Fa
Exam
13
us
For
22 The diagram shows a parallelogram ABCD.
(a) Find the gradient of DC.
(b) Calculate the area of the parallelogram ABCD.
(c) T h e parallelogram is reflected in the line y = 2.
Find the coordinates of the image of the point D .
(a) Find the coordinates of the point about which ABCD has rotational symmetry of
order 2.
Answer ( a )
(b)
4004/1. m4/1/w97
....................................................
..........................................
unit
PI
PI
(c)
( .................).................)
(11
(d)
( .................,................. 1
U1
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16
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runiinrr s:
Use
Use
24
.
)
X
In the diagram, the points A , B and C have co-ordinates (0, l),(1,3) and (- 1,- 1).
(a) Calculate the gradient of the line AB.
(b) Find the equation of the line AB.
(c)
The point (10, k ) lies on the line AB produced.
Find the value of k.
(a) The length of the line segment AB is fl .
Calculate the value of 1.
( e ) The triangle BCD has line of symmetry y = x .
Write down the co-ordinates of D .
Answer ( a ) ........................................................
111
( b ) ........................................................
PI
(c) k = ..................................................
PI
( d )1 = ...................................................
PI
(e) D is
(............... , ...............1
121
10
11 (a)
The co-ordinates of the points A, B and C are (-7, l), (3, 6) and (0, 2) respectively.
(i) Find the gradient of BC.
(ii) Write down the equation of the line BC.
(iii) Given that B is the mid-point of the line AX, find the co-ordinates of X.
@)
PI
PI
PI
Numbers are written out in five columns to make the following pattern:
c1
c
2
c3
c
4
c5
Row 1
2
3
4
5
6
Row 3
Row 4
12
13
14
17
18
19
15 16
20 21
k
x
Y
Z
..
..
Row n
Rown+l
..
..
..
It is noticed that the products 2 x 8 = 16 and 3 x 7 = 21 both lie in column C5.
The four numbers k, x , y, z are next to each other in the pattern as shown in the diagram.
(i) (a)
Using the patterns in the table above, express x in terms of k, y in terms of k
and z in terms of k.
(b) Find and simplify xy - kz.
(c) Explain why x y and k z lie in the same column.
[I1
PI
[11
(ii) The number k is in Row n of column C2.
Find an expression for k in terms of n.
(iii) By also expressing the numbers x, y and z in terms of n, or otherwise,
show that both the products X J Jand kz lie in column CI.
402412 Nov 99
9506
[31
7
For
Examiner’.
Use
For
Examinerf
use
Simplify
(i)
(i)
+ 2),
n2 - (n - a)(. +
n2 - (n - 2)(n
U).
Using your answer to part (a)(ii), write down the value of
169472- 16944 x 16950.
Answer
16
(a)(i) ........................................
PI
(ii) ........................................
[I1
(b) ............................................
PI
The diagram shows three points A(-2, -l), B(1,-1) and C(4,3).
Calculate
(a)
the area of triangle ABC,
yt
(b) the cosine of A k .
Answer
(a) ............................................
h
(b) cos ABC
=
........................
PI
[21
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9505
For
Examiner's
4
For
Exammeri
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Use
7
The diagram in the answer space shows the line y = m x .
On the same diagram, sketch and label
(a)
y =rnx+2,
(b) y = -mx.
3
--
2 -Answer
PI
-2 --
8
(a)
The volume of a cube is 200 cubic centimetres.
Find the length of an edge of the cube, correct to the nearest centimetre.
(b) The volume of a sphere of radius r is $ n r 3 .
Find, correct to one significant figure, the volume of a sphere with radius 10 cm.
Answer
(a) ......................................
(b) ....................................
9
(a)
Write down the positive square root of 24.
(b)
Express 1.65 hours in minutes.
Answer
4004/1.4024/1 Nov 99
PI
cm3 [I]
(a) ............................................
(b) ..............................
9505
cm
U1
minutes [I]
3
For
Examiner's
For
Examiner's
Use
USr
4
Evaluate
(a)
72 - 7l
(b)
16-'.
+ 7O,
Answer
5
(a) ............................................
PI
(b) ............................................
U1
The temperature on the surface of the moon in the middle of the day was 126°C.
The temperature on the surface of the moon in the middle of the night was -154°C.
(a)
By how much did the temperature decrease during this period?
(b) Find the average of the temperatures in the middle of the day and the middle of
the night.
Answer
~~~~~~
6
~
(a) .......................................
"C P I
(b) .......................................
"C U1
~
(a)
Find the gradient of the straight line 5x
@)
The point (p,2p) lies on the straight line
+ y = 14.
x + 4y = 36.
Calculate the value of p.
Answer
402411
s 99
(a) ............................................
PI
(b) p = ......................................
[21
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7
For
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12 A is the point (-1,O) and B is the point ( 2 , 6 ) .
Find
(a) the coordinates of the midpoint of AB,
(b) the gradient of the line AB,
(c) the equation of the line parallel to AB,through the point (0,4).
(.............., ..............1
HI
(b)................................................
[I1
(c).................................................
111
Answer ( a )
13 [The value of n is 3.14 correct to three significant figures.]
In the diagram, the circle, centre 0,passes through A and B .
The radius of the circle is 4 cm and A d B = 45".
(a) Find the area of the minor sector AOB.
(b) The tangent at A meets OB produced at T.
Find the shaded area.
A
T
0505
Answer ( a )..........................................
cm2 [2]
( b )..........................................
cm2 [ I ]
40041I , 4241 I
W 2000
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For
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NEITHER ELECTRONIC CALCULATORS NOR MATHEMATICAL TABLES
MAY BE USED IN THIS PAPER.
1
(a) Write down the remainder when 365 is divided by 7.
(b) There were 365 days in the year 1993.
The fist day of the year was a Friday.
On what day of the week did 1994 begin?
Answer (a) ...............................................
(b) ...............................................
2
U1
The price of a box of soap powder increased from $2.50 to $2.65.
Calculate the percentage increase in the price.
Answer
3
PI
...................................................
%PI
A is the point (7,3) and B is the point (5,ll).
Find
(a) the coordinates of the midpoint of AB,
(b) the vector E .
Answer (a)
4004/1.4024/1 S 2000
(.............. , .............1
E11
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Examiner’s
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For
Examiner’s
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12
21
y
8
A
7
6
5
4
3
2
B
1
–3 –2 –1 0
–1
1
2
3
4
5
6
7
x
–2
–3
–4
C
–5
The diagram shows three points A(02, 7), B(02, 2) and C(6, 04).
Find
(a) the length BC,
(b) the area of triangle ABC,
(c) the value of sin ABC.
Answer (a) .........................................units [2]
(b) ........................................units2 [2]
(c).................................................. [1]
4024/1/O/N/01
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For
Examiner’s
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14
Three lines, l 1, l 2 and l 3, are drawn on the diagram in the answer space.
The equation of the line l 1 is y = x + 5.
The equation of the line l 2 is 3y + x = −3.
23
(a) Use the diagram to solve the simultaneous equations
y = x + 5 and 3y + x = −3 .
Answer (a) x # ................, y # ...............[2]
(b) Write down the equation of the line l 3.
Answer (b) .............................................[1]
(c) The equation of another line, l 4, is y = −1.
Draw this line on the diagram in the answer space.
Answer (c)
l3
y
l1
4
2
l2
06
04
02
0
2
4
6
8 x
02
[1]
04
(d) The region enclosed by these four lines is defined by four inequalities.
One of these is 3y + x ≥ −3.
Write down the other three inequalities.
Answer (d) .............................................
.............................................
.............................................[2]
4024/01/O/N 2003
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13
For
Examiner’s
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22
The points A(05, 5), B(1, 03) and C(4, 03) are shown in the diagram.
A (–5, 5)
y
x
0
B (1, –3)
C (4, –3)
Find
(a) the coordinates of the midpoint of AC,
(b) the gradient of the line AB,
(c) the equation of the line which passes through (0, 3) and is parallel to AB,
(d) the length of AB,
(e) the value of cosine ABC.
Answer (a) (................, ..................) [1]
(b) .............................................. [1]
(c) .............................................. [1]
(d) .................................... units[1]
(e) .............................................. [1]
4024/01/O/N 2003
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8
15
The lines x ! y # 2 and x 0 3y # 6 are shown on the diagram in the answer space.
(a) Find the gradient of the line x 0 3y # 6.
Answer (a) .............................................. [1]
(b) On the diagram in the answer space, shade the region defined by the inequalities
x ! y ≤ 2 , x 0 3y ≤ 6 and x ! 1 ≥ 0.
o
Answer (b)
g
O 03390
y
4
2
–4
–2
0
2
4
6 x
–2
[2]
–4
4024/01/M/J 2003
3
2
The points A, B and C are (9, 8), (12, 4) and (4, –2) respectively.
(a) Find
(i) the gradient of the line through A and B,
[1]
(ii) the equation of the line through C which is parallel to AB.
[2]
(b) Calculate the length of the line segment
3
(i) AB,
[1]
(ii) BC.
[1]
(c) Show that AB is perpendicular to BC.
[1]
(d) Calculate the area of triangle ABC.
[1]
Mr Smith bought three companies, Alpha, Beta and Gamma, for a total of $80 000 000.
The amounts he paid for these companies were in the ratio 4 : 5 : 7.
(a) Calculate how much he paid for each company.
[2]
(b) When he sold the companies, he made a profit of 12% on the $80 000 000 he paid for them.
Calculate the profit he made on the sale of the three companies.
[1]
(c) When he sold the companies, he made a profit of 25% on Alpha and a loss of 10% on Beta.
Calculate
(i) the profit he made on Alpha,
[1]
(ii) the percentage profit that he made on Gamma.
[3]
(d) When the previous owner, Mr Jones, sold the companies to Mr Smith for $80 000 000, he made a
profit of 60%.
Calculate the total amount Mr Jones had paid for the companies.
© UCLES 2004
4024/02/O/N/04
[2]
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For
Examiner’s
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12
21
y
R
4
3
2
1
P
–4
Q
–3
–2
–1
0
1
2
3
4
5
6
7
8
9 x
–1
PQRS is a parallelogram.
P is (–4, 0), Q is (1, 0) and R is (9, 4).
(a) Find the coordinates of S.
(b) Find the coordinates of the midpoint of PR.
(c) Find the equation of the line RS.
(d) Find the equation of the line QR.
(e) Calculate the area of the parallelogram PQRS.
Answer (a)
(............ , ............)
[1]
(b)
(............ , ............)
[1]
(c) ............................................ [1]
(d) ........................................... [2]
(e) ....................................unit2 [1]
© UCLES 2005
4024/01/O/N/05
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Examiner’s
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For
Examiner’s
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7
12
(a) A TV programme list shows that a film begins at 21 55.
The film lasts for 100 minutes.
At what time will it end?
Express your answer using the 24 hour clock.
(b) The times taken by an athlete to run three races were 3 minutes 59.1 seconds,
4 minutes 3.8 seconds and 4 minutes 1.6 seconds.
Calculate the mean time.
Answer (a) ............................................................. [1]
(b) ................ minutes ................. seconds [2]
13
(a) P is the point (–3, 3) and Q is the point (13, –2).
Find the coordinates of the midpoint of PQ.
Answer (a)
(.................. , ..................)
[1]
(b) The line x –3y = 2 is shown on the diagram in the answer space.
The line x – 3y = k cuts the y-axis at the point (0, –4).
(i)
Draw the line x – 3y = k on the diagram.
(ii)
Calculate the value of k.
Answer (b)(i)
y
4
2
–2
0
2
4
6
8
10
12
14 x
–2
–4
[1]
(ii) k = ........................................ [1]
© UCLES 2005
4024/01/M/J/05
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2
Section A [52 marks]
Answer all the questions in this section.
1
5x2 – 20.
(a) (i) Factorise completely
(ii) Simplify
5x2 – 20
.
+ 10x – 20
[2]
[2]
10x2
(b) Express as a single fraction in its simplest form
4
3 .
–
y–3
y+5
(c) Given that
T = 2π
L
,
g
express g in terms of π, T and L.
2
[3]
[3]
The points A and B are (5, 3) and (13, 9) respectively.
(a) Find
(i) the midpoint of AB,
[1]
(ii) the gradient of the line through A and B,
[1]
(iii) the length of the line AB.
[1]
(b) C is the point (–8, 5).
冢冣
4
The point D is such that DC = 3 .
(i) Find the coordinates of D.
[2]
(ii) What type of quadrilateral is ABCD?
[1]
© UCLES 2006
4024/02/O/N/06
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Examiner’s
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For
Examiner’s
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14
22
y
10
8
6
A
4
2
–4
–2
0
–2
2
6
4
C
8
x
10
–4
B
–6
The triangle with vertices A(4, 4), B(–2, –6) and C(4, –1) is shown in the diagram.
Find
(a) (i) the area of ∆ ABC,
(ii) the coordinates of the point P such that ABCP is a parallelogram,
(iii) the area of the parallelogram ABCP,
^
(iv) tan BAC.
(b) It is given that the length of BC = k units.
^
Write down cos BCA, giving your answer in terms of k.
Answer (a) (i) .................................unit2 [1]
(ii)
(............ , ............)
[1]
(iii) .................................unit2 [1]
^
(iv) tan BAC = ........................[1]
^
(b) cos BCA = ............................[1]
___________________________________________________________________________
© UCLES 2006
4024/01/O/N/06
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For
Examinerʼs
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14
25
The diagram shows the points A (1, 2), B (4, 6) and D (–5, 2).
y
B(4,6)
D(–5,2)
A(1,2)
x
0
(a) Find the coordinates of the midpoint of AB.
(b) Calculate the length of AB.
(c) Calculate the gradient of the line AB.
(d) Find the equation of the line AB.
(e) The triangle ABC has line of symmetry x = 4.
Find the coordinates of C.
(f)
Find the value of cosine DÂB.
Answer (a)
(.................. , ..................)
[1]
(b) ................................................... [1]
(c) ....................................................[1]
(d) ....................................................[2]
(e)
(.................. , ..................)
[1]
(f) cos DÂB = .................................[1]
© UCLES 2006
4024/01/M/J/06
15
For
Examiner’s
Use
22 The equation of a line 艎 is x + 2y = –1.
For
Examiner’s
Use
(a) Write down the gradient of the line 艎.
(b) Find the equation of the line parallel to 艎 that passes through the point (0 , 5).
Answer (a) ............................................[1]
(b) ............................................[2]
(c) The diagram in the answer space shows the line 艎 and the line y = 2x + 1.
On this diagram,
(i) draw the line y = –2,
(ii) shade and label the region, R, defined by the three inequalities
y –2
x + 2y –1
y 2x + 1 .
Answer (c)(i)(ii)
y
4
y = 2x + 1
3
2
ᐉ
1
–4
–3
–2
–1
0
1
2
3
4 x
–1
–2
–3
–4
[2]
Question 23 is printed on the following page
© UCLES 2008
4024/01/O/N/08
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10
17 A straight line passes through the points P (1, 2) and Q (5, –14).
Find
(a) the coordinates of the midpoint of PQ,
(b) the gradient of PQ,
(c) the equation of PQ.
Answer (a)
(............. , .............)
[1]
(b) ............................................[1]
(c) ............................................[2]
18 The Earth is 1.5 × 108 kilometres from the Sun.
(a) Mercury is 5.81 × 107 kilometres from the Sun.
How much nearer is the Sun to Mercury than to the Earth?
Give your answer in standard form.
(b) A terametre is 1012 metres.
Find the distance of the Earth from the Sun in terametres.
Answer (a) ....................................... km [2]
(b) ............................ terametres [2]
© UCLES 2008
4024/01/M/J/08
11
14 (a) Find the coordinates of the point where the line 2y = 3x + 15 crosses the y-axis.
Answer (a)
(............ , ............)
For
Examiner’s
Use
[1]
(b) The coordinates of the points P and Q are (–1 , 10) and (3 , 4) respectively.
Find
(i) the gradient of PQ,
Answer (b)(i) ...................................[1]
(ii) the midpoint of PQ.
Answer (b)(ii)
© UCLES 2009
4024/01/O/N/09
(............ , ............) [1]
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14 A straight line passes through the points P (–8, 10) and Q (4, 1).
For
Examiner’s
Use
Find
(a) the coordinates of the midpoint of PQ,
Answer (a) (............... , ................) [1]
(b) the equation of PQ.
Answer (b) ..................................... [ 2]
© UCLES 2010
4024/12/M/J/10
www.XtremePapers.net
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13
(iii) The point D lies on the line AB.
The line CD is parallel to the y-axis.
Do not
write in this
margin
(a) Find the coordinates of D.
Answer
(........... , ...........)
[2]
→
→
(b) Express AD in terms of AB .
Answer
© UCLES 2011
4024/22/O/N/11
→
AD = ........................... [1]
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12
6
You may use the graph paper on the next page to help answer this question.
Do not
write in this
margin
The point A is (0, 7), and the point B is (6, 9).
→
(a) Express AB as a column vector.
Answer
...................................... [1]
Answer
...................................... [1]
Answer
P = ............................... [2]
(b) Find the gradient of AB.
(c) The equation of the line AB is x + Py + Q = 0 .
Find P and Q.
Q = ............................... [2]
(d) The point C is (12, 2).
(i) Given that C is the midpoint of BM, find the coordinates of M.
Answer
(........... , ...........)
[1]
(ii) Calculate AC.
Answer
© UCLES 2011
4024/22/O/N/11
.............................. units [1]
18
24 P is the point (–2, 1) and Q is the point (3, 7).
(a) M is the midpoint of PQ.
Find the coordinates of M.
Answer
(............, ............)
[1]
(b) Find the gradient of the line PQ.
Answer
........................................ [1]
(c) The line with equation 2y + 3x + k = 0 passes through the point P.
(i) Find k.
Answer
k = .................................. [2]
Answer
........................................ [1]
(ii) Find the gradient of this line.
© UCLES 2011
4024/12/M/J/11
www.XtremePapers.net
10
14 A is the point (0, 4) and B is the point (–6, 1).
For
Examiner’s
Use
(a) M is the midpoint of the line AB.
Find the coordinates of M.
Answer
(............, ............)
[1]
(b) Find the equation of the line AB.
Answer
..................................... [2]
15
A
4
B
65
D
C
ABCD is a rectangle with AC = 65 cm and AD = 4 cm.
Calculate the area of ABCD.
Answer
© UCLES 2012
4024/12/M/J/12
..............................cm2 [3]
13
20 The diagram shows 10 points, with
coordinates ( h, k ), where h and k are
integers.
For
Examiner’s
Use
y
5
4
3
2
1
0
1
2
3
4
x
(a) For these 10 points find
(i) the maximum value of k – h,
Answer
..................................... [1]
(ii) the value of k, for the point that lies on the line y = 12 x.
Answer k = ............................... [1]
(b) The coordinates of the 10 points satisfy the inequalities
h
a,
k
b,
h+k
c.
Write down the values of a, b and c.
Answer a = ....................................
b = ....................................
c = ............................... [2]
© UCLES 2012
4024/12/O/N/12
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18
23
For
Examiner’s
Use
y
7
T
6
5
4
3
2 S
R
1
0
1
2
3
4
5
6
7
8
x
The diagram shows a triangle RST.
(a) Write down
(i) the gradient of the line ST,
Answer
............................................ [1]
Answer
............................................ [1]
(ii) the equation of a line that is parallel to ST,
(iii) the equation of the line with gradient 3 that passes through S.
Answer
............................................ [1]
(b) One of the inequalities that defines the shaded region RST is x G 6 .
Write down the other two inequalities that define this region.
Answer
.................................................
............................................ [2]
© UCLES 2013
4024/11/M/J/13