Mathematicsbtrcc.pbworks.com/w/file/fetch/107106156/Stage 5 Algebra.pdfx8ux3 b) 7 5 r ur c) g) b4ub...

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Mathematics

Stage 5

Algebra

S.Cooper

This book is on loan to ……………………………………………………………………………………………………………………….

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Algebra (1) 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

2. Simplify each of the following:

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.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

2. Solve each of the following equations

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.Cooper

Algebra (3) More equations 1. Solve each of the following equations

a) 1243 x

b) 2565 a

c) 2827 y

d) 854 b

e) 1836 r

f) 14322 x

g) 36724 x

h) 15523 x

i) 77137 x

j) 93513 g

k) 36214 x

l) 565 m

m) 16738 h

n) 65495 i

o) 60133 x

p) 72354 y

q) 271149 e

r) 46652 k

s) 78346 c

t) 485410 s


a) 152233 xx

b) 3862124 xx

c) 842423 xx

d) 51122435 mm

e) 41433252 yy

3. Find the value of x for each of the following

a) 5243 xx

b) 4234 xx

c) 8413 xx

d) 6378 xx

e) 523122 xx

f) 652735 xx

g) 522214 xx

h) 143382 xx

i)

25

122

x

j)

32

533

x

S.Cooper

Algebra (4) Factorisation 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

2. 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

3. Factorise each of the following:

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

S J Cooper

5. (a) Factorise 342 xx

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

6. (a) Factorise 442 xx

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

S J Cooper

Algebra (5) 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 (6) 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 20,4

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 (7) 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

(b) Hence draw the graph of xxy 32 .


x –4 –3 –2 –1 0 1 2 3

x2

–x

y

(b) Hence draw the graph of xxy 2.

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 (8) 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

6. a) Copy and complete the table below for the graph of 62 xy

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

4. (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

S J Cooper

Algebra (9) Drawing graphs 1. a) Copy and complete the table below for the graph of 43 xy

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(10) Perpendicular & Parallel lines 1. Work out the equation of the line which passes through (1, 3) and is parallel to the line

12 xy

2. Work out the equation of the line which passes through (–1, 4) and is perpendicular to the

line 52 xy

3. Work out the equation of the line which passes through (2,–3) and is parallel to the line

42 yx

4. Work out the equation of the line which passes through (–3,–1) and is perpendicular to the

line 13 yx

5. Obtain the equation of the line through R which is perpendicular to the line PQ.

2 4 6 0

2

4

y

x

P

Q

R

S J Cooper

6. Obtain the equation of the line through C which is perpendicular to the line AB.

7. Obtain the equation of the perpendicular bisector between the points A(2, 1) and B(6, 7)

2 4 6 0

2

4

y

x

A

B

C

S J Cooper

Algebra (11) Inequalities 1. Represent each of the following inequalities on a number line

a) 4x

b) 7x

c) 1x

d) 3x

e) 2x

f) 2x

g) 3x

h) 1x

2. State what inequality is represented by each of the following diagrams

a)

b)

c)

d)

e)

f)

g)

3. Solve each of the following inequalities

a) 1312 x

b) 1023 x

c) 392 x

d) 1343 x

e) 18175 x

f) 11233 x

g) 592 x

h) 3125 x

i) 7114 x

j) 1596 x

k) 4295 x

l) 183 x

m) 13154 x

n) 23127 x

o) 1373 x

– 2 –1 0 1 2 3 4 x

– 2 –1 0 1 2 3 4 x

– 4 –3 –2 –1 0 1 2 x

– 5 –4 –3 –2 –1 0 1 x

– 2 –1 0 1 2 3 4 x

– 5 –4 –3 –2 –1 0 1 x

– 2 –1 0 1 2 3 4 x

S J Cooper

4. Solve each of the following inequalities:

a) 14225 xx

b) 8214 aa

c) 25447 yy

d) gg 4365

e) 1315 ee

f) pp 2184

g) rr 2817

h) tt 2253

i) uu 5375

j) yy 44383

k) 36129 ee

l) 68125 ww

m) 14792 ff

n) 79410 kk

o) hh 4249

p) vv 61932

q) 225211 ss

r) 218710 zz

s) 4881213 yy

t) xx 21992

5. 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

S J Cooper

Algebra (12) 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 (13) Sequences II 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

3. Find an expression for the nth term of the following quadratic sequences

a) 0, 5, 12, 21, 32, 45, …

b) 7, 15, 25, 37, 51, 67, …

c) 20, 36, 56, 80, 108, 140, …

d) 11, 16, 25, 38, 55, 76, …

e) 10, 21, 38, 61, 90, 125, …

4. 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?

5. Obtain the nth term for the sequence below

4, 6, 6, 4

6. Obtain the nth term for the quadratic sequence below

241, 494, 749, 1006, 1265, 1526, …

S J Cooper

Algebra (14) Simultaneous Equations I Exercise 1

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

Exercise 2

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.

S J Cooper

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 (15) Simultaneous Equations II Exercise 1

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

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


(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

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

12 xy

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