IGCSE Higher Sheets 2 - Weeblyrunnymedemathematics.weebly.com/uploads/4/0/6/7/40678035/... · 2019....

For use only in Whitgift School IGCSE Higher Sheets 2

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IGCSE Higher

Sheet H2-1 1-08-1 Bounds Sheet H2-2 1-08-2 Bounds Sheet H2-3 1-09a-1 Standard Form Sheet H2-4 1-09a-2 Standard Form Sheet H2-5 1-09a-3 Standard Form-Non Calculator Sheet H2-6 1-09a-4 Standard Form- Non Calculator Sheet H2-7 1-09a-5 Standard Form Sheet H2-8 2-01a-1 Indices Sheet H2-9 2-02a-1 Quadratic Expansion Sheet H2-10 2-02a-2 Quadratic Expansion Sheet H2-11 2-02a-3 Quadratic Expansion Sheet H2-12 2-02b-1 Quadratic Factorisation Sheet H2-13 2-02b-2 Quadratic Factorisation Sheet H2-14 2-02b-3 Quadratic Factorisation Sheet H2-15 2-02c-1 Algebraic Fractions Sheet H2-16 2-02c-2 Algebraic Fractions Sheet H2-17 2-02c-3 Fractional Equations Sheet H2-18 2-02c-4 Simplifying Fractions Sheet H2-19 2-02c-5 Quadratic Fractions



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Sheet H2-1 1-08-1 Bounds NB: If 6.7x = (to 1dp) then we can say that the upper bound is 6.75 (not 6.749 ).

1. Write down the upper and lower bounds for the following:

(a) 73.43 (to 2dp)(b) 7320 (to 3sf)(c) 7320 (to 4sf)(d) 147.037 (to 3dp)(e) 100 (to 3sf)(f ) 100 (to 1sf)

wxyzab

======

2. If 3.1a = (to 1dp), 8.6b = (to 1dp) and 7.9c = (to 1dp) then calculate:

(a) The greatest value of a b+ . (b) The smallest value of ab .

(c) The largest value (to 3sf) of ac

.

(d) The smallest value (to 3sf) of bca

(e) The largest value of c a−

3. A man runs a 100m race and his time has been measured as 10.3s. If the track is accurate to the nearest metre and his time is accurate to the nearest 0.1s then find the upper and lower bounds (to 1dp) for his average speed?

4. The area of a rugby field is given as 6950 2m , correct to 3sf. The length of the field is

given as 95m, correct to 2sf. (a) Find the upper and lower bounds for the area of the field. (b) Find the upper and lower bounds for the length of the field. (c) Use these to calculate the upper and lower bounds (to 3sf) for the width of the field.

5. The formula for the distance, s, travelled by a body with initial speed u, constant

acceleration a after a time t is given by 221 atuts += . Find the greatest and least possible

values (to 3sf) of s when 6.1u = , 4.5a = , 13.6t = all correct to 1dp. 6. Pythagoras’ theorem states that 222 cba =+ where a, b and c are the three lengths of a

right- angled triangle and c is the hypotenuse. If 4.3a = cm and 12.1c = cm, both correct to 1dp find the smallest and largest values for b (to 1dp).

7. The formula auvs

2

22 −= is used to find the distance travelled by an object whose initial

speed is v, whose final speed is u and whose acceleration is a. Find an inequality for s (to 2sf) if 15v = , 11u = and 2.3a = , all correct to 2sf.



Sheet H2-2 1-08-2 Bounds NB: If 6.7x = (to 1dp) then we can say that the upper bound is 6.75 (not 6.749 ).

1. Write down inequalities which show the possible values for the following:

(a) 7.36=x (to 3sf) (b) 107.53=y (to 3dp) (c) 120=z (to 3sf) (d) 120=w (to 2sf) (e) 0.147=k (to 1dp)

2. A rectangular piece of metal measures 7m by 6m (both to nearest metre). A circle of

radius 2m (to nearest metre) is cut out from the sheet. (a) Calculate an inequality for R, the area of the rectangle (b) Calculate an inequality for C, the area of the circle (to 3sf). (c) Use (a) and (b) to calculate an inequality for A, the area of the remaining shape (to 3sf).

3. Consider the formula 2 2

2v ua

s−

= .

(a) Calculate a (to 3sf) in the above if v =11.5, u = 8.7 and s = 2.3 where all these are exact. (b) Write down the maximum and minimum values for v, u and s if the three numbers in are correct to 1 dp. (c) Hence calculate the maximum and minimum values for a (to 3sf) if the three numbers in are correct to 1 dp.

4. The total resistance R, in an electrical circuit in which there are two resistors in parallel (of

resistance R1 and R2 respectively) is given by the formula

1 2

11 1R

R R

=+

. R1 is measured

as 7.16 ± 0.005 ohms and R2 as 4.8 ± 0.05 ohms . (a) What is the least possible value for R (to 3sf)? (b) What is the greatest possible value for R (to 3sf)?

5. The time period, T seconds, of a pendulum is calculated using the formula

T = 6.283 × gL

where L metres is the length of the pendulum and g m/s2 is the acceleration due to gravity.

L = 1.36 correct to 2 decimal places. g = 9.8 correct to 1 decimal place. Find the difference (to 3dp) between the lower bound of T and the upper bound of T.



Sheet H2-3 1-09a-1 Standard Form 1. Evaluate the following, leaving your answers either as whole numbers or as decimals:

3 5

2 4

1 6

7 1

(a) 10 (b) 10(c) 10 (d) 10(e) 10 (f ) 10(g) 10 (h) 10

−

− −

2. Write the following as powers of 10:

(a) 1000000 (b) 100(c) 0.001 (d) 0.0000001(e) 100000000 (f ) 0.0001(g) one million (h) one thousandth(i) a thousand million (j) one millionth

3. Evaluate the following, leaving your answers either as whole numbers or as decimals:

2 3

9 1

3 4

5 4

(a) 3 10 (b) 5 10(c) 7 10 (d) 2 10(e) 9 10 (f ) 7 10(g) 8 10 (h) 6 10

−

−

−

× ×

× ×

× ×

× ×

4. Evaluate the following:

5 4

3 2

5 3

4 6

(a) 3.2 10 (b) 5.01 10(c) 2.34 10 (d) 3.72 10(e) 8.92 10 (f ) 3.57 10(g) 1.17 10 (h) 8.32 10

− −

−

× ×

× ×

× ×

× ×

5. Write the following in the form 10na× where 1 10a≤ < and n is a whole number:

(a) 5000 (b) 70000(c) 0.003 (d) 0.00004(e) 20000 (f ) 0.00001(g) four thousand (h) three millionths(i) two thousand million (j) five thousandths

6. Write the following in the form 10na× where 1 10a≤ < and n is a whole number:

(a) 7600 (b) 3720000(c) 0.0132 (d) 0.00417(e) 203000 (f ) 0.0000276(g) 87600000000 (h) 0.000000000281



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Sheet H2-4 1-09a-2 Standard Form 1. Express the following in standard form:

(a) 532,000 (b) 678,100,000 (c) 78,120,000,000 (d) 0.003234 (e) 0.00915 (f) 0.00000145 (g) 0.0002607 (h) 517 (i) 19.3 (j) 0.871 (k) 9.2 (l) 0.97 (m) 10.2 (n) three million (o) five millionths (p) five thousand million million

2. Write the following in decimal form:

3

5

1017.2)b(1062.3)a(

−×

×

3. Calculate the following in decimal form (to 3sf):

6 15

7 8

(a) 0.2 (b) 0.3(c) 0.15 (d) 0.22

4. Calculate the following in standard form (to 3sf):

630 20

3 2

1013

1(a) 2 (b) 3 (c)2

1 1(d) (e) 0.005 (f )3 11

3(g) 5 (h) 0.007 (i)4

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

5. Calculate the following in standard form (to 3sf where necessary):

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )9557

7514

71557

1024.11021.1)f(1031.21021.9)e(1035.21091.3)d(103.1102.7)c(

101.2109.2)b(102.4104.3)a(

×÷××÷×

×÷××××

××××××

−−

−−−



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Sheet H2-5 1-09a-3 Standard Form-Non Calculator

1. Express the following in standard form:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

2 8 5 7

12 2 2 3

2 4 5 2

8 5 3 2

4 6 6 3

8 2 2 5

(a) 2 10 4 10 (b) 4 10 5 10

(c) 3 10 8 10 (d) 6 10 5 10

(e) 7 10 8 10 (f ) 8 10 2 10

(g) 8 10 2 10 (h) 1.6 10 2 10

(i) 1.8 10 3 10 ( j) 8.4 10 4 10

(k) 9.6 10 3.2 10 (l) 8.8 10 1.1 10

− −

− −

−

− −

− − − −

× × × × × ×

× × × × × ×

× × × × ÷ ×

× ÷ × × ÷ ×

× ÷ × × ÷ ×

× ÷ × × ÷ ×


( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

7 6 9 8

3 2 6 5

14 13 5 6

9 10 11 11

9 8 8 7

12 11 10 9

(a) 5 10 3 10 (b) 8 10 4 10

(c) 5 10 2 10 (d) 9 10 4 10

(e) 6 10 3 10 (f ) 5 10 2 10

(g) 8 10 4 10 (h) 9 10 5 10

(i) 4 10 7 10 ( j) 2.5 10 1.5 10

(k) 1.2 10 4.5 10 (l) 7 10 4 10

− −

− −

− −

× + × × + ×

× + × × − ×

× − × × − ×

× − × × + ×

× + × × + ×

× + × × − ×

3. The density of a certain type of stone is 4105.3 × kg/ 3m . Find the mass of this type of

stone where its volume is 20 3m . Leave your answer in standard form. 4. The average speed of a plane is 6103.2 × metres per hour. How long will it take to travel a

distance of 8106.4 × metres. 5. The area of a country is given as 4102.9 × 2km . Express this in 2m in standard form. 6. Express the following in standard form:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

2 11 4 5

6 7 5 2

5 2 13 4

8 3 12 5

4 5 2 9

5 12 3

(a) 3 10 2 10 (b) 3 10 3 10

(c) 4 10 3 10 (d) 8 10 2 10

(e) 9 10 2 10 (f ) 6 10 3 10

(g) 2.0 10 2.1 10 (h) 1.9 10 3.0 10

(i) 3.6 10 1.5 10 ( j) 7.2 10 3.6 10

(k) 3.20 10 1.60 10 (l) 4.40 10 1.1

− − −

− −

× × × × × ×

× × × × ÷ ×

× ÷ × × ÷ ×

× × × × × ×

× × × × ÷ ×

× ÷ × × ÷ ( )70 10×

PTO



Sheet H2-5 1-09a-3 Standard Form-Non Calculator (cont.)


( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )91268

8945

16171312

651415

3456

101178

102.1104.5)l(103.1102.4)k(106.5103.1)j(101.5108.9)i(105.9102.1)h(104.1102.3)g(109.3103.8)f(102.1104.7)e(

105.2103.5)d(103.2107.6)c(107.3101.7)b(102.1105.3)a(

×+××+×

×−××+×

×−××−×

×−××−×

×−××+×

×+××+×

−−

−−

−−



Sheet H2-6 1-09a-4 Standard Form-Non Calculator 1. Evaluate the following, leaving your answers in standard form: f

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

5 4 6 2

4 7 6 3

7 2 5 1

11 6 12 5

(a) 3.4 10 2 10 (b) 2.1 10 3 10

(c) 5.1 10 4 10 (d) 6.5 10 2 10

(e) 8.6 10 2 10 (f ) 8.1 10 3 10

(g) 1.5 10 3 10 (h) 1.8 10 6 10

−

−

× × × × × ×

× × × × × ×

× ÷ × × ÷ ×

× ÷ × × ÷ ×

2. Evaluate the following, leaving your answers in standard form:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

5 5 11 10

7 5 5 4

6 4 3 3

4 3 8 7

(a) 5.1 10 2.3 10 (b) 2 10 3 10

(c) 9 10 4 10 (d) 9.5 10 4.5 10

(e) 8.3 10 2.1 10 (f ) 7.5 10 2.1 10

(g) 9 10 3 10 (h) 7.2 10 3 10

× + × × + ×

× + × × + ×

× + × × − ×

× − × × − ×

3. Double the following numbers and leave your answer in standard form:

757

254

8

101.9)i(104.7)h(101.5)g(106.3)f(102.6)e(106.2)d(105.3)c(032.0)b(250)a(

−−

−−

×××

×××

×

4. Square the following numbers and leave your answer in standard form:

1037

945

2

101.1)i(102.1)h(106)g(108)f(107)e(102)d(103)c(1.0)b(25)a(

−−

−

×××

×××

×

5. At a time when Jupiter, Pluto and the Sun are in a line, the distances of Jupiter and Pluto

from the Sun are respectively 87.9 10 km× and 96 10 km× . What is the distance (in standard form) between Pluto and Jupiter when the two planets and the Sun are in a line with:

(a) The planets on opposite sides of the Sun? (b) The planets on the same side of the Sun?

PTO

96 10 km× 87.9 10 km×

Sun

Jupiter Pluto

NOT TO SCALE



Sheet H2-6 1-09a-4 Standard Form-Non Calculator (cont.)

6. Write the following in standard form: (a) fifty three million (b) four hundred thousand (c) seven million six hundred thousand (d) one hundred and twenty five thousand million (e) seventy thousandths (f) eleven millionths

7. Calculate the following, leaving your answers in standard form.

( ) ( )( ) ( )( ) ( )

8 7

8 7

8 7

(a) 6 10 1.5 10

(b) 6 10 1.5 10

(c) 6 10 1.5 10

× × ×

× ÷ ×

× + × 8. Calculate the following, leaving your answers in standard form:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

8 9 6 4

5 2 11 2

7 6 12 11

3 4 7 8

4 3 9 8

(a) 3.8 10 2 10 (b) 2.8 10 2 10

(c) 5.2 10 3 10 (d) 3.6 10 4 10

(e) 2.3 10 1.5 10 (f) 7.2 10 5 10

(g) 1.5 10 2.6 10 (h) 5.2 10 1.8 10

(i) 9.1 10 7.3 10 (j) 1.2 10 9 10

−

− − − −

× × × × ÷ ×

× × × × ÷ ×

× + × × − ×

× + × × − ×

× + × × − × 9. (a) Write 500 000 000 in standard form.

(b) Write 7 × 10–3 as an ordinary number. (c) Work out the value of 7 × 10–3 × 500 000 000. Give your answer in standard form.



Sheet H2-7 1-09a-5 Standard Form 1. The speed of light is approximately 300000 km/s. Calculate how, in km (standard form to

2sf) light travels in on year (365 days). 2. A book says that the Earth is known to be 149,600,000km from the Sun. It then says that

the earth is 51.58 10−× light years from the Earth. Use these two pieces of information to find what distance (in km, to 2sf) a light year represents. Leave your answer in standard form.

3. The centre of the Milky Way is 2 6 104. × light years from earth, and the nearest galaxy is 16 105. × light years from the Earth. (a) (i) Which of these distances is greater?

(ii) By how many light years? (b) If one light year is km1046.9 12× then find the distance of the nearest galaxy from the Earth in km. Leave your answer in standard form correct to 3 significant figures.

4. A certain company produces three million pads of paper per year. Each pad holds 250

sheets of paper. (a) If these three million pads weigh 61021.4 × kg then how much does each sheet of

paper weigh? (Give your answer in kg and in standard form to 3sf) Another pad weighs 1.5kg and measures 210mm by 297mm by 15mm. (b) Find the volume of this pad of paper (Give your answer in 3cm and in standard form

to 3sf). (c) Show that the density of the paper is 1.60 g/ 3cm (to 3sf).

5. A thunderstorm is taking place 6km away.

Light travels 5103× km in one second. Sound travels 310226.1 × km in one hour.

(a) How long does it take for the light from the lightning to travel 6km? Give your answer in standard form.

(b) Show that sound travels approximately 340m in one second. (c) How long (to the nearest second) does it take for the sound of the thunder to travel

6km?

PTO



Sheet H2-7 1-09a-5 Standard Form (cont.) 6. The population of the world at the end of 1995 was 9102.5 × people.

(a) It was projected to grow by 4% in 1996. Calculate the projected population at the end of 1996, giving your answer in standard form (to 2sf).

(b) In fact the population at the end of 1996 was 9105.5 × . What was the percentage increase (to 2sf) in the population over 1996?

(c) The projected population at the end of 2020 is 10108.1 × . How many more people is this than at the end of 1995? (Give your answer in standard form to 2sf.)

7. The density of water is 31 10× kg/ 3m . Find the following (all in standard form):

(a) the mass of water (in kg) in a cuboid measuring 2m by 3m by 5m (b) the volume (in 3m ) of water whose mass is 85 10× tonnes (one tonne is 1000kg). (c) The volume (in 3cm ) of 1 3m of water. (d) The mass (in g) of 1 3m of water. (e) The density of water in g/ 3cm . (f) the mass of water (in g) in a cuboid measuring 6cm by 3cm by 10cm.

8. The population of a certain country is 85.7 10× and its area is 107.21 10× 2m . Find the

population density (people per 2m ) of this country in standard form to 3sf. 9. The diameter of the earth is 71.3 10× m. Using a model that the earth is a perfect sphere,

find the circumference of the earth in km. Write the answer in standard form (to 2sf). 10. The adult population of a country is 60 million. The average annual income per adult is

$43,000. Find in standard form the total annual income from the adult population. 11. The speed of light is approximately 82.8 10× m/s. Express this in km/h in standard form to

3sf. 12. In 1999 the population of a country was 99.2 10× . Over the next five years the population

rose by 15%. Find the population in 2004. 13. The population of a certain type of bird increased from 37.8 10× to 41.2 10× over a ten

year period. Find the percentage increase over that period (to 3sf).



Sheet H2-8 2-01a-1 Indices 1. Simplify the following:

( ) ( ) ( )( )

2 3 7 5

12 5 5 4

3 9 62 2 3

32 12 3

(a) (b)(c) (d)

(e) (f )

(g) (h)

x x y yz z a a

b b c c

d d e e

−

−

−−

× ×

÷ ÷

× ÷

÷ ×

2. Simplify the following:

( )1

34 5 63

5 152 3

3

3 7 3 2 13

(a) (b)

1(c) (d)

1(e) (f )

f f r

sw w

eq q e e

×

×

× ×


( ) ( )

( ) ( )

( ) ( )

( )

2 63

1 112 202 4

1 210 95 3

1 28 182 3

423

3 6 39

3 34 24 2

8 10

(a) 4 (b) 27

(c) 64 (d) 16

(e) 32 (f ) 8

(g) 4 (h) 27

8(i) ( j) 8

16(k) (l)25

x y

z w

q p

k j

a bf

a ab b

− −

−

−

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

22 3 3 2 2

14 2 5 3 2 42

34 2 3 6

6 2

2 5 3 2 3 4 5 3

a 5 b 12 3

c 16 d 3 5

e 9 f 8

12 24g h3 2 2 4

a b x y z xy

m n p q q r

a b h k

m nc d c d m n m n−

÷

×

× ×



Sheet H2-9 2-02a-1 Quadratic Expansion 1. Expand and simplify the following (i.e. get rid of brackets):

(a) 3(2 1) (b) 5(2 1)(c) 3(3 5) (d) 4(5 2)(e) 5(2 3) 2( 3) (f ) 2(5 3) 3(2 1)(g) 2(7 5) 3(2 3) (h) 3(2 5) 2(4 3)

x yz ss s y yu u d d

+ +− ++ + + + − ++ − + + − +

2. Expand and simplify the following (i.e. get rid of brackets):

(a) (2 3) (b) (5 2)(c) (5 4) (d) (3 1)(e) 3 ( 2) (f ) 4 (2 1)(g) 5 ( 1) (h) 6 (2 1)

x x x xy y z zk k h hr r t t

+ +− +− −+ +

3. Expand and simplify the following (i.e. get rid of brackets):

(a) (3 2) (b) 2 (5 3)(c) 2 (5 2) (d) 6 (2 1)(e) 2 ( 5) 3 ( 2) (f ) 5 ( 1) ( 3)(g) 4 ( 2) 3( 1) (h) 6 ( 3) (3 1)

x x x xy y u uk k k k g g g gv v v h h h h

+ ++ +

+ + + − − ++ − + + − +

4. Multiply out the following brackets:

(a) ( 2)( 3) (b) ( 1)( 4)(c) ( 2)( 6) (d) ( 5)( 3)(e) ( 5)( 4) (f ) ( 2)( 3)(g) ( 1)( 7) (h) ( 1)( 3)(i) ( 5)( 2) ( j) ( 2)( 2)

x x x xr r e ew r y yz z q qs s d d

+ + + ++ + + ++ + − +− − − +− + − −

5. Multiply out the following:

(a) (3 1)( 1) (b) (4 1)( 1)(c) (5 1)( 1) (d) (2 1)( 1)(e) (2 1)( 1) (f ) (9 1)( 3)( ) (5 2)( 4) (h) (2 5)( 7)g(i) (2 5)( 5 ) ( j) (5 1)( 6)

x x x xt t z zw w r re e q qp p y y

+ + + ++ − − −+ + − +− + − −− + + −



Sheet H2-10 2-02a-2 Quadratic Expansion 1. Multiply out the following brackets:

(a) ( 1)( 5) (b) ( 2)( 6)(c) ( 3)( 4) (d) (2 3)( 4)(e) (3 1)( 2) (f ) (3 1)(2 3)(g) (2 1)( 5) (h) (2 3)(3 4)(i) ( 1)(2 3) ( j) (5 1)(3 2)

x x x xt t w wr r e et t s s

y y y y

+ + + ++ + + ++ + + +− + − +− − − −


(a) (2 1)(3 2) (b) (5 2)(3 4)(c) (6 1)(2 3) (d) (2 9)(3 1)(e) (7 1)(2 3) (f ) (9 2)(3 2)( ) (7 11)(2 1) (h) (8 1)(5 3)g(i) (3 1)(3 1) ( j) (7 2)(7 2)(k) (2 1)( 3) (l) (2 1)(5 1)

x x x xt t y yz z r re e q qp p y yk k v v

+ + + +− − − −− + − +− + + −− + + −+ + − +

3. Multiply out and simplify the following brackets:

2

2

(a) ( 2)( 3) (b) ( 5)( 4)d(c) ( 1)( 2) ( ) (3 1)(2 1)

(e) (5 2)(2 3) (f ) (5 1)(5 1)(g) (2 1)(2 1) (h) (3 2)(i) (2 1) ( j) (5 2 )(2 3 )(k) (5 3 )(4 ) (l) (7 3 )(2 )

x x x xt t q q

y y m my y pq d e d ep q p q s t s t

+ + + ++ + + −+ − − +

+ − +

− + −+ + − −



Sheet H2-11 2-02a-3 Quadratic Expansion

1. Expand and simplify the following:

2 2

2 2

2 2

2 2

(a) ( 3) (b) ( 5)(c) ( 4) (d) ( 6)(e) (2 3) (f ) (5 2)(g) (3 )(2 ) (h) (3 2 )(5 )(i) (5 2 )(3 4 ) ( j) (2 3 )(5 2 )(k) (3 2 ) (l) (5 3 )

x yy zw ta b a b m n m np q p q x y x yc d p q

+ +

− −

− −+ + − −+ − − −

+ −

2. Find a in the following:

2

2

2

2

2

(a) 7 12 ( )( 3)(b) 9 20 ( )( 4)(c) 10 21 ( )( 3)(d) 5 14 ( )( 7)(e) 11 18 ( )( 9)

x x x a xx x x a xx x x a xx x x a xx x x a x

+ + = + +

+ + = + +

+ + = + +

− − = + −

− + = + −

3. (a) Find m and n where ( )( )2 3 5x mx n x x+ + = + + .

(b) What is the connection between m the numbers 3 and 5? (c) What is the connection between n the numbers 3 and 5?

4. (a) Find a and b where ( )( )bxaxxx ++=++ 1072 .

(b) Find a and b where ( )( )bxaxxx ++=++ 2092 . (c) Find a and b where ( )( )bxaxxx ++=++ 1492 . (d) Find a and b where ( )( )bxaxxx ++=++ 782 .

5. Write the following in the form ( )( )bxax ++ :

2 2

2 2

(a) 6 5 (b) 14 33(c) 10 16 (d) 11 24

x x x xx x x x

+ + + +

+ + + +



Sheet H2-12 2-02b-1 Quadratic Factorisation

1. Factorise the following quadratics:

2 2

2 2

2 2

2 2

2 2

(a) 10 21 (b) 11 30(c) 7 12 (d) 8 15(e) 12 20 (f ) 5 6(g) 4 4 (h) 10 25(i) 5 4 ( j) 3 2

x x x xx x x xx x x xx x x xx x x x

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +


2 2

2 2

2 2

2 2

2 2

2 2

(a) 7 10 (b) 6 5(c) 6 8 (d) 7 12(e) 17 30 (f ) 10 24(g) 11 28 (h) 9 8(i) 8 16 ( j) 2 1(k) 16 15 (l) 14 13

x x x xx x x xx x x xx x x xx x x xx x x x

− + − +

+ + − +

− + + +

+ + − +

− + + +

+ + − +


2 2

2 2

2 2

2 2

(a) 3 10 (b) 5 24(c) 3 28 (d) 2 35(e) 4 12 (f ) 5 14(g) 6 (h) 4

x x x xx x x xx x x xx x x

+ − + −

− − − −

− − + −

+ − −

4. Factorise the following:

2 2

2 2

2 2

2 2

2 2

(a) 9 18 (b) 20(c) 7 10 (d) 3 40(e) 42 (f ) 7 12(g) 2 24 (h) 16(i) 3 ( j) 25

x x x xx x x xx x x xx x xx x x

+ + − −

− + + −

− − + +

+ − −

+ −

5. Factorise the following:

2 2

2 2

2 2

2 2

2 2

2 2

(a) 5 6 (b) 5 6(c) 5 6 (d) 5 6(e) 4 60 (f ) 5 36(g) 20 99 (h) 1(i) 132 ( j) 6 9(k) 10 25 (l) 100

x x x xx x x xx x x xx x xx x x xx x x

− − + +

+ − − +

− − + −

− + −

+ − + +

− + −





(a) ( 2)( 3) (b) ( 4)( 7)(c) ( 2)( 2) (d) ( 4)( 5)(e) ( 8)( 3) (f ) ( 1)( 1)(g) ( 11)( 12) (h) ( 7)( 9)

x x x xx x x xx x x xx x x x

+ + + +− − − −− + − +− + + −


2 2

2 2

2 2

2 2

2 2

(a) 5 6 (b) 7 12(c) 3 2 (d) 8 12(e) 12 35 (f ) 13 42(g) 6 9 (h) 10 24(i) 12 32 ( j) 2 1

x x x xx x x xx x x xy y z zh h t t

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +


2 2

2 2

2 2

2 2

2 2

2 2

(a) 7 6 (b) 3 28(c) 14 24 (d) 17 70(e) 15 56 (f ) 6(g) 13 48 (h) 18 40(i) 15 26 ( j) 3 2(k) 3 130 (l) 7 60

x x x xx x x xa a b by y z zh h t tm m q q

− + + −

− + − +

− + + −

− − − −

+ + + +

− − + −


2 2

2 2

2 2

2 2

2 2

2 2

2 2

(a) 2 1 (b) 110(c) 19 48 (d) 2 8(e) 54 (f ) 3 40(g) 42 (h) 20(i) 30 ( j) 27(k) 1 (l) 42(m) 100 (n) 36

s s h hz z u uw w v vt t x xx x p py r rk r

+ + − −

+ + + −

− − −

+ − − −

+ − +

− − −

− −

5. Factorise the following: 2 2

2 2

2 2

2 2

(a) 11 30 (b) 9 10(c) 2 8 (d) 7 8(e) 42 (f ) 3(g) 4 (h) 9

x x x xt t x xy y z zu w

+ + − −

− − − −

− − +

− −





(a) (2 1)(3 2) (b) (5 2)(3 4)(c) (6 1)(2 3) (d) (2 9)(3 1)(e) (7 1)(2 3) (f ) (9 2)(3 2)(g) (7 11)(2 1) (h) (8 1)(5 3)(i) (3 1)(3 1) ( j) (7 2)(7 2)(k) (12 1)(11 3) (l) (13 1)(7 5)

x x x xt t y yz z r re e q qp p y yk k v v

+ + + +− − − −− + − +− + + −− + + −+ + − +


2 2

(a) ( )(2 3 ) (b) (2 )( 5 )(c) (2 )(3 5 ) (d) (5 2 )(3 )(e) (7 3 )(2 5 ) (f ) (5 3 )(5 3 )(g) (2 3 ) (h) (2 5 )

x y x y x y x yx y x y x y x yx y x y x y x yx y x y

+ + + +− + − −− + − +

+ −

3. Factorise the following quadratics (showing the intermediate step in each case):

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

2

(a) 2 7 6 (b) 3 14 8(c) 6 11 4 (d) 4 20 9(e) 18 27 4 (f ) 4 13 3(g) 12 13 4 (h) 15 17 4(i) 6 11 4 ( j) 4 13 9(k) 8 5 3 (l) 10 13 3(m) 25 49 (n) 16 25(o) 32 44 9 (p) 8 19 15(q) 16

x x x xa a y yd d z zr r u u

q q p ph h t t

e sd d w wc

+ + + +

+ + + +

+ + + +

− − − −

− + − +

+ − + −

− −

+ + − −

− 267 12 (r) 42 5 3c x x+ − −

4. Factorise the following quadratic expressions:

2 2

2 2

2 2

(a) 8 15 (b) 20(c) 3 14 8 (d) 6 17 7(e) 4 15 4 (f ) 6 19 10

x x x xx x x xx x x x

+ + − −

− + − +

− − + +


8135)h(5124)g(6113)f(1252)e(

65)d(65)c(107)b(127)a(

22

22

22

22

++++

++−+

+−−−

++++

xxxxxxxx

xxxxxxxx

PTO



Sheet H2-14 2-02b-3 Quadratic Factorisation (cont.) 6. Factorise the following as far as possible:

2 2

2 2

2 2

(a) 5 (b) 2(c) 36 (d) 11 24(e) 4 20 9 (f) 5 8 4

a ab r rt b b

p p q q

+ +

− + +

+ + − −

7. Factorise the following as far as possible:

483(f)584(e)

6(d)2(c)9(b)23(a)

22

22

22

+−−−

−−+

−++

qqppnnzz

yxx


5176)f(5138)e(5136)d(6114)c(103)b(209)a(

222

222

+++−−−

+−−−+−

xxxxxxxxxxxx



Sheet H2-15 2-02c-1 Algebraic Fractions

1. Calculate the following, leaving your answers as fractions in their lowest form:

3 1 3 1(a) (b)5 4 4 76 2 2 1(c) (d)

11 5 3 84 1 1 7(e) (f )9 6 3 122 5 7 3(g) (h)9 12 10 811 1 11 2(i) ( j)18 4 36 3

+ +

+ +

+ +

+ +

+ +


1 1 2 1 2(a) (b)2 3 3 4

3 1 1 2 3 4(c) (d)4 5 2 5

2 3 2 3 2 2 1(e) (f )7 3 3 51 2 3 3 4 3 1(g) (h)

12 3 4 20

x x x x

x x x x

x x x x

x x x x

+ − − ++ +

+ + + ++ +

+ + + −+ +

− + − ++ +


3 1 2 1 5 2 3 1(a) (b)2 3 3 4

2 1 1 3 1 2 1(c) (d)2 5 3 5

2 3 2 5 1 3 2(e) (f )2 6 3 41 2 5 4 1 3 4(g) (h)

4 12 9 122 1 3 5 6 2 2 1(i) ( j)

8 12 15 6

x x x x

x x x x

x x x x

x x x x

x x x x

+ + + +− −

+ + + +− −

+ − + −− −

− − − −− −

+ − − −− −



Sheet H2-16 2-02c-2 Algebraic Fractions


2 1 1 2 5 3(a) (b)3 5 2 3

2 3 3 2 3 1 2(c) (d)4 6 2 7

3 5 2 5 1 3 2(e) (f )3 4 5 61 5 3 6 1 2 5(g) (h)

6 12 4 5

x x x x

x x x x

x x x x

x x x x

+ + − ++ +

+ − + ++ +

+ + + −− −

− + − −− −


3 1 2 1 1 4 1 2 3(a) (b)2 5 10 4 5 20

3 2 1 1 2(c) (d) (write as )2 3 6 2 2 2

2 1 3 2 2(e) (f ) 12 3 3 4

x x x x x x

x x x x x xx

x x x xx x

+ − − + ++ + + +

− − ++ + + +

+ + −+ + + + +

3. Calculate the following, leaving your answers as fractions in their lowest form: For example

( )( )( )

( )( )( ) ( )( ) ( )( )

3 41 33 3 4 1 3 9 4 4 7 13

1 3 3 1 1 3 1 3

x xx x x x x

x x x x x x x x

++ +

+ + + + + += + = =

+ + + + + + + +

.

2 3 3 4(a) (b)1 2 2 3

5 7 5 2(c) (d)1 1 3 2

3 4 5 4(e) (f )2 3 3 1 3 1 4 3

x x x x

x x x x

x x x x

+ ++ + + +

+ −+ − + +

+ ++ − + −



Sheet H2-17 2-02c-3 Fractional Equations 1. Solve the following:

2 4 8(a) (b)3 6 1 3 2

4 6(c) (d) 61 2 1 3

1 2 4(e) (f )4 5 7 53 2 2 1 5 1(g) (h)

3 6 4 21

x xx xx

x xx x x x

x x x

+= =

+ −

= =+ −

+ + += =

− + −= =


2 3 1 2(a) (b)4 5 1 21 1 1 3(c) (d)1 1 2 2

x x x xx x x xx x x xx x x x

+ + + ++ +

+ + − −+ − + −

− −− + − +


1 2 1 1 2 1 3 2(a) (b)2 3 4 3 4 5

2 1 5 2 1 3 1 2 3(c) (d)4 5 6 3 4 12

x x x x x x

x x x x x x

+ + − − ++ + + +

+ − + − −+ − + −



Sheet H2-18 2-02c-4 Simplifying Fractions

1. Simplify the following as far as possible:

2

2

2 2

2

2 2

2 4 3 3(a) (b)2 2 2

3 4(c) (d)4

5 2 1(e) (f )2 13 2 1(g) (h)

2 1

x xx

x x xx x x

x x x xx x xx x x

x x

+ ++

+ ++

+ + ++ ++ + −+ +

2. Simplify the following as far as possible:

2 2

2 2

2 2

2 2

2 2

2 2

2 2 2

2

3 2 5 6(a) (b)5 4 6 86 5 4(c) (d)7 10 4 4

2 7 12(e) (f )3 2 2 155 6(g) (h)6 8

x x x xx x x xx x xx x x xx x x xx x x xx x x yx x x y

+ + + ++ + + ++ + −+ + + ++ − − +− + + −− + −− + +



Sheet H2-19 2-02c-5 Quadratic Fractions


2 2

2 2

2 2

2 2

3 2 5 6(a) (b)2 3

2 3 1 12(c) (d)1 4

7 10 4 8 3(e) (f )2 2 3

1 25 1(g) (h)1 5 1

x x x xx x

x x x xx x

x x x xx x

x xx x

+ + + ++ ++ + + −+ +

− + − +− −

− −− +


2 2

2 2

3 2 1(a) (b)5 4 4 3

x x xx x x x+ + −+ + + +

3. Simplify 2 23 25 4 7 6x x x x

++ + + +

by first factorizing the denominators and finding the

lowest common denominator. 4. Simplify the following

2 2

2 2

2 2

4 7(a)6 8 5 65 2(b)7 12 9 20

3 4(c)1 4 3

x x x x

x x x x

x x x

++ + + +

++ + + +

+− + +



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