AQA Level 2 Certificate in Further Mathematics

94 Pages·2012·0.56 MB·English


AQA Level 2 Certificate in
Further Mathematics
Worksheets - Teacher Booklet

Level 2
Specification
Level 2 Certificate in
Further Mathematics
8360
Worksheets - Teacher Booklet
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Level2CertificateinFurtherMathematics–8360–Worksheets–TeacherBooklet
Contents
1 CoordinateGeometry- Circles 7
2 GeometricProblems andProof 17
3 Algebraic Proof 26
4 Trigonometry 31
5 Matrices - 1 37
6 Matrices - 2 41
7 Inequalities 46
8 Functions 53
9 CoordinateGeometry- Calculus 59
10 FactorTheorem 68
11 Sequences 75
12 Algebraic Problems – including ratio 84
5
Worksheets–Teacher–Booklet–8360–AQALevel2CertificateinFurtherMathematics
Glossary for Mark Schemes
These examinations aremarkedin sucha wayas toaward positive achievement wherever possible.
Thus,for these papers,marksareawarded under various categories.
M Method marksareawardedfor acorrectmethodwhich could lead
to acorrect answer.
A Accuracymarksareawarded whenfollowing onfrom a correct
method. It is not necessaryto always see themethod. Thiscan be
implied.
B Marks awarded independent of method.
MDep A methodmark dependent ona previous methodmark being
awarded.
BDep A mark that canonlybe awarded if a previous independent mark
has been awarded.
ft Followthroughmarks. Marks awardedfollowing amistake in an
earlier step.
SC Special case. Marks awarded within the schemefor acommon
misinterpretation which has somemathematical worth.
oe Or equivalent. Accept answers that areequivalent.
1
eg, accept 0.5 aswell as
2
6
Level2CertificateinFurtherMathematics–8360–Worksheets–TeacherBooklet
1 Coordinate Geometry - Circles
Question 1
Writedown the equationof eachof these circles.
(a) Centre(0, 3) radius2 (2marks)
(b) Centre(1,5) radius 4 (2marks)
(c) Centre(3, 4) radius 7 (2marks)
(d) Centre(8, 15) radius17
Does thiscircle passthroughtheorigin?
Show working tosupport your answer. (4marks)
MarkScheme
(a) x2 + (y  3)2 = 4 B2 B1 LHS, B1 RHS
(b) (x  1)2 + (y + 5)2 = 16 B2 B1 LHS, B1 RHS
(c) (x + 3)2 + (y  4)2 = 7 B2 B1 LHS, B1 RHS
(d) (x  8)2 + (y 15)2 = 289 B2 B1 LHS, B1 RHS
(8)2 + (15)2 M1 oe
64 + 225= 289, Yes A1
Question 2
Writedown the centre and radius of each of thesecircles.
(a) x2 + y2 = 36 (2marks)
(b) (x  3)2 +(y 4)2 = 100 (2marks)
(c) (x + 5)2 + y2 = 3 (2marks)
MarkScheme
(a) (r) = 6 (centre =) (0, 0) B2 B1 For each
(b) (r) = 10 (centre =) (3, 4) B2 B1 For each
(c) (r) = 3 (centre =) (5, 0) B2 B1 For each
7
Worksheets–Teacher–Booklet–8360–AQALevel2CertificateinFurtherMathematics
Question 3(non-calculator)
AB is thediameter of a circle.
A is (3, 6) and B is(5, 12).
Workout theequationof thecircle. (5marks)
MarkScheme
3  5 6 12 M1
or
2 2
(1, 9) A1
(5 1)2  (12  9)2 M1 oe
ft Their centre
5 A1
(x  1)2 + (y 9)2 = 25 A1ft ft Their centreandradius
8
Level2CertificateinFurtherMathematics–8360–Worksheets–TeacherBooklet
Question 4(non-calculator)
PQ is adiameter of acircle, centre C.
y Not drawn
accurately
Q
C (1, 2)
P (1, 1)
O x
(a) Workout thecoordinatesof Q. (1mark)
(b) Workout theequationof thecircle. (3marks)
MarkScheme
(a) (3, 3) B1
(b) M1 oe
22  12
5 A1
(x  1)2 + (y 2)2 = 5 B1ft ft Their radius
9
Worksheets–Teacher–Booklet–8360–AQALevel2CertificateinFurtherMathematics
Question 5(non-calculator)
A (12, 6) and B (14, 4) are two points onacircle,centre C(20, 12).
y
Not drawn
accurately
C (20, 12)
A (12, 6)
M
B (14, 4)
O x
(a) Workout thecoordinatesof themidpoint M, of AB. (2marks)
(b) Showthat thelength CM = 7 2 (3marks)
(c) Workout theradiusof the circle. (2marks)
MarkScheme
(a) 1214 6 4 M1
or
2 2
(13, 5) A1
(b) (2013)2 (125)2 M1 ft Their M
98 A1 72 72
492 = 7 2 A1 72 (1 1) = 7 2
(c) (2012)2 (126)2 M1 oe
10 A1
10