KCSE 2015 Mathematics Alt B Paper 1 with Marking Scheme

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SECTION I (50 marks)
Answer all the questions in this section in the spaces provided.

The population of a certain county was three hundred sixty eight thousand eight hundred and forty nine. Two reporters, A and B gave this figure correct to 3 significant figures and to the nearest hundreds respectively. Find the difference between the numbers given by the reporters. (2 marks)
A tailor had a ribbon which he intended to cut into equal lengths of either 28 cm, 16cm or 40 cm. Determine:
1. the shortest length of ribbon that he would use to obtain an exact number of pieces; (2 marks)
2. the number of 16 cm pieces that can be obtained from the ribbon. (1 mark)
A metal bar with a cross-sectional area of 44 cm² has a mass of 5.06 kg. The density of the bar is 2.3 g/cm³. Calculate the length, in cm, of the bar. (3 marks)
Use logarithms, correct to 4 significant figures to evaluate: $1$ (3 marks)
Given that A = x² + 2xh, find the positive value of x when A = 360 and h = 13. (3 marks)
A rectangular field ABCD is such that AB = 90m and BC = 70m. A well, w in the field is located 30 m from A and angle WAB = 45°. A tree, T also in the field is located 40 m from B and angle TBA = 40°. Using a scale drawing:
1. Draw the field and show the positions of W and T; (3 marks)
2. find the distance, in metres, between the well and the tree. (1 mark)
A chord subtends an angle of 60° at the circumference of a circle of radius 3.5 cm. Determine the area of the minor segment correct to 4 significant figures. (Take π = ²²/₇) (4 marks)
Given the inequalities 3 ≥ 5 and x > -4, find the integral values of x that satisfy the inequalities. (3 marks)
2x-1
Kassam spent ¹/₅ of his monthly salary on house rent, ³/₈ on loan repayment, Ksh 26000 on domestic expenses and saved the rest. If he saved Ksh 8000, calculate his monthly salary. (3 marks)
A line L₁ passes through (3, -4) and is perpendicular to the line y = 3x – 2. Determine:
1. the equation of L₁ in the form ax + by = c where a, b and c are constants. (3 marks)
2. the x intercept of line L₁ (1 mark)
The vertices of a triangle JKL are J(2,1), K(5,1) and L(3,3). The triangle JKL is mapped onto J'K'L' by a rotation of +90° about point (1,-1). On the grid provided below, draw triangle JKL and its image J'K'L'. (3 marks)
Given that cos 60° = ½ without using mathematical tables or a calculator, find:
1. sin 60° (2 marks)
2. tan 30° (1 mark)
A builder used square tiles of side 34.2 cm to cover a rectangular floor of length 307.8 cm and width 236.7 cm. Determine the number of whole square tiles that were fitted on the floor. (3 marks)
The figure below represents a triangular prism. AB = 5cm, BC = 30cm, BF = 12 cm and angle ABF is a right angle.
$2$
Calculate the surface area of the prism. (4 marks)
The volume of a hemisphere is 41.2 cm2. Calculate, correct to one decimal place, the radius of the hemisphere. (2 marks)
The diagram below is a speed-time graph for a car that travelled between two stations in 90 seconds.
$3$
Calculate the average speed of the car correct to one decimal place. (3 marks)

SECTION II (50 marks)
Answer any five questions from this section in the spaces provided.

A businessman transported bags of rice using 2 lorries and 5 pickups. Each lorry made 4 trips carrying 132 bags per trip. 3 pickups made 7 trips each and the remaining pickups made 6 trips each. The pickups carried the same number of bags each per trip. The ratio of the number of bags transported by the lorries to that transported by the pickups was 8:3.
1. Calculate:
  1. the total number of bags transported; (3 marks)
  2. the number of bags carried by a pickup per trip. (4 marks)
2. The cost per trip for each lorry was Ksh 5000 and that of a pickup was Ksh 1 500 Calculate the total cost of transporting the rice. (3 marks)
A rectangular piece of land has a perimeter of 84 m. Its length is 12 m longer than its width.
1. If the width is xm:
  1. write an expression for the perimeter of the land; (1 mark)
  2. find the ratio of the length to the width. (4 marks)
2. The piece of land was fenced with 4 strands of barbed wire allowing 30 cm for binding at each end of a strand. A space of 3 m on one side of the land was left for the gate.
  1. Determine the length of barbed wire that was used. (3 marks)
  2. If the fencing poles were 3 m apart, determine the number of poles used. (2 marks)
The base of an open rectangular tank is 3 m by 2.5 m and its height is 4m.
1. Calculate:
  1. the capacity of the tank in litres. (3 marks)
  2. the surface area in m² of the tank. (2 marks)
2. An open cylindrical tank has an equal capacity and same height as the rectangular tank in (a) above. Calculate, correct to one decimal place:
  1. the radius of the cylindrical tank; (3 marks)
  2. the surface area in m² of the tank. (2 marks)
1. Using a ruler and a pair of compasses only, construct triangles ABC and ABD on opposite sides of line AB below, such that DAB = DBA = ABC =BAC = 60°. (2 marks)
  $9$
2. 1. Name the quadrilateral ADBC. (1 mark)
  2. State two properties of the quadrilateral. (2 marks)
3. Construct a circle touching all the sides of the quadrilateral. (2 marks)
4. Determine, correct to one decimal place, the area of the region enclosed by the quadrilateral but outside the circle. (3 marks)
In the figure below WXYZ is a cyclic quadrilateral and TYN is a tangent to the circle at Y. Angle TYZ = 35°, angle YZX = 70° and angle WPZ = 130°.

Giving reasons, find the size of:
1. ZXYN; (2 marks)
2. ZZYX; (2 marks)
3. ZZXY; (2 marks)
4. ZZWY; (2 marks)
5. ZWXZ. (2 marks)
A salesman sold 300 bags of animal feeds to a retailer at Ksh 1700 each. He was given a commission of 3% on 120 of the bags and 2% on the remaining bags. The salesman allowed a discount of 1.2% on of the bags sold. This discount was deducted from his commission.
1. Calculate:
  1. the discount allowed; (2 marks)
  2. the net commission the salesman got. (3 marks)
2. The retailer sold all the bags at Ksh 1850 each and paid Ksh 3000 for transport. Calculate the profit made by the retailer. (3 marks)
3. In addition, a value added tax (VAT) of 16% was charged on the profit made by the retailer. Calculate the amount of tax collected. (2 marks)
A piece of wire is bent to form the sides of a rectangle whose length is 6 cm more than the width. The area of the rectangle formed is 567 cm.
1. Determine the length of the wire. (4 marks)
2. The same piece of wire could be bent to form a semi-circle. Determine the area that would be enclosed by the semi-circle, correct to one decimal place. (4 marks)
3. Express the area of the semi-circle as a percentage of the area of the rectangle, correct to 3 significant figures. (2 marks)
A cattle ranch is in the shape of a parallelogram. The length of one side is 600m and the other is 500 m. The smaller angle between the sides is 30°.
1. Calculate the area, in hectares, of the ranch. (3 marks)
2. A water point in the shape of a regular hexagon of side 10 m is built in the ranch. Calculate:
  1. the grazing area, in hectares, correct to 3 decimal places. (4 marks)
  2. the number of cattle that can be kept in the ranch if one animal requires a grazing area of 0.625 ha. (3 marks)

MARKING SCHEME

368800 and 369000
Difference: 369000 - 368000
= 200
For all factors
1. 28 = 2² x 7
  16 = 2⁴
  40 = 2³ x 5
  Length of ribbon = 2⁴ x 5 x 7
  = 560
2. 560 = 35
  16
Volume = 5.06 x 1000
2.3
= 2200 cm³
Length = 2200 = 50 cm
44
Number Log

0.04068
35.72 2.6094
1.5529 +

0.2799 0.1623
1.4470

0.7153 x ¹/₃

1.732 0.2384

M1 Operations
M1 All logs
A1 Accept 1.731
360=x² +26x
x² +26x - 360=0
(x +36) (x -10)=0
x = 10
$5$
B1 Suitable scale used
B1 Rectangular fielu ABCD correctly drawn.
B1 Location of Wand
Distance WT = (3.8 +0.1) x 10
= 38 ± 1m
Angle at the centre = 120º
120 x 22 x 3.5 x 3.5 - ½ x (3.5)² sin120°
360 7
= 12.83 - 5.304
= 7.526
Area of sector
Area of triangle
3 ≥ 10x - 5
8 ≥ 10x
x ≤⁴/₅ , x > -4
Integral values; -3, -2,-1,0.
Let x be monthly salary
x -{¹/₅x + 26000 +³/₈ x}= 8000
¹⁷/₄₀x = 34000
x =Ksh. 80000
1. Gradient of L₁ =^-1/₃
  Equation of L₁y + 4 = -1
  x - 3 3
  -3y - 12=1 - 3
  3y +12=-2 + 3
  x + 3y=-9
2. x intercept
  x =-9
$6$
Drawing object JKL
Locating image points J', K' and L'
Image J' K'L' drawn
B1 Image J'K'L' drawn
1. 3rd side of Δ=√2²- 1² = √3
  sin 60°= √3
  2
2. tan 30º = 1
  √3
  $7$
Number of whole tile along the length
= 307.8 = 9
34.2
Number of whole tiles along the width
236.7 = 6
34. 2
Total number of tiles fitted
= 9 X 6 = 54
Hypotenuse of Δ = √5² + 12²= 13
Area of Δs= ½ x 12 x 5x 2
= 60
Area of rectangles:
= 5 x 30 + 12 x 30 + 13 x 30
= 900
Total S.A = 900 + 60 = 960 cm²
Volume of hemisphere
½ x ⁴/₃ πr³= 41.2
r= ³√41.2 x 3
2π
= 2.7
Distance covered
= ½ (16 x 25) + 34 x 25 + ½ x 25 x 40
= 1550
Speed = 1550
90
= 17.2 m/s
1. 1. No of bags transported by lorry
    2 x 4 x 132 = 1056
    Total number of bags transported
    x = 1056 x 11
    8
    = 1452
  2. Bags transported by pick-up
    1452 x 3
    11
    No of bags carried by each pickup per trip.
    = 396
    3 x 7 + 2 x 6
    396 - 12
    33
2. Lorry transport cost:
  2 x 4 x 5000 = 40000
  Pickup transport cost:
  (3 x 7+2 x 6) X 1500
  = 49500
  Total cost:
  40000 + 49500 = Ksh. 89500
1. 1. Expression:
    2(x + x + 12) = 4x + 24
  2. 4x + 24 = 84
    4x = 60
    x = 15
    Length of plot 15 + 12 = 27m
    Ratio of length to width
    27: 15 = 9:5
2. 1. Length of 1 strand with allowance
    84 - 3+0.3
    = 81.3
    Total length:
    81.3 x 4
    = 325.2 m
  2. Number of poles:
    84
    3
    = 28
1. 1. 3 x 2.5 x 4 x 100 x 100 x 100
    1000
    = 30000 litres
    Volume Conversion to litre
  2. 3 x 4 x 2 + 2.5 x 4 x 2 +3 x 2.5
    = 24 +20 + 7.5 = 51.5 m²
2. 1. πr² x 4 = 3 x 2.5 x 4
    r= √7.5
    π
    = 1.5
  2. π(1.5 x 1.5)+ 2 x 1.5 x π x 4
    = 44.8 m²
1. $8$
  Construction of 60°
  Construction of quad ADBC
2. 1. ADBC is a rhombus
  2. - All sides equal
    - Opposite sides equal and parallel
    - Opposite Zs equal
    - Diagonals bisect each other at right angles
3. Construction of circle:
  BI Any two correct 1 mark each.
  B1 Identifying centre
  B1 Circle drawn
4. Area inside quad, and outside circle.
  8² sin 60° - π x 3.5²
  55.4 - 2 x 3.5²
  = 55.4 - 38.5
  = 16.9 cm²
  Area of circle (radius 3.5+0.1)
  Area of rhombus
1. XYN = 70°
  angle in alternate segment
  NB Reason should be relevant to situation.
2. ZYX =180-(70+35)= 75°
  Sum of angles on a straight line
3. ZXY = 35°
  angle in alternate segment.
4. ZWY = 35°
  angle in alternate segment or angle subtended by chord ZY
5. WXZ=130 - 70=60°
  exterior angle in triangle equal to sum of apposite interior angles
1. 1. Discount = 12 x 3 x 300 x1700
    100 5
    = Ksh 3672
  2. 3 x 120 x 1700+ 2 x 180 x 1700
    100 100
    = 6120 + 6120 = Ksh 12 240
    Net commission:
    = 12240 - 3672 = Ksh 8568
2. Profit:
  = 300 (1850 - 1700) + 3672 - 3000
  = Ksh 45 672
3. Tax:
  16 x 45672
  100
  =Ksh 7307.52
1. (x+6)=567
  x² + 6x - 567 = 0
  (x + 27) (x - 21) = 0
  x = 21 or -27
  Length of wire
  =[(21+6)+21] x 2
  = 96 cm
2. Let radius be r
  2r + πr =96
  r = 96
  2+π
  =18.7
  Area of semicircle
  ½π x 18.7
  = 549.3
3. Area of semicircle as percentage of area of rectangle:
  549.3 x 100
  567
  = 96.9%
1. ½ x 600 X 500 X sin30 x 2
  10000
  = 15 ha
  M1 Conversion to hectares
  area of 1 triangle
2. 1. 6 x ½ + x 10²sin60
    10000
    =0.026 ha
    grazing area=15-0.026
    =14.974 ha
    M1 mult. by 6
  2. Number of animals
    = 14.974 ÷ 0.0625
    =239.584
    = 239 animals