WCGS Mathematics Lower Sixth Bridging Work 2017 1. To be successful in Mathematics, students need to be confident in certain aspects of algebra, coordinate geometry and trigonometry before starting the course. All students wishing to study Mathematics in the Lower Sixth are required to have completed this pack before joining the course. 2. Students are often rusty after the summer, so this pack should be completed between Results Day and the first day back to ensure your skills are up to scratch right at the start of the course. We do not advise students to do it earlier than this unless they will be away during the last few weeks of the holidays. 3. Students need to be able to write their solutions clearly and in full, with the work well set out. Work should be written on A4 lined paper and your name written at the top. Full working must be shown, with each line of working below the previous one, that is, you need to work down the page. Question numbers should be put in the margin. Only one column is to be used on a page. Failure to show full working will result in your being asked to redo the pack. 4. Answers are given at the end and students must MARK their work before it is handed in. Marking should be done clearly in RED. Any questions that are not initially solved correctly should be retried until the correct answer is obtained. If you cannot obtain the correct answer, after retrying the question, a red circle should be put around that question number. Work that is not marked in this way will not be accepted. 5. The topics covered in this pack will be considered assumed knowledge when the course begins. If there are topics in this pack that you are uncertain of then you should visit the BBC Bitesize website (http://www.bbc.co.uk/education) for support. 6. There will be a TEST on this work at the beginning of the term. The results of this test, together with public exam results, will be used to put you into sets. These sets will be adjusted after a further assessment that will take place 3 – 4 weeks into the term.
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WCGS Mathematics Lower Sixth Bridging Work 2017
1. To be successful in Mathematics, students need to be confident in certain aspects of algebra, coordinate geometry and trigonometry before starting the course. All students wishing to study Mathematics in the Lower Sixth are required to have completed this pack before joining the course.
2. Students are often rusty after the summer, so this pack should be completed between Results Day and the first day back to ensure your skills are up to scratch right at the start of the course. We do not advise students to do it earlier than this unless they will be away during the last few weeks of the holidays.
3. Students need to be able to write their solutions clearly and in full, with the work well set out. Work should be written on A4 lined paper and your name written at the top. Full working must be shown, with each line of working below the previous one, that is, you need to work down the page. Question numbers should be put in the margin. Only one column is to be used on a page. Failure to show full working will result in your being asked to redo the pack.
4. Answers are given at the end and students must MARK their work before it is handed in. Marking should be done clearly in RED. Any questions that are not initially solved correctly should be retried until the correct answer is obtained. If you cannot obtain the correct answer, after retrying the question, a red circle should be put around that question number. Work that is not marked in this way will not be accepted.
5. The topics covered in this pack will be considered assumed knowledge when the course begins. If there are topics in this pack that you are uncertain of then you should visit the BBC Bitesize website (http://www.bbc.co.uk/education) for support.
6. There will be a TEST on this work at the beginning of the term. The results of this test, together with public exam results, will be used to put you into sets. These sets will be adjusted after a further assessment that will take place 3 – 4 weeks into the term.
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
2
The rules of indices
Exercise 1 – Simplify these expressions
1. 43 xx 2. 23 32 xx
3. pp 24 3 4. 243 xx
5. 52)(y 6. 35 210 xx
7. 4)3 2
( pp 8. 322 )(39 xx
9. 244 )3(7 aa 10.
333 2)4( yy
Expanding brackets
Exercise 2 – expand and simplify
1. 3(x + 2) + (x – 7) 2. 5x – 6 – (3x – 2)
3. -2y²(5 – 7y + 3y²) 4. 7(x – 2) + 3(x + 4) – 6(x – 2)
5. 5x – 3(4 – 2x) + 6 6. 3x² - x(3 – 4x) + 7
7. 4x(x + 3) -2(3x – 7) 8. 3x²(2x + 1) – 5x²(3x – 4)
9. (a – b + c)(2a + b – 2c) 10. (2x² + x + 2)( x² – 3x – 2)
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
3
Factorising expressions
Exercise 3 – factorise completely
1. 10y² - 5y 2. 35x² - 28x
3. xy² - x²y 4. 12y² - 4yx
5. x2 + 14x – 51 6. 2x2 – 5x + 3
7. (x + 3)2 – 7(x + 3) 8. 8x2 + 19x + 6
Solving quadratics
Exercise 4 – solve using factorising. Some will need to be rearranged first.
1) 2)
3) – 4) –
5) 6) –
7) – 8)
9) 10)
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
4
Using the Quadratic Formula
Exercise 5 – solve using the Quadratic Formula (give your answers to 3sf)
1) 2) –
3) – 4)
Completing the square
Exercise 5a – Convert these expressions to completed square format
1) 2)
3)
4)
5) 6)
Extending the rules for indices
Exercise 6 - Evaluate without using a calculator
1. 2
1
9 2. 3
1
64
3. 2
3
49 4. 2
1
25
Simplify
5. 23 xx 6. 75 xx
7. 2
5
2
3
xx 8. 2
32x
9. 6
1
3
2
39 xx 10. 5
2
5
21
5 xx
Evaluate without using a calculator
11. 2
1
25 12. 2
1
81
13.
0
4
3
14. 4
1
1296
15. 2
3
16
91
16.
3
2
8
27
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
5
Using surds and irrational numbers
Exercise 7 - simplify
1. 28 2. 72
3. 3
27 4. 8020
5. 72418200 6. 28263175
7. √ √ √ 8. √ √
9. Find the value of when √
Rationalising the denominator
Examples
a. 3
1 =
3
3
3
1
= 3
3
b. 23
1
=
23
23
23
1
= 2323
23
= 223239
23
= 7
23
c.25
25
=
25
25
25
25
= 25
252255
= 3
1027
Exercise 8 – rationalise the denominators
1. 2
1 2.
15
3
3. 31
1
4.
52
1
5. 35
1
6.
54
23
7. 1117
1117
8.
23
32
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
6
Changing the subject of a formula
Exercise 9 – Make y the subject.
1. )(2)(5 xypy 2. )3()3( ypyx
3. bya
ya
4. y
x
xay
4
5. 2
xy
xy 6. p
y
nym
)(
Inequalities
Exercise 10 – Solve these inequalities
1. 5213 xx 2. )1(2)1(3 xx
3. - 35
2
x 4.
Algebraic fractions
Exercise 11 – Simplify as far as possible
1. xx
xx
3
22
2
2.
145
2142
2
xx
xx
3. 55
2 xx write as a single fraction 4.
xx
12
5. 5
2
2
13
xx 6. 5
2
2
13
xx
7. 5
2
2
13
xx
8. xx
x
5
5
10
52
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
7
Simultaneous Equations
Exercise 12 – Solve to find both x and y.
1. 2034
2452
yx
yx 2.
952
53
yx
yx
3. 62
4
xy
xy 4.
1
1322
yx
yx
Coordinate Geometry
Exercise 13 –Straight Lines
1.
2.
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
8
3.
4.
5.
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
9
Trigonometry
Exercise 14 – Basic Trigonometry
Find the lengths marked with the letter x and y
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
10
Exercise 15 – Advanced Trigonometry
A.
B.
C.
D.
E.
End of bridging work.
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
11
Lower Sixth Mathematics
Bridging Work 2017 – Solutions
Exercise 1 Exercise 2 Exercise 3 Exercise 4
1 7x 1 4x – 1 1 5y( 2y – 1 ) 1 0, -4
2 56x 2 2x – 4 2 7x( 5x – 4 ) 2 0, -3
3 22 p 3 -10y² + 14y³ - 6y 4 3 xy( y – x ) 3 -8, 5
4 23 x 4 4x + 10 4 4y( 3y – x ) 4 2, 6
5 y10
5 11x – 6 5 ( x + 17 )( x – 3 ) 5 -2, -1/2
6 85x 6 7x² - 3x + 7 6 ( 2x – 3 )( x – 1 ) 6 -4, 2/3
7 2p 7 4x² + 6x + 14 7 ( x + 3 )( x – 4 ) 7 1/5, 3
8 827x 8 -9x³ + 23x² 8 ( 8x + 3 )( x + 2 ) 8 -2/3, 2
9 1263a 9 9 ±2
10 632y 10 10 - 5/2, 5/2
Exercise 5 Exercise 5a Exercise 6
1 -0.438, -4.56 1 1 3 9 2
1
3x 2 -4.14, 3.14 2 2 4 10 5x
3 -1.76, 0.0946 3 (
)
3 343 11
3
1
4 -0.773, 1.94 4 4 5
1 12 -0.008
5 5 5x 13 1
6 (
)
6 2x 14 6
7 4x 15 4
125
8 3x 16 9
4
Wallington County Grammar School Lower Sixth Maths Bridging Work 2017
12
Exercise 7 Exercise 8
1 72 6 √ 1 2
2 5
2
35
2 26 7 √ 2 5
5 6 11
10245312
3 3 8 √ 3
2
31 7
3
171114
4 56 9 √
5 √ 4 25 8 -1
Exercise 9
1. 3/)52( pxy 2. 3y 3. )1/()( babay
4. )/(3 xax 5. 3/5x 6. )/( 2 mpmn
Exercise 10
1. 4x 2. 1x 3. x < 7.5 4. 41 x
Exercise 11
1. )3/()2( xx 2. )2/()3( xx 3. 5/3x 4. x/3 ,
5. 10/)117( x 6. 10/)913( x 7. 10/)253( 2 xx 8. x2/1
Exercise 12
1. 4,2 yx 2. 1,2 yx
3. (-4, 1) and (-2,2) 4. )3,2()2,3( and
Exercise 13
1. a: DCGF &;& b: EAHB &;& c: I
2. 52 xy 3. a: 5 b: √ c: 17 4. a: (4, 3) b: (4, 7) c: (-1, -3)
5. a: b: c:
Exercise 14
9: 6.17 10: 4.11 11: 55.5&00.5 12: 8.27&1.13
13: cmx 26.4 14: cm50.3 15: cm2.26
Exercise 15
A a: 34.2cm b: 4.39m c: 11.8cm
B a: 37.1° b: 91.7° c: 56.8°
C a: 3.26m2 b: 1070cm2 c: 14.3cm2
D a: 7.27miles b: 039
E 12.9cm
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