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For use only in Whitgift School IGCSE Higher Sheets 5

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IGCSE Higher

Sheet H5-1 H3-01-1 Linear Sequences Sheet H5-2 H3-01-2 Linear Sequences Sheet H5-3 H3-01-3 Linear Sequences Sheet H5-4 H3-02a-1 Functions Sheet H5-5 H3-02a-2 Functions Sheet H5-6 H3-02c-1 Functions Sheet H5-7 H3-02d-1 Functions Sheet H5-8 H3-02d-2 Functions Sheet H5-9 H3-02d-3 Functions Sheet H5-10 H3-02d-4 Functions Sheet H5-11 H3-02d-5 Functions Sheet H5-12 H3-02d-6 Functions Sheet H5-13 H3-03a-1 Graphs of Quadratics Sheet H5-14 H3-03a-2 Graphs Sheet H5-15 H3-03c-1 Graphs Sheet H5-16 H3-03c-2 Graphs Sheet H5-17 H3-03d-1 Straight Lines Sheet H5-18 H3-03d-2 Straight Lines Sheet H5-19 H3-03d-3 Straight Lines Sheet H5-20 H3-03d-4 Straight Lines Sheet H5-21 H3-03d-5 Straight Lines



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Sheet H5-1 H3-01-1 Linear Sequences

1. Find the next two terms in the following sequence

1 12 2

1 12 2

(a) 5, 7, 9, 11... (b) 1, 11, 21, 31... (c) 15, 22, 29, 36...1 2(d) 2, 11, 20, 29... (e) 14, 21, 28, 35... (f ) 1, , 2, ...

(g) 15, 12, 9, 6... (h) 101, 99, 97, 95... (i) 17, 11, 5, 1...( j) 3, 7, 11, 15... (k) 9, 2, 5, 12... (l) 3, 1 , 0, 1 ,..

−− − − − − − − −

3 7 3 5 31 1 1 12 4 4 8 4 8 2 4 4

.(m) , , 1, 1 ,... (n) 1, , , ... (o) , 1 , 2, 2 ,...

2. Find the nth term, nt , of the sequences in question 1. 3. (a) Find the nth term, nt of ,...23,17,11,5 .

(b) Find the 40th term of ,...23,17,11,5 . (c) Which term of ,...23,17,11,5 is equal to 479?

4. (a) Find the nth term, nt of ,...24,17,10,3

(b) Find the 71st term of ,...24,17,10,3 . (c) Which term of ,...24,17,10,3 is equal to 710?

5. (a) Find the nth term, nt of ,...4,9,14,19 .

(b) Find the 23rd term of ,...4,9,14,19 . (c) Which term of ,...4,9,14,19 is equal to –346?

6. (a) Find the nth term, nt of ,...9,7,5,3 −−−−

(b) Find the 15th term of ,...9,7,5,3 −−−− . (c) Which term of ,...9,7,5,3 −−−− is equal to -287?


(b) Find the 150th term of ,...13,10,7,4 . (c) Which is the first term of this sequence to be greater than 1000?


(b) Find the 100th term of ,...1061,1068,1075,1082 . (c) Which is the first negative term of this sequence?




1. A series of rows of tins are piled on top of one another. The bottom row has 35 tins and each row has two fewer tins than the row below it. (a) How many tins are there in the nth row from the bottom? (b) What is the maximum number of rows of tins?

2. A farmer has to put fence posts in a straight line. The fence posts are initially in a pile. The

first fence post is put into a hole 50m from the pile and there is a gap of 3m between the posts so the second post is 53m away from the pile, the third is 56m away etc. (a) How far from the pile will the nth post be? (b) Which post is 83m from the pile?

3. A French teacher gives his class a set of words to learn each week. On the first week of the

term he asked them to learn 25 words and then increased this by 5 words each week. (a) How many words would the class have to learn in the nth week of the term? (b) How many words would the class have to learn in the 9th week of the term? (c) In which week would the class have to learn 90 words?

4. A ladder is such that each rung is 0.75cm shorter than the one below it. If

the bottom rung is 32cm long then (a) How long is the nth rung from the bottom? (b) Which rung is 18.5cm long?

5. A football club had 35,000 supporters at its first home match. The attendance increased by

250 at each home game. (a) How many supporters would be at its nth home game? (b) If there were 37,750 at its last home game of the season then how many home games

did it play? 6. Joshua decides to save money in the following way: He saves £1 in the first week, £1.20 in the second week, £1.40 in the third week, and so on.

(a) How much would he save in the n th week ? (b) How much would he save in the 8th week? (c) In which week would he first save at least £5? (d) After 10 weeks, Joshua wants to buy a tennis racquet which costs £19.99. He realises that he hasn’t saved quite enough, but by how much is he short?




1. (a) Find the nth, nt , of ,...13,9,5,1 (b) Find the 18th term of ,...13,9,5,1 (c) Which term of ,...13,9,5,1 is equal to 113?

2. (a) Find the nth, nt , of ,...41,30,19,8 (b) Find the 12th term of ,...41,30,19,8 (c) Which term of ,...41,30,19,8 is equal to 250? 3. (a) Find the nth, nt , of ,...18,13,8,3

(b) Find the 9th term of ,...18,13,8,3 (c) Which term of ,...18,13,8,3 is equal to 513?

4. (a) Find the nth, nt , of ,...55,48,41,34

(b) Find the 12th term of ,...55,48,41,34 (c) Which term is the first term to be greater than 37 ?

5. (a) Find the nth, nt , of ,..14,22,30,38

(b) Find the nth term of ,..60,71,82,93 (c) After how many terms will the nth term of ,..60,71,82,93 be smaller than the nth term of ,..14,22,30,38 .

6. (a) Find the nth, nt , of ,....85,

21,

83,

41

(b) Find the 7th term of ,....85,

21,

83,

41

(c) What is the first term of ,....85,

21,

83,

41 to be greater than 5?

7. (a) A man wants to invest £10000 into a bank for two years. Bank A offers him 10% per

year. How much will he have in his account after two years? (b) Bank B offers him £P per year where P is the amount such that after two years he would have exactly the same amount in Bank A or B. Find P. (c) If this value of P is kept the same and he then decides to invest for more than two years, which bank should he opt for?

8. (a) A sequence is such that its first term is a and it increases by d. If the fourth term is 23 and the seventh term 38 find write down two simultaneous equations involving a and d. (b) Solve these to find a and d.



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Sheet H5-4 H3-02a-1 Functions

1. Given that f ( ) 5 1x x= + , find the following : (a) ( )f 1

(b) ( )f 0

(c) ( )f 1− (d) the value of x such that f ( ) 11x = .

2. Given that 2 1g( )

3xx +

= , find the following :

(a) ( )g 4

(b) ( )g 7

(c) ( )g 2− (d) the value of x such that g( ) 7x = .

3. Given that 2h( ) 1x x= + find the following : (a) h(1) (b) h( 1)− (c) h(2) (d) the two values of x such that h( ) 101x = .

4. Given that 1f ( ) xxx+

= find the following :

(a) f ( 1)− (b) f (2)

(c) 1f2

⎛ ⎞⎜ ⎟⎝ ⎠

(d) a value of x such that f ( ) 2x = . 5. Given that 2f ( ) 2 8x x x= − − calculate the following :

(a) ( )f 2

(b) ( )f 3−

(c) 1f2

⎛ ⎞⎜ ⎟⎝ ⎠

(d) a value of x such that f ( ) 0x = PTO



Sheet H5-4 H3-02a-1 Functions (cont.)

6. Given that the function h( )x is defined by 3 2h( )

2xxx+

=−

, find the following :

(a) ( )h 1 (b) the value of a such that h( ) 11a = (c) the value of a such that h( ) 7a = (d) the value of x such that h(x) has no value.

7. Given that 2 4h( )

2 1zz

z−

=−

calculate the following :

(a) h(3) (b) h( 2)− (c) h(0)

(d) 1h2

⎛ ⎞−⎜ ⎟⎝ ⎠

(e) the value of x such that h(x) has no value.



Sheet H5-5 H3-02a-2 Functions

1. Given that 2f ( ) 5 6x x x= − + , find the following : (a) ( )f 2−

(b) ( )f 0

(c) ( )f 1 (d) two values of x such that f ( ) 0x = .

2. Given that 3 1g( ) xx

x+

= , find the following :

(a) ( )g 1

(b) ( )g 0.5

(c) ( )g 2

(d) if g( ) 2x = then solve 3 1 2x

x+

= to find x.

3. Given that 2

2

1h( )1

xxx

−=

+ find the following :

(a) h( 1)− (b) h(2.5) (to 3sf) (c) h(3.5) (to 3sf)

(d) if 3h( )5

x = then solve 2

2

1 351

xx

−=

+ to find x.

4. Given that ( )( )g( ) 1 2x x x x= + + find the following :

( )( )

( )( )

(a) g 1

(b) g 1

1(c) g2

(d) g 2

(e) three values of for which g 0x x

−

⎛ ⎞⎜ ⎟⎝ ⎠−

=



Sheet H5-6 H3-02c-1 Functions

1. Find the range of the following functions: (a) 2f ( )x x= where the domain is { }1 , 2 ,3 , 4 .

(b) 2f ( )x x= where the domain is { }2, 1, 0, 1 , 2− − .

(c) f ( ) 2 1x x= + where the domain is { }3, 4, 5, 6 .

(d) f ( ) 4x x= − where the domain is { }1, 2, 3, 4 .

(e) 3f ( )x x= where the domain is { }2, 1, 1, 2− − .

(f) 24f ( )xx

= where the domain is { }1, 2, 3, 4, 6, 12, 24 .

(g) 12f ( ) 2xx

= + where the domain is { }2, 1, 1, 2, 3, 4− − .

(h) 2f ( ) 1x x= + where the domain is all the real numbers (that is all numbers on the number line).

(i) 1f ( )xx

= where the domain is { }: 1x x ≥ .

2. Find the values which must be excluded from the domains of the following functions :

(a) ( ) 1f xx

= (b) ( ) 3f1

xx

=−

(c) ( )f2

xxx

=−

(d) ( ) 2f2 1

xx

=−

(e) ( ) 2

1f4

xx

=−

(f) ( )f x x=

(g) ( )f 1x x= − (h) ( ) 1f x xx

= +

(i) ( ) 1f2

xx

=−

(j) ( ) ( )( )1f

1 2x

x x=

− −



Sheet H5-7 H3-02d-1 Functions

1. Given that the function )(f x is defined by 3 12f ( )

2xxx+

=+


(a) f ( 5)− (b) f (0) (c) f ( 1)− (d) the value of x such that f ( ) 4.5x = (e) a value of x such that f ( )x x= .

2. Given that the function g( )x is defined by 2 3g( )

2xx

x+

=+


(a) g(0) (b) the value of x such that g( ) 3x = (c) the value of x such that g( ) 1x = (d) the value of x such that g( ) 2x = − .

3. If 2g( ) 3 5x x x= − then find:

(a) g(1)

(b) 1g2

⎛ ⎞⎜ ⎟⎝ ⎠

(c) g( 2)−

4. If 3 1f ( )2 7

xxx−

=+

then find x such that f ( ) 1x = .

5. If ( )2h( ) 7x x= + , for what values of x is h( ) 81x = ? 6. If 3 2f ( ) 2 5 8x x x x= + + − and 3g( ) 2 3 2x x x= − + then find the following:

(a) f (2) (b) ( )g 3− (c) f ( 2)− (d) ( )g 0

(e) ( )f 0 (f) a value of a such that f ( ) 0a = .




1. The function f ( ) 2 3x x= + maps 1 to 5, 2 to 7, 3 to 9 etc. Find the function g( )x (called the inverse function of )(f x ) which does the reverse process, i.e. which maps 9 to 3, 7 to 2, 5 to 1 etc..

2. (a) The function f ( ) 5 2x x= + then find x such that:

(i) f ( ) 12(ii) f ( ) 7(iii) f ( ) 72

xxx

===

(b) If 5 2y x= + then find x in terms of y (that is make x the subject of the formula). (c) Hence write down the inverse function ( )1f x− . (d) Calculate the following:

-1

-1

-1

(i) f (12)(ii) f (7)(iii) f (72)

3. (a) The function 1f ( ) , 0xx xx+

= ≠ then find x such that:

(i) f ( ) 2(ii) f ( ) 1.5(iii) f ( ) 0

xxx

===

(b) If 1xyx+

= then find x in terms of y.

(c) Hence write down the inverse function ( )1f y− . (d) Calculate the following:

-1

-1

-1

(i) f (2)(ii) f (1.5)(iii) f (0)




Use the following steps:

-1 -1

(i) Write f ( ) e.g. 5 2- 2(ii) Make the subject of the formula e.g.5

- 2(iii) Write f ( ) e.g. f ( )5

y x y xyx x

yx y y

= = +

=

= =

1. Find ( )1f y− for the following functions

2

(a) f ( ) 2 for all 3(b) f ( ) for all 43 1(c) f ( ) for all

413(d) f ( ) 02

(e) f ( ) 3 4 for all (f ) f ( ) 1 for 0

2 1(g) f ( ) for all 3

(h) f ( ) 3 5 for all 2 5(i) f ( ) 7

73 5 9( j) f ( )2 9 2

x x x

x x x

xx x

x xx

x x xx x x

xx x

x x xxx x

xxx xx

= −

=

−=

= ≠

= +

= + >+

=

= −+

= ≠−−

= ≠ −+

2. State the values for which the domain of ( )1f y− must be limited in question 1.




1. If f ( ) 2 1x x= + and 2g( )x x= then: (a) Calculate the following

(i) f (2) (ii) f ( 1)(iii) f (3) (iv) f (0.5)(v) f (4) (vi) f (1)(vii) f (9) (viii) f (0.25)(ix) g(2) (x) g( 1)(xi) g(3) (xii) g(0.5)(xiii) g(5) (xiv) g(7)

−

−

(b) Use part (a) to find f(g(2)) and g(f (2)) . (c) Use part (a) to find f(g( 1))− and g(f ( 1))− . (d) Use part (a) to find f(g(3)) andg(f (3)) . (e) Use part (a) to find f(g(0.5)) and g(f (0.5)) .

3. If f ( ) 3 1x x= − and g( ) 5 1x x= + then: (a) Calculate the following

(i) f (1) (ii) f (4)(iii) f (2) (iv) f ( 2)(v) f (6) (vi) f (21)(vii) f (11) (viii) f ( 9)(ix) g(1) (x) g(4)(xi) g(2) (xii) g( 2)(xiii) g(11) (xiv) g(5)(xv) g( 7) (xvi) g(0)

−

−

−

−

(b) Use part (a) to find f(g(1)) and g(f (1)) . (c) Use part (a) to find f(g(4)) and g(f (4)) . (d) Use part (a) to find f(g(2)) and g(f (2)) . (e) Use part (a) to find f(g( 2))− and g(f ( 2))− .

4. If f ( ) 3 1x x= − and 7)g( += xx then:

(a) Calculate 1 1f ( ) and g ( )x x− − (b) Show that ( )( ) ( )( )1 1f g 4 =4 and find g f 4− − .

(c) What are the values of ( )( ) ( )( )1 1f g 7 and g f 2− − ?




1. (a) If f ( ) 3x x= + then find f ( 1)x + . (b) If f ( ) 2 1x x= + then find f (3 2)x + . (c) If f ( ) 1x x= − and 2g( )x x= then find ( )( )f g x and ( )( )g f x .

(d) If 2f ( ) 1x x= − and ( )g 2 1x x= + then find ( )( )f g x and ( )( )g f x .

(e) If f ( ) 25xx = + and ( )g 5 1x x= + then find ( )( )f g x and ( )( )g f x .

(f) If f ( ) 2 1x x= + and ( ) 1g2

xx −= then find ( )( )f g x and ( )( )g f x .

2. If f ( ) 5 1x x= + and g( ) 2 3x x= + then find ( )( ) ( )( )f g g fx x− . 3. (a) If f ( ) 2 1x x= + and g( ) 3 7x x= − then find ( )( )g f x .

(b) Find the value of x such that ( )( )g f 26x = . 4. (a) If f ( ) 1x x= + and 2g( )x x= then find ( )( )g f x .

(b) Find the values of x such that ( )( )g f 9x = . 5. A function is defined by 13)(f −= xx for all x.

(a) Show that ( )( )f f 9 4x x= − .

(b) Find ( )1f y−

(c) Show that ( )( )( ) 1f f y

− is identically equal to ( )( )( )1 1f f y− − .

6. A function is defined by10f( ) , 88

xx xx+

= ≠−

.

(a) Find a value of x such that f( ) 5x = . (b) Find a positive integer p such that f( )p p= by solving a suitable equation. (c) Find also 1f ( )p− for the positive integer found in (b).

7. The one to one functions f and g are defined by ( )f 2x x= − for ax ≥ and ( ) 2

1g xx

=

for bx > . (a) Find the smallest possible values of a and b. (b) Find ( ) ( )1 1f and gx x− − .

PTO



Sheet H5-11 H3-02d-5 Functions (cont.)

8. The functions f and g are defined by ( )f 2 1x x= + for all x and ( ) 5g , 33

x xx

= ≠−

.

(a) Find ( ) ( )1 1f and gx x− − . (b) State the restrictions, if any, on the domains of these inverse functions.

9. The function )(f x is defined by ( ) 2f 1, 0x x x= + ≥ . On the same diagram draw a sketch

of f( )y x= and 1f ( )y x−= .




1. If 2f ( ) 3x x= + then find the following:

(a) f (2)(b) f ( 1)(c) a value of such that f ( ) 3x x

−=

2. If 1f ( )2

xxx+

=−

then find the following:

1

(a) the value of which cannot be in the domain of f( )(b) f ( )

x xy−

3. If 2f ( )x x= and g( ) 1x x= + then find the following:

( )( )

( )

(a) f g(2)

(b) f g( )

(c) a value of such that f g( ) 16

x

x x =

4. f and g are functions as follows:

f : 2 5

g : 2

x x

x x

→ +

→ +

(a) Calculate ( )f 3− .

(b) Given that ( )f 17a = then find the value of a. (c) Which values of x cannot be in the domain of g? (d) Find the inverse function of g.

5. f and g are functions as follows:

f : 2 5g : 3 2

x xx x→ +→ +

(a) Find ( )f g( )x .

(b) Given that ( )f g( ) g( )a a= then find the value of a.



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Sheet H5-13 H3-03a-1 Graphs of Quadratics

1. The table of values below is for the graph 2 6 1y x x= − +

x 0 1 2 3 4 5 6 y -7

(a) Copy and complete the above table. (b) Draw a scale from 0 to 6 on the x-axis (2cm per unit) and from -10 to 5 on the y-axis

(1cm per unit). (c) Draw a sketch of the curve 2 6 1y x x= − + . (d) Draw the line 2 3y x= − − on the graph. (e) At what point is the line 2 3y x= − − is a tangent to the curve 2 6 1y x x= − + .

2. The table of values below is for the graph 2 2 4y x x= − −

x -3 -2 -1 0 1 2 3 4 5 y 4

(a) Copy and complete the above table. (b) Draw a scale from –3 to 5 on the x-axis (2cm per unit) and from –8 to 12 on the y-axis

(1cm per unit). (c) Draw a sketch of the curve 2 2 4y x x= − − . (d) What line do you need to draw to solve the equation 2 2 4 5x x− − = ? (e) What are the solutions (to 1dp) to the equation 2 2 4 5x x− − = ? (f) What is the smallest value of 2 2 4x x− − and which value of x achieves this smallest

value? 3. The table of values below is for the graph 21 4y x x= − −

x -6 -5 -4 -3 -2 -1 0 1 2 y -4

(a) Copy and complete the above table. (b) Draw a scale from –6 to 2 on the x-axis (2cm per unit) and from –12 to 6 on the y-axis

(1cm per unit). (c) Draw a sketch of the curve 21 4y x x= − − . (d) Use your graph to solve (to 1dp) the equation 21 4 2x x− − = (e) What is the largest value of 21 4x x− − and which value of x achieves this largest

value? PTO



Sheet H5-13 H3-03a-1 Graphs of Quadratics (cont.)

4. (a) Solve the equation 2 5 6 0x x+ + = by factorising. (b) Explain why the equation 2 5 5 0x x+ + = cannot be solved in this way. (c) Copy and complete the table of values below is for the graph 2 5 5y x x= + +

x -6 -5 -4 -3 -2 -1 0 1 y 1

(d) Copy and complete the above table. (e) Draw a scale from –6 to 1 on the x-axis (2cm per unit) and from –3 to 11 on the y-axis

(1cm per unit). (f) Use your graph to solve (to 1dp) the equation 2 5 5 0x x+ + = .



Sheet H5-14 H3-03a-2 Graphs 1. The table of values below is for the graph 12 5y x

x= + −

x 0.5 1 2 3 4 5 10 15 20 y 2.4

(f) Fill in the above table, to 1dp.

(g) Draw a sketch of the curve 12 5y x

x= + − .

(h) Draw the tangent to the curve at the point where 5x = . (i) Find the gradient of this tangent using two points on the line. (j) Find the gradient of this tangent this using differentiation. (k) What is the minimum value of y?

(l) Use differentiation to find the minimum value. PTO



Sheet H5-14 H3-03a-2 Graphs (cont.) 2. The table of values below is for the graph 200100y x

x= − −

x 3 5 10 20 30 40 50 60 70 80 90 100 y 30 46

(a) Fill in the above table, to the nearest whole number.

(b) Draw a sketch of the curve 200100y xx

= − − .

(c) Draw the tangent to the curve at the point where 50x = . (d) Find the gradient of this tangent using two points on the line. (e) Find the gradient of this tangent this using differentiation.

(f) What is the maximum value of y?

(g) Use differentiation to find the maximum value.

PTO



Sheet H5-14 H3-03a-2 Graphs (cont.)

3. The graph of 3 23 2 1y x x x= − + + is shown below.

(a) Draw on the above the line 2y x= − + . (b) Draw on the above the line 2 3y x= − . (c) At what point is the line 2y x= − + the tangent to the curve? (d) At what point is the line 2 3y x= − the tangent to the curve? (e) Draw on the above the line 3y = . (f) Use this to solve 3 23 2 1 3x x x− + + = .

PTO



Sheet H5-14 H3-03a-2 Graphs (cont.) 4. The table of values below is for the graph 3 22 3 1y x x x= + − −

x -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 y 3.4 3 2.4

(a) Copy and complete the above table, to 1dp. (b) Draw a sketch of the curve 3 22 3 1y x x x= + − − on the axes below.

(c) Draw the tangent to the curve at the point where 1x = . (d) Find the gradient of this tangent.



Sheet H5-15 H3-03c-1 Graphs 1. The table of values below is for the graph 3 4 1y x x= − − .

x -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 y 1.6 -4

(g) Copy and complete the above table, to 1dp. (h) Draw a sketch of the curve 3 4 1y x x= − − .

(i) Draw the tangent to the curve at the point where 2x = . (j) Find the gradient of this tangent using two points on the line.

PTO



Sheet H5-15 H3-03c-1 Graphs (cont.)

2. The table of values below is for the graph 3 24y x x= − x -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 y -12.4 -0.9 -8 10.1 (g) Copy and complete the above table. (h) Draw a sketch of the curve 3 24y x x= − .

(i) On the graph draw the straight line 12y = . (j) Use your graph to solve (to 1dp) the equation 3 24 12x x− = . (k) On the graph draw the straight line 20 4y x= − .

(l) Hence solve (to 1dp) the equation 3 24 20 4x x x− = − .

PTO




3. The table of values below is for the graph 2

120 30y xx

= + −

x 1 2 3 4 5 10 15 25 30 y -18.8 -4.8

(a) Copy and complete the above table (to 1dp).

(b) Draw a sketch of the curve 2

120 30y xx

= + − .

(c) On the graph draw the straight line 100 5y x= − .

(d) Use your graph to find (to 1dp) the two solutions to the equation

2

120 30 100 5x xx

+ − = − .

(e) Find the minimum value of the curve (to the nearest whole number).




4. The graph of 2 3 2y x x= + − is shown below.

a. What line must be drawn on to solve 2 3 2 15x x+ − = ? b. Draw this line to solve (to 1dp) 2 3 2 15x x+ − = .

c. What line must be drawn on to solve 2 3 2 10 2x x x+ − = − ? d. Draw this line to solve (to 1dp) 2 3 2 10 2x x x+ − = − . e. If 2 2 7 0x x− − = then fill in the right hand side of the following: 2 3 2 ...............x x+ − = f. What line must be drawn on to solve 2 2 7 0x x− − = ? g. Draw this line to solve (to 1dp) 2 2 7 0x x− − = .



Sheet H5-16 H3-03c-2 Graphs

1. The graph of 3 24 1y x x= − + is shown below.

(a) What is the equation of the straight line that needs to be drawn on the above graph to solve the equation 3 24 1 3x x− + = ? (b) Draw this to find the solution (to 1dp) to 3 24 1 3x x− + = . (c) If 3 24 11 0x x− + = then rearrange to fill in the right hand side of the following: 3 24 1 ....x x− + = (d) Hence draw an appropriate straight line and find the solution (to 1dp) to

3 24 11 0x x− + = .

PTO



Sheet H5-16 H3-03c-2 Graphs (cont.) 5. The graph of 5y x

x= + is shown below.

(a) What is the equation of the straight line that needs to be drawn on the above graph to

solve the equation 5 7xx

+ = ?

(b) Draw this to find the two solutions (to 1dp) to 5 7xx

+ = .

(c) If 53 14xx

+ = then rearrange to fill in the right hand side of the following:

5 ...xx

+ =

(d) Hence draw an appropriate straight line and find the two solutions (to 1dp) to

53 14xx

+ = .

PTO




6. The graph of 3 2 3y x x= − − is shown below.

By drawing suitable straight lines find the solutions to the following equations (to 1dp) NB In each case state which line you need to draw: (a) 3 2 3 5x x− − = (one solution) (b) 3 2 11 0x x+ − = (one solution) (c) 3 4 2 0x x− − = (three solutions)

PTO




7. The graph of 32 3y x x= + − is shown below.

By drawing suitable straight lines find the solutions to the following equations (to 1dp) NB In each case state which line you need to draw: (a) 32 3 2x x+ − = − (one solution) (b) 31 4 0x x− + − = (three solutions) (c) 36 2 0x x− − = (one solution)



Sheet H5-17 H3-03d-1 Straight Lines

1. A triangle has vertices A(1, 7), B(2, 11), C(5, -1). Calculate the gradients of the three lines which form this triangle.

2. Write the following exactly in the form y = mx + c where m and c are fractions

(a) 3 4 9(b) 5 4 10(c) 3 2 4(d) 12 9 4

y xy xx yx y

− =+ = −− == −

3. Write the following as equations involving only integers (e.g. 632 =+ yx ):

23)d(

2)c(

4)b(

2)a(

53

41

32

2134

+=

−=

=−

−=

xy

xy

xy

xy

4. Find the equations, in the form cmxy += , of the four straight lines A, B, C and D shown

below.

5. Use question 4 to solve the following simultaneous equations: (a) xy =+ 4 and 42 =+ xy (b) xy =+ 4 and 0=+ xy (c) 0=+ xy and 42 =+ xy

A

B

C

D




1. Draw a set of axes from –8 to 8 on both axes using 1cm per unit. On this set of axes draw and label the following straight lines: (a) xy = (b) 12 += xy (c) xy −= 8

(d) 221

−= xy

(e) xy 35−= 2. Use the graph in question 1 to find the point of intersection of the following pairs of lines:

(a) 12 += xy and xy =

(b) xy 35−= and 221

−= xy

(c) 221

−= xy and 12 += xy

(d) xy = and xy −= 8 3. Draw a set of axes from –8 to 8 on both axes using 1cm per unit. On this set of axes draw

and label the following straight lines:

1(a) 12

(b) 6(c) 6(d) 8 3

1(e)2

y x

y xy xy x

y x

= +

= −= += −

=

4. Use the graph in question 3 to find the point of intersection of the following lines:

(a) 121

+= xy and xy 38 −=

(b) 6y x= − and 12

y x=

(c) xy 38 −= and 6y x= + (d) 6y x= − and 6y x= +

5. The straight line L1 has equation y = 3x +13

The straight line L2 is parallel to the straight line L1. The straight line L2 passes through the point (5, 1). Find an equation of the straight line L2.




1. Find the gradient of AB, AC and BC in the following: (a) (b)

2. Draw a set of axes from –8 to 8 on both axes using 1cm per unit. On this set of axes draw

and label the following straight lines: (a) xy 2= (b) 2+= xy (c) xy −= 6 (d) 4−= xy (e) xy 25 −= (f) 6−=y

3. (a) Use question 2 to find the point of intersection of the following pairs of lines:

(i) xy 25 −= and 4−= xy (ii) xy −= 6 and 4−= xy (iii) xy 2= and 2+= xy (iv) xy 25 −= and xy −= 6

(b) Which two lines are parallel to each other? 4. Draw a set of axes from –8 to 8 on both axes using 1cm per unit. On this set of axes draw

and label the following straight lines: (a) 3 5y x= − (b) 8 2y x= − (c) 6y x= − (d) 22 += xy

(e) 221

+= xy (f) xy214 −=

5. (a) Use question 4 to find the point of intersection of 221

+= xy and xy214 −= .

(a) Which two lines meet at the point (0, 2)?

A

B

C C

A

B




1. Find the gradients of the straight lines which pass through the following pairs of points: (a) (1, 12) and (3, 8) (b) (2, 7) and (5, 13) (c) (-4, 8) and (2, -10) (d) (-2, -11) and (13, 4) (e) (-3, 7) and (-1, 11) (f) (-5, -1) and (-1, -17) (g) (-1, 9) and (7, -15) (h) (11, -1) and (11, 9) (i) (8, 12) and (1, 12)

2. Find the gradients of the four straight lines A, B, C and D shown below.

3. Find the equations of the lines in the above diagram in the form cmxy += . 4. A triangle has vertices A(3, 5), B(7, -3) and C(9, 8).

(a) Draw an axis from 0 to 10 on the x-axis and from –5 to 10 on the y –axis (1cm per unit on both axes). Draw and label the triangle ABC.

(b) What sort of triangle is it? (c) Calculate the gradients of the three lines which form this triangle. (d) What is the result of multiplying the gradients of AB and AC together? (e) What does this signify?

A B

C

D




1. Find the gradients of the straight lines which pass through the following pairs of points: (a) (2, 9) and (4, 13) (b) (7, 11) and (12, 31) (c) (-2, 8) and (2, 16) (d) (-3, -5) and (-5, -11) (e) (2, 13) and (7, 13) (f) (3, 4) and (3, 9)

2. Draw and label the following straight lines (with an x-axis from –8 to 8 and a y-axis from

–8 to 12 using 1cm per unit on both axes). (a) xy = (b) xy −= 8 (c) 22 += xy (d) xy 26 −= (e) 2054 −=− xy (f) 1232 −=+ xy

3. Use your graph in question 2 to find the solutions to the following simultaneous equations:

(a) xy = and xy −= 8 (b) xy = and xy 26 −= (c) 22 += xy and xy 26 −= (d) xy −= 8 and 22 += xy (HINT: Find where it crosses the two axes) (e) xy = and 22 += xy (HINT: Find where it crosses the two axes)

4. Draw and label the following straight lines (with an x-axis from –6 to 6 and a y-axis from

–8to 8 using 1cm per unit on both axes). (a) 2=− xy (b) 6=+ yx (c) 42 −=− xy (d) 22 −=+ yx (e) 84 =− xy

5. Use your graph in question 4 to find the solutions to the following simultaneous equations:

(a) 42 −= xy and 2+= xy (b) 22 −=+ yx and 2+= xy (c) xy −= 6 and 2+= xy (d) 84 =− xy and 22 −=+ yx



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