All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Related Documents

Share

Description

hi

Transcript

FREE-STANDING MATHEMATICS QUALIFICATIONADVANCED LEVEL
Additional Mathematics
6993
QUESTION PAPER
Candidates answer on the printed answer book.
OCR supplied materials:
ã Printed answer book 6993
Other materials required:
ã Scientiﬁc or graphical calculator
Monday 13 June 2011Morning
Duration:
2 hours
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the printed answer book and the question paper.ã The question paper will be found in the centre of the printed answer book.ã Write your name, centre number and candidate number in the spaces provided on the printedanswer book. Please write clearly and in capital letters.
ã Write your answer to each question in the space provided in the printed answer book.
Additional paper may be used if necessary but you must clearly show your candidate number,centre number and question number(s).ã Use black ink. Pencil may be used for graphs and diagrams only.ã Read each question carefully. Make sure you know what you have to do before starting youranswer.ã Answer
all
the questions.ã Do
not
write in the bar codes.ã You are permitted to use a scientiﬁc or graphical calculator in this paper.ã Final answers should be given correct to three signiﬁcant ﬁgures where appropriate.
INFORMATION FOR CANDIDATES
This information is the same on the printed answer book and the question paper.ã The number of marks is given in brackets
[ ]
at the end of each question or part question on thequestion paper.ã You are advised that an answer may receive
no marks
unless you show sufﬁcient detail of theworking to indicate that a correct method is being used.ã The total number of marks for this paper is
100
.ã The printed answer book consists of
20
pages. The question paper consists of
8
pages. Any blankpages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
ã Do not send this question paper for marking; it should be retained in the centre or destroyed.
© OCR 2011 [100/2548/0] OCR is an exempt Charity5R–0G15
Turn over
2Formulae Sheet: 6993 Additional MathematicsIn any triangle
ABC
Cosine rule
a
2
=
b
2
+
c
2
−
2
bc
cos
A
A BC b ac
Binomial expansion
When
n
is a positive integer
(
a
+
b
)
n
=
a
n
+
n
1
a
n
−
1
b
+
n
2
a
n
−
2
b
2
+
...
+
nr
a
n
−
r
b
r
+
...
+
b
n
where
nr
=
n
C
r
=
n
!
r
!
(
n
−
r
)
!
© OCR 2011 6993 Jun11
3Answer all questions on the Printed Answer Book provided.Section A1
Determine whether the point
(
5, 2
)
lies inside or outside the circle whose equation is
x
2
+
y
2
=
30.You must show your working.
[3]2
The equation of a curve is
y
=
x
3
−
x
2
−
2
x
−
3.Find the equation of the tangent to this curve at the point
(
3, 9
)
.
[5]3
In the triangle PQR, PQ
=
8cm, RQ
=
9cm and RP
=
7cm.
(i)
Find the size of the largest angle.
[4](ii)
Calculate the area of the triangle.
[3]4
Solve the equation 5sin2
x
=
2cos2
x
in the interval 0
◦
≤
x
≤
360
◦
.Give your answers correct to 1 decimal place.
[5]5
The coordinates of the points A, B and C are
(−
2, 1
)
,
(
5, 2
)
and
(
4, 9
)
respectively.
(a)
Find the coordinates of the midpoint, M, of the line AC.
[1](b)
Show that BM is perpendicular to AC.
[3](c) (i)
Use the result of part
(b)
to state the mathematical name of the triangle ABC.
[1](ii)
Prove this by another method.
[2]6
Solve the inequality
x
2
−
12
x
+
35
≤
0.
[4]7 (a)
Determine whether or not each of the following is a factor of the expression
x
3
−
7
x
+
6.You must show your working.
(i)
(
x
−
2
)
[2](ii)
(
x
+
1
)
[1](b) (i)
Factorise the function f
(
x
) =
x
3
−
7
x
+
6.
[3](ii)
Solve the equation f
(
x
) =
0.
[1]
© OCR 2011 6993 Jun11
Turn over
48 (i)
On the axes given, indicate the region for which the following inequalities hold. You shouldshade the region which is
not
required.5
x
+
3
y
≥
303
x
+
y
≥
12
y
≥
0
x
≥
0
[5](ii)
Find the minimum value of 6
x
+
y
subject to these conditions.
[2]9
The gradient function of a curve is given by d
y
d
x
=
3
x
2
−
2
x
+
4.Find the equation of the curve, given that it passes through the point
(
2, 2
)
.
[4]10
You are given that sin
θ
=
25
with 0
◦
≤
θ
≤
90
◦
.Using the identity sin
2
θ
+
cos
2
θ
=
1, ﬁnd an exact value for cos
θ
.
[3]
© OCR 2011 6993 Jun11

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks