SECTION A
QUESTION 1
1.1In the diagram, B is the midpoint of the line segment OA. =
and = 30.
Determine:
1.1.1 the co-ordinates of B. (2)
1.1.2 the gradient of OA.(2)
1.1.3 the inclination of OA, correct to one decimal place.(2)
1.1.4 the gradient of PQ, correct to two decimal places.(3)
1.2The points A (4; -3), B (-5; 0) and C(-3; ) are given. Determine the value
of if:
1.2.1 A, B and C are collinear;(3)
1.2.2 AB BC.(2)
1.2.3 BC = .(4)
[18]
QUESTION 2
Given the circle with equation.
2.1 Show that the co-ordinates of the centre of the circle are (-1;2).(4)
2.2 Find the length of the diameter of the circle in simplified surd form. (2)
2.3 Prove that the point (-3,6) is a point on the circle. (3)
[9]
QUESTION 3
3.1 The diagram below shows quadrilaterals ABCD, WXYZ and PQRS.
3.1.1 Give the co-ordinates of , if point B is reflected about the
line (2)
3.1.2Describe the transformation of PQRS to WXYZ in terms of
a rule(s) in the form ( ; ).(4)
3.1.3If the original quadrilateral ABCD is rotated 90in a clockwise
direction about the origin, write down the co-ordinates of for
this rotation.(2)
3.2 The circle is given.
3.2 1 The equation of the new circle when the original circle is translated
under a given rule is Explain the translation
in words.(2)
3.2.2Write down the equation of the new circle if the original circle is
enlarged by a factor of 2,5 through the origin.(2)
3.2.3By what scale factor will the area of the new circle increase?(2)
[14]
QUESTION 4
On the Game of SURVIVAL, a piece of music was played to a group of 80 contestants
and they were asked to estimate how long the piece of music had lasted. The estimates
in seconds are given in the cumulative frequency graph below:
`Using the graph, determine:
4.1 the median (1)
4.2the interquartile range (2)
4.3the number of contestants that overestimated the playing time of the piece of
music, if in fact the music lasted 17,5 minutes. (2)
[5]
QUESTION 5
5.1Use your calculator to simplify correct to four decimal places
if and (2)
5.2 Simplify to a single trigonometric identity without a calculator:
(5)
5.3 Solve the following equations:
5.3.1 for (2)
5.3.2 for (4)
[13]
QUESTION 6
The sketch graph which is not drawn to scale, shows the curves of and
defined by and
6.1 Write down the values of and using the key co-ordinates from the graph. (2)
6.2 If P(; 0,51) and Q (165,36; ) are the points of intersection of the two graphs
then write down the values of and . (answer to two decimal digits)(2)
6.3 For which value(s) of is
6.3.1 ?(4)
6.3.2 ?(4)
6.4 If then write down the range of (2)
6.5 Graph is shifted 30 to the right to form graph .
What is the equation of ? (2)
[16]
SECTION B
QUESTION 7
The circle touches the at A(2;0) and passes through B(4;-6).
7.1 What is the co-ordinate of M? (1)
7.2 Hence, find the equation of the circle with centre M.(6)
7.3 If the co-ordinates of M are , find the equation of the tangent to the
circle at B.(5)
[12]
QUESTION 8
An ellipse with equation is drawn with centre the origin.
8.1 If an ellipse has equation then its area is given by the formula
if . Use the formula to find the area of the ellipse drawn above
in terms of .(2)
8.2 If an ellipse has equation then its perimeter can be approximated
by the formula when
Use this formula to approximate the perimeter of the ellipse drawn above in
terms of .(3)
8.3 If the and values of the ellipse is reduced through the origin by a factor
of by what factor will the area of the ellipse be reduced.(2)
[7]
QUESTION 9
In the diagram,
9.1 Determine, leaving answers in simplified surd form if necessary:
9.1.1 the length of OM.(2)
9.1.2 (2)
9.1.3 (1)
9.1.4 (2)
9.2 If is rotated about the origin through , determine
the value of (2)
[9]
QUESTION 10
Given:
10.1 Prove the above identity.(6)
10.2 Without using a calculator, prove that the identity is not valid for (3) [9]
QUESTION 11
In the diagrsm P ( - 2 ; 4 ) is a point on OP and R is a point in the third quadrant so that PR QO. .
Determine, leaving your answer,
correct to one decimal digit, if
necessary, the value of sin 2.
[8]
QUESTION 12
The Great Pyramid of Giza, also known as the Pyramid of Khufu, was built in 2600BC
and is one of the remaining wonders of the world. When it was first built, its height PT,
as shown in the diagram, scaled to an amazing 146 metres. The slope of the pyramid
is 51,9
12.1 Show by calculation that the slant height, PQ, of the pyramid is 185,53 metres.(2)
12.2 A tourist stands at a point R, which is 100 metres from the foot Q of the
pyramid. Calculate the distance, PR, to the top of the pyramid,
if. (3)
12.3 If the volume of the Khufu Pyramid is 2 590 000 m3
12.3.1 determine the area of its base, if it is given that:
VOLUME OF PYRAMID = area of base height
(2)
12.3.2 On average, the soccer stadiums for the 2010 World Cup Soccer will
have an area of 7 700 m2. To the nearest integer, determine how many soccer stadiums make up the base of the Khufu Pyramid. (2)
[9]
QUESTION 13
Jenny keeps a record of the time that she and her peers work in their Mathematics
Class on Mondays, which is a 90 minute period. The results, to the nearest minute,
for 10 Monday lessons are given below:
43 44 46 48 48 49 52 58 62 90
13.1 Find the five number summary for the data above.(5)
13.2Construct a box and whisker plot for the data.(3)
13.3If the mean for the data is above 45 minutes, give a reason as to why you
think Jenny worked for 90 minutes in one of the 10 Monday lessons.(1)
[9]
QUESTION 14
The test results and various calculations for 20 students appear in the table below:
14.1 Determine the standard deviation of the test results.(2)
14.2 Determine what percent of the students scored within one standard
deviation of the mean.(3)
14.3 Hence, or otherwise, determine whether the test results approximate a
normal distribution. Justify your answer.(2)
[7]
QUESTION 15
Let , , and be integers such that and = .
- The mode of these four numbers is 11.
- The range of these four numbers is 8.
- The mean of these four numbers is 8.
Calculate the values of , , and .[5]
1