SECTION I (50 marks)
Answer all the questions in this section
 Use logarithms to evaluate (4 marks)
 Solve for x in the equation given below (3 marks)
9^{2x+1} = 30 − 34^{4x}  The lengths of a triangle are in the ratio 5:6:9. if its area is 250√2, calculate its perimeter (4 marks)
 Simplify completely without using mathematical tables or calculator (3 marks)
 Four metal rods of length 15x^{3}h^{2}y, 3x^{2}yh^{5} and 9x^{4}y^{2}h are to be cut into smaller pieces of the same length such that there is no metal left. What is the maximum length of one of the pieces (1 mark)
 Given that the log 2 = 0.30103 and log 7 = 0.84510, find without using mathematical tables or calculator, log 9.8 (3 marks)
 Given that 3x − y = 17, find the value of y^{2} + 9x^{2} − 6xy − 9 (2 marks)
 The center of a circle is C (6,7). A tangent to the circle passes through the point P (8,3) lying on the circle. Find the eqaution of the tangent (4 marks)
 Coffee grade 1 at sh 30 per 50 g is blended with coffe grade 2 at sh 20 per 25g and the mixture is sold at sh 96 per 100g at a profit of 22%. Calculate the ratio in which the two brands of coffee were mixed. (4 marks)
 In the figure below, O is the centre of the circle which passes through thepoints C, T and D. CT is parallel to OD and line ATB is tangent to the circle at T. If the angle BTC is 44^{o}, find the size of angle TOD. (3 marks)
 Two similar solids have masses of 1000g and 1728g. if it costs Ksh 1080 to paint the outside of the larger solid, how much will it cost to paint the outside of the smaller solid (3 marks)
 The figure below shows part of a circle. AB = 12 cm and CD= 6 cm. AC = CB. Calculate the shaded area. (4 marks)
 Three interior angles of a polygon are 155^{o}, 153^{o} and 160^{o}. Each of the other interior angles is 148o. How many sides does the polygon have (3 marks)
 Write down the inequalities that describe the set of points in the unshaded region P. (3 marks)
 The position vectors respectively. Given that the length of AB is 7√2, find the value of b. (3 marks)
 A line segment AB is shown below. Construct a triangle CBA = 30^{o}. Hence use the constructed line AC to find a point T such that B divides At in the ratio 5: −2 (3 marks)
SECTION II (50 Marks)
Answer Five questions only from this section
 A cylindrical tank is to be constructed. A model of the tank is made such that it is similar to the actual tank. The curved surface area of the model is 2160 cm^{2 }and that of the proposed tank is 135m^{2}
 Given that the length of the model is 6cm, calculate the height of the tank in metres. (3 marks)
 Caculate the volume of the model given that the diameter of the actual tank is 14 m. (3 marks)
 Determine the volume of the actual tank in m^{3} (2 marks)
 The actual tank is used to store some liquids whose density is 0.82 g/cm^{3}. If the tank is half full, determine the mass of the liquid in kg. (2 marks)
 Given that the length of the model is 6cm, calculate the height of the tank in metres. (3 marks)

 A bus travelling at 99km/hr passes a check point at 10.00 a.m and a matatu travelling at 132 km/h in the same direction passes through the check point at 10:15 a.m. If the bus and the matatu continue at their uniform speeds, find the time the matatu will overtake the bus. (6 marks)
 Two passenger trains A and B which are 240 m apart and travelling in opposite directiosn at 164 km/h and 88km/h respectively approach one another on a straight railway line. Train A is 150 metres long and train B is 100 metres long. Determine time in seconds that elapses before the two trains completely pass each other. (4 marks)
 A bus travelling at 99km/hr passes a check point at 10.00 a.m and a matatu travelling at 132 km/h in the same direction passes through the check point at 10:15 a.m. If the bus and the matatu continue at their uniform speeds, find the time the matatu will overtake the bus. (6 marks)

 On the grid provided, draw the garph of the function y = x^{2} + x + 9 for −3 ≤ x ≤3
X −3 −2 −1 0 1 2 3 Y 15 11 9 9 11 15 21  Calculate the midordinate for 5 strips between x = −2 and x = 3 and hence use the midordinate rule to aapproximate the area under the curve between x = −2, x = 3 and the x  axis
X −1.5 −0.5 0.5 1.5 2.5 Y 9.75 8.75 9.75 12.75 17.5  Assuming that the area determined by inetgration to be the actual area, caluate the percentage error in using the midordinate rule. (2 marks)
 On the grid provided, draw the garph of the function y = x^{2} + x + 9 for −3 ≤ x ≤3
 On the graph paper provided plots the points P (2,2) Q (2,5) and R(4,4).
 Join them to form a triangle PQR (1 mark)
 Reflect the triangle PQR in the line X = 0 and label the image P'Q'R'. (2 marks)
 Triangle PQR is given a translation by vector T(^{2}_{2}) to P"Q"R". Plot the triangle P"Q"R". (3 marks)
 Rotate triangle P"Q"R" about the origin through −90^{o}. State the coordinates of P'''Q'''R'''. (3 marks)
 Identify two pair of triangles that are direct congruence (1 mark)
 Join them to form a triangle PQR (1 mark)
 Three warshisps P,Q,R are at sea such that ship Q is 400 km on a bearing of 030^{o} from ship P. Ship R is 750 km from ship Q and on a bearing of S60^{o}E from ship Q. Ship Q is 100km and to the north of an emeny warship S.
 Taking a scale of 1 cm to represent 100 km, locate the position of ships P,Q,R and S. (4 marks)
 Find the compass bearing of:
 Ship P from ship S (1 mark)
 N15^{o}W
 Ship S from ship R (1 mark)
 S45^{o}W
 Ship P from ship S (1 mark)
 Use the scale drawing to determine
 The distance of S from P (1 mark)
 6.8 cm × 100 = 680 km ± 10 km
 The distance of R from S (1 mark)
 8.8 cm × 100 = 880 km ± 10 km
 The distance of S from P (1 mark)
 Find the bearing of
 Q from R (1 mark)
 N60^{o}W or 300^{o}
 P from R (1 mark)
 270^{o}
 Q from R (1 mark)
 A certain number of people agreed to contribute equally to buy books worths shs. 1200 for a school library. Five people pulled out and so the others agreed to contribute an extra sh. 40 each. Their contribution enabled them to raise the sh. 1200 expected
 If the original number of people was x, write an expression of how much each was originally going to contribute (1 mark)
 ^{1200}/_{x}
 Write down the expression of how much each contributed after the five people pulled out. (1 mark)
 ^{1200}/_{x−5}
 Calculate how many people made the contributions (5 marks)
 If the prices of books before buying went up in the ratio 5:4, how much extra did each contributor give. (3 marks)
 If the original number of people was x, write an expression of how much each was originally going to contribute (1 mark)
 PQRS is a trapezium where PQ is parallel to SR. PR and SQ intersect at X, so that SX = KSQ and PX = hPR wher k and h are constants. Vectors PQ = 3q and PS = s,SR = q
 Show this information on a diagram (1 mark)
 Express vector SQ in terms of s and q (1 mark)
 SQ = SP + PQ
= −S + 3Q
= 3Q − S
 SQ = SP + PQ
 Express SX in terms of k, q and s (1 mark)
 SX = KSQ
=K(3Q − S )
 SX = KSQ
 Express SX in terms of h, q and s (1 mark)
 S = spp + px
= −s + hpr
= −s + h(s+q)
 S = spp + px
 Obtain h and k (4 marks)
 In what ratio does X divide SQ? (2 marks)
 Show this information on a diagram (1 mark)
 A solid cylinder has a radius of 21cm and a height of 18 cm. A conical hole of radius r is drilled in the cylinder on one of the end faces. The conical hole is 12 cm deep. If the material removed from the hole is 2^{2}/_{3}% of the volume of the cylinder, find: (Use π = ^{22}/_{7})
 The surface area of the hole(5 marks)
 The radius of a spherical ball made out of the material (3 marks)
volume = ^{4}/_{3}πr^{3}  The surface area of the spherical ball (2 marks)
 The surface area of the hole(5 marks)
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