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KCSE PAST PAPERS AND ANSWERS-MATHEMATICS PAPER 1 AND 2 1995-2014

MATHEMATICS PAPER 1 K.C.S.E 1995 QUESTIONS

SECTION 1 (52 MARKS)

Answer all the questions in this section

1.         Without using logarithms tables evaluate                                           ( 3 marks)

                                    384.16 x 0.0625

                                          96.04

2.         Simplify                                                                                              ( 3 marks)

                                    2x – 2          ÷         x – 1

                                    6x2 – x – 12           2x – 3

3.         Every week the number of absentees in a school was recorded.  This was done for 39 weeks these observations were tabulated as shown below

Number of absentees 0.3 4 -7 8 -11 12 – 15 16 – 19 20 – 23
(Number of weeks) 6 9 8 11 3 2

            Estimate the median absentee rate per week in the school                 ( 2 marks)

4.         Manyatta village is 74 km North West  of Nyangata village. Chamwe village is 42 km  west of Nyangate. By using an appropriate scale drawing, find the bearing of Chamwe from Manyatta                                                                 ( 2 marks)

5.         A perpendicular to the line -4x + 3 = 0 passes through the point ( 8, 5) Determine its  equation                                                                  ( 2 marks)

6.         The volume Vcm3 of an object is given by

            V = 2 π r3    1 – 2

                   3            sc2

Express in term of π r, s and V                                                           ( 3 marks)

8.         Two baskets A and B each contains a mixture  of oranges and lemons. Basket A  contains 26 oranges and 13 lemons. Basket B contains 18 oranges and 15 lemons.  A child selected basket at random and picked at random a fruit from it.

            Determine the probability that the fruit picked was an orange.

9.         A solid cone  of height 12cm and radius 9 cm is recast into a solid sphere. Calculate the surface area  of the  sphere.                                     ( 4 marks)

10.       The first, the third  and the seventh terms  of an increasing arithmetic progression are three consecutive terms of a geometric progression. In  the first term of the arithmetic progression is  10 find the common difference of the arithmetic progression.                                                                                     ( 4 marks)

  1. Akinyi bought and beans from  a wholesaler. She then mixed the maize  and beans the ratio 4:3 she brought the maize as Kshs. 12 per kg and the beans 4 per kg.  If she was to make a profit of 30% what should be the selling price  of 1 kg  of the mixture?                                                                         ( 4 marks)

  1. A clothes dealer sold 3 shirts and 2 trousers for Kshs. 840 and 4 shirts and 5 trousers for Kshs 1680.

Form a matrix equation to represent the above information. Hence find the cost of 1 shirt and the cost of 1 trouser.                                                            ( 4 marks)

  1. Water flows from a tap. At the rate 27cm3 per second, into a rectangular container  of  length 60cm, breath 30 cm and height 40 cm. If at 6.00 p.m. the container was half full, what will be the height of water at 6.04 pm? ( 3 marks)
  2. In the diagram below < CAD, = 200, < AFE = 1200 and BCDF is a cyclic quadrilateral. Find < FED.                                                                   ( 3 marks)
  1. The cash prize of a television is Kshs 25000. A customer paid a deposits of Kshs 3750. He repaid the amount owing in 24 equal monthly installments.

If hw was charged simple interest at the rate of 40% p.a, how much was each installment?                                                                                                ( 4 marks)

  1. A bus takes 195 minutes to travel a distance of ( 2x + 30) km at  an average speed of ( x – 20) km/h

Calculate the actual distance  traveled. Give your  answers  in kilometers.

( 3 marks)

SECTION II ( 48 MARKS)

Answer any six questions from this section

  1. At the beginning of every year, a man deposited Kshs 10,000 in a financial institution which paid compound interest at the rate of 20% p.a. He stopped further deposits after three years. The Money remained invested in the financial institution for a further eight years.

(a) How much money did he have at the end of the first three years            ( 4 marks)

(b) How much interest did the money generate in the entire period ( 4 marks)

  1. The figure below is a right pyramid with a rectangular base ABCD and VO as the height. The vectors AD= a, AB = b and  DV = v
  1. Express

(i) AV in terms of a and c                                                       ( 1 mark)

(ii) BV in terms of a, b and c                                                  ( 2 marks)

(b) M is point on OV such that OM: MV = 3:4, Express BM in terms of a, b and 

      c.

      Simplify your answer as far as possible                                        ( 5 marks)

  1. (a) In the figure below O is the centre of a circle whose radius is 5 cm AB = 8 cm and < AOB is obtuse.

           Calculate the area of the major segment                                             ( 6 marks)

(b) A wheel rotates at 300 revolutions per minute. Calculate the angle in radians through which a point on the wheel turns  in one  second.

  • The table shows the height metres of  an object thrown vertically upwards varies with the time t seconds

The relationship between s and t is represented by the equations s = at2 + bt + 10 where b are constants.

t 0 1 2 3 4 5 6 7 8 9 10
s   45.1                  
  • (i) Using the information in the table, determine the values of a and b  

( 2 marks)

                  (ii) Complete the table                                                                  ( 1 mark)

            (b) (i) Draw a graph to represent the relationship between s and t     ( 3 marks)

                 (ii) Using the graph determine the velocity of the object when t = 5 seconds

                                                                                                                        (2 marks)

  • (a) Construct a table of values for the function y = x2 – 6 for -3 < x <4                                                                                                                                ( 2 marks)
  • By drawing a suitable line on the  same grid estimate the roots of the equation

X2 + 2x – 2 =0                                                                              ( 3 marks)

  • The figure below represents a plot of land ABCD, where BC = CD = 60 metres, < BCD = 1200 < ABC = 750 and  < ADC = 850

(a) Calculate the distance from B to through D                                 ( 5 marks)

(b) The plot is to be fenced using poles that are 3 metres apart except at corner A, where the two poles next to the  corner pole are each less than 3 metres from A. Calculate the distance from the pole at  corner at corner A to each of the poles next to it.

  • On the grid provided on the opposite  page ABCE is a trapezium

(a) ABCD is mapped onto A’B’C’D’ by a positive quarter turn. Draw the image A’B’C’D’ on the grid.                                                                           ( 1 mark)

(b) A transformation  maps     -2    -1 A’B’C’D onto A” B” C” D”

                                                 1     -1

(i) Obtain the coordinates of A” B” C” D” on the grid                      ( 2 marks)

(ii) Plot the image A” B” C” D” on the grid                                       (1mark)

(c) Determine a single matrix that maps A” B” C” D”                       ( 4 marks)

MATHEMATICS  PAPER 2 K.C.S.E 1995 QUESTIONS

SECTION 1 ( 52 MARKS)

1.         Use logarithms to evaluate        (0.07284)2                                         ( 4 marks)

                                                            3    0.06195

2.        Solve the simultaneous equations                                                       ( 4 marks)

                                               2x – y = 3

                                               X2 – xy = -4

3.        The tables shows the yearly percentage taxations rates.

Year 1987 1988 1989 1990 1991 1992 1993 1994
Percentage taxation rate 65 50 50 45 45 45 40 40

Calculate three- yearly moving averages for the data giving answers to s.f              

 ( 3 marks)

4.         Calculate volume of a prism whose length is 25cm and whose cross- section is an equilateral triangles of 3 cm

5.         Find the value of x in the following equations:

                                                49x + 1 + 72x = 350                                            ( 4 marks)

6.         A translation maps a point ( 1, 2) onto) (-2, 2). What would be the coordinates of the object whose image is ( -3 , -) under the same   translation?

7.         The ratio of the lengths of the corresponding sides of  two similar rectangular water tanks is 3:5. The volume of the smaller tank is 8.1 m3. Calculate the volume of the larger tank.                                                                                  ( 3 marks)

8.         Simplify completely

                                                3x2 – 1 – 2x + 1

                                                X2 – 1      x + 1

9.         A boat moves 27 km/h in still water. It is to move from point A to a point B which is directly east of A. If the river flows from south to North at 9 km/ h, calculate the track of the boat.

10.       The second and fifth terms of a geometric progressions are 16 and 2 respectively. Determine the common ratio and the first term

11.       In the figure below CP= CQ and <CQP = 1600. If ABCD is a cyclic quadrilateral, find < BAD.

12.       In the figure below, OA = 3i + 3J ABD OB = 8i –j, C is a point on AB such that AC: CB = 3:2, and D is a point such that OB // CD and 2 OB = CD.

            Determine the vector DA in terms of i and j.                                     ( 4 marks)

13.       Without using logarithm tables, find the value of x in the equation

                                    Log x3 + log 5x = 5 log 2 – log 2                                (3 marks)

                                                                                      5

14.       Two containers, one cylindrical and one spherical, have the same volume. The height of the cylindrical container is 50 cm and its radius is 11 cm. Find the radius of the spherical container.                                                                      ( 2 marks)

15.       Two variables P and L are such that P varies partly as  L and partly as the square root of L.

            Determine the relationship between P and L when L = 16, P = 500 and when L = 25, P = 800.                                                                                          ( 5 marks)

16.       The shaded region below represents a forest. The region has been drawn to scale where 1 cm represents 5 km. Use the mid – ordinate rule with six strips to estimate the area of forest in hectares.                                               ( 4 marks)

SECTION II (48 Marks)

Answer any six questions from this section

17.       A circular path of width 14 metres surrounds a field of diameter 70 metres. The path is to be carpeted and the field is to have a concrete slab with an exception of four rectangular holes each measuring 4 metres by 3 metres.

            A contractor estimated the cost of carpeting the path at Kshs. 300 per square metre and the cost of putting the concrete slab at Kshs 400 per square metre. He then made a quotation which was 15% more than the total estimate. After completing the job, he realized that 20% of the quotation was not spent.

            (a) How much money was not spent?

            (b) What was the actual cost of the contract?

18.       The table below shows high altitude wind speeds recorded at a weather station in a period of 100 days.

Wind speed ( knots) 0 – 19 20 – 39 40 – 59 60-79 80- 99 100- 119 120-139 140-159 160-179
Frequency (days) 9 19 22 18 13 11 5 2 1

(a) On the grid provided draw a cumulative frequency graph for the data    ( 4 marks)

(b) Use the graph to estimate

            (i) The interquartile range                                                                    ( 3 marks)

            (ii) The number of days when the wind speed exceeded 125 knots  ( 1 mark)

19.       The probabilities that a husband and wife will be alive 25  years from now are 0.7 and 0.9 respectively.

            Find the probability that in 25 years time,

  • Both will be alive
  • Neither will be alive
  • One will be  alive
  • At least one will be alive

20.       A hillside is in the form of a plane inclined at an angle of 300 to the horizontal. A straight section of road 800 metres long lies along the line of greatest slope from a point A to a point B further up the hillside.

  • If a vehicle moves from A and B, what vertical height does it rise?
  • D is another point on the hillside and is on the same height as B. Another height straight road joins and D and makes an angle of 600 with AB. C is a point on AD such that AC = ¾ AD.

Calculate

  • The length of the road from A to C
    • The distance of CB

A ship leaves B and moves directly southwards to an island P, which is on a bearing of 140 from A. The submarine at D on realizing that the ship was heading fro the island P, decides to head straight for the island to intercept the ship

Using a scale 0f 1 cm to represent 10 km, make a scale  drawing showing the  relative positions  of A, B, D, P.                                                          ( 2 marks)

Hence find

(i) The distance from A to D                                                        ( 2 marks)

(ii) The bearing of the submarine from the ship was setting off from B

                                                                                                      ( 1mark)

(iii) The bearing of the island P from D                                        ( 1 mark)

(iv) The distance the submarine had to cover to reach the island P

                                                                                                      ( 2 marks)

  • Using ruler and compasses only, construct a parallelogram ABCD such that AB = 10cm, BC = 7cm and < ABC = 1050. Also construct the loci of P and Q within the parallel such that AP ≤ 4 cm, and BC  ≤ 6 cm. Calculate the area  within the parallelogram and outside the regions bounded by the loci.
  • (a) Complete the table for the function y = 2 sin x                       ( 2 marks)
x 00 100 200 300 400 500 600 700 800 900 1000 1100 1200
Sin 3x 0 0.5000                      
y 0 1.00                      

(b)  (i) Using the values in the completed table, draw the graph of y = 2 sin 3x for 00 ≤ x ≤ 1200 on the grid provided

      (ii) Hence solve the equation 2 sin 3x = -1.5                          ( 3 marks)

  • A manufacture of jam has 720 kg of strawberry syrup and 800 kg of mango syrup for making two types of jam, grade A and B. Each types is made by mixing strawberry and mango syrups as follows:

Grade A: 60% strawberry and 40% mango

Grade B: 30% strawberry and 70% mango

             The jam is sold in 400 gram jars. The selling prices are as follows:

                        Grade A: Kshs. 48 per jar

                        Grade B: Kshs 30 per jar.

  • Form inequalities to represent the given information                   ( 3 marks)
  • (i)   On the grid provided draw  the inequalities                           ( 3 marks)

(ii) From your, graph, determine the number  of jars of each grade the manufacturer should produce to maximize his profit                 ( 1 mark)

(iii) Calculate the total amount of money realized if all the jars are sold

                                                                                                ( 1 mark)

MATHEMATICS PAPER 1 K.C.S.E 1996 QUESTIONS

SECTION 1 ( 52 Marks)

1.         Use logarithms to evaluate                                                                  ( 3 marks)

    3 36.15 x 0.02573

          1,938

2.         Factorize completely 3x2 – 2xy – y2                                                   ( 2 marks)

3.         The cost of 5 skirts and 3 blouses is Kshs 1750. Mueni bought three of the skirts and one of the blouses for Kshs 850.

            Find the cost of each item                                                                  ( 3 marks)

4.         A man walks directly from point A towards the foot of a tall building 240m away. After covering 180m, he observes that the angle of the top of the building is 45. Determine the angle of elevation of the top of the building from A.         ( 3 marks)

5.         In the figure below, ABCD is a cyclic quadrilateral and BD is a diagonal. EADF is a straight line. <CDF = 680, < BDC = 450 and < BAE = 980.

            Calculate the size of

            (a)        < ABD                                                                                                ( 2 marks)

            (b)        < CBD                                                                                                ( 2 marks)

6.         An employee started on a salary of  £ 6,000 per annum and received a constant annual increment. If he earned a total of  £ 32,400 by the end of five years, calculate his annual increment.                                                            ( 3 marks)

7.         Mr. Ngeny borrowed Kshs. 560,000 from a bank to buy a piece of land. He was required to repay the loan with simple interest for a period of 48 months. The repayment amounted to Kshs 21000 per  month.

            Calculate

            (a) The interest paid to the bank                                                         ( 2 marks)

            (b) The rate per annum of the simple interest                         ( 4 marks)

8.         A rectangular tank of base 2.4 m by 2.8 m and a height of 3 m contains 3,600 liters of water initially. Water flows into the tank at the rate of 0.5 litres per second

            Calculate the time in hours and minutes, required to fill the tank      ( 4 marks)

9.         A car dealer charges 5% commission for selling a car. He received a commission of Kshs 17,500 for selling a car. How much money did the owner receive from the sale of his car?                                                                                 ( 2 marks)

10.       Five pupils A, B, C, D and E obtained the marks 53,  41, 60, 80 and 56 respectively. The table below shows part of the work to find the standard deviation.

Pupil Mark x x – x ( x-x)2
A B C D E 53 41 60 80 56 -5 -17 2 22 -2  

(a) Complete the table                                                                         ( 1 mark)

(b) Find the standard deviation                                                          ( 3 marks)

11.       A and B are two matrices. If A = 1     2   find B given that A2 = A + B ( 4 marks)

                                                                  4      3

12.       Solve the equation

                                                Sin 5 θ = -1 for 00 ≤ 0 ≤ 1800                          ( 2 marks)

                                                      2          2

13.       A fruiterer bought 144 pineapples at Kshs 100 for every six pineapples. She sold some of them at Kshs. 72 for every three and the rest at Kshs 60 for every two.

If she made a 65% profit, calculate the number of pineapples sold at Kshs 72 for every three                                                                                             ( 3 marks)

14.       Make V the subject of the formula

                                                T = 1 m (u2 – v2)

                                                       2                                                               ( 3 marks)

15.       The figure below represents a hollow cylinder. The internal and external radii are estimated to be 6 cm and 8 cm respectively, to the nearest whole number. The height of the cylinder is exactly 14 cm.

(a)        Determine the exact values for internal and external radii which will give maximum volume of the material used.                          ( 1 mark)

(b)        Calculate the maximum possible volume of the material used

            Take the value of to be 22/7                                                   ( 2 marks)

16.       Two lorries A and B ferry goods between tow towns which are 3120 km apart. Lorry A traveled at km/h faster than lorry B and B takes 4 hours more than lorry A to cover the distance.

            Calculate the speed of lorry B                                                            ( 5 marks)

SECTION II (48 MARKS)

Answer any six questions from this section

17.       The data given below represents the average monthly expenditure, E  in K £, on food in a certain village. The expenditure varies with number of dependants, D in the family.

Dependants 3 7 12 25 32
Expenditure E ( K£) -210 250 305 440 500
  • Using the grid provided, plot E against D and draw the line of the best fit

                                                                                                                  ( 2 marks)

  • Find the gradient  and the  E- intercept of the graph                         ( 3 marks)
  • Write down  an equation connecting  E and D                                  ( 1 mark)
  • Estimate the cost of  feeding a family with 9 dependants                 ( 2 marks)

18.       The table below shows the income tax rates

Total income per month in Kenya Rate in shillings per pound
1 – 325 326 – 650 651-975 976 – 1300 1301 – 1625 Over 1625 2 3 4 5 7 7.50

Mr. Otiende earned a basic salary of Kshs 13,120 and a house allowance of Kshs 3,000 per month. He claimed a tax relief for a married person of Kshs 455 per month

(a) Calculate

(i) The tax payable without the relief

(ii) The tax paid after the relief

(b)        Apart from the income  tax, the following monthly deductions are made. A service charge of Kshs 100, a health insurance fund  of Kshs 280 and 2% of his basic salary as widow and children pension scheme.

            Calculate

(i) The total monthly deductions made from Mr. Otiende’s income                                                                                                              ( 2 marks)

(ii) Mr. Otiende’s net income from his employment               ( 2 marks)

19.       The equation of a curve us y = 3x2 –  4 x + 1

            (a) Find the gradient function of the curve  and its value when x = 2 ( 2 marks)

            (b) Determine

            (i) The equation of the tangent to the curve at the  point (2, 5)          ( 2 marks)

(ii) The angle which the tangent to the curves at the  point ( 2, 5) makes with the horizontal                                                                                               ( 1 mark)

(iii) The equation of the line through the point ( 2, 5) which is perpendicular to the tangent in (b) (i)

20.       The position of two A and B on the earth’s surface are ( 360 N, 490E) and ( 3600N, 1310 W) respectively.

            (a) Find the difference in longitude between  town A and town B   ( 2 marks)

(b) Given that the radius of the earth is 6370, calculate the distance  between town A and town B.

(c) Another town, C  is 840 east of town B and  on the same  latitude as towns A   and B. Find the longitude of town C.

21.       The table below shows some values of the function y = x2 + 2x – 3

x -6 -6.75 -5.5 -5 -4.75 -4.5 4.25 -4.0 -3.75 -3.75 -3.5 -3.25 -3
y 21 18.56   14.06   10.06 8.25   5   2.25 1.06 0

a)         Complete the table

b)         Using the completed table and the mid- ordinate rule with six ordinates, estimate the area of the region bounded by the y = x2 + 2 x – 3 and the line y = 0, x = -6 and x = -3                                                            ( 3 marks)

(i) By integration find the actual area of the region in (b) above       2 marks)

(ii) Calculate the percentage error arising from the estimate in (b)     (2 marks)

22.       In the diagram below OABC is a parallelogram, OA = a and AB = b. N is a point on OA such that ON: NA = 1: 2

  • Find
    • AC in terms of a  and b
    • BN in terms of a and b
    • The lines AC and BN intersect at X, AX = hAC and BX = kBN
    • By expressing OX in two ways, find the values of h and k
    • Express OX in terms of a and b                                  ( 1 mark)
  • Use ruler and compasses only in this question

The diagram below shows three  points A, B and D

(a) Construct the angle bisector of acute angle  BAD                  ( 1 mark)

(b) A point P, on the  same side of AB and D, moves in such a way that < APB = 22 ½ 0 construct the locus of P                                      ( 6 marks)

(c) The locus of P meets the angle bisector of < BAD  at C measure < ABC

                                                                                                      ( 1 mark)

      Hence find area of the image A” B” C”                                 ( 2 marks)

SECTION II (48 Marks)

Answer any six questions from this section

17.       Two businessmen jointly bought a minibus which could ferry 25 paying passengers when full. The fare between two towns A and B was Kshs 80 per passengers for one way. The minibus made three round trips between two towns daily. The cost of fuel was Kshs 1500 per day. The driver and the conductor were paid daily allowances of 200 and Kshs 150 respectively.

A further Kshs 4,000 per day was set aside for maintenance, insurance and loan payment.

  • One day, the minibus was full on every trip.
    • How much money was collected from the passengers that day?
    • How much was the net profit?
  • On another day, the minibus was 80% full on the average for the three round trips, how  much and each businessman  get  if the day’s profit  was shared in the ratio of 2: 3
  • In the figure below AOC is a diameter of the circle centre O; AB = BC and < ACD = 250, EBF is a tangent to the circle at B.G is a point on the minor arc CD.

      (a) Calculate the  size of

                  (i) < BAD                                                                                ( 3  marks)

                  (ii) the Obtuse < BOD                                                                        ( 3 marks)

                  (iii) < BGD                                                                              ( 1 mark)

      (b) Show the < ABE = < CBF. Give reasons                                     ( 2 marks)

  1. In an agricultural research centre, the length of a sample of 50  maize cobs were  measured and recorded as shown in the frequency distribution table below.
Length in cm Number  of  cobs
8 – 10 11 – 13 14 – 16 17 – 19 20 – 22 23 – 25 4 7 11 15 8 5

Calculate

  • The mean
    • (i) the variance

(ii) The standard deviation                                                ( 8 marks)

  • Four towns R, T, K and G are such  that T  is 84  km  directly to the north R, and K is on a bearing of 2950 from R at a distance  of 60 km. G is on a bearing of 3400 from K and a distance of 30 km. Using a scale of 1  cm  to represent 10 km, make  an accurate scale drawing  to show  the relative positions of the town.

Find

  • The distance  and the bearing of  T  from K
    • The  distance  and the bearing G from T
    • The bearing of R from G
  • Kubai saved Kshs 2,000 during the first year of employment. In each subsequent year, he saved 15% more than the preceding year until he retired.
    • How much did he save in the second year?                           ( 1 mark)
    • How much did he save in the third year?                               ( 1 mark)
    • Find the common ratio between the savings in two consecutive years

 ( 3 marks)

  • How many years did he take to save the savings a sum of Kshs 58,000?

( 3 marks)

  • How much had he saved after 20 years of service?               ( 2 marks)
  •  A school has to take 384 people for a tour. There are two types of buses available, type X and type Y. Type X can carry 64 passengers and type Y can carry 48 passengers. They have to use at least 7 buses.
    • Form all the linear equalities which will represent the above information

( 3 marks)

  • On the grid provided, draw the inequalities and shade the

Unwanted region                                                                          ( 3 marks)

  • the charges  for  hiring the buses are

Type X. Kshs 25000

Type y: Kshs 20000

Use your graph to determine the number of buses of each type that should be  hired to minimize the cost.

  • Complete the table given below using the functions.

Y = -3 cos 2x0 and y = 2 sin ( 3/2 x0 + 300) for 0 < x < 1800

X0 00 200 400 600 800 1000 1200 1400 1600 1800
– 3 cos 2 x0 -3.00     1.50 2.82 2.82   0.52 -2.30  
2 sin ( 3/2 x0 + 300) 1.00   2.00 1.73   0.00 -1.00     -1.73
  • Using the grid provided, draw the graphs y = -3 cos 2xo and y = 2 sin  ( 3/2 x0 + 300) on the same axes.

Take 1 cm to represent 200 on the x  – axis and  2 cm to represent one unit on the y – axis.                                                 ( 4 marks)

  • From  your  graphs, find the roots of 3 cos 2x + sin ( 3/2 x0 + 300) = 0
  • Data collected  form  an experiment involving two variables X and Y was recorded as  shown in the table below
x 1.1 1.2 1.3 1.4 1.5 1.6
y -0.3 0.5 1.4 2.5 3.8 5.2

The variables are  known to satisfy a relation of the form y = ax3 + b where  a and b are constants

  • For each value of x  in the table above,  write  down the value  of  x3( 2 marks)
  • (i) By drawing a suitable straight line graph, estimate the values of a and b

( 2 marks)

            (ii) Write down the relationship connecting y and x

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