on a bearing of 3000 and R of 550m directly south of P.
    1. Using a scale of 1 cm to represent 100m on the ground, draw a diagram to show the position of the points. (3mks)
    2. Use the scale drawing to determine;
      1. The distance and bearing of Q from R. (3mks)
      2. The bearing and distance of point S from P given that point S is directly 600m East of R. (3mks)
      3. The bearing and distance of Q from S. (3mks)
  2. A bus travelling at a speed of 80km/hr left Mombasa at 8.00am for Nairobi. Two hours later, a car travelling at a speed of 100km/hr left Nairobi for Mombasa.
    Given that the distance between both cities is 500km, find the: -
    1. distance covered by bus at the time the car is starting. (1mk)
    2. Distance between the two vehicles at the time the is starting (1mk)
    3. Relative speed of the two vehicles (1mk)
    4. Time taken for the two vehicles to meet (2mks)
    5. Time the met. (1mk)
    6. After meeting, the speed of both vehicles dropped to 60km/hr due to traffic jam. At what time did each vehicle arrive at its destination? (3mks)
  3. The kite shown on the fig below has vertices, A(-2,4)B(5,7)C (7,4) D(5,1).The kite ABCD is rotated about the origin through -90º to obtain its first image A’B’C’D’.The kite A’B’C’D’ is then reflected along the line = X = 0 to obtain A”B’’C’’D’’.Find
    1. The equation of the Mirror line that reflects point A to itself, B on to D and C onto itself
    2. Draw a kite A’B’C’D and state its coordinates.
      4
  4. A given solid is made up of a cone and three quarters of hemisphere as shown in the diagram below .The radius and vertical height of the cone are 4cm and 6.2 cm respectively
    5
    1. Calculate the volume of the solid (4mks)
    2. Calculate the surface area of the solid (4mks)
    3. The material used for making the solid costs shs 40 per cm3.What is the cost of making ten such solids (2mks)
  5. In the diagram below, two circles, centres A and C and radii 7cm and 24cm respectively intersect at B and D. AC = 25cm.
    6
    1. Find angle ABC = 90º. (3mks)
    2. Calculate
      1. the size of obtuse angle BAD (3mks)
      2. the area of the shaded part (4Mks)
    3. Find the area of the farm in hectares (3mks)
    4. Karis wishes to irrigate his farm using a sprinkler machine situated in the farm such that it is equidistant from points A, B and C.
      1. Calculate the distance of the sprinkler from point C. (2 Marks)
      2. The sprinkler rotates in a circular motion so that the maximum point reached by the water jets is the vertices A, B and C. Calculate the area outside his farm that will be irrigated. (3 Marks)
  6. Two lines L1=2y-3x-6 and L2=3y+x-20=0 intersect at point A.
    1. Find the coordinates of A (3marks)
      1. A third line L3 is perpendicular to L2 at point A. Find the equation of L3 in form of y=mx+c, where m and c are constants. (3marks)
    2. Another line L4 is parallel to l1 and passes through (-1,3). Find the x-intercept and the y-intercept of L4. (4marks)
  7.            
    1. PQRS is a quadrilateral with vertices P(1,4), Q(2,1), R(2,3) and S(6,4). On the grid provided, plot the quadrilateral. (2mark)
    2. Draw P’Q’R’S’ the image of PQRS under a positive quarter turn about the origin and write down its co-ordinates. (3 marks)
    3. Draw P”Q”R”S” the image of P’Q’R’S’ under an enlargement scale factor -1 and center (0,0) and write down its co-ordinates. (3 marks)
      4Determine the single transformation that maps PQRS onto P”Q”R”S (2marks)
  8. The heights of 40 pupils in a class were measured to the nearest centimeter and recorded as below.
    175 154 157 180 165 150 152 162
    173 168 169 181 177 179 175 169
    151 153 156 158 163 169 179 180
    145 149 150 156 171 175 176 178
    169 160 155 174 170 176 182 170
    1. Use the class interval of 5 to group the data start from 145-149, 150 - 154 . . . (3 marks)
    2. Estimate the mean (4 marks)
    3. Calculate the median height of the class. (3 marks)


MARKING SCHEME

SECTION A: ATTEMPT ALL THE QUESTIONS

  1. Evaluate
    (32÷4 of (-4)×6+28)  (4mks)
             -40÷5×2
    32 ÷ 16 x 6 + 28
          - 8 - 2 
    -2 x 6 + 28
         -16
    (-12 x 28)
        -16
    =(16/-16)
    = -½
  2. The angle of depression of a hen on a horizontal surface on the ground to a crow perched on top of a building is 28 degrees. The building is 8m tall vertically. When the hen sees the crow, the hen moves directly towards base of the building to a point such that the angle of elevation of the cow from B is 320.
    Calculate the distance hen moved. (3mks)
    7Tan28 =  8  
                  AC
    AC =   8   
            tan28
    = 15.045
    BC =    8   
            Tan32
    = 12.803
    AB = 2.242m
  3. Simplify the expression. (3 marks)
    15a²b - 10ab²
    3a²-5ab+2b² 
    Numerator
    5ab(3a - 2b)
    5ab (3a - 2b) 
    (3a - 2b)(a -b)
    Denominator
    3a²-3ab-2ab+2b²
    (3a - 2b)(a -b)
    =5ab
      a-b
  4. The position vectors of points P and Q are
    1
    Find the position vector of point X, which divides line PQ internally in ratio, 2:3 (3mks)
    819
  5. Three Villages X,Y and Z are such that y is 12 km south of X and Z is 15km from X..Village. Z is on a bearing of 330º from Y. Calculate the bearing of Z from X. (3mks)
    930 + 23.58 = 53.58
    180 - 53.58 = 126.42
    Bearing of Z from X is 
    180º + 126.42 = 306.42º
  6. A line AB is formed by coordinates A (1,2) and B (4,6). Another line 2 is the perpendicular bisector to line AB/. Find the equation of line 2. (3mks)
    h. of AB = 6 - 2
                     4 -1
    = 4/
    Line 2 M1 x M2 = -1
    ¾M2 = -1
    M2 = - 4/3
    Mid AB = 1 + 4 6 + 2
                       2         2 
    (2.5, 4)
      y - 4  
    = -¾
    x - 2.5
    4(y - 4) = -3x + 7.5
    4y = 3x + 23
  7. Ann bought four pens and three exercise books for a total of sh.17 while peter bought five pens and two similar pens and two exercise books for a total of sh.16.
    Find the cost of a pen and an exercise book (3mks)
    4x + 3y = 17 x 2
    5x + 2y = 16 x 3
    8x + 6y = 34
    15x + 6y = 48
    7x = 14
    x = 2

    8 + 3y = 17
    3y = 9
    y = 3
    Pen costs 2 shillings
    Exercise book costs 3 shillings
  8. Three villages are situated such that they form 3 sides of a triangle. The distance of villages P to R is 40km while R to Q is 60km.When represented on a paper angle PRQ is 78º.Find the distance of village P to Q. (4mks)
    10
    Cosine rule a2 = b2 + c2 - 2bcosA
    (PQ)2 = 402 + 602 - 240 .60cos78
    = 1600 + 3600 - 4800 x 0.2079
    = 5200 - 997.97
    = 4202km
    PQ = √4202 = 64.82
  9. A water tank is cylindrical in shape is filled by a pipe p in 6 hours. This same tank is emptied when filled in 12 hours by a pipe. When someone starts with the tank empty, both pipes are opened at the same time and then pipe Q turned off after 4 hours. Find the time taken to fill the tank. (3mks)
    P = 1/6   E = 1/12
    1/6 - 1/12 =  1/12 x 4
    4/12
    7/12 ÷ 1/
    7/12 ÷ 6/= 6.5h
    Time taken 
    4 + 3.5 = 7½
  10. The figure below shows triangle PQR in which PR = 12cm. T is a point on PR such that TR =4cm. Line ST is parallel to QR. If the area of triangle PQR is 336cm². Find the area of the quadrilateral STQR. (4 marks)
    11
    16:12
    LSF 4:3
    ASF = 16:9
    Area of PST 16 ⇒ 336
    336 x 9 = 189
       16
    Area of STRQ = 336 -189
    = 147cm2
  11. Kamau spent 3/5 of his money on buying food,and ½ of what was left on rent.After that,he finds that he has sh.1400 left in his pocket. Determine the amount money spent on buying food. (3mks)
    Food = 3/5
    Rent = 2/5 x ½
    = 1/5
    1/5x = 1400
    Food = 1400 x 3
    = 4800 shillings
  12. Find the centre of enlargement if P(0,-1) is the image of P(2,3) under enlargement scale factor 3 (3mks)
    12
  13. Use the reciprocal tables to evaluate to determine 3 decimal places, (4mks)
         2     -    3    
    0.4821   24.71.
    2(       1        ) - 3(        3        )
       4.821 x 101       2.471 x 10
    2(0.2074 x 10) - 3(0.407 x 10-1)
    4.148 - 0.12141
    4.02659
  14. Three students shared a certain amount of money. The first got sh.120 more than the second thrice what the third student got. If the second got sh.300, how much did each of the other two get? (3mks)
    If the second got sh x then the first got sh(x + 120)
    3rd got sh(x + 120)
                         3
    Then x = sh300
    1st got 420 x 1/3 = 140
  15. In the figure below, BT is tangent to the circle at B. AYCT and BYD are straight lines. AY=6cm, CT=8cm, BY=4.8cm and YD = 5cm
    13
    Find length of:
    1. CY
      4.8 x 5 = 6 x CY
      CY = 4
    2. BT
      BT2 = AT x CT
      = 18 x 8
      = 144
      BT = 12
  16. Use the method of completing the square to solve for xo(3 mks)
    4x2 + 12x - 9 = 0
      15ab2 - 10ab2
    3a2 - 5ab + 2b2
    Num
    5ab(3a - 2b)
    Den 
    3a2 - 3ab - 2ab + 2b2
    (3a - 2b)(a - b)
      5ab (3a - 2b)  
    (3a - 2b)(a - b)
    = 5ab 
       a - b

SECTION B: ATTEMPT ANY FIVE QUESTIONS IN THIS SECTION

  1. Three points P, Q and S are pm the vertices of a triangular plain field. P is 400m from Q on a bearing of 3000 and R of 550m directly south of P.
    1. Using a scale of 1 cm to represent 100m on the ground, draw a diagram to show the position of the points. (3mks)
      14
    2. Use the scale drawing to determine;
      1. The distance and bearing of Q from R. (3mks)
        360 - 45 = 315º
        045º ± 1
        490 ± 10m
      2. The bearing and distance of point S from P given that point S is directly 600m East of R. (3mks)
        8 ± 1 x 100 = 80cm
         131º ± 1 
      3. The bearing and distance of Q from S. (3mks)
        4.15 ± 1 x 100 = 415
        323º ± 1
  2. A bus travelling at a speed of 80km/hr left Mombasa at 8.00am for Nairobi. Two hours later, a car travelling at a speed of 100km/hr left Nairobi for Mombasa.
    Given that the distance between both cities is 500km, find the: -
    1. distance covered by bus at the time the car is starting. (1mk)
      80 x 2 = 160km
    2. Distance between the two vehicles at the time the is starting (1mk)
      500 - 160 = 340km
    3. Relative speed of the two vehicles (1mk)
      100 + 80 = 180km/hr
    4. Time taken for the two vehicles to meet (2mks)
      TK = 340
               180
      = 1.9
    5. Time the met. (1mk)
      8 + 2 1.54mm
      11.54am
    6. After meeting, the speed of both vehicles dropped to 60km/hr due to traffic jam. At what time did each vehicle arrive at its destination? (3mks)
      3.9 x 80 = 312   500 - 312
      188
       60
  3. The kite shown on the fig below has vertices, A(-2,4)B(5,7)C (7,4) D(5,1).The kite ABCD is rotated about the origin through -90º to obtain its first image A’B’C’D’.The kite A’B’C’D’ is then reflected along the line = X = 0 to obtain A”B’’C’’D’’.Find
    1. The equation of the Mirror line that reflects point A to itself, B on to D and C onto itself
    2. Draw a kite A’B’C’D and state its coordinates.
  4. A given solid is made up of a cone and three quarters of hemisphere as shown in the diagram below .The radius and vertical height of the cone are 4cm and 6.2 cm respectively
    5
    1. Calculate the volume of the solid (4mks)
      Volume sphere = 4/3 πr
      4/x 3.15 x 4 x 4 x ¾ = 200.96
      Cone 1/3πr2h = 1/x 3.14 x 4 x 4 x 6.2 = 103.83
      Total 304.89cm3
    2. Calculate the surface area of the solid (4mks)
      Sphere 4πrx ¾ = 4 x 3.14 x 4 x 4 x ¾ 
      = 150.816cm2
      Cone πrL = 3.142 x 7.348
      = 92.349 cm2
      Total = 243.166
    3. The material used for making the solid costs shs 40 per cm3.What is the cost of making ten such solids (2mks)
      304.789 x 40 = 12191.56
  5. In the diagram below, two circles, centres A and C and radii 7cm and 24cm respectively intersect at B and D. AC = 25cm.
    16
    1. Show that angle ABC = 90º. (3mks)
      b2 = a2 + 62 - 2aCosB
      252 = 242 + 72 - 2.24 . 7CosB
      625 = 576 + 49 - 336CosB
      CosB = 623 - 625
      CosB = 0
      B = 90
    2. Calculate
      1. the size of obtuse angle BAD (3mks)
        ½ x 7 x 24 = 84 x 2
        = 168
        17
        242 = 252 + 72 - 24 x 7cosA
        576 = 674 - 168 CosA
        +98 = +168CosA
        CosA = 0.5833
        A = 54.31º
        BAD = 108.62
      2. the area of the shaded part (4Mks)
        ½ x 7 x 24 x 2 = 168
        =108.62 x 3.142 x 2 x 7
             360
        = 13.27
        Shaded = 168 - 13.27
        = 154.73
  6. Two lines L1=2y-3x-6 and L2=3y+x-20=0 intersect at point A.
    1. Find the coordinates of A (3marks)
      (2y - 3x = 6) x 3
      (3y + x = 20) x 2
      6y - 9x = 18
      6y + 2x = 40
      -11x = -22
      x = 2
      1. A third line L3 is perpendicular to L2 at point A. Find the equation of L3 in form of y=mx+c, where m and c are constants. (3marks)
        3y = -x + 20
        y = - 1/3 x + 20/3
        m = 3
        y - 6 = 3
        x - 2
        y - 6 = 3x - 6
        y = 3x + 0
    2. Another line L4 is parallel to l1 and passes through (-1,3). Find the x-intercept and the y-intercept of L4. (4marks)
      2y = 3x + 6
      y = 3/2x + 3
      y - 33/2
      x + 1
      2(y - 3) = 3(x + y)
      2y - 6 = 3x + 3
      2y = 3x + 9
      y = 3/2x + 9
      x - intersept y =0
      2/3x x 3/2x = -9/22/3
      x = -3
      y intercept 3a = 0
      y = 9/2
      = 4.5
  7.            
    1. PQRS is a quadrilateral with vertices P(1,4), Q(2,1), R(2,3) and S(6,4). On the grid provided, plot the quadrilateral. (2mark)
    2. Draw P’Q’R’S’ the image of PQRS under a positive quarter turn about the origin and write down its co-ordinates. (3 marks)
    3. Draw P”Q”R”S” the image of P’Q’R’S’ under an enlargement scale factor -1 and center (0,0) and write down its co-ordinates. (3 marks)
      18
      Determine the single transformation that maps PQRS onto P”Q”R”S (2marks)

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