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
6x^{2} – 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 Vcm^{3} of an object is given by
V = 2 π r^{3} 1 – 2
3 sc^{2}
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)
- 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)
- 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)
- Water flows from a tap. At the rate 27cm^{3} 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)
- In the diagram below < CAD, = 20^{0}, < AFE = 120^{0} and BCDF is a cyclic quadrilateral. Find < FED. ( 3 marks)
- 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)
- 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
- 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)
- 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
- 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)
- (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 = at^{2} + 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 = x^{2} – 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 = 120^{0} < ABC = 75^{0} and < ADC = 85^{0}
(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
X^{2} – 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:
49^{x + 1 }+ 7^{2x} = 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 m^{3}. Calculate the volume of the larger tank. ( 3 marks)
8. Simplify completely
3x^{2} – 1 – 2x + 1
X^{2} – 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 = 160^{0}. 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 x^{3} + 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 30^{0} 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 60^{0} 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 = 105^{0}. 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 | 0^{0} | 10^{0} | 20^{0} | 30^{0} | 40^{0} | 50^{0} | 60^{0} | 70^{0} | 80^{0} | 90^{0} | 100^{0} | 110^{0} | 120^{0} |
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 0^{0} ≤ x ≤ 120^{0} 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 3x^{2} – 2xy – y^{2} ( 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 = 68^{0}, < BDC = 45^{0} and < BAE = 98^{0}.
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 A^{2} = A + B ( 4 marks)
4 3
12. Solve the equation
Sin 5 θ = -1 for 0^{0} ≤ 0 ≤ 180^{0} ( 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 – v^{2})
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 = 3x^{2} – 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 ( 36^{0} N, 49^{0}E) and ( 360^{0}N, 131^{0} 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 = x^{2} + 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 = x^{2} + 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 = 25^{0}, 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)
- 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 295^{0} from R at a distance of 60 km. G is on a bearing of 340^{0} 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 2x^{0} and y = 2 sin ( ^{3}/_{2} x^{0} + 30^{0}) for 0 < x < 180^{0}
X^{0} | 0^{0} | 20^{0} | 40^{0} | 60^{0} | 80^{0} | 100^{0} | 120^{0} | 140^{0} | 160^{0} | 180^{0} |
– 3 cos 2 x^{0} | -3.00 | 1.50 | 2.82 | 2.82 | 0.52 | -2.30 | ||||
2 sin ( ^{3}/_{2} x^{0} + 30^{0}) | 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} x^{0} + 30^{0}) on the same axes.
Take 1 cm to represent 20^{0} 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 x^{0} + 30^{0}) = 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 = ax^{3} + b where a and b are constants
- For each value of x in the table above, write down the value of x^{3}( 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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