INSTRUCTIONS TO CANDIDATES.
- This paper consists of two sections; section I and section II
- Answer ALL questions in sections I and only FIVE sections in section II
- Show all the steps in your calculations; giving your answers at each stage in the spaces provided below each question.
- Marks may be given for correct working even if the answer is wrong
- Non-programmable silent electronic calculators and KNEC mathematical tables may be used.
SECTION I (50 marks)
Answer all the questions in this section in the spaces provided.
- Evaluate −4{(−4+−15÷5)+−3−4÷2} (3 marks)
84÷−7+3−−5 - Simplify completely the expression: (3 marks)
- Given that cos θ = 3/5, find sin θ − tan (90°− θ) without using tables or calculator. (2 marks)
- Under an enlargement, the images of points A(3,1) and B(1,2) are A1(3,7) and B1 (7,5). Without construction, find the centre and the scale factor of enlargement. (4 marks)
- List all the integral values of x that satisfy the inequalities; (3 marks)
x − 3/2 ≤ 2x+1 < 5 - A bus travelling at an average speed of x km/h left station at 8.15 am. A car, travelling at an average speed of 80km/h left the same station at 9.00 am and caught up with the bus at 10.45 am. Find the value of x. (3 marks)
- The interior angle of a regular polygon with 3x sides exceeds the interior angle of another regular polygon having x sides by 40°. Determine the value of x. (3 marks)
- Use squares, cubes and reciprocals tables to evaluate, to 4 significant figures, the expression: (3 marks)
1 + 3
3√27.56 (0.0712)2 - From a point 20m away on a level ground the angle of elevation to the bottom of the window is 27° and the angle of elevation of the top of the window is 32°. Calculate the height of the window. (3 marks)
- Solve for x in the equation: 53y+3 + 53y−1=125. 2 (4 marks)
- Mr. Kanja, Miss Kanene and Mrs. Nyaga have to mark a form three mathematics contest for 160 students. They take 5 minutes, 4 minutes and 12 minutes respectively to mark a script. If they all start to mark at 9.00 am non-stop, determine the earliest time they will complete the marking. (4 marks)
- Evaluate (2 marks)
- Two similar cylinders have diameter of 7cm and 21cm. If the larger cylinder has a volume of 6237cm3, find the heights of the two cylinders. (take π = 22/7) (3 marks)
- The cost of providing a commodity consists of transport, labour and raw materials in the ratio 8:4:12 respectively. If the transport cost increases by 12%, labour cost by 18% and raw materials by 40%, find the percentage increase of producing the new commodity. (3 marks)
- Given that 4p − 3q = and p+2q = , find value of p and q (4 marks)
- In the figure below ABCDE is a cross-section of a solid. The solid has a uniform cross-section. Given that AP is an edge of the solid, complete the sketch showing the hidden edges with a broken lines. (3 marks)
SECTION II (50 Marks)
Answer any five questions from this section in the spaces provided.
- The figure below represents a sector of a circle radius r units. The area of the sector is 61.6 cm2 and the length of the arc AB is one tenth of the circumference of the circle from which the sector was obtained. ( Take π= 22/7)
- Calculate;
- the angle subtended by the sector at the centre. (2 marks)
- The radius r of the circle. (3 marks)
- If the sector above is folded to form a cone;
- Calculate the base radius of the cone. (2 marks)
- The volume of the cone. (3 marks)
- Calculate;
- Two factories A and B produce both chocolate bars and eclairs. In factory A, it costs Kshs x and Kshs y to produce 1 kg of chocolate bars and 1 kg of eclares respectively. The cost of producing 1 kg of chocolate bars and 1 kg of eclairs in factory B increases by the ratio 6:5 and reduce by the ratio 4:5 respectively.
- Given that it costs Kshs 460 000 to produce 1 tonne of chocolate bars and 800kg of eclares in factory A and Kshs 534 000 to produce the same quantities in factory B, form two simplified simultaneous equations representing this information. (3 marks)
- Use matrix method to find the cost of producing 1 kg of chocolate bars and 1 kg of eclaires in factory A. (5 marks)
- Find the cost of producing 100 kg of chocolate bars and 50 kg of eclaires in factory B. (2 marks)
- The vertices of triangle ABC are A(6,2), B(8,2) and C(6,0).
- On the grid provided below, draw triangle ABC. (1 mark)
- Triangle A’B’C’ is the image of triangle ABC under a reflection in the line y = x. On the same grid draw triangle A’B’C’ and state its coordinates (2 marks)
- Triangle A”B”C” is the image of triangle A’B’C’ under and enlargement scale factor 2 about the centre (−1,9). On the same grid, draw triangle A”B”C” and states its coordinates. (2 marks)
- By construction, find and write down the co-ordinates of the centre and angle of rotation which can be used to rotate triangle A”B”C” onto triangle A’’’B’’’C’’’ shown on the grid above. (3 marks)
- State any pair of triangles that are:\
- Oppositely congruent. (1 mark)
- Directly congruent. (1 mark)
- On the grid provided below, draw triangle ABC. (1 mark)
- The figure below shows a velocity-time graph of an object a which accelerates from rest to a velocity of V ms−1 then decelerated to rest in a total time of 54 seconds.
- If it covered a distance of 810 metres;
- Find the value of V. (2 marks)
- Calculate its deceleration, given that its initial acceleration was 12/3 ms−2 (2 marks)
- A bus left town X at 10.45 am and travelled toward town Y at an average speed of 60 km/h. A car left town X at 11.45 am on the same day and travelled along the same road toward Y at an average speed of 100km/h. The distance between town X and town Y is 500km.
- Determine the time of the day when the car overtook the bus. (3 marks)
- Both vehicles continued towards town Y at their original speeds. Find how long the car had to wait in town Y before the bus arrived. (3 marks)
- If it covered a distance of 810 metres;
- The masses to the nearest kilogram of some students were recorded in table below.
Mass(kg) 41-50 51-55 56-65 66-70 71-85 Frequency 8 12 16 10 6 Height of rectangle - Complete the table above to 1 decimal place. (2 marks)
- On the grid provided below, draw a histogram to represent the above information. (3 marks)
- Use the histogram to:
- State the class in which the median mark lies. (1 mark)
- Estimate the median mark. (2 marks)
- The percentage number of students with masses of at least 74kg. (2 marks)
-
- a straight line L1 whose equation is 9y − 6x=- −6 meets the x-axis at Z. Determine the coordinates of Z. (2 marks)
- A second line L2 is perpendicular to L1 at Z. Find the equation of L2 in the form ax+by=c, where ,b and c are integers. (3 marks)
- a third line L3 passes through the point (2,5) and is parallel to L1. Find:
- The equation of L3 in the form ax+by=c, where a, b and c are integers. (2 marks)
- The coordinate of point R at which L2 intersects L3. (3 marks)
- In the diagram below, the coordinates of points O, P and Q are (0,0), (2,8) and (12,8) respectively. A is a point on OQ such that 4OA=3OQ. Line OP produced to R is such as OR=5OP.
- Find vector RA. (3 marks)
- Given that point L is on PQ such that PL: LQ=12:5, find vector RL. (4 marks)
- Show that R, L and A are collinear. (2 marks)
- Find the ratio of RL:LA. (1 marks)
- Five points, P, Q, R, V and T lie on the same plane. Point Q is 53km on the bearing of 055° of P. Point R lies 162° of Q at a distance of 58km. Given that point T is west of P and 114km from R and V is directly south of P and S40°E from T.
- Using a scale of 1:1,000,000, show the above information in a scale drawing. (3 marks)
- From the scale drawing determine:
- The distance in km of point V from R. (2 marks)
- The bearing of V from Q. (2 marks)
- Calculate the area enclosed by the points PQRVT in squares kilometers. (3 marks)
MARKING SCHEME
1 |
−4{(−4−5)−3−2} ✓ |
M1 |
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2 |
2(3xy−4)(xy−2) ✓ |
M1 M1 A1 |
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3 |
4/5 − ¾ = 1/20 |
M1 A1 |
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4 |
let centre be (x,y)
3−x = 7−x ✓ |
M1 M1 A1 B1 |
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5 |
x − 3/2 ≤ 2x+1
x ≥ −5/6 ✓ |
M1 A1 B1 |
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6 |
time taken to catch up = 10.45 − 9.00 = 1.45=1¾ h ✓ x × ¾ = 7 ✓ x−80 4 x=140 km/h ✓ |
M1 M1 A1 |
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7 |
(3x−2)180 = (x−2)180 +40 ✓ |
M1 |
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8 |
= 1 + 3 = 1 + 3 ✓
3.021 (0.071)2 3.021 50.41 ×10-2 = 1 + 3 × 1 ✓ =0.3310 + 3 × 0.1984 × 10 =6.283✓ |
M1 M1 A1 |
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9 |
=20 tan 32 − 20 tan27✓ =12.50 − 10.19✓ =2.31m✓ |
M1 M1 A1 |
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10 |
53y × 53 + 53y = 125.2✓ 5 let 53y=x 125x + X/5 = 125.2 625x + x = 626✓ x=1✓ 53y = 1 = 50 3y = 0 ⇒ y = 0✓ |
M1 M1 A1 B1 |
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11 |
LCM of 5,4,12=60 minutes✓ no of scripts marked in 60 minutes = 60/5 + 60/4 + 60/12 = 32 scripts✓ time to mark 160 scripts=(160×60)/32=300 minutes.✓ time to complete marking=9.00+5.00=1400=2.00pm✓ |
M1 M1 A1 B1 |
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12 |
437 − 21 ✓ 99 99 = 416 = 4 20/99 or ✓ |
M1 A1 |
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13 |
height of larger cylinder = 6237 = 18 cm✓
22/7 × 10.52 |
M1 M1 A1 |
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14 |
%increase=(8/24 × 12/100 + 4/24 × 18/100 + 12/24 × 40/100) × 100✓
=27/100 ×100✓ |
M1 M1 A1 |
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15 |
M1 |
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16 |
B1 - continuous lines |
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17 |
(a) (i) 1/10 ×360✓
=36° ✓ |
M1 |
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18 |
(a) 1000x + 800y = 460,000✓
1000 × 6/5 x + 800 × 4/5y = 534,000✓ |
B1 B1 B1 M1 M1 M1 M1 A1 M1 A1 |
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19 |
(a) (b) A’(2,6), B’(2,8), C’(0,6) ✓ (c) A”(5,3), B”(5,7), C”(1,3) ✓ |
B1 - Δ ABC |
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20 |
(a) (i) ½ × 54 × V=810✓
V = 810 = 30 m/s✓ |
M1 |
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21 |
(a)
(c) (i) 56-65 (ii) 55.5 + 6 2 × 0.8 = 55.5 + 37.5 = 59.25 (iii) 5/26 × 100 = 8.929% |
B1B1 |
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22 |
(a) −6x = −6✓
x=1 |
M1 |
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23 |
M1 |
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24 |
(a) (b) (i) (7 ± 0.1) × 10km ✓ = 70km ✓ (ii) (180° + 25°) ± 1° ✓ =205° ✓ (iii) QV = 96km; PV = 56km; PT = 48km ✓ area of region = area of ∆QRV + area of ∆PQV + area of ∆TPV =½ × 58 × 96 sin 43 + ½ × 53 × 56 sin 125 + ½ × 48 × 56 ✓ |
B1 - Q |
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