4024/02/M/J/06 © UCLES 2006 MATHEMATICS (SYLLABUS D) 4024/02
Section A [52 marks]
Answer all questions in this section.
1 (a) Solve the equation 3x2 – 4x – 5 = 0, giving your answers correct to two decimal places. [4]
(b) Remove the brackets and simplify (3a – 4b)2. [2]
(c) Factorise completely 12 + 8t – 3y – 2ty. [2]
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2 (a) A solid cuboid measures 7 cm by 5 cm by 3 cm.
(i) Calculate the total surface area of the cuboid. [2]
(ii) A cube has the same volume as the cuboid.
Calculate the length of an edge of this cube. [2]
(b) [The volume of a cone is × base area × height.]
[The area of the curved surface of a cone of radius r and slant height l is πrl.]
A solid cone has a base radius of 8 cm and a height of 15 cm.
Calculate
(i) its volume, [2]
(ii) its slant height, [1]
(iii) its curved surface area, [2]
(iv) its total surface area. [1]
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4024/02/M/J/06 © UCLES 2006 [Turn over]
3 (a) In the diagram, the points A, B, C and D lie on a circle, centre O.
DO∧B = 124° and CD∧O = 36°.
Calculate
(i) DC∧B, [1]
(ii) DA∧B, [1]
(iii) OD∧B, [1]
(iv) CB∧O. [1]
(b) The diagram shows a circle, centre O,with the sector POQ shaded.
Given that PO∧Q = 140° and the radius
of the circle is 8 cm, calculate
(i) the area of the shaded region, [2]
(ii) the total perimeter of the unshaded region. [3]
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4 (a) These are the prices for a ride in an amusement park.
(i) A family of two adults and three children went on the ride.
They paid with a $20 note.
Calculate the change they received. [1]
(ii) Express $2.25 as a percentage of $3.60. [1]
(b) Diagram I represents part of the framework of the ride.
The points A, B, C, D, E and F are on the framework.
The points H, C, G, E and F lie on a horizontal line.
The lines BH and DG are vertical.
BC = 80 m, HC = 60 m, DG = 40 m, GE = 35 m and DC∧G = 32°.
Diagram I
Calculate
(i) HC∧B, [2]
(ii) CD, [3]
(iii) the angle of depression of E from D. [2]
4024/02/M/J/06 © UCLES 2006 [Turn over]
(c)
Diagram II
Diagram II shows part of the ride.
The carriage that carried the family was 4.6m long.
It was travelling at a constant speed of 15 m/s as it passed the point F.
(i) Calculate, correct to the nearest hundredth of a second, the time taken for the carriage to pass
the point F. [2]
(ii) Express 15 m/s in kilometres per hour. [1]
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5
Pattern 1 Pattern 2 Pattern 3
Counters are used to make patterns as shown above.
Pattern 1 contains 6 counters.
The numbers of counters needed to make each pattern form a sequence.
(a) Write down the first four terms of this sequence. [1]
(b) The number of counters needed to make Pattern n is An + 2.
Find the value of A. [1]
(c) Mary has 500 counters.
She uses as many of these counters as she can to make one pattern.
Given that this is Pattern m, find
(i) the value of m, [1]
(ii) how many counters are not used. [1]
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4024/02/M/J/06 © UCLES 2006
6 (a) The results of a survey of 31 students are shown in the Venn diagram.
= {students questioned in the survey}
M= {students who study Mathematics}
P = {students who study Physics}
S = {students who study Spanish}
(i) Write down the value of
(a) x, [1]
(b) n(M ∩ P), [1]
(c) n(M ∪ S), [1]
(d) n(P′). [1]
(ii) Write down a description, in words, of the set that has 16 members. [1]
(b) In the diagram, triangle AQR is similar to triangle ABC.
AQ = 8 cm, QB = 6 cm and AR = 10 cm.
(i) Calculate the length of RC. [2]
(ii) Given that the area of triangle AQR is 32cm2, calculate the area of triangle ABC. [2]
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4024/02/M/J/06 © UCLES 2006 [Turn over
Section B [48 marks]
Answer four questions in this section.
Each question in this section carries 12 marks.
7 James and Dan are partners in a small company.
From each year’s profit, James is paid a bonus of $15 000 and the remainder is shared between James and Dan in the ratio 2 : 3.
(a) In 1996 the profit was $20 000.
Show that Dan’s share was $3000. [1]
(b) In 1997 the profit was $21 800.
Calculate
(i) the percentage increase in the profit in 1997 compared to 1996, [2]
(ii) the total amount, including his bonus, that James received in 1997. [2]
(c) In 1998 Dan received $7500.
Calculate the profit in 1998. [3]
(d) In 1999, the profit was $x, where x > 15 000.
(i) Write down an expression, in terms of x, for the amount Dan received. [1]
(ii) Given that Dan received half the profit, write down an equation in x and hence find the
amount that Dan received. [3]
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8
4024/02/M/J/06 © UCLES 2006
8 Answer the whole of this question on a sheet of graph paper.
The table below gives some values of x and the corresponding values of y, correct to one decimal place,where y= + – 5.
x 1 1.5 2 2.5 3 4 5 6 7 8
y 13.1 7.3 4.5 3.0 2.1 1.5 1.7 p 3.7 5.3
(a) Find the value of p. [1]
(b) Using a scale of 2 cm to 1 unit, draw a horizontal x-axis for 0 x 8.
Using a scale of 1 cm to 1 unit, draw a vertical y-axis for 0 y 14.
On your axes, plot the points given in the table and join them with a smooth curve. [3]
(c) Use your graph to find
(i) the value of x when y = 8, [1]
(ii) the least value of + for values of x in the range 0 x 8. [1]
(d) By drawing a tangent, find the gradient of the curve at the point where x = 2.5. [2]
(e) On the axes used in part (b), draw the graph of y = 12 – x. [2]
(f) The x coordinates of the points where the two graphs intersect are solutions of the equation
x3 + Ax2 + Bx + 144 = 0.
Find the value of A and the value of B. [2]
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9
4024/02/M/J/06 © UCLES 2006 [Turn over
9 In the diagram, A and B are two points on a straight coastline.
B is due east of A and AB = 7 km.
The position of a boat at different times was noted.
(a) At 8 a.m., the boat was at C, where
AC∧B = 66° and AB∧C = 48°.
Calculate
(i) the bearing of B from C, [1]
(ii) the distance AC. [3]
(b) At 9 a.m., the boat was at D, where AD = 6.3 km and DA∧B = 41°.
Calculate
(i) the area of triangle ADB, [2]
(ii) the shortest distance from the boat to the coastline. [2]
(c) At 11 a.m., the boat was at E, where AE = 9 km and BE = 5 km.
Calculate the bearing of E from A. [4]
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10
4024/02/M/J/06 © UCLES 2006
10 (a) The lengths of 120 leaves were measured.
The cumulative frequency graph shows the distribution of their lengths.
Use this graph to estimate
(i) the median, [1]
(ii) the interquartile range, [2]
(iii) the number of leaves whose length is more than 31.5 cm. [1]
(b) Each member of a group of 16 children solved a puzzle.
The times they took are summarised in the table below.
(i) Write down an estimate of the number of children who took less than 13 minutes. [1]
(ii) Calculate an estimate of the mean time taken to solve the puzzle. [3]
(iii) Two children are chosen at random.
Calculate, as a fraction in its simplest form, the probability that one of these children took
more than 10 minutes and the other took 10 minutes or less. [2]
(iv) A histogram is drawn to illustrate this information.
The height of the rectangle representing the number of children in the interval 10 < t 12
is 8 cm.
Calculate the height of the rectangle representing the number of children in the interval
5 < t 10. [2]
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11
4024/02/M/J/06 © UCLES 2006
11 (a) A = (1 -3) B=(-2p 3p) C = (-1 0)
3 -2 -3p p 0 1
(i) Evaluate 4C – 2A. [2]
(ii) Given that B = A–1, find the value of p. [2]
(iii) Find the 2 × 2 matrix X, where AX = C. [2]
(iv) The matrix C represents the single transformation T.
Describe, fully, the transformation T. [2]
(b) PQ→=(3) PR→=(h) QU→=(7) PS→= (17)
-9 -6 2 k
(i) Given that R lies on PQ, find the value of h. [1]
(ii) Express PU→as a column vector. [1]
(iii) Given that U is the midpoint of QS, find the value of k. [2]
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4024/02/M/J/06
BLANK PAGE
1 (a) For numerical p = 4 and r = 6 B1
q = 76 or q = 8.71
x = 2.12 or −0.79
(b) 9 a2 + 16b2− 24ab (c) (4−y)(3+2t)
2 (a) (i) 2(7×5 + 7×3 + 3×5), 142 cm2
(ii) x3=7×5 ×3 soi, 4.7 to 4.72 cm
(b) (i)1/3 π64× 15
1005 to 1010 cm3
(ii) 17 cm (iii) 8 ×17 × π M1
427 to 427.3 cm2 (iv) 628 to 628.6 f.t. cm2 B1 1 f.t. 201 + their 427
3 (a) (i) DĈB = 62 o
(ii) B A D ˆ = 118 o f.t. B1 (f.t. 180 – their 62)
(iii) B D O ˆ = 28 o B1
(iv) B O C ˆ = 26 o B1
(b) (i)140/360soi
78.1 to 78.25 cm2
(ii) 220 o B1 2 ×8 ×π×220/360 M1
46.7 to 46.73 cm
4 (a) (i) $6.05
(ii) 62.5%
(b) (i) cos B C H ˆ =60/80 oe
41.4 o to 41.41 o
(ii) sin 32 = 40/CD
CD=40/ sin 32
75.48 to 75.5 m
(iii) tan d = 40/35
d = 48.8o to 49o
(c) (i) 4.6/15
0.31 s
(ii) 54 km/h
5 (a) 6, 10, 14, 18
(b) 4
(c) (i) 124
(ii) 2
6 (a) (i) (a) 8
(b) 4
(c) 21
(d) 19 f.t.
(b)(i) 8/6 = 10/RC or 8/14= 10/10+RC oe
7.5cm
(ii) (8/14)2 or (14/8)2 oe
98cm2
© University of Cambridge International Examinations 2006
7 (a) 3/5 × 5000
(b)(i) 1800/20000
9%
(ii) ( 2/5) × 21800 -15000
$17 720
(c) (5/3) × 7500
$12 500
$27 500 f.t.
(d)(i) (3/5 ) (x-15000) oe
(ii) their 3/5(x-15000 ) =x/2 f.t
x = 90 000 ⇒ $45000
8 (a) 2.5 B1 1
(c) (i) 1.4 < x < 1.5
(ii) 6.4 to 6.5
(d) Negative value
2.0 to 2.5
(e) Line with negative slope thro’ (0,12)
Also through (6,6)
(f) Attempt to simplify
x2/8 + 18/x -5 = 12-x
Allow M1 for attempt to sub
x = 1.2 and 7.5 and solve
A = 8 AND B = −136
GCE O Level – May/June 2006 4024 02
© University of Cambridge International Examinations 2006
9 (a) (i) 138o
(ii) 7/66 sin=AC/48 sin
AC= 7 sin 48 /sin 66
5.69 to 5.7 km
(b)(i) 1/2× 7×6.3 sin 41
14.46 to 14.5 km2
(ii) 6.3 sin 41 or area/3.5
4.13 to 4.15 km
(c) Cosine Rule involving B ˆ AE
cos A =
= 105/126
33.5 o to 34 o
(0)56o – 56.5o f.t.
10 (a) (i) 31.8 cm
(ii) 32.1 – 31.65 cm
0.42 to 0.48 cm
(iii) 108
(b) (i) 9
(ii) (2 x 7.5) + (4 x 11) + (6 x 13) +
(3 x 15) + (1 x 18) ÷ 16
12.5 min
(iii) 7/30 cao
(iv) 1.6 cm
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