SUKELLEMO JOINT MOCK
Kenya Certificate of Secondary Education
MATHEMATICS PAPER 2
2 ½ hours
Instructions to Candidates
 This paper consists of TWO sections; Section I and II
 Answer ALL the questions in Section 1 and only five questions from Section II.
 Show all the steps in your calculation, giving your answer at each stage
 Marks may be given for correct working even if the answer is wrong.
 Nonprogrammable silent electronic calculators and KNEC mathematical tables may be used except where stated otherwise.
SECTION I (50 MARKS).
 Find the selling price of 6 kg of a mixture of maize flour and millet flour if 4 kg of maize flour costing sh. 60 per kg is mixed with 6 kg of millet flour costing sh. 45 per kg and a profit of 20% is realized. (3 marks)
 If x = 9.6, y= 3.60 and z = 5, Find the percentage error in the calculation of ^{(x+y)}/_{z}, giving your answer to three significant figures. (3 marks)
 Solve the equation log_{2} (x^{2}4)  log_{2} (x+2) = 4 (3 marks)
 Form a quadratic equation whose roots are 2.5 + √3 and 2.5  √3 giving your answer in the form ax^{2}+bx+c=0 where a, b and c are integers (3 marks)
 Make b the subject of the formula. (3 marks)
x = a
√(ab)(a+b)  Express in surd form and simplify by rationalizing the denominator. (3 marks)
3 sin 45°  2 cos 30°
tan 30°  Triangle ABC is such that AB = 8 cm, BC = 11 cm and AC = 15 cm. calculate correct to 2 decimal places the;
 Angle ABC (2 marks)
 Radius of the circum circle (2 marks)
 In the figure below, the chords CD and AB intersect externally at T. DT = 4 cm, BT = 3 cm and CD = 5 cm. calculate the length AB. (3 marks)
 Solve the following equation for 0^{0 }≤ x ≤ 360^{0} 2cosx = sin^{2} x + 2 (4 marks)
 Given that A (4, 2, 6) and B (3, 1, 2) and that a point N divides AB in the ratio 2:7. Find the vector ON in terms of i, j and k. (3 marks)
 The equation of a circle is given by ^{2}/_{3} x^{2 }+ ^{2}/_{3} y^{2 } 4x+ 2 ^{2}/_{3} y  2 = 0. Determine the centre and the radius of the circle. (3 marks)
 Expand (1+^{2x}/_{3})^{8 }in ascending powers of x up to the fourth term. Hence use your expansion to evaluate (0.98)^{8} to three significant figures. (3 marks)
 Find the base radius of a cylindrical hole with maximum volume which can be drilled into a cone of height 16 cm and radius 12 cm as shown below. (3 marks)
 ABCD is a regular tetrahedron. AB=BC=CA=AD=BD=CD= 8cm.
 Calculate the angle between line AD and plane ABC. (2 marks)
 Calculate the angle between planes ABD and ABC (2 marks)
 A contractor intends to transport 1,000 bags of cement using a lorry and a pick up. The lorry can carry a maximum of 80 bags while a pick up can carry a maximum of 20 bags. The pickup has to make more than twice the number of trips the lorry makes and the total number of trips has to be less than 30. The cost per trip is sh 2000 for the lorry and sh 900 for the pickup and the contractor wishes to minimize cost. Let x and y be the trips for the lorry and pickup respectively.
 State the objective function. (1 mark)
 Write down all the inequalities which govern the condition above. (2 marks)
 The first, second and fifth terms of an arithmetic sequence are the first three consecutive terms of a geometric sequence. Find the common ratio (3 marks)
SECTION II (50 marks)
 Using a ruler and a pair of compasses only for all constructions in this question.
 Construct triangle ABC in which AB = 6 cm, BC = 7 cm and angle ABC = 75^{o}(3 marks)
 Find locus X such that AX = 3 cm (1 mark)
 On the same side of BC as A, construct the locus of P such that angle BPC = 120^{o}(3marks)
 Show by shading the locus of Q inside triangle ABC such that angle BPC≥angle BQC. (1 mark)
 On the side of AB opposite C, construct the locus of T such that the area of triangle ATB = 6 cm^{2} (2 marks)
 The data below shows the heights of students in a class.
Height 130 139 140 149 150159 160169 170179 180189 190199 Frequency 8 10 11 18 14 12 7  Using assumed mean of 164.5, calculate the mean and the standard deviation(5 marks)
 Draw a cumulative frequency curve on a grid and use it to estimate the median (5 marks)
 P varies directly as the cube of Q and inversely as the square root of R
 Given that P = 35 when Q = 8 and R = 144, find P when Q = 20 and R = 22 (5 marks)
 If Q decreases by 24% and R increases by 40% find the percentage change in P.(5 marks)
 The triangle ABC has vertices A (1, 2), B (2, 1) and C (2, 3). A^{I }B^{I }C^{I }is the image of ABC under transformation given by the matrix
 What are the coordinates of A^{I }B^{I }C^{I }? Plot ABC and A^{I }B^{I }C^{I } on the same axis (3 marks)
 State the ratio of the areas of the two triangles and use the area of ABC to calculate the area of A^{II }B^{II }C^{II } (3 marks)
 If A^{II }B^{II }C^{II } is the image of ABC under transformation given by the matrix . Find the coordinates of A^{II }B^{II }C^{II}. Plot A^{II }B^{II }C^{II} and describe the transformation fully. (4 marks)

 An industrialist has 460 litres of a chemical which is 75% pure. She mixes it with a chemical of the same type but 90% pure so as to obtain a mixture which is 78% pure. Find the amount of the 90% pure chemical used. (3 marks)
 Three machines A, B and C are set to work together. A working alone takes 6 hours to complete the work; B takes 8 hours while C takes 12 hours. All the three machines started working at the same time. 40 minutes later machine A broke down. B and C continued for another 1 hour before B ran out of fuel and therefore stopped working for 20 minutes while C continued. If B resumed working after 20 minutes, calculate the:
 Fraction of the work left after machine A broke down. (2 marks)
 Fraction of the work done by C working alone for 20 minutes (2 marks)
 Total time taken for the work to be completed. (3 marks)
 The table below shows taxation rates in Kenya
Monthly taxable income (kshs p.m) Tax rate % 19680 10 968118800 15 1880127920 20 2792137040 25 37041 and above 30  Taxable income per month. (6 marks)
 Basic salary per month (2 marks)
 Net monthly pay. (2 marks)
 The probability of James, Tyson and David passing an examination are ^{4}/_{5},^{3}/_{4} and ^{2}/_{3} respectively. Find the probability that in one attempt:
 only one passes the examination. (2 marks)
 All the three passes the examination. (2 marks)
 Two pass the examination. (2 marks)
 None passes the examination. (2 marks)
 At least one passes the examination. (2 marks)
 When aero plane left town P (65º N,15º E) to another town Q (65º N,165º W) at a speed of 200 knots using the shortest route. Take π = ^{22}/_{7} and radius of the earth R= 6370 km.

 Calculate the distance travelled in nautical miles. (2 marks)
 Calculate the time taken to travel from P to Q in hours. (2 marks)
 Another plane left P at 1.30 pm local time and travelled to T (65ºN,60ºE) along a parallel of latitude. Calculate the:
 Distance between P and T to the nearest km (3 marks)
 Local time of arrival at town T if the plane flew at the speed of 470 km/h(3 marks)

MARKING SCHEME
SECTION I (50 MARKS).
 Find the selling price of 6 kg of a mixture of maize flour and millet flour if 4 kg of maize flour costing sh. 60 per kg is mixed with 6 kg of millet flour costing sh. 45 per kg and a profit of 20% is realized. (3 marks)
BP per kg = (4 x 60) + (6 x 45)
4 + 6
= Sh. 51 _{}BP of 6kg = 51 x 6 = Sh. 306
SP per kg = ^{120}/_{100} x 306
= Sh. 367.20  If x = 9.6, y= 3.60 and z = 5, Find the percentage error in the calculation of ^{(x+y)}/_{z}, giving your answer to three significant figures. (3 marks)
maximum value = 9.65 + 3.605 = 2.9456
4.5
minimum value = 9.55 + 3.595 = 2.39
5.5
actual value = 9.6 + 3.60 = 2.64
5
absolute error = 2.9456  2.39 = 0.2778
2
percentage error = ^{0.2778}/_{2.64} x 100% = 10.5%
 Solve the equation log_{2} (x^{2}4)  log_{2} (x+2) = 4 (3 marks)
log_{2} x^{2} 4 = Log_{2}2^{4}
(x+2)
(x+2)(x2) = 2^{4}
(x+2)
x2 = ^{1}/_{16}
x = 2 + ^{1}/_{16}
x= 2^{1}/_{16}  Form a quadratic equation whose roots are 2.5 + √3 and 2.5  √3 giving your answer in the form ax^{2}+bx+c=0 where a, b and c are integers (3 marks)
x = 2.5 + √3 , x = 2.5  √3
(x  2.5  √3)(x  2.5 + √3) = 0
x^{2}  2.5x +x√3  2.5x + 6.25  2.5√3  x√3 + 2.5√3 3 =0
= 4x^{2}  20x + 13 = 0  Make b the subject of the formula. (3 marks)
x = a
√(ab)(a+b)
x = a
√(a^{2} b^{2})
x^{2} = a^{2}
a^{2} b^{2}
a^{2} = a^{2}x^{2}  b^{2}x^{2}
b^{2}x^{2} = a^{2}x^{2}  a^{2}
b^{2} = a^{2}x^{2}  a^{2 }/x^{2}
b^{2} = a^{2}  a^{2}
x^{2}
b = ±√( a^{2}  a^{2}/_{x2})  Express in surd form and simplify by rationalizing the denominator. (3 marks)
3 sin 45°  2 cos 30°
tan 30°
sin 45º = ^{1}/_{√2} × ^{√2}/_{√2} = ^{√2}/_{√2}
Cos 30º = ^{√3}/_{2}
Tan 30º = ^{1}/_{√3} × ^{√3}/_{√3} = ^{√3}/_{3}
(3 ^{√2}/_{2}  2 ^{√3}/_{2 })/^{√3}/_{3}
3 √2  2√3
2^{√3}/_{3}
(9√2  6√3) √3
2√3 (√3)
9√6  18
6
3√6  6
2
or 3√6  3
2  Triangle ABC is such that AB = 8 cm, BC = 11 cm and AC = 15 cm. calculate correct to 2 decimal places the;
 Angle ABC (2 marks)
b^{2} = a^{2} + c^{2}  2ac Cos B
15^{2} = 11^{2} + 8^{2}  2(11 x 8)Cos B
225 = 185  176 Cos B
176 Cos B =  40
Cos B =  0.2273
B = 103.14º  Radius of the circum circle (2 marks)
15 = 2r
sin 103.14º
R = 7.70cm
 Angle ABC (2 marks)
 In the figure below, the chords CD and AB intersect externally at T. DT = 4 cm, BT = 3 cm and CD = 5 cm. calculate the length AB. (3 marks)
AT.BT = CT.DT
(AB + 3).3 = 9 x 4
AB + 3 = ^{36}/_{3}AB = 12  3
9cm  Solve the following equation for 0^{0 }≤ x ≤ 360^{0} 2cosx = sin^{2} x + 2 (4 marks)
2 cos x = 1  cos^{2}x + 2
cos^{2}x + 2cosx  3 = 0
p = cos x
p^{2} + 2p 3 = 0
p^{2} + 3p p 3 =0
p(p+3)  1(p+3) =0
p = 1 or 3
cos x = 1
x=0º, 360º  Given that A (4, 2, 6) and B (3, 1, 2) and that a point N divides AB in the ratio 2:7. Find the vector ON in terms of i, j and k. (3 marks)
 The equation of a circle is given by ^{2}/_{3} x^{2 }+ ^{2}/_{3} y^{2 } 4x+ 2 ^{2}/_{3} y  2 = 0. Determine the centre and the radius of the circle. (3 marks)
^{3}/_{2} x ( ^{2}/_{3}x^{2} + ^{2}/_{3}y^{2}  4x + 2 ^{2}/_{3}y  2=0)
= x^{2} + y^{2}  6x + 4y 3 = 0
x^{2}  6x + 9 + y^{2} + 4y + 4 = 3 + 9 + 4
(x  3)^{2} ( y +2)^{2} = 4^{2}
centre = (3,2)
Radius = 4 units  Expand (1+^{2x}/_{3})^{8 }in ascending powers of x up to the fourth term. Hence use your expansion to evaluate (0.98)^{8} to three significant figures. (3 marks)
1 + 8(^{2x}/_{3}) + 28(^{2x}/_{3})2 + 56(^{2x}/_{3})^{3}
= 1 + ^{16x}/_{3} + 112x^{2}/9 + 448x^{3}/27
(1 + ^{2x}/_{3}) = [ 1 + (0.02)]
^{2x}/_{3} = 0.02
x = 0.03
0.988
1 + ^{16}/_{3}(0.03) + ^{112}/_{9}(0.03)^{2}
+ ^{448}/_{27}( 0.02)^{3}
1  0.16 + 0.0112  0.000448
= 0.851  Find the base radius of a cylindrical hole with maximum volume which can be drilled into a cone of height 16 cm and radius 12 cm as shown below. (3 marks)
^{r}/_{12} = ^{16h}/_{16}
16r = 12(16 h)
^{4}/_{3}r 16 h
h = 16  ^{4}/_{3} r
V = πr^{2}h
=πr^{2}( 16  ^{4}/_{3} r)
= (16πr^{2}  ^{4}/_{3}πr^{3})
^{dv}/_{dr} = 32πr  4πr^{2}
at max, ^{dv}/_{dr} = 0
4πr(8 r) = 0
r=0 or 8
r=8cm  ABCD is a regular tetrahedron. AB=BC=CA=AD=BD=CD= 8cm.
 Calculate the angle between line AD and plane ABC. (2 marks)
Sin 60º = ^{4}/_{AO}
AO = ^{4}/_{Sin60º}
=4.619cm
Cosθ = ^{4.619}/_{8}
θ = 54.73º  Calculate the angle between planes ABD and ABC (2 marks)
h = √(8^{2}  4^{2})
= 6.928cm
Cos θ = ^{2.309}/_{6.928}
θ = 70.53º
 Calculate the angle between line AD and plane ABC. (2 marks)
 A contractor intends to transport 1,000 bags of cement using a lorry and a pick up. The lorry can carry a maximum of 80 bags while a pick up can carry a maximum of 20 bags. The pickup has to make more than twice the number of trips the lorry makes and the total number of trips has to be less than 30. The cost per trip is sh 2000 for the lorry and sh 900 for the pickup and the contractor wishes to minimize cost. Let x and y be the trips for the lorry and pickup respectively.
 State the objective function. (1 mark)
2000x + 900y  Write down all the inequalities which govern the condition above. (2 marks)
y > 2x
x + y <30
80x + 20y ≥ 1000, 4x + y ≥ 50
 State the objective function. (1 mark)
 The first, second and fifth terms of an arithmetic sequence are the first three consecutive terms of a geometric sequence. Find the common ratio (3 marks)
a, a+d, a+ 4d
^{a + d}/_{a} = ^{ a + 4d}/_{a+d}
a(a + 4d) = (a+d)^{2}
a^{2} + 4ad = a^{2} +2ad+d^{2}
2ad = d^{2}
2a=d
common ratio (r)
^{a+2d}/_{a} =^{ a+2a}/_{a}
=^{3a}/_{a} =3
SECTION II (50 marks)
 Using a ruler and a pair of compasses only for all constructions in this question.
 Construct triangle ABC in which AB = 6 cm, BC = 7 cm and angle ABC = 75^{o}(3 marks)
 Find locus X such that AX = 3 cm (1 mark)
 On the same side of BC as A, construct the locus of P such that angle BPC = 120^{o}(3marks)
 Show by shading the locus of Q inside triangle ABC such that angle BPC≥angle BQC. (1 mark)
 On the side of AB opposite C, construct the locus of T such that the area of triangle ATB = 6 cm^{2} (2 marks)
 The data below shows the heights of students in a class.
Height 130 139 140 149 150159 160169 170179 180189 190199 Frequency 8 10 11 18 14 12 7  Using assumed mean of 164.5, calculate the mean and the standard deviation(5 marks)
Class x d=xA f fd fd^{2} cf 130139 134.5 30 8 240 7200 8 140149 144.5 20 10 200 4000 18 150159 154.5 10 11 110 1100 29 160169 164.5 0 18 0 0 47 170179 174.5 10 14 140 1400 61 180189 184.5 20 12 240 4800 73 190199 194.5 30 7 210 6300 80
∑fd=40
∑fd^{2 }= 24,800
mean = A + ∑fd
∑f
164.5 + 40/80 =165
standard dev = √(∑fd^{2}/∑f)  (^{∑fd}/_{∑f})^{2 }=√(^{24800}/_{80})  (^{40}/_{80})^{2}
= 17.160  Draw a cumulative frequency curve on a grid and use it to estimate the median (5 marks)
median = ½ x 80 =40
From cumulative frequency
curve, median = 165.5
 Using assumed mean of 164.5, calculate the mean and the standard deviation(5 marks)
 P varies directly as the cube of Q and inversely as the square root of R
 Given that P = 35 when Q = 8 and R = 144, find P when Q = 20 and R = 22 (5 marks)
P=KQ^{3}
√R
35 = K x 8 x 8 x 8
√144
35=^{512k}/_{12}
K= ^{105}/_{128}
P = 105Q^{3}
128√R
P= 105 x 20^{3}
128 x √225
= 105 x 8000m
128 x 15
=437.5  If Q decreases by 24% and R increases by 40% find the percentage change in P.(5 marks)
Original value of P =KQ^{3}
√R
new value = K x (0.76Q)^{3}
√1.4R
= 0.438976KQ^{3 }
1.1832√R
= 0.3710 KQ^{3}
√R
change = (0.3710 1) KQ^{3}
√R
0.6290KQ^{3}
√R
%change = 0.6290 x 100
62.90%
P has decreased by 62.90%
 Given that P = 35 when Q = 8 and R = 144, find P when Q = 20 and R = 22 (5 marks)
 The triangle ABC has vertices A (1, 2), B (2, 1) and C (2, 3). A^{I }B^{I }C^{I }is the image of ABC under transformation given by the matrix
 What are the coordinates of A^{I }B^{I }C^{I }? Plot ABC and A^{I }B^{I }C^{I } on the same axis #(3 marks)
 State the ratio of the areas of the two triangles and use the area of ABC to calculate the area of A^{II }B^{II }C^{II } (3 marks)
Ratio of the 2 triangles = determinant of matrix
(2 x 3) = (0 x 0) =6
thus ratio = 6:1
Area of ABC = ½ x 2 x 1= 1 square unit
Area of A^{I}B^{I}C^{I }= 1 x 6= 6 square units  If A^{II }B^{II }C^{II } is the image of ABC under transformation given by the matrix . Find the coordinates of A^{II }B^{II }C^{II}. Plot A^{II }B^{II }C^{II} and describe the transformation fully. (4 marks)
A^{II} (3,2) ^{ }B^{II} (0,1)^{ }C^{II}(4,3)
The transformation is a shear parallel to the x axis with the x axis invariant shear factor. The image of A(1,2) is A^{1}(3,2)
 What are the coordinates of A^{I }B^{I }C^{I }? Plot ABC and A^{I }B^{I }C^{I } on the same axis #(3 marks)

 An industrialist has 460 litres of a chemical which is 75% pure. She mixes it with a chemical of the same type but 90% pure so as to obtain a mixture which is 78% pure. Find the amount of the 90% pure chemical used. (3 marks)
^{75}/_{100} x 460 = 345
345 + 0.9x = 358.8 + 0.78x
Let 90% pure chemical use x 0.12x=13.8
345 + ^{90}/_{100}x =^{ 75}/_{100}(460 + x)
x = 115 litres  Three machines A, B and C are set to work together. A working alone takes 6 hours to complete the work; B takes 8 hours while C takes 12 hours. All the three machines started working at the same time. 40 minutes later machine A broke down. B and C continued for another 1 hour before B ran out of fuel and therefore stopped working for 20 minutes while C continued. If B resumed working after 20 minutes, calculate the:
 Fraction of the work left after machine A broke down. (2 marks)
^{1}/_{6} + ^{1}/_{8} + ^{1}/_{12} = ^{3}/_{8}
^{40}/_{60} x ^{3}/_{8} = ^{1}/_{4}
= 1  ^{1}/_{4} = ^{3}/_{4}  Fraction of the work done by C working alone for 20 minutes (2 marks)
1hr = ^{1}/_{12}
20min
^{20/}_{60} x ^{1}/_{12} = ^{1}/_{36}  Total time taken for the work to be completed. (3 marks)
B & C = ^{1}/_{8} + ^{1}/_{12}= ^{5}/_{24}
^{3}/_{4}  ^{5}/_{24} = ^{13}/_{24}, ^{13}/_{24}  ^{1}/_{36} =^{ 37}/_{72}
^{5}/_{24} = 1hr
^{37}/_{72} = ^{37}/_{72} x ^{24}/_{5} = 2^{7}/_{15}
time= 40minutes + 1hr 20 mins + 2hrs 28mins
=4hrs 28min
 Fraction of the work left after machine A broke down. (2 marks)
 An industrialist has 460 litres of a chemical which is 75% pure. She mixes it with a chemical of the same type but 90% pure so as to obtain a mixture which is 78% pure. Find the amount of the 90% pure chemical used. (3 marks)
 The table below shows taxation rates in Kenya
Monthly taxable income (kshs p.m) Tax rate % 19680 10 968118800 15 1880127920 20 2792137040 25 37041 and above 30  Taxable income per month. (6 marks)
Tax= net tax + relief = 6900 + 1520 + (3/20x1200) = Sh.8600
8600  6440=2160
9680 x 0.1 = 968
9120 x 0.15=1368
9120 x 0.20=1824
9120 x 0.25=2280
6440
y x 0.30 = 2160
y = 7200
taxable income = 7200 +(9120 x 3)+ 9680
= 44,240  Basic salary per month (2 marks)
Taxable income = (1.15 of BS) + allowance
44240 = 1.15x + 16000  6260
1.15x = 34500
x= 30,000  Net monthly pay. (2 marks)
Total deductions = 6900 + 1200 (^{12}/_{100} x 30,000) + 4000 + 600
=13,300
Net monthly pay = 44,240  (0.15 x 30,000)  13,300
=26,440
 Taxable income per month. (6 marks)
 The probability of James, Tyson and David passing an examination are ^{4}/_{5},^{3}/_{4} and ^{2}/_{3} respectively. Find the probability that in one attempt:
 only one passes the examination. (2 marks)
(^{1}/_{3} x ^{4}/_{5} x ^{1}/_{4}) + (^{3}/_{4} x ^{1}/_{5} x ^{1}/_{3}) + (^{2}/_{3} x ^{1}/_{5} x ^{1}/_{4})
= ^{1}/_{15} + ^{1}/_{20} + ^{1}/_{30}= ^{3}/_{20}  All the three passes the examination. (2 marks)
^{4}/_{5} x ^{3}/_{4} x ^{2}/_{3} = ^{2}/_{5}  Two pass the examination. (2 marks)
(^{4}/_{5} x ^{3}/_{4} x^{ 1}/_{3}) + (^{3}/_{4} x ^{2}/_{3} x ^{1}/_{5}) + (^{4}/_{5} x ^{2}/_{3} x ^{1}/_{4})
= ^{1}/_{5} + ^{1}/_{10} +^{ 2}/_{15}
=^{13}/_{20}  None passes the examination. (2 marks)
^{1}/_{5} x ^{1}/_{4} x^{ 1}/_{3} = ^{1}/_{60}  At least one passes the examination. (2 marks)
1  ^{1}/_{60} = ^{59}/_{60}
 only one passes the examination. (2 marks)
 When aero plane left town P (65º N,15º E) to another town Q (65º N,165º W) at a speed of 200 knots using the shortest route. Take π = ^{22}/_{7} and radius of the earth R= 6370 km.

 Calculate the distance travelled in nautical miles. (2 marks)
θ = 180º  (65 + 65) =50º
D= 60x50 = 3000nm  Calculate the time taken to travel from P to Q in hours. (2 marks)
Time = ^{d}/_{s} = ^{3000nm}/_{200knots} = 15hours
 Calculate the distance travelled in nautical miles. (2 marks)
 Another plane left P at 1.30 pm local time and travelled to T (65ºN,60ºE) along a parallel of latitude. Calculate the:
 Distance between P and T to the nearest km (3 marks)
Longitude difference = 6015 = 45º
Distance(km) = ^{θ}/_{360}2πRcos α
^{45}/_{360} x 2 x ^{22}/_{7} x 6370 x cos65
=2115km  Local time of arrival at town T if the plane flew at the speed of 470 km/h(3 marks)
Time = ^{2115.2}/_{470} = 4hrs 30 minutes
^{45 x 4}/_{60} = 3hrs
time at T = 1:30 + 4^{1}/_{2} + 3hrs
=9:00pm
 Distance between P and T to the nearest km (3 marks)

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