KCSE MATHEMATICS QUESTIONS AND SOLUTIONS ~ Topically Analyzed
Form 3 Mathematics
In this question use a ruler and a pair of compasses only
In the figure below, AB and PQ are straight lines
(a) Use the figure to:
(i) Find a point R on AB such that R is equidistant from P and Q (ii) Complete a polygon PQRST with AB as its line of symmetry and hence measure the distance of R from TS. (b) Shade the region within the polygon in which a variable point X must lie given that X satisfies the following conditions I: X is nearer to PT than to PQ II: RX is not more than 4.5 cm III. angle PXT > 90
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Form 3 Mathematics
The gradient function of a curve is given by the expression 2x + 1. If the curve passes through the point ( -4, 6);
(a) Find: (i) The equation of the curve (ii) The vales of x, at which the curve cuts the x- axis (b) Determine the area enclosed by the curve and the x- axis Form 1 Mathematics
A tank has two inlet taps P and Q and an outlet tap R. when empty, the tank can be filled by tap P alone in 4 ½ hours or by tap Q alone in 3 hours. When full, the tank can be emptied in 2 hours by tap R.
(a) The tank is initially empty. Find how long it would take to fill up the tank (i) If tap R is closed and taps P and Q are opened at the same time (ii) If all the three taps are opened at the same time (b) The tank is initially empty and the three taps are opened as follows P at 8.00 a.m Q at 8.45 a.m R at 9.00 a.m (i) Find the fraction of the tank that would be filled by 9.00 a.m (ii) Find the time the tank would be fully filled up Form 4 Mathematics
The diagram on the grid below represents as extract of a survey map showing two adjacent plots belonging to Kazungu and Ndoe.
The two dispute the common boundary with each claiming boundary along different smooth curves coordinates ( x, y) and (x, y2) in the table below, represents points on the boundaries as claimed by Kazungu Ndoe respectively.
(a) On the grid provided above draw and label the boundaries as claimed by Kazungu and Ndoe
Form 4 Mathematics
(b) In a certain week a businessman bought 36 bicycles and 32 radios for total of Kshs 227 280. In the following week, he bought 28 bicycles and 24 radios for a total of Kshs 174 960
Using matrix method, find the price of each bicycle and each radio that he bought (c) In the third week, the price of each bicycle was reduced by 10% while the price of each radio was raised by 10%. The businessman bought as many bicycles and as many radios as he had bought in the first two weeks. Find by matrix method, the total cost of the bicycles and radios that the businessman bought in the third week. Form 1 Mathematics
Two cylindrical containers are similar. The larger one has internal cross- section area of 45cm2 and can hold 0.945 litres of liquid when full. The smaller container has internal cross- section area of 20cm2
(a) Calculate the capacity of the smaller container (b) The larger container is filled with juice to a height of 13 cm. Juice is then drawn from is and emptied into the smaller container until the depths of the juice in both containers are equal. Calculate the depths of juice in each container. (c) On fifth of the juice in the larger container in part (b) above is further drawn and emptied into the smaller container. Find the difference in the depths of the juice in the two containers. Form 3 Mathematics
In the figure below, OQ = q and OR = r. Point X divides OQ in the ratio 1: 2 and Y divides OR in the ratio 3: 4 lines XR and YQ intersect at E.
(a) Express in terms of q and r
(i) XR (ii) YQ (b) If XE = m XR and YE = n YQ, express OE in terms of: (i) r, q and m (ii) r, q and n (c) Using the results in (b) above, find the values of m and n. Form 1 Mathematics
A retailer planned to buy some computers form a wholesaler for a total of Kshs 1,800,000. Before the retailer could buy the computers the price per unit was reduced by Kshs 4,000. This reduction in price enabled the retailer to buy five more computers using the same amount of money as originally planned.
(a) Determine the number of computers the retailer bought (b) Two of the computers purchased got damaged while in store, the rest were sold and the retailer made a 15% profit Calculate the profit made by the retailer on each computer sold Form 3 Mathematics
A frequency distribution of marks obtained by 120 candidates is to be represented in a histogram. The table below shows the grouped marks. Frequencies for all the groups and also the area and height of the rectangle for the group 30 – 60 marks.
(a) (i) Complete the table
(ii) On the grid provided below, draw the histogram (b) (i) State the group in which the median mark lies (ii) A vertical line drawn through the median mark divides the total area of the histogram into two equal parts Using this information or otherwise, estimate the median mark Form 2 Mathematics
In the diagram below PA represents an electricity post of height 9.6 m. BB and RC represents two storey buildings of heights 15.4 m and 33.4 m respectively.
The angle of depression of A from B is 5.50 While the angle of elevation of C from B is 30.50 and BC = 35m.
(a) Calculate, to the nearest metre, the distance AB
(b) By scale drawing find, (i) The distance AC in metres (ii) Angle BCA and hence determine the angle of depression of A from C Form 1 Mathematics
Three business partners: Asha Nangila and Cherop contributed Kshs 60,000, Kshs 85,000 and Kshs 105 000 respectively. They agreed to put 25% of the profit back into business each year. Thay also agreed to put aside 40% of the remaining profit to cater for taxes and insurance. The rest of the profit would then be shared among the partners in the ration of their contributions. At the end of the first year, the business realized a gross profit of Kshs 225 000
(a) Calculate the amount of money Cherop received more than Asha at the end of the first year (b) Nangila further invested Kshs 25,000 into the business at the beginning of the second year. Given that the gross profit at the end of the second year increased in the ratio 10: 9, calculate Nangila's share of the profit at the end of the second year. ​Related Questions and Answers on Commercial Arithmetic I
Form 4 Mathematics
The diagram below shows a sketch of the line y = 3x and the curve y = 4 – x2 intersecting at points P and Q.
a) Find the coordinates of P and Q
(b) Given that QN is perpendicular to the x- axis at N, calculate (i) The area bounded by the curve y = 4 – x2, the x- axis and the line QN (ii) The area of the shaded region that lies below the x- axis (iii)The area of the region enclosed by the curve y = 4-x2, the line y – 3x and the y axis Form 4 Mathematics
Mwanjoki flying company operates a flying service. It has two types of aeroplanes. The smaller one uses 180 litres of fuel per hour while the bigger one uses 300 litres per hour.
The fuel available per week is 18,000 litres. The company is allowed 80 flying hours per week while the smaller aeroplane must be flown for y hours per week. (a) Write down all the inequalities representing the above information (b) On the grid provided on page 21, draw all the inequalities in a) above by shading the unwanted regions (c) The profits on the smaller aeroplane is Kshs 4000 per hour while that on the bigger one is Kshs 6000 per hour Use the graph drawn in (b) above to determine the maximum profit that the company made per week. Form 3 Mathematics
The product of the first three terms of geometric progression is 64. If the first term is a, and the common ration is r.
(a) Express r in terms of a (b) Given that the sum of the three terms is 14 (i) Find the value of a and r and hence write down two possible sequence each up to the 4th term. (ii) Find the product of the 50th terms of two sequences Form 1 Mathematics
A solution whose volume is 80 litres is made 40% of water and 60% of alcohol. When litres of water are added, the percentage of alcohol drops to 40%
(a) Find the value of x (b) Thirty litres of water is added to the new solution. Calculate the percentage (c) If 5 litres of the solution in (b) is added to 2 litres of the original solution, calculate in the simplest form, the ratio of water to that of alcohol in the resulting solution Form 3 Mathematics
(a) Two integers x and y are selected at random from the integers 1 to 8. If the
same integer may be selected twice, find the probability that (i) x – y = 2 (ii) x – y is more (iii) x>y (b) A die is biased so that when tossed, the probability of a number r showing up, is given by p ® = Kr where K is a constant and r = 1, 2,3,4,5 and 6 (the number on the faces of the die (i) Find the value of K (ii) if the die is tossed twice, calculate the probability that the total score is 11 Form 4 Mathematics
Triangle ABC is shown on the coordinates plane below
(a) Given that A (-6, 5) is mapped onto A (6,-4) by a shear with y- axis invariant
(i) draw triangle A"B"B", the image of triangle ABC under the shear (ii) Determine the matrix representing this shear (b) Triangle A B C is mapped on to A" B" C" by a transformation defined by the matrix (1 1) (i) Draw triangle A" B" C" (ii) Describe fully a single transformation that maps ABC onto A"B" C" Form 2 Mathematics
A garden measures 10m long and 8 m wide. A path of uniform width is made all round the garden. The total area of the garden and the paths is 168 m2.
(a) Find the width of the path (b) The path is to covered with square concrete slabs. Each corner of the path is covered with a slab whose side is equal to the width of the path.
The rest of the path is covered with slabs of side 50 cm. The cost of making each corner slab is Kshs 600 while the cost of making each smaller slab is Kshs 50.
Calculate (i) The number of smaller slabs used Form 1 Mathematics
A certain sum of money is deposited in a bank that pays simple interest at a certain rate.
After 5 years the total amount of money in an account is Kshs 358 400. The interest earned each year is 12 800 Calculate (i) the amount of money which was deposited (ii) the annual rate of interest that the bank paid (b) A computer whose marked price is Kshs 40,000 is sold at Kshs 56,000 on hire purchase terms. (i) Kioko bought the computer on hire purchase term. He paid a deposit of 25% of the hire purchase price and cleared the balance by equal monthly installments of Kshs 2625 Calculate the number of installments (ii) Had Kioko bought the computer on cash terms he would have been allowed a discount of 12 ½ % on marked price. Calculate the difference between the cash price and the hire purchase price and express as a percentage of the cash price. Form 4 Mathematics
A particle moves along straight line such that its displacement S metres from a given point is S = t3 – 5t2 + 4 where t is time in seconds
Find
(a) the displacement of particle at t = 5 (b) the velocity of the particle when t = 5 (c) the values of t when the particle is momentarily at rest (d) The acceleration of the particle when t = 2 Form 2 Mathematics
The figure below is a model representing a storage container. The model whose total height is 15cm is made up of a conical top, a hemispherical bottom and the middle part is cylindrical. The radius of the base of the cone and that of the hemisphere are each 3cm. The height of the cylindrical part is 8cm.
(a) Calculate the external surface area of the model
(b) The actual storage container has a total height of 6 metres. The outside of the actual storage container is to be painted. Calculate the amount of paint required if an area of 20m2 requires 0.75 litres of the paint Form 3 Mathematics
In the diagram below, the coordinates of points A and B are ( 1,6) and (15,6) respectively)
Point N is on OB such that 3 ON = 2OB. Line OA is produced to L such that OL = 3 OA
(a) Find vector LN
(b) Given that a point M is on LN such that LM: MN = 3: 4, find the coordinates of M (c) If line OM is produced to T such that OM: MT = 6:1 (i) Find the position vector of T (ii) Show that points L, T and B are collinear Form 2 Mathematics
The figure below represents a quadrilateral piece of land ABCD divided into three triangular plots The lengths BE and CD are 100m and 80m respectively. Angle ABE = 300,angle ACE = 450and angle ACD = 1000
Find to four significant figures:
(i) The length of AE (ii) The length of AD (iii) the perimeter of the piece of land (b) The plots are to be fenced with five strands of barbed wire leaving an entrance of 2.8 m wide to each plot. The type of barbed wire to be used is sold in rolls of lengths 480m. Calculate the number of rolls of barbed wire that must be bought to complete the fencing of the plots Form 4 Mathematics
Bot juice Company has two types of machines, A and B, for juice production. Type A machine can produce 800 litres per day while type B machine produces 1,600 litres per day.
Type A machine needs 4 operators and type B machine needs 7 operators. At least 8,000 litres must be produced daily and the total number of operators should not exceed 41. There should be 2 more machines of each type. Let x be the number of machines of type A and Y the number of machines for type B,
Form 2 Mathematics
The diagram below represents a pillar made of cylindrical and regular hexagonal parts. The diameter and height of the cylindrical part are 1.4m and 1m respectively. The side of the regular hexagonal face is 0.4m and height of hexagonal part is 4m.
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