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Student Teacher
AS MATHEMATICS HOMEWORK C1
City and Islington Sixth Form College Mathematics Department
September 2014
Homework Introduction Aim to complete most questions and attempt some extension work. If you find the work
difficult then get help [lunchtime workshops room 216, online, friends, teacher etc].
Homework
You should expect to spend 4 hours on maths homework per week.
• Complete homework task set by the teacher
• Review notes, read your textbook, consult websites
• revise for exams and progress tests
To learn effectively you should check your work carefully and mark answers � x ? If you
have questions or comments, please write these on your homework. Your teacher will then
review and mark your Mathematics.
Topic Date
completed
Comment
HW1 Linear and Quadratic Equations
HW2 Indices
HW3 Fractions and Negative Numbers
HW4 Surds and Irrational Numbers
HW5 Simultaneous Equations
HW6 Practice Test
HW7 Arithmetic Sequences
HW8 General Sequences
HW9 Coordinate Geometry
HW10 Transformations & Inequalities
HW11 Differentiation 1 (Methods)
HW12 Differentiation 2 (Tangents and
Normals)
HW13 Integration 1
HW14 Edexcel Exam C1 May 2010
1 | p a g e
HW1 Linear and Quadratic Equations Your first Maths homework! It is important to get into good habits right away. Read the
notes you made in class and try to remember what was said about the background to the
topic.
Use lined or squared paper. Write your name, title of homework, date, teacher. Work in
pen or pencil. Show appropriate working then mark your answers with a tick, cross or
? This way you will learn more effectively. If you have any questions or feedback write this
on the homework. This is your chance to make a good first impression – thoughtful work
clearly presented please.
Key words: solve, factors, factorise, equation, linear, quadratic
Now solve these linear equations. They are not hard but you should think very carefully
about the logic of what you are doing. This is also a good opportunity to brush up your
mental arithmetic.
1. 1732 =+x 2. 59258 +=+ xx
3. 84)4(7 =+x 4. )1(39)1(2 −−=+ xx
5. 11)5(32 =+x 6. 4
13
52=
+
−
x
x
Now solve the following quadratic equations by factorising.
For example solve
7. 01272 =++ xx 8. 01282 =++ xx
9. 024112 =+− xx 10. 042232 =+− xx
11. 02422 =−+ xx 12. 039102 =−− xx
13. 0642 =−x 14. 092 =− xx
Extension
1. 010173 2 =++ xx 2. )5(3112 −=+ xxx
3. 3
95
−=+
xx 4. xx 938)5(2
2 −=−
Answers
Q1-6 7, 9, 8, 2, 223
, 109−
7. -3, -4 8. -2, -6 9. 3, 8 10. 2, 21 11. 4, -6 12. -3, 13
13. -8, 8 14. 0, 9
1. 5,32 −− 2. -3, -5 3. -6, 4 4. 4,2
3
5
05012
0)5)(12(
05112
21
2
==
=−=−
=−−
=+−
xorx
xorx
xx
xx
HW1
2 | p a g e
HW2 Indices
Complete on a separate sheet of paper and show clear working. Mark using the
answers below.
Key words: Indices, index, power, exponent
Write out the main rules of indices – understand and learn them. Next try to answer the
questions below.
1. Write the following as powers of 5.
a) 625 b) 1 c) 125
1 d) 5 e) 5
2. Write the following as powers of 4
a) 64 b) 4
1 c)
16
1 d) 2 e) 32
3. Write the following as whole numbers or fractions.
a) 52 b)
43− c) 2
1
100 d) 5
1
243 e) 025
4. Simplify the following
a) 2
3
5
5−
b) 32 )7( c)
2)13( d) 3525 × e)
42 34 ×
5. Simplify
a) 1850
35 52
×
× b)
3 64 c) 20 55 −− d) 2
1
4 )11(−
− e)
210 333 −− ++
6. Make sure you are well equipped for the course. Do you have: lined paper, pencil
case, ruler, rubber, loose leaf folders, lots of pens and pencils, coloured pencils,
highlighters?
Extension Questions
7. Calculate some coordinates and draw graphs of x
y 2= and x
y−= 2 on the same axes
with 33 ≤≤− x
8. (a) (i) Is this true? 8134 =− , (ii) What is ?3100 4 =−
(b) Evaluate ,32
xy = for: (i) 4=x , (ii) 5−=x
(c) Is this true? yxyx +=+ 22, try substituting some numbers for x and y,
9. Which is bigger, 6002 or
4003 , and why? Write out a short proof!
Answers
1. ]55555[ 12
1
304 − 2. ]44444[ 2
5
2
1
213 −−
3. ]131081
132[ 4. ]3651375[
465
5. ]9
13121
25
244
4
27[
8. (a) (ii) 19
(b) (i) 48 (ii) 75
HW2
3 | p a g e
HW3 Fractions and Negative Numbers
Complete on a separate sheet of paper and show clear working. Mark using the
answers below.
Key words: substitute, values, fraction, negative number
It is very important that you become fluent in the use of fractions and negative numbers.
You will ONLY be successful at AS maths if you regularly practice this. Take every
opportunity to work out simple sums in your head. The exercise below is designed to show
you the level of work you should easily be able to do without a calculator.
Substitute the values of x into the equations. Show appropriate working.
1. 232 ++= xxy
43
21 ,3, −=x
[Ans. 1677
415 2 ]
2. 7532 ++= xxy
23
21 ,2, −−=x
[Ans. 425
441 9 ]
3. 422 −−= xxy 3,,
83
21 −−=x
[Ans. 1164295
411 −− ]
4. 1
32 −
+=
x
xy
21,5,3 −−=x
[Ans. 3
10121
43 −− ]
BIDMAS - Brackets, Indices, Divide/multiply, add/subtract
Evaluate the following. Show sufficient working to demonstrate that you can do these sums
without a calculator. (Do use a calculator to check them) Don’t be surprised if I give you a
short test on this in class.
5. (i) 2811 ÷+ (ii) 6)23(8 ÷×−
(iii) 6537 ×+× (iv) 234 432 ++
(v) 23
218
+
+ (vi)
334
6312
÷+
×+
(vii) )38(30 −− (viii) ]3)113(76[2
5−−×
[Ans. 15, 7, 51, 59, 4, 6, 25, 15]
Extension - taken from Essential Mathematics qmul
[Ans.20
7]
−+
÷×
nx
)(
HW3
4 | p a g e
HW4 Surds and Irrational Numbers
Complete on a separate sheet of paper and show clear working. Mark using the
answers below.
Key words: N – Natural 1 2 3 … , Z – integers-2 -1 0 1 2 3 , Q – rational (fractions),
R – real numbers (includes irrational) eg 2
Simplify (take out factors, make it look simpler)
1. 45 2. 48
3. 162 4. 121
64
5. 2
18 6. 3250 −
7. 1275 − 8. 457203 +
9. )34)(35( ++ 10. )37)(37( −+
Rationalise the denominator (make the denominator into a rational number)
11. 5
1 12.
3
12
13. )37(
4
+ 14.
)34(
)27(
−
+
Exam questions
Give your answers in the form 2ba +
15. 2)83( − 16.
)84(
1
−
17. Express )53(
)53(2
−
+ in the form 5cb + where b, c are integers
Extension
Find out more about irrational numbers like 2 and π . How can you prove 2 is
irrational? Who was Hippasus? (google root two) Find out about different infinities
http://youtu.be/A-QoutHCu4o
Answers
1. 53 2. 34 3. 29 4. 11
8
5. 3 6. 2 7. 33 8. 527
9. 3923+ 10. 46 11. 5
5 12. 34
13. 23
3214 − 14.
13
6372428 +++ 15. 21217 −
16. 241
21 + 17. 537 +
Old Babylonian Tablet [1900-1700 BC] illustrating Pythagoras' Theorem and the square root of 2
HW4
5 | p a g e
HW5 Simultaneous Equations
Complete on a separate sheet of paper and show clear working. Mark using the
answers below.
Key words: elimination, substitution, quadratic, linear
1. Solve the pairs of simultaneous equations. (use elimination or substitution)
a) 323
1325
=+
=−
yx
yx b)
135
5
=−
=+
yx
yx
c) 32
823
=−
=+
yx
yx d)
1023
06
−=−
=+
yx
yx
2. Solve the pairs of simultaneous equations. They have one linear and one quadratic
factor. (use substitution)
a) yx
yx
2
2022
=
=+ b)
32
65 2
=−
=−
xy
xyx
Exam Question
3. Solve the simultaneous equations
2=+ yx
11422 =− xy
giving your answers in rational form. (5)
Answers
1. (a) (2, -1.5) (b) (2, 3)
(c) (2, 1) (d) (-3, 0.5)
2. (a) (-4,-2) (4, 2) (b) (-1, 1) (2, 7)
3. )3
5,
3
1()3,5( − http://youtu.be/Nx9H9NDcW6E
HW5
6 | p a g e
HW6 Practice Test Your test will be very similar to this. Please use is as a guide for revision.
1. (a) Simplify 98
[1]
(b) Simplify 1275−
[1]
(c) Simplify )512)(512( −+
[2]
(d) Rationalise the denominator 75
8−
[2]
2. Write down the value of
(a) 213− (b) 012
[2]
(c) Find the value of 279
33 45
×
× as a power of 3
[2]
(d) Find the value of 2
3
16−
[2]
3. Solve the following equations, leave answers as fractions
(a) )1(39)1(2 −−=+ xx
[3]
(b) 413
52=
+
−
x
x
[3]
4. Solve by factorizing
(a) 01282 =++ xx
[2]
(b) 0642 =−x
[2]
(c) xx 938)5(2 2 −=−
[2]
5. Solve by completing the square – leave answers as surds
(a) 018102 =+− xx [3]
(b) 0652 =+− xx [must be solved with fractions] [3]
HW6
7 | p a g e
6. Factorise. Then sketch the graph showing the intercepts with the x and y axes
(a) 28112 +−= xxy [3]
Express in completed square form. Then sketch the graph showing the position of
the vertex and the intercept with the y axis
(b) 2362 ++= xxy [3]
7 Evaluate and write in simplest form
(a) 74
123
+
(b) 100287149825
××
×× [2]
(c) Substitute into 7532 ++= xxy
(i) 2−=x
(ii) 23=x [4]
8. Solve the simultaneous equations
(a) x + 6y = 0
3x – 2y = –10 [3]
(b) x – 2y = 1
x2 + y
2 = 29 [5]
Answers
1. (a) 27 (b) 33 (c) 139 (d) 9
7420 + 6. (a)
2. (a) 169
1 (b) 1 (c) 43 (d)
64
1
3. (a) 2 (b) 10
9−
4. (a) -2,-6 (b) -8, 8 (c) 3/2, 4 (b)
5. (a) 75 ±=x (b) 3,2=x
7. (a) 28
23 (b) 4
7 (c) (i) 9 (ii) 4
85
8. (a) 2
1,3 =−= yx (b) 2,5 == yx ,
5
14,
5
23 −=−= yx
7 4
28
(-3, 14)
23
8 | p a g e
HW7 Arithmetic Sequences and Series
Complete on a separate sheet of paper and show clear working. Mark using the
answers below.
Key words: Sequence, Arithmetic, Series, Sum, Common difference
Formulae: dnaU n )1( −+= , [ ]dnan
Sn )1(22
−+= , [ ]lan
S n +=2
1. Write out the proof for the sum of an Arithmetic Sequence (and commit to memory)
[ ]dnan
Sn )1(22
−+= see page 107 in text book.
2. Given the sequence 3, 7, 11, …. Calculate the 9th term 9U and the sum of 9 terms 9S
3. In the arithmetic sequence 21, 17,13,…
a) Calculate 10u b) Calculate 20S
4. In the arithmetic sequence 2, 5, 8, …. 50. Calculate the number of terms.
5. The nth term of an arithmetic sequence is given by nun 35+=
a) Write down the first 3 terms of the sequence.
b) Calculate the sum of the first 12 terms.
6. The 5th term of an arithmetic sequence is 31 and the 12th term is 80[ Hint - form two
simultaneous equations and solve them]
a) Calculate the first term a, and the common difference d.
b) Calculate the sum of the first 12 terms 12S
7. There are 20 terms in an arithmetic sequence. First term is -7, last term 240
a) Calculate the common difference d.
b) Calculate the sum of the first 10 terms 10S
8. On Alice’s 11th birthday she started to receive an annual allowance. The first annual
allowance was £500 and on each following birthday the allowance was increased by £200.
(a) Show that, immediately after her 12th birthday, the total of the allowances that Alice
had received was £1200. [Find 2S ] (1)
(b) Find the amount of Alice’s annual allowance on her 18th birthday. [Find 8U ] (2)
(c) Find the total of the allowances that Alice had received up to and including her 18th
birthday. [Find 8S ] (3)
When the total of the allowances that Alice had received reached £32 000 the allowance
stopped.
(d) Find how old Alice was when she received her last allowance. [If 32000=nS , Solve to
find n, then work out Alice’s age, see answers for more help] (7)
[C1 Jan 2006]
HW7 HW7
9 | p a g e
Extension
1. Sue is training for a marathon. Her training includes a run every Saturday starting with
a run of 5 km on the first Saturday. Each Saturday she increases the length of her run
from the previous Saturday by 2 km.
(a) Show that on the 4th Saturday of training she runs 11 km. (1)
(b) Find an expression, in terms of n, for the length of her training run on the nth
Saturday. (2)
(c) Show that the total distance she runs on Saturdays in n weeks of training is
n(n + 4) km. (3)
On the nth Saturday Sue runs 43 km.
(d) Find the value of n. (2)
(e) Find the total distance, in km, Sue runs on Saturdays in n weeks of training. (2)
[C1 May 2008]
2. Jill gave money to a charity over a 20-year period, from Year 1 to Year 20 inclusive. She
gave £150 in Year 1, £160 in Year 2, £170 in Year 3, and son on, so that the amounts of
money she gave each year formed an arithmetic sequence.
(a) Find the amount of money she gave in Year 10. (2)
(b) Calculate the total amount of money she gave over the 20-year period. (3)
Kevin also gave money to charity over the same 20-year period.
He gave £A in Year 1 and the amounts of money he gave each year increased, forming an
arithmetic sequence with common difference £30.
The total amount of money that Kevin gave over the 20-year period was twice the total
amount of money that Jill gave.
(c) Calculate the value of A. (4)
Answers
1. See text book 2. 35, 171
3. -15, -340 4. n= 17
5. 8, 11, 14; 294 6. a=3, d= 7; 498
7. 13 515
8. b) 1900£8 =U c) 9600£8 =S d) ])1(2[2
32000 dnan
−+=
032042 =−+ nn
0)20)(16( =+− nn
16=n So Alice 26 years
Extension 1. b) nUn 23 += d) 20=n e) 480km http://youtu.be/tBOcXpkiyEg
2. a) £240, b) £4900, c) £205 http://youtu.be/pOuRQ6MUwpU
10 | p a g e
HW8 General Sequences
Complete on a separate sheet of paper and show clear working. Mark using the
answers below.
Keywords: Sequence, Arithmetic, Geometric, Converge, Diverge, Oscillating,
Periodic, Increasing, Decreasing, Recurrence relation
Read the chapter on Sequences in your text book (p93 – 104),
1. Write down the first five terms in each sequence
a) 231 +=+ nn uu 41 =u b) nn uu 231 −=+ 51 =u
c) 32
1 −=+ nn uu 21 =u d) 2
1n
n
uu =+ 101 =u
2. Calculate the follow sums written in ‘Sigma’ notation
a) ∑=
5
1
4r
r b) ∑=
7
5
2
r
r c) ∑=
−10
5
35r
r d) ∑=
−5
2
1)1(
r
r
r
3. In the recurrence relations below find 6,2,1 uuu K for different starting values of 1u .
Use your calculator with ANS button!
a) 8210
2311
2
1 ==+
=+ uoruu
u nn
.
b) 822
2
111 ==
+
=+ uoruu
u
un
n
n. Do you recognise this number?
Extension
1. A sequence of numbers a1, a2, a3… is defined by
an + 1 = 3an – 5, n ≥ 1.
Given that a2 = 10,
(a) find the value of a1. (2)
(b) Find the value of ∑=
4
1r
ra . (3) [adapted from C1 May 2014]
2. A sequence u1, u2, u3, ..., satisfies
un + 1 = 2un − 1, n ≥ 1.
Given that u2 = 9,
(a) find the value of u3 and the value of u4 , (2)
(b) evaluate ∑=
4
1r
ru . (3) [C1 January 2013]
Answers 1. a) 4, 14, 44, 134, 404 b) 5, -7, 17, -31, 65 c) 2, 1, -2, 1, -2, 1 d) 10,5,2.5,1.25, 0.625 2. a) 60 b) 110 c) 38 d) 13/60 3. a) 2, 2.7, 3.029, 3.217, 3.335, 3.412 b) 8, 8.7, 9.869, 12.040, 16.795, 30.509 c) 2, 1.5, 1.417, 1.414, 1.414, 1.414 d) 8, 4.125, 2.305, 1.586, 1.424, 1.414 Ext. 1) a) 5 b) 110 2) a) 17, 33 b) 64
HW7 HW8
11 | p a g e
HW9 Coordinate Geometry
Complete on a separate sheet of paper and show clear working. Mark using the
answers below.
Key words: gradient, perpendicular, normal y=mx +c, ax + by +c =0
)( 11 xxmyy −=− equation of line, reciprocal, midpoint, distance, gradient,
perpendicular, normal
This work is covered in your text book pages 65-84. If you are in difficulties read the book,
talk to a friend, visit Maths workshop (lunchtime room 216).
1. Using the points A(2,1) B(6,4) calculate the following. You may find a sketch
helpful.
a) Coordinates of the Midpoint
b) Distance between points
c) Gradient of the line
d) Gradient of perpendicular line
2. The lines below have the following equations. = �, = −�, = 3, � = −4, = −3�, =
�
�� Which is which?
3. Find the equations of the lines from a given gradient and a point. Make sure you
rearrange the equations so that they match the answers given.
a) m=3, (2, 8) b) m=5, (1, 1)
c) m=-3 (1, 2) d) m= -2/3 (7, 0)
e) m=7 (2, - 4)
HW9
A
D
B C
E
F
12 | p a g e
4. Find the equations of the lines from two given points.
a) A(2, 4) B(6, 6) b) A(-2,11) B(1, -1)
c) A(2, -2) B(4, 8) d) A(-3, 0) B(9, 2)
5. Calculate the point of intersection of the two lines [Hint: try simultaneous equations]
52
153
=−
=+
yx
yx
Extension
1. The points A(2, 3) B(10, 7) C(5, -3) are vertices of a triangle.
Plot the points on square paper
a) Calculate gradient AB and gradient AC. What do you notice?
b) Find the exact length of the shortest side.
c) Calculate the area of the triangle as an exact answer.
2. The line l1 has equation 3x + 5y – 2 = 0.
a) Find the gradient of l1.
The line l2 is perpendicular to l1 and passes through the point (3, 1).
b) Find the equation of l2 in the form y = mx + c, where m and c are constants.
[Jan 2010]
Answers
1. a) (4,�
�) b) 5
c) �
� d) −
�
�
2. Teacher to mark
3. a) 23 += xy b) 45 −= xy
c) 53 =+ yx d) 1432 =+ yx
e) 187 −= xy
4. a) 321 += xy b) 34 +−= xy
c) 0125 =−− yx d) 036 =+− yx
5. (4, 3)
Ext: 1. a) 1/2, -2 b) 3√5 c) 30 sq units 2. a) � = −
�
� b) =
�
�� − 4
13 | p a g e
HW 10 Tranformations and Inequalities
Complete on a separate sheet of paper and show clear working. Mark using the
answers below. Remember to write your Name, Title, Date.
Keywords: Transformation, scale factor, translation, reflection, inequality, less
than or equal, more than. OR AND.
Transformations [typical exam questions]
Read pages 51-59 . Write a short summary to remind you of the key points.
1. This diagram shows a sketch of the curve with
equation y = f(x). The curve crosses the x-axis at
the points (2, 0) and (4, 0). The minimum point on
the curve is P(3, –2).
In separate diagrams sketch the curve with equation
(a) y = –f(x), (3)
(b) y = f(2x). (3)
On each diagram, give the coordinates of the points at which the curve crosses the x-
axis, and the coordinates of the image of P under the given transformation.
2. The figure above shows a sketch of the
curve with equation y = f(x). The curve
passes through the points (0, 3) and (4,
0) and touches the x-axis at the point (1,
0).
On separate diagrams sketch the curve
with equation
(a) y = f(x + 1), (3)
(b) y = 2f(x), (3)
(c)
(3) On each diagram show clearly the coordinates of all the points where the curve meets
the axes.
3. Given that f(x) =x
1, x ≠ 0,
(a) sketch the graph of 3)(f += xy and state the equations of the asymptotes. (3)
(b) find the coordinates of the point where 3)(f += xy crosses a coordinate
axis. (2)
Inequalities
Read page 37
1. What do the following diagrams show? Write them as inequalities
See next page
.2
1f
= xy
HW10
O 2 4
P(3, –2)
x
y
(0, 3)
(4, 0)
(1, 0)O x
y
14 | p a g e
−6 −4 −2 2 4 6
2
4
6
x
y
a) b) c)
d) e) f)
2. Solve the following linear inequalities.
a) 83 >+x b) 2243 ≤+x c) x9826 −≥
d) xx 2133)73(2 −>+− e) 623
4<+
+x
Solve the following quadratic inequalities. Draw a sketch of each graph.
3. a) 022 >−− xx b) 0342 <+− xx c) 042 >+− x
d) xx 523 2 −≥ e) 0)5)(2)(1( <−−+ xxx
Extension
1. Solve the inequalities.
a) 33789 <−≤ x b) 20)2(5 −>−x AND 0)3(7 <−x
2. [C1 May 2007]The equation x2 + kx + (k + 3) = 0, where k is a constant, has different real
roots.
(a) Show that 01242 >−− kk .
(2)
(b) Find the set of possible values of k.
(4) Answers Transformations
1.a) b)
2. a) b) c)
3. b) (-3
1,0) a) y=3, x=0
Inequalities
1. a) 3<x b) 4−≤x c) 9≥x
d) 3<x OR 7>x e) 6−≤x OR 1−≥x f) 52 ≤≤ x
2. a) 5>x b) 6≤x c) 2−≥x
d) 3>x e) 8<x
3. a) 1−<x OR 2>x b) 31 << x c) 22 <<− x
d) 2−≤x OR 3
1≥x e) 1−<x OR 52 << x
Extension
1. a) 52 <≤ x b) 32 <<− x 2. 2−<k OR 6>k
2 4
(3, 2)
11
2
2(1 , –2)
3 4
6
1
3
2 8
3 -4 9
3 -1 7 -6 2 5
15 | p a g e
HW 10 Differentiation 1
Complete on a separate sheet of paper and show clear working. Mark using the
answers below.
Keywords: Function, derivative, gradient, calculus, differentiate
Remember to sketch graphs in pencil. Do you have a well-stocked pencil case? Get
organised! Read the chapter in your book and look at the examples.
1. Do some research - who invented (or discovered) calculus? When did they do it?
Why did they do it? Read pages 115-122 in your book.
2. Differentiate the following functions to find dx
dy
a) 29
3xxy += b) 53 += xy
c) 4−= xy d) 3
4
18xy =
e) 1176
31 ++= xxy f)
2
4 15
xxy +=
3. Expand, simplify then find )(' xf
a) 2)4()( −= xxf b) )6)(5)(3()( −++= xxxxf
c) 2
36 57)(
x
xxxf
+= d) )52()( 2 xxxxf +=
4. 2
)2)(1( −+= xxy
a) sketch the curve
b) expand the function
c) find the derivative of the function
d) complete the table (right).
e) what do you notice about the graph when 0=dx
dy
Do your answers for the gradient make sense in terms of your sketch. Are they likely to be
correct?
Extension
1. Differentiate with respect to x
(a) x4 + 6√x,
(b)
(c) x
xx376 +
2. Go to www.mathsnetalevel.com (username: cityisli , password: ask teacher)
Click C1 -> 6. Differentiation
Try: Matching O-test and have a look at some of the other resources.
x
x2)4( +
x -2 -1 0 1 2 3 4
y
dx
dy
HW11
16 | p a g e
Answers
2. a) xxdx
dy69 8 += b) 3=
dx
dy c)
5
4
xdx
dy −= d) 3
1
24xdx
dy=
e) 72 5 += xdx
dy f)
3
3 220
xx
dx
dy−=
3. a) 82)(' −= xxf b) 3343)(' 2 −+= xxxf c) 528)(' 3 += xxf
d) 2
1
2
3
2
155)(' xxxf +=
4. a) b) 43 23 +−= xxy
c) xxdx
dy63 2 −=
d)
e) 0=dx
dy at the turning points of the graph.
Extension
1. a) 2
1
3 34−
+= xxdx
dy b) 2161 −−= x
dx
dy c)
23
21
2
353 xx
dx
dy+=
−
x -2 -1 0 1 2 3 4
y -16 0 4 2 0 4 20
dx
dy 24 9 0 -3 0 9 24
17 | p a g e
HW12 Differentiation 2
Complete on a separate sheet of paper and show clear working. Mark using the
answers below.
Keywords: Derivative, gradient, tangent, normal, differentiate
Remember to check your work as you go along. Mark your own work using the following
symbols: correct �, wrong (but don’t know why) �, no idea????????
1. 108)(2 +−= xxxf
a) Find )(' xf at the point x = 4 [This is the same as )4('f ]
b) Write )(xf in the form )(xf = bax ++ 2)(
c) What do your answers to parts a and b tell you about the graph?
2. )1)(2)(23( −++= xxxy
a) Sketch the graph of y, giving the coordinates of where the curve crosses the axes.
[Your sketch should be half A4 page and carefully drawn]
b) Find the y coordinate where x = −1 and show it on your graph.
c) Expand the expression for y.
d) Differentiate y in terms of x.
e) Find the gradient at x = −1.
f) Work out the equation of the tangent at x = −1 and write it in the form
y = mx + c. g) Draw the tangent on the graph using the equation you have just found.
3. a) Find the equation of the tangent to the graph 354
−+= xx
y at the point x = 2.
b) Find the equation of the normal to the graph 174624 ++−= xxxy at
the point x = −2.
Extension
1. The curve C has equation y = x3 – 4x2 + 8x + 3.
The point P has coordinates (3, 0).
(a) Show that P lies on C. (1)
(b) Find the equation of the tangent to C at P, giving your answer in the form
y = mx + c, where m and c are constants. (5)
Another point Q also lies on C. The tangent to C at Q is parallel to the tangent to
C at P.
(c) Find the coordinates of Q. (5)
[C1 May 2005]
31
HW12
18 | p a g e
2. The curve C has equation
y = (x + 3)(x – 1)2.
(a) Sketch C, showing clearly the coordinates of the points where the curve meets the
coordinate axes.
(4)
(b) Show that the equation of C can be written in the form
y = x3 + x
2 – 5x + k,
where k is a positive integer, and state the value of k. (2)
There are two points on C where the gradient of the tangent to C is equal to 3.
(c) Find the x-coordinates of these two points.
(6)
[C1 Jan 2008]
Answers
1. a) 0
b) 6)4()( 2 −−= xxf
c) The gradient at the minimum point is 0
2. a) curve, g)straight line
3. a) ( )9,2
14 += xy b)
( )1,2−
064 =+− yx
Extension
1. b) y = –7x + 21 c)
−
3
46,5
2. a)
b) = �� � �� − 5� � 3
c) � � �� , �2 [see examsolutions.net C1 Jan 2008]
420-2-4
10
8
6
4
2
0
-2
-4
420-2-4
10
8
6
4
2
0
-2
-4
420-2-4
10
8
6
4
2
0
-2
-4
420-2-4
10
8
6
4
2
0
-2
-4
b) y = 2
c) y = 3x3 + 5x
2 − 4x − 4
d) 4109 2 −+= xxdx
dy
e) 5−=dx
dy
f) y = −5x −3
y
x
19 | p a g e
HW13 Integration
Complete on a separate sheet of paper and show clear working. Mark using the
answers below.
Keywords: Differential equation, integrate, anti-derivative, general solution,
particular solution
Read pages 142-147.
This topic is called Integration but is also about solving differential equations. This is some
of the notation you will see.
���� � ��, �ℎ!" � �
�#� ��#� � $ %′(�) � ��, �ℎ!"%(�) � 1
" � 1��#� � $ '��(� � 1
" � 1��#� � $
Exercise A
Find an expression for y.
1. ���� � �) 2. ���� � 15��
3. ���� � 8�*� 4.
���� � 6�+
,
Solve to find %(�) 5. %-(�) � 12� � 5 6. %-(�) � 10 ���.
Integrate the following
7. /(� � 5)� (� 8. / 7�*+. (�
Exercise B
Solve to find the particular solution y passing through the given point.
1. ���� � 8�, (�3, 15) 2.
���� � 5, (2, 3)
2. ���� � 3�� � 5, (2, 7) 2.
���� � �� � 8� � ��
�. , (3, 8)
Extension
1. The curve C has equation y = f(x), x > 0, and f ′(x) = 4x – 6√x + 2
8
x.
Given that the point P(4, 1) lies on C,
(a) find f(x) and simplify your answer. (6)
(b) find an equation of the normal to C at the point P(4, 1).
(4)
[C1 Jan 2008]
http://youtu.be/i4jKwj3aXEg
http://youtu.be/N_BYWaJhw8Y
HW13
family of
solutions
20 | p a g e
2. The gradient of a curve C is given by x
y
d
d =
2
22 )3(
x
x +, x ≠ 0.
(a) Show that x
y
d
d = x
2 + 6 + 9x
–2.
(2)
The point (3, 20) lies on C.
(b) Find an equation for the curve C in the form y = f(x). (6)
[C1 June 2008]
http://youtu.be/boKDsAa7bXk
Further Study:
Differential Equations
You have just been solving your first differential equations. More complicated versions of
this are used for modelling in Engineering, Physics, Chemistry, Economics and Meteorology.
Here are some important examples that you can read about.
Physics and engineering
Maxwell’s equations in electromagnetism
Navier-Stokes equations in fluid mechanics
Newtons law of cooling
Simple harmonic motion
Economics and Finance
The Black–Scholes PDE
Malthusian population growth model
Use a computer graph package like Omnigraph to investigate the family of curves generated
by the following differential equations. Ask your teacher!
1. ���� � 3�� � 4� 2. ���� � � �
� 3. ���� � � �
Answers
ExA 1) � �01 � $ 2) � 3�� � $ 3) � � �
�. � $ 4) � 2� �
3, � $
5) f(x) � 6x� � 5x � c 6) %(�) � 10� � �� � $ 7)
�,� � 5�� � 25� � $ 8) 14�+
. � $ ExB 1) y � 4x� � 21 2) y � 5x � 7 3) y � x� � 5x � 11
4) y � 8,� � 4x� � ��
8 � 30
Ext 1) a) %(�) � 2�� � 4�,. � 9
� � 3 b) 2� � 9 � 17 � 0 2) b) � �,
� � 6� � 2� � 4
To practise further questions in text book, see Exercise 5D page 150.
21 | p a g e
Edexcel C1 May 2010 Here is a C1 exam for you to use as revision for your exam week w/b 8th December 2014.
Please refer to www.examsolutions.net for answers and video solutions. You can also find
more past papers here.
May 2010 C1 EDEXCEL TIME 1 HOUR 30 MINUTES TOTAL 75 MARKS
1. Write
√(75) – √(27)
in the form k √x, where k and x are integers.
(2)
2. Find
⌡
⌠−+ xxx d)568( 2
1
3 ,
giving each term in its simplest form.
(4)
3. Find the set of values of x for which
(a) 3(x – 2) < 8 – 2x,
(2)
(b) (2x – 7)(1 + x) < 0,
(3)
(c) both 3(x – 2) < 8 – 2x and (2x – 7)(1 + x) < 0.
(1)
4. (a) Show that x2 + 6x + 11 can be written as
(x + p)2 + q,
where p and q are integers to be found.
(2)
(b) Sketch the curve with equation y = x2 + 6x + 11, showing clearly any intersections with the
coordinate axes.
(2)
(c) Find the value of the discriminant of x2 + 6x + 11.
(2)
5. A sequence of positive numbers is defined by
1+na = √( 2
na + 3), n ≥ 1,
1a = 2.
HW14
22 | p a g e
(a) Find 2a and 3a , leaving your answers in surd form. (2)
(b) Show that 5a = 4.
(2)
6.
Figure 1
Figure 1 shows a sketch of the curve with equation y = f(x). The curve has a maximum point A at
(–2, 3) and a minimum point B at (3, – 5).
On separate diagrams sketch the curve with equation
(a) y = f (x + 3),
(3)
(b) y = 2f(x).
(3)
On each diagram show clearly the coordinates of the maximum and minimum points.
The graph of y = f(x) + a has a minimum at (3, 0), where a is a constant.
(c) Write down the value of a.
(1)
7. Given that
y = 8x3 – 4√x +
x
x 23 2 +, x > 0,
find x
y
d
d.
(6)
23 | p a g e
8. (a) Find an equation of the line joining A(7, 4) and B(2, 0), giving your answer in the form
ax + by + c = 0, where a, b and c are integers.
(3)
(b) Find the length of AB, leaving your answer in surd form.
(2)
The point C has coordinates (2, t), where t > 0, and AC = AB.
(c) Find the value of t.
(1)
(d) Find the area of triangle ABC.
(2)
9. A farmer has a pay scheme to keep fruit pickers working throughout the 30 day season. He pays
£a for their first day, £(a + d ) for their second day, £(a + 2d ) for their third day, and so on, thus
increasing the daily payment by £d for each extra day they work.
A picker who works for all 30 days will earn £40.75 on the final day.
(a) Use this information to form an equation in a and d.
(2)
A picker who works for all 30 days will earn a total of £1005.
(b) Show that 15(a + 40.75) = 1005.
(2)
(c) Hence find the value of a and the value of d.
(4)
10. (a) On the axes below sketch the graphs of
(i) y = x (4 – x),
(ii) y = x2
(7 – x),
showing clearly the coordinates of the points where the curves cross the coordinate axes.
(5)
(b) Show that the x-coordinates of the points of intersection of
y = x (4 – x) and y = x2
(7 – x)
are given by the solutions to the equation x(x2 – 8x + 4) = 0.
(3)
The point A lies on both of the curves and the x and y coordinates of A are both positive.
(c) Find the exact coordinates of A, leaving your answer in the form (p + q√3, r + s√3), where p,
q, r and s are integers.
(7)
24 | p a g e
11. The curve C has equation y = f(x), x > 0, where
x
y
d
d = 3x –
x√
5 – 2.
Given that the point P (4, 5) lies on C, find
(a) f(x),
(5)
(b) an equation of the tangent to C at the point P, giving your answer in the form
ax + by + c = 0, where a, b and c are integers.
(4)
TOTAL FOR PAPER: 75 MARKS
END
