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ALGEBRA S J Cooper

Foundation &

Higher Tier

thomaswhitham.pbworks.com

S J Cooper

Algebra (1) Collection of like terms

1. Simplify each of the following expressions

a) aaa 53

b) mmm 73

c) xxx 573

d) ddd 627

e) nnn 735

f) ppp 79

g) ttt 342

h) jj 4

i) kkkk 836

j) rrrr 4104

2. Simplify each of the following expressions

a) baba 432

b) dcdc 434

c) yxyx 37

d) 654 ee

e) 1782 ww

f) stst 3235

g) jiji 5236

h) qpqp 638

i) hggh 43610

j) vuvu 5722

3. Simplify

a) xyxyx 34236

b) aaa 65275

c) nmnm 253

d) qpqp 51025

e) uvuv 736

f) cccc 437

g) tsst 3549

h) efef 5234

i) srsr 364

j) hkhk 812715

4. Simplify

a) 5342 xxx

b) xxxx 423 22

c) 22 3523 xxxx

d) 52364 22 xxxx

e) 72495 22 pppp

f) 1354 22 yyyy

g) 53145 22 xxxx

h) 32344 22 rrrr

i) 74587 2 eee

j) 22 623 hhhh

S J Cooper

Algebra(2) Solving simple equations

1. Solve each of the following equations

a) 1752 x

b) 1923 x

c) 4275 p

d) 482 y

e) 2514 m

f) 12186 a

g) 10163 h

h) 3172 q

i) 3554 b

j) 5949 x

k) 573 d

l) 4165 t

m) 872 k

n) 3156 y

o) 1257 n

p) 1594 r

q) 1173 e

r) 194 w

s) 1878 j

t) 1635 x


a) xx 3534

b) 9375 pp

c) 8518 mm

d) 7332 ff

e) 8347 uu

f) xx 953

g) xx 33872

h) 12764 xx

i) 17335 rr

j) 1431410 ee

k) 25258 yy

l) 16231 xx

m) 7638 xx

n) 653 xx

o) 25912 bb

p) 43631 tt

q) mm 3815

r) 2185 kk

s) uu 4367

t) 241126 hh

3. Solve the following equations

a) 72

x

b) 43

y

c) 67

2

p

d) 25

3

a

e) 34

6

b

f) 92

5

m

g) 53

32

x

h) 44

13

x

i) 23

104

x

j) 34

183

p

k) 927

x

l) 654

y

m) 243

x

x

n) 395

r

r

o) 23

87

v

v

p) 8

658

b

q) 3

217

f

r) 67

42

g

s) 44

310

r

t) 87

29

t

S J Cooper

Algebra (3) Removal of brackets

1. Remove the brackets for each of the following:

a) 134 x

b) 325 x

c) 156 m

d) 743 y

e) 762 p

f) x545

g) 5210 x

h) ba 27

i) yx 324

j) pnm 253

k) tr 3215

l) 2xx

m) 32 xx

n) 54 aa

o) xx 212

p) 53 mm

q) 437 xx

r) bb 45

s) yxy 2

t) cbaa 232

u) pqp 25

v) tsrr 546

w) 533 nmm

x) 4532 fee

y) 38 yx

z) jih 73

2. Remove the brackets and simplify each of the following:

a) 23 xx

b) 14 xx

c) 25 xx

d) 36 xx

e) 110 xx

f) 34 xx

g) 21 xx

h) 33 xx

i) 76 xx

j) 22 xx

k) 61 xx

l) 18 xx

m) 14 xx

n) 77 xx

o) 52 xx

p) 312 xx

q) 212 xx

r) 123 xx

s) 133 xx

t) 3223 xx

S J Cooper

Algebra (4) Factorisation common factor

Factorise each of the following

(1) 93 a

(2) 164 m

(3) 128 p

(4) 2012 x

(5) 615 r

(6) ba 123

(7) gf 814

(8) ts 144

(9) zy 216

(10) ji 3528

(11) nm 6327

(12) wvu 482

(13) pnm 251510

(14) gfe 1239

(15) cba 442016

(16) xx 22

(17) yy 42

(18) xx 42 2

(19) yy 93 2

(20) 22 20155 mnmnnm

S. J. Cooper

Algebra (5) Factorisation

1. Factorise each of the following:

a. 342 xx

b. 862 xx

c. 1582 xx

d. 652 xx

e. 862 xx

f. 35122 xx

g. 1282 xx

h. 1452 xx

i. 1522 xx

j. 202 xx

k. 442 xx

l. 452 xx

m. 652 xx

n. 862 xx

o. 42132 xx

p. 1662 xx

q. 202 xx

r. 2142 xx


a) 132 2 xx

b) 443 2 xx

c) 4133 2 xx

d) 168 2 xx

e) 672 2 xx

f) 656 2 xx

g) 994 2 xx

h) 384 2 xx

i) 14 2 x

j) 122512 2 xx


a. 076 xx

b. 022 xx

c. 0612 xx

d. 018 xx

e. 01243 xx

4. (a) Factorise 342 xx

(b) Hence, or otherwise solve the equation 0342 xx

5. (a) Factorise 442 xx

(b) Hence, or otherwise solve the equation 0442 xx

6. Solve each of the following equations:

a)

b)

c)

S. J. Cooper

7. (a) Factorise 15812 2 xx

(b) Hence, or otherwise solve the equation 015812 2 xx






a) 023 2 xx

b) 0992 2 xx

c) 0274 2 xx

d) 092110 2 xx

e) 08143 2 xx

f) 01544 2 xx

g) 05136 2 xx

h) 0212 2 xx

i) 0299 2 xx


a) 24 4 15x x

b) 23 2 6 4x x

c) 212 1 13 4x x

d) 22 5 5 30 4x x x

e) 28 8 15 6x x x

S J Cooper

Algebra (6)


a) 2 9x

b) 2 49x

c) 24 49x

d) 29 25x

e) 216 25y

f) 2 281 16a b

g) 2 136

4x

h) 2 425

9k

i) 2 250 32u v

j) 2 227 108c d

2. Complete the square for each of the following:

a) 2 4 8x x

b) 2 6 1x x

c) 2 6 3x x

d) 2 10 7x x

e) 2 8 5y y

f) 2 3 4x x

g) 2 5 7a a

h) 22 4 5x x

i) 23 6 4x x

j) 22 8 7x x

3. Solve each of the following equations by completing the square, leaving your answer as a surd:

a) 2 4 9 0x x

b) 2 10 1 0x x

c) 2 3 1 0x x

d) 2 4 8 0b b

e) 2 6 2 0e e

f) 2 14 10 0y y

g) 2 8 12 0x x

h) 2 10 24 0c c

i) 2 12 27 0x x

j) 2 20 44 0x x

4. Using the formula, giving answers to 2 decimal places where appropriate, solve each of the following:

a) 2 5 4 0x x

b) 2 4 2 0x x

c) 2 9 7 0x x

d) 22 2 3 0x x

e) 22 3 6 0y y

f) 23 7 2 0x x

g) 24 3 5 0a a

h) 27 5 6 0b b

i) 23 10 6 0r r

j) 22 9 4 0u u

S J Cooper

Algebra (7) Substitution into formulae

Exercise 1

Answer each of the following questions without the use of a calculator.

1. If TSD , find the value of D when 4S and 15T .

2. Using the formula blA , find (i) A when 7l and 9b

(ii) b when 6l and 102A

3. If 53 nS , find the value of S when (i) 3n (ii) 6n (iii) 20n

4. Given blP 22 , find (i) P when 7l and 5b

(ii) b when 2l and 18P

5. Using the formula atuv find v when (i) 3u , 10a and 4t

(ii) 20u , 2a and 6t

6. Given V

MD find (i) D when 120M and 8V

(ii) V when 91M and 7D

7. When xy 20 , find y when (i) 4x (ii) 25x (iii) 81x

8. Evaluate 25 n when (i) 1n (ii) 8n (iii) 15n

9. Find the value of 3

25 h when (i) 8h (ii) 11h

10. Find the value of 390 f when (i) 4f (ii) 8f

S J Cooper

Exercise 2

Answer each of the following questions with the use of a calculator.

1. Find the value of a when

t

vua

, given that 6.5u , 6.3v and 4t

2. Using the formula 2mcE , find the value of E given that 15m and 7c

3. Using the formula

2

tvuS

, find S when 17u , 23v and 5t .

4. Using the formula b

axz

find the value of z when 3.97x , 95a and 5.2b

5. If l

mxT find the value of T when 50x , 6.19m and 120l

6. If mumvC find C when 3m , 9v and 3u

7. Find the value of T when

6

21

nnnT and (i) 4n (ii) 7n

8. Where 22 yxP find P when (i) 12x and 5y

(ii) 8x and 14y

9. Using the formula asuv 22 obtain the value of v when 10u , 5.2a and 25s

10. Work out the value 2

2

1atutS when 1.4u , 8.9a and 10t

S J Cooper

Algebra (8) Forming Equations

1. I think of a number double it and subtract three. If I then have five what was number was I

thinking of?

2. Najid thinks of a number, multiplies it by five and subtracts 12 his answer is 23. What number

did Najid start with?

3. Susan was told to think of a number. Then she was asked to multiply her number by four and

add six. When asked what was her answer she gave 34. What number did Susan think off?

4. The perimeter of the rectangle below is 62cm work out the value of a.

5. Given that the perimeter of the triangle below is 71cm work out the value of x.

6. If the perimeter of the regular pentagon below has the same perimeter as the rectangle find the

value of a.

7. John is 3 years older than Janet. However in 10 years time John will be double Janet’s age. If

John is x years old. Find x.

8. Given that the area of the rectangle opposite is 36cm2, find y.

3x+6

5x

x – 7

a

2a

17

y

4

a

S J Cooper

9. Given that the area of the triangle opposite is 48m2, find p.

10. Given that the area of the 40cm2 find the two possible values for x.

11. Given that the area of the rectangle below is 27cm2 find the one possible values for x.

12. Given that the volume of the cuboid is 240cm3, find the one

possible values for x.

p

8

x

(x – 13)

x

(x – 6)

x

5

S J Cooper

Algebra (9) Simultaneous Equations I

Exercise 1

Solve each of the following sets of simultaneous equations. Show your working.

1. 92

133

yx

yx

2. 423

825

yx

yx

3. 73

144

yx

yx

4. 532

835

ba

ba

5. 22

04

dc

dc

6. 443

1247

ih

ih

7. 115

1858

yx

yx

8. 1632

237

nm

nm

9. 92

113

tu

tu

10. 1974

875

yx

yx

Exercise 2

Solve each of the following sets of simultaneous equations. Remember to show your working.

1. 72

52

ba

ba

2. 534

73

yx

yx

3. 82

2045

dc

dc

4. 02

856

ut

ut

5. 1352

534

fe

fe

6. 1965

732

sr

sr

7. 1665

823

qp

qp

8. 2

1623

kh

kh

9. 1658

1234

hg

hg

10. 495

132

yx

yx

11. 1123

197

ba

ba

12. 249

7815

vu

vu

13. 2665

1123

ji

ji

14. 3485

1229

qp

qp

15. 1257

23

kk

jk

16. 11103

452

ed

ed

17. 287

923

nm

nm

18. 867

829

yx

yx

S J Cooper

Exercise 3

1. Two numbers have a sum of 18 and a difference of 6. Find these two numbers.

2. Find two numbers which add to give 17 and have a difference of 8.

3. Three times one number added to twice another gives 21. If the difference between the two

numbers is 7 find the two numbers.

4. Alan and Sarah have £15 saved between them. If Alan has £3 more than Sarah how much

does each person have?

5. A bag contains 30 counters red and blue. If there are five more blue counters than red how

many red counters are there?

6. The cost of six pens and four pencils in the print room is £1.26, whereas the cost of five pens

and two pencils is 93pence. Work out the cost of a pen and a pencil.

7. For a recent concert Mohammed sold 30 tickets raising £79 for school funds. If the tickets cost

£2 per child and £3 per adult, how many tickets did he sell to adults?

8. The cost of three pears and seven apples is £1.41. If the difference between the two prices is

3pence find the cost of each item.

9. For a recent holiday the cost of two adults and three children was

quoted for £1220. Whereas for two adults and six children the price

was £1760. What is the price per child and per adult?

10. Angela has £10 spending money.

In a big department store Angela can by four lipsticks and three

different nail polishes for £9.75. Or alternatively she could buy

one lipstick and six nail polishes for £9. What is the cost of each

item?

S J Cooper

Algebra (10) Simultaneous Equations II

Exercise 1


1. 3 5 1

5 2 8

x y

x y

2. 2 3 14

3 4 19

a b

a b

3. 4 3 1

3 2 5

c d

c d

4. 5 3 1

4 2 14

t u

t u

5. 3 2 0

4 3 1

e f

e f

6. 7 3 3

5 2 1

r s

r s

7. 3 5 4

4 7 4

p q

p q

8. 6 3 30

4 5 6

h k

h k

9. 4 3 6

3 4 1

g h

g h

10. 5 6 7

3 8 2

x y

x y

Exercise 2

1. (a) Draw the graph of 122 xxy for values of x between –5 and 3

(b) On the same set of axes draw the straight line with equation 62 xy

(c) Hence solve the set of simultaneous equations 122 xxy and 62 xy


(b) On the same set of axes draw the straight line with equation 4 yx

(c) Hence solve the set of simultaneous equations 322 xxy and 4 yx

3. By drawing suitable graphs solve the set of simultaneous equations given by 2 2 3y x x and

12 xy

S J Cooper

Algebra (11) Simultaneous Equations III


1. 2 3 1

2

y x x

y x

2. 12

7

xy

yx

3. 113

032

xy

yx

4.

xy

yx

34

152

5. 62

8

xy

xy

6. 11

12

yx

xy

7. 22

132 2

xy

xxy

8. xy

yx

21

1322

9. 432

2922

yx

yx

10. 23

2022

xy

yx

11. 42

4

yx

xy

S J Cooper

Algebra (12) Drawing a straight line graph

1. For each of the following

(i) Copy and complete the table

(ii) draw the graph for the straight line.

(a)

x –4 0 4

13 xy

(b)

x –3 0 4

52 xy

(c)

x –3 0 3

35 xy

(d)

x –4 0 4

43 xy

(e)

x –4 0 4

14 xy

2. Draw the graphs for each of the following:

(a) 42 xy

(b) 52 xy

(c) 5y

(d) 26 xy

(e) 64 xy

(f) 108 xy

(g) 8 yx

(h) 122 yx

(i) 82 yx

(j) 1243 yx

(k) 3x

(l) 321 xy

S J Cooper

3. a) Draw on the same set of axes the graphs of 32 xy and 34 xy

b) Hence solve the simultaneous equations 32 xy and 34 xy

4. a) Draw on the same set of axes the graphs of 7 yx and 43 xy

b) Hence solve the simultaneous equations 7 yx and 43 xy

5. a) Draw on the same set of axes the graphs of 75 xy and 62 yx

b) Hence solve the simultaneous equations 75 xy and 62 yx

6. The graph drawn opposite represents 42 xy

Work out the coordinates of A and B

7. The graph drawn below represents 93 yx

Work out the coordinates of P and Q

8. The graph drawn below represents 62 yx

Work out the coordinates of E and F

A

B x

y

0

P

Q x

y

0

E

F

x

y

0

S J Cooper

Algebra (13) Gradients and Equations of lines

1. The graph below represents the graph of 63 xy

a) Find the coordinates of the points A and B

b) Hence find the gradient of the line.

2. a) Find the gradient of a line which passes through points 6,2 and 12,5

b) Find the gradient of a line which passes through the points 3,2 and 15,3

3. Write down the gradient and value of the y intercept for each of the following graphs.

a) 42 xy

b) 73 xy

c) 85 xy

d) 23 xy

e) 47 xy

f) 7 yx

g) 12 yx

h) 62 yx

i) 122 yx

j) 321 xy

4. Work out the gradient and the value for the y intercept of the graph with equation 123 xy

5. A straight line passes through the point 4,3 and has gradient equal to 2.

a) On a set of axes draw this graph.

b) Write down an expression for the gradient of the line

6. A straight line passes the points 3,1 and 18,4

a) Find the gradient of the line

b) Find an expression for the equation of the line

A

B x

y

0

S J Cooper







9. Find an expression for the equation of the straight line drawn below.

10. Find the equation of the line drawn below.

x

y

0

x

y

0

S J Cooper

Algebra (14) Drawing quadratics curves

1. (a) Copy and complete the table below for the graph of 42 xy for values of x from –3 to 3.

x –3 –2 –1 0 1 2 3

x2

– 4

y

(b) Hence draw the graph of 42 xy .

2. (a) Copy and complete the table below for the graph of xxy 42 for values of x from –3 to 3.

x –3 –2 –1 0 1 2 3

x2

–4x

y

(b) Hence draw the graph of xxy 42 .


x –3 –2 –1 0 1 2 3

x2

+3x

y



x –4 –3 –2 –1 0 1 2 3

x2

–x

y


S J Cooper

5. (a) Copy and complete the table below for the graph of 32 xy for values of x from –4 to 3.

x –4 –3 –2 –1 0 1 2 3

x2

–3

y

(b) Hence draw the graph of 32 xy .

6. (a) Copy and complete the table below for the graph of 62 xxy for x from –4 to 3.

x –4 –3 –2 –1 0 1 2 3

x2

+x

–6

y

(b) Draw the graph of 62 xxy . (c) Hence state the values at which 0y .


x –4 –3 –2 –1 0 1 2 3

x2

–5x

+4

y

(b) Draw the graph of 452 xxy .


x –3 –2 –1 0 1 2 3 4

x2

+4x

+3

y

(b) Draw the graph of 342 xxy .

(c) Hence state the coordinates where the curve meets the x-axis.

S J Cooper

Algebra (15) Quadratic equations

1. a) Copy and complete the table below for the graph of 52 xy

x –4 –3 –2 –1 0 1 2 3 4

y

b) Draw the graph of 52 xy for values of x between –4 and 4

2. a) Copy and complete the table below for the graph of 12 xxy

x –4 –3 –2 –1 0 1 2 3 4

y

b) Draw the graph of 12 xxy for values of x between –4 and 4


x –3 –2 –1 0 1 2 3 4

y


4. a) Copy and complete the table below for the graph of xxy 42

x –4 –3 –2 –1 0 1 2 3

y

b) Draw the graph of xxy 42 for values of x between –4 and 3


x –5 –4 –3 –2 –1 0 1 2 3

y


S J Cooper


x –4 –3 –2 –1 0 1 2 3 4

y

b) Draw the graph of 62 xy for values of x between –4 and 4

c) Use your graph to determine values of x when 2y

d) State the minimum value of y on the graph of 62 xy


x –4 –3 –2 –1 0 1 2 3 4

y


c) Use your graph to determine values of x when 5y

d) State the minimum value of y on the graph of 122 xxy


x –4 –3 –2 –1 0 1 2 3 4

y


c) On the same set of axes draw the graph of 12 xy

d) Using the graph solve the set of simultaneous equation 322 xxy and 12 xy

e) Hence show that the two solutions are the same for the equation 0242 xx


(d) On the same set of axes draw the straight line with equation 62 xy

(e) Hence solve the set of simultaneous equations 122 xxy and 62 xy


(d) On the same set of axes draw the straight line with equation 4 yx

(e) Hence solve the set of simultaneous equations 322 xxy and 4 yx

S J Cooper

Algebra (16) Drawing graphs


x – 3 –2 – 1 0 1 2 3

y 4 12

b) Hence draw the graph of 43 xy in the range 33 x

2. a) Copy and complete the table below for the graph of x

y10

x 0.1 0.5 1 2 3 4 5 6 7 8 9 10 20

y 20 1

b) Hence draw the graph of x

y10

in the range 201.0 x

c) Explain why it is not possible to find a value for y when 0x

3. a) Copy and complete the table below for the graph of xxy 33

x – 3 –2 – 1 0 1 2 3

y 0 – 2

b) Hence draw the graph of xxy 33 in the range 33 x

4. a) Copy and complete the table below for the graph of 2

1

xy

x 0.1 0.5 1 2 3 4 5 6 7 8 9 10 20

y 4 0.01

b) Hence draw the graph of 2

1

xy in the range 201.0 x

5. a) Copy and complete the table below for the graph of siny x

x 0 20 40 60 80 90 100 120 140 160 180 200 220 240 260 270 280 300 320 340 360

y 0 -0.6

b) Hence draw the graph of siny x in the range 0 360x

S J Cooper

6. Pair of the following equations with the graphs drawn below

a) 32 xy

b) 32 yx

c) x

y1

d) 3xy

e) 43 xy

A B C

D E

7. Draw a rough sketch to represent each of the following graphs

a) 52 xy

b) 73 yx

c) 73 xy

d) 103 xy

e) 2

4

xy

f) 1622 yx

g) 522 yx

x

y

x

y

x

y

x

y

x

y

S J Cooper

Algebra (17) Inequalities

1. Solve each of the following inequalities

a) 1312 x

b) 1023 x

c) 392 x

d) 1343 x

e) 7114 x

f) 4295 x

g) 183 x

h) 13154 x

i) 23127 x

j) 1373 x

2. Solve each of the following inequalities:

a) 14225 xx

b) rr 2817

c) tt 2253

d) uu 5375

e) yy 44383

f) 68125 ww

g) 14792 ff

h) 79410 kk

i) hh 4249

j) vv 61932

k) 218710 zz

l) xx 21992

3. Solve the following inequalities:

a) 18122 x

b) 21323 x

c) 9743 x

d) 20924 x

e) 65435 x

f) 21727 x

g) 10835 x

h) 22762 x

i) 1901410 x

j) 152033 x

4. Solve the following inequalities:

a) 2 25x

b) 2 81x

c) 2 49 0x

d) 21 0x

e) 2 4

9x

S J Cooper

Algebra (18) Inequalities II

11. Write down the inequality shaded for each of the following:

a) b)

c) d)

e) f)

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

S J Cooper

g) h)

i) j)

k) l)

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

S J Cooper

12. For each of the following draw the graph which best each inequality.

(a) 1x (b) 4y (c) 3x (d) 2y (e) 31 x

(f) 53 x (g) 02 y (h) 64 y (i) 14 x (j) 51 y

13. Write down an equation which fits each of the following shaded regions

a) b)

c) d)

e) f)

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

S J Cooper

g) h)

i) j)

k) l)

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

S J Cooper

14. Describe the shaded region in the diagram drawn below.

15. (a) Write down the equation of the three lines drawn around the shaded region below.

(b) Write down three inequalities which best describe the shaded region.

16. (a) On one set of axes draw the graphs of (i) 2x

(ii) 3y

(iii) 1 xy

(b) Shade in the region defined by the set of inequalities 2x , 3y , 1 xy

–5 –4 –3 –2 –1 1 2 3 4 x 0

1

2

3

4

y

–1

–2

–3

–4

–5

5 1 2 3 4 6 7 8 9 x

5

6

7

8

9

y

4

3

2

1

0

S J Cooper

17. (a) On one set of axes draw the graphs of (i) 4x

(ii) 1y

(iii) 12 xy

(b) Shade in the region defined by the set of inequalities 4x , 1y , 12 xy

S J Cooper

Algebra (19) Rearranging formulae

Make x the subject for each of the following

1. cmxy

2. 52

3

ax

3. yt

cax

4. e

d

x

t

5. tyx 22

6. cbxa

7. kgxfw

8. bx

M

9. 3

52

2

13

xx

10. dcxbax

Make y the subject for each of the following

11. myrtxy

12. a

f

y

cb

13. d

tayc

14. gwy 82

15. vuwy

16. gx

y

17. Vy

3

4 3

18. y

TGyE

19. tyxy 252

20. 11

y

A

y

BA

S J Cooper

Algebra (20) Index Notation

1. Simplify each of the following:

a) 38 xx

b) 57 rr

c) bb 4

d) 626 iii

e) 244 eee

f) 313 hhh

g) 35 nnn

h) 327 xxx

i) aaaa 342

j) 222 yyy


a) baba 253

b) 3243 nmnm

c) 3542 effe

d) 2463 dcdc

e) 5739 qpqp

f) yxyx 472

g) 968 khkh

h) 106511 tsts

i) 654 fefe

j) 3752 vuuu


a) 49 hh

b) 715 tt

c) 67 mm

d) 412 ww

e) 69 yy

f) 88 jj

g) 1620 xx

h) 1114 dd

i) 5

8

t

t

j) 8

13

x

x


a) 22x

b) 26t

c) 34y

d) 53a

e) 232b

f) 352x

g) 373c

h) 423qp

5. Simplify each of the following expressions:

a) 5

8

r

r

b) 8

9

y

y

c) 7

612

x

xx

d) 7

45

f

ff

e) 3

56

xy

yx

f)

43

324

vu

vvu

g) 2 43 7 5 4r s r s

h) fef

efe45

275

S J Cooper

Algebra (21) Continuing a sequence

1. Write down the next two terms for each of the following sequences and a rule in words.

(a) 2, 9, 16, 23

(b) 1, 6, 11, 16

(c) 8, 14, 20, 26

(d) 5, 8, 11, 14

(e) 6, 15, 24, 33

(f) 4, 8, 12, 16

(g) 9, 11, 13, 15

(h) 7, 15, 23, 31

(i) 7, 5, 3, 1

(j) 13, 9, 5, 1

2. Write down the next two terms for each of the following and give a rule for continuing the

sequence.

3. Write down the first five terms for each of the following described sequences.

a) Add 3 to the previous term: 2, …….

b) Add six to the previous term: 7, …….

c) Subtract 3 from the last term: 10, ……

d) Subtract 9 from the previous term: 61, ……

e) Multiply the previous term by 3: 1, …..

f) Multiply the previous term by 2: 3, ……

g) Divide the last term by 2: 32, ……

h) Add the next even number each time: 3, 5, ……

i) Add a number that increases by 2 each time: 1, 2, …….

(a) 20, 17, 14, 11

(b) 37, 31, 25, 19

(c) 99, 88, 77, 66

(d) 47, 37, 27, 17

(e) 56, 49, 42, 35

(f) 80, 71, 62, 53

(g) 63, 55, 47, 39

(h) 7, 3, -1, -5

(i) 1, 2, 4, 8

(j) 2, 4, 16, 256

S J Cooper

4. Fill in the missing gaps in the following sequences:

(a) 3, ………, 11, 15, ………, ………,

(b) ………, 14, 20, ………, ………, 38

(c) 2, ………, ………, 23, 30, ………,

(d) ………, 15, 23, 31 ………, ………,

(e) 19, ………, 11 ………, 3 ………,

(f) 1, 2 ………, 8 16 ………,

(g) ………, 42, ………, 32, ………, 22

(h) 8, ………, 18, ………, 28, 33

(i) 3, 9, ………, ………, 243, 729

(j) ………, 14, 26, ………, ………, 62

5. Adding together the previous two terms on the line above generates the PASCAL’s triangle.

The first four rows of the PASCAL’s triangle are shown below

(a) Write down the next three rows in this sequence.

(b) Form a new PASCAL’s triangle starting with 2, 2 and write down the next five rows.

6. Using the squares in your books draw the next two arrangements for each of the following

patterns:

a)

b)

1 1

1 2 1

1 3 3 1

1 4 6 4 1

S J Cooper

c)

d)

e)

f)

S J Cooper

Algebra (22) The nth term

1. Give the first four terms of the sequence for which

a) 13 nU n

b) 45 nU n

c) nU n 2

d) 14 nU n

e) 57 nU n

f) 45 nU n

g) 2nU n

h) 12 nU n

i) 3nU n

j) nnU n 2

k) nU n 35

l) 86 nU n

2. The following collections of dots suggest another sequence of numbers.

The sequence is started below. Write down the

next seven terms in this sequence.

1, 3, 6, 10, etc.

a) Write down the values of (i) the 2nd term (ii) the 10th term.

b) What sorts of numbers belong to this sequence?

c) Which term of this sequence has a value of (i) 55? (ii) 231?

3. For each of the following sequences write the next two terms in the sequence and then find a

formula for the nth term in the sequence.

a) 3, 6, 9, 12, 15

b) 2, 5, 8, 11, 14

c) 7, 11, 15, 19, 23

d) 2, 6, 10, 14, 18

e) 7, 13, 19, 25, 31

f) 2, 7, 12, 17, 22

g) 10, 14, 18, 22, 26

h) 2, 9, 16, 23, 30

i) 11, 19, 27, 35, 43

j) 0, 5, 10, 15, 20

k) 3, 13, 23, 33, 43

l) 18, 30, 42, 54, 66

m) 1, 7, 13, 19, 25

n) 2, 11, 20, 29, 38

o) 3, 14, 25, 36, 47

p) 9, 13, 17, 21, 25

q) 1, 4, 9, 16, 25

r) -2, 1, 6, 13, 22

s) 1, 8, 27, 64, 125

t) 2, 4, 8, 16, 32

S J Cooper

4. Find an expression for the nth term of the sequence ...... , 16

10 ,

13

8 ,

10

6 ,

7

4 ,

4

2

Hence obtain the 100th term for this sequence.

5. For each of the following give a formula for the nth term of the sequence

a) 3, 9, 27, 81, ….. b) 0, 3, 8, 15, 24, …. c) ... ,

16

1 ,

9

1 ,

4

1 , 1

6.

This is the beginning of a sequence of tile patterns.

a) Draw the next two patterns

b) Write down the first seven terms of the sequence given by the number of tiles in each pattern.

c) What sort of numbers does the pattern generate?

d) Give the nth term of this sequence.

7. The diagrams below are made up using matchsticks

Diagram 1 Diagram 2 Diagram 3

a. Draw the next pattern in the sequence

b. Copy and complete the table below

c. Write down an expression for the number of matchsticks (m) in the nth diagram.

d. How many matchsticks are their in the 20th diagram?

S J Cooper

Algebra (23) Sequences concluded

1. The nth term of the sequence

4, 10, 18, 28, 40

is given by ann 2 .

Find the value of a.

2. Obtain the nth term for each of the following sequences

a) 0, 2, 6, 12, 20

b) 4, 9, 16, 25, 36

c) 0, 4, 10, 18, 28

d) 3, 3, 5, 9, 15

e) 5, 12, 21, 32, 45

3. a) write down the first six terms for the sequence of diagrams below.

b) Hence obtain the nth term for this sequence.

c) What name is given to this special sequence of numbers?

4. Obtain the nth term for the sequence below

4, 6, 6, 4

S J Cooper

Algebra (24) Trial & Improvement

1. Obtain the value of x, correct to one decimal place given that 1742 xx

2. Helen is using trial and improvement to find a solution to the equation

x3 + 3x = 65

The table shows her first two tries. Continue the table to find the solution to the

equation, Giving your answer correct to 1 decimal place

x x3 + 3x Comment

3 36 too small

4 76 too big

3. Find the value of x, correct to 2 decimal places for the equation 56372 xx

4. Alan is using trial and improvement to find a solution to the equation

x – = 4

The table shows his first trial.

x x –

Comment

4 3.75 Too low

Continue the table to find a solution to the equation correct to 1 decimal place.

x

1

x

1

S J Cooper

5. Given that 1033 xx , find the value of x correct to one decimal places.

6. A rectangle has width p and a length 5cm longer than its width. Given that its area

is equal to 56cm2

a) Show that the 05652 pp

b) Using trial and improvement find the value of p correct to

three decimal places.

7. Find the value of x correct to 3 decimal places such that 152 23 xx

8. A semicircle has radius r and perimeter 22cm.

(i) Show that 222 rr

(ii) Using trial and improvement find the value of r

correct to 3 decimal places.

9. A cuboid has a square base area with side x and length 3cm

more than its height. Given the volume of the cuboid is

500cm3.

a) Show that 05003 23 xx

b) Using trial and improvement find a value for x

correct to 2 decimal places.

10. Asif and Ben are working out this question.

A solution of the equation x3 + 2x = 500 lies between 7 and 8.

Use trial and improvement to find this solution, correct to one decimal place.

Asif’s answer is 7.6 Ben’s answer is 7.5

Which answer is correct? You must show all your working.

S J Cooper

Algebra(25) transformations of curves I

1. On the same set of axes draw the graphs of

(a) 2xy (b) 12 xy (c) 32 xy (d) 52 xy

2. on the same set of axes drawn the graphs of

(a) 2xy (b) 21 xy (c) 2

3 xy (d) 22 xy

3. The graph drawn is of xfy , with points A and B labelled.

Sketch on separate axes each of the

following, paying particular attention to the

points A and B.

i) 1 xfy

ii) 2 xfy

iii) 2 xfy

iv) 4 xfy

4. The graph drawn is of xy cos

Sketch each of the following on

separate sets of axes

a) xy cos1

b) 2cos xy

c) 90cos xy

d) 180cos xy

y

x 0

0.5

1

-0.5

-1

180 360 0

x

y

S J Cooper

Algebra(26) transformations of curves II

1. On the same set of axes draw the graphs of

(a) 2xy (b) 22xy (c) 2

2

1xy

2. on the same set of axes drawn the graphs of 43 xxf and

(a) xfy 2 (b) xfy (c) xfy (d) xfy 3

3. The graph drawn is of xfy , with points A, B and C labelled.

Sketch on separate axes each of the

following, paying particular attention to the

points A and B.

i) xfy 3

ii) xfy

iii) xfy

iv) xfy 2

4. The graph drawn is of xy sin

Sketch each of the following on

separate sets of axes

a) xy sin5

b) xy sin

y

x 0

0.5

1

-0.5

-1

180 360 0

x

y

S J Cooper

Algebra(27) Algebraic Fractions

Simplify each of the following fractions

1. yx

x24

2

2. ab

ba

5

10 23

3. 32

25

3

21

yx

yx

4. xx

x

22

5. xx

x

3

32

6. 32

12

xx

x

7. 6

232

2

xx

xx

8. 43

122

2

xx

xx

9. 472

162

2

xx

x

10. 12

3522

2

xx

xx

11. 352

2532

2

xx

xx

12. 4

2742

2

x

xx

By initially factorising where possible express each of the following as a single fraction

in its simplest form.

13. 152

2

2

252

2

xx

x

x

x

14. xx

xx

x

xx

5

34

5

62

22

15. xx

x

xx

xx

2

1

12

422

2

16. 12

123422

2

xx

x

xx

xx

17. 67

2

4

622

2

xx

x

x

xx

18. 12

12 22

x

xx

x

xx

19. 32

66

3

652

2

xx

xx

x

xx

20. 3145

9

15

32

22

xx

x

x

xx

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