Substitution
Q1.          Given that a = 7, b = 3 and c = 5
(a)     Work out the value of a + 2b + 3c
..................................................................................................................
..................................................................................................................
Answer ..................................................
(2)

(b)     Work out the value of abc
..................................................................................................................
..................................................................................................................
Answer ..................................................
(2)

(c)     abcd = 0
Write down the value of d.
..................................................................................................................
Answer ..................................................
(1)
(Total 5 marks)

Q2.          A shopkeeper uses this formula to calculate the total cost when customers pay by monthly installments.
C = d + 24 × m
          C is the total cost in pounds.
          d is the deposit in pounds.
          m is the monthly installment in pounds.
(a)     The deposit for a wardrobe is £16.
          The monthly payments are £10.
          What is the total cost?
…..................................................................................................
....................................................................................................................
....................................................................................................................
Answer £ ............................................
(2)
(b)     How many years does it take to finish paying for goods using this formula?
....................................................................................................................
Answer ...................................... years
(1)
(c)     The total cost of a sofa is £600.
The deposit is £120.
Work out the value of the monthly instalment.
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer £ ............................................
(3)
(Total 6 marks)
Q3.          Complete this table.
 
Expression	Value	
2x	8	
5x	 	
2x + 3y	5	
y	 	
3x – y	 	

....................................................................................................................
....................................................................................................................
(Total 5 marks)
Q4.    (a)     Simplify 5x + 3y – 2x + 4y
................................................................................................
Answer .................................................
(2)
(b)     Find the value of 5p + 2q when p = 4 and q = – 7
................................................................................................
................................................................................................
Answer .................................................
(2)
(c)     Find the value of u2 – v2 when u = 5 and v = 3
...............................................................................................
................................................................................................
Answer .................................................
(2)
(Total 6 marks)
Q5.   This is the payment plan for Donal’s mobile phone.
He receives a bill every month.

(a)     In January, Donal did not make any calls.
How much was his bill?
....................................................................................................................
Answer £ .................................................
(1)
(b)     In February, Donal made 100 minutes of calls. How much was his bill?
....................................................................................................................
....................................................................................................................
Answer £ .................................................
(2)
(c) In March, Donal’s bill was £7.50
How many minutes of calls did he make?
....................................................................................................................
....................................................................................................................
Answer ................................................. minutes
(2)
(Total 5 marks)
 Q6.          ABCD is a quadrilateral.
      
Not drawn accurately
(a)  Write down an expression for the perimeter of the quadrilateral in terms of x and y. Simplify your answer.
....................................................................................................................
....................................................................................................................
Answer ..............................................
(2)
(b)     When x = 4 cm, the perimeter of the quadrilateral is 68 cm.
Find the value of y.
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer ........................................cm
(3)
(Total 5 marks)
 
Collecting like terms & brackets
Q7.    (a)     Simplify 5x + 3y – 2x + 4y
................................................................................................
Answer .................................................
(2)
(b)     Find the value of 5p + 2q when p = 4 and q = – 7
................................................................................................
................................................................................................
Answer .................................................
(2)
(c)     Find the value of u2 – v2 when u = 5 and v = 3
...............................................................................................
................................................................................................
Answer .................................................
(2)
(Total 6 marks)
 Q8.          (a)     Simplify                5c + 3c – c
................................................................................................
Answer .............................................
(1)
(b)     Simplify                7x + 2y – 3x – 5y
................................................................................................
................................................................................................
Answer .............................................
(2)
(Total 3 marks)
Q9.          (a)     Simplify      8a – b + 4a – 2b
....................................................................................................................
....................................................................................................................
Answer ..................................................
(2)
(b)     Multiply out        x(x + 7)
....................................................................................................................
Answer ..................................................
(1)
(Total 3 marks)

Q10.          (a)     Simplify      a + a + a
Answer .......................................................................
(1)
(b)     Simplify      8b + 3 – 2b + 7
....................................................................................................................
....................................................................................................................
Answer .......................................................................
(2)
(c)     Solve the equation        = 6
....................................................................................................................
Answer x = ................................................................
(1)
(Total 4 marks)

Q11. (a) Expand and simplify  4(2x – 1) + 3(x + 6)
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer .................................................
(2)
(b)     Expand      x2(4 – 2x)
....................................................................................................................
....................................................................................................................
Answer .................................................
(2)
(Total 4 marks)

Q12. Simplify 2x + 8 + 4x – 3 
....................................................................................................................
....................................................................................................................
Answer .................................................
(2)
 (Total 2 marks)

Rules of Indices
Q13.          (a)     Simplify a2 × a4
....................................................................................................................
Answer .......................................................................
(1)
(b)     Simplify b9 ÷ b3
....................................................................................................................
Answer .......................................................................
(1)
(c)     Simplify c5 × c × c5
....................................................................................................................
Answer .......................................................................
(1)
(Total 3 marks)
 Q14. (a)     Simplify             x2 × x3
Answer .........................................
(1)
(b)     Simplify             y8 ÷ y5
Answer .........................................
(1)
(c)     Expand              x(x3 – 3)
....................................................................................................................
....................................................................................................................
Answer .........................................
(2)
(Total 4 marks)
 Q15.          (a)     Simplify
(i)      y7 × y2
................................................................................................................
Answer .................................................
(1)
(ii)     y7 ÷  y2
................................................................................................................
Answer .................................................
(1)
(iii)     (y7)2
...............................................................................................................
Answer .................................................
(1)
(b)     (i)      If y = –1 which answer in part (a) is positive?
................................................................................................................
Answer .................................................
(1)
(ii)     If y = 0.5 which answer in part (a) has the greatest value?
................................................................................................................
Answer .................................................
(1)
(Total 5 marks)

 Sequences
Q16.         Here is a sequence of triangle patterns.

          Complete the table.
 
 	Pattern 1	Pattern 2	Pattern 3	Pattern 4	
Number of shaded triangles	1	2	3	 	
Total number of triangles	3	5	 	 	

(Total 2 marks)

Q17.          Here is a sequence of numbers
4        7        10        13
(a)     Write down the next term in the sequence.
…………................................................................................................
Answer .......................................................................
(1)
(b)     Write down the rule for continuing the sequence.
....................................................................................................................
Answer .......................................................................
(1)
(Total 2 marks)

Q18. (a)     The rule for the next term of a sequence is
Multiply the previous term by three and subtract one.
The first two terms of the sequence are 2 and 5.
Write down the next two terms.
....................................................................................................................
....................................................................................................................
Answer             2            5           ...............          ...............
(2)
(b)     The nth term of a different sequence is 5n.
The first term is 5
Write down the next three terms.
Answer             5        ............         ...........         ............
(1)
(c)     Work out the nth term of this sequence.
7           10           13           16           19
....................................................................................................................
Answer ...........................................................................
(2)
(Total 5 marks)

 
Q19. This question is about sequences that starts 1, 4 …
(a)     Here are the first three terms of a sequence
1        4        16        …
The rule for continuing this sequence is
 
Multiply by 4	
What is the next term?
....................................................................................................................
Answer .......................................................................
(1)
(b)     Another sequence uses a pattern of dots.
Here are the first three patterns.

Pattern 1                    Pattern 2                  Pattern 3
(i)      Draw Pattern 4
(1)

 

 (ii)     How many dots are in Pattern 5?
................................................................................................................
Answer .......................................................................
(1)
(c)     Here are the first five terms of a different sequence
1            4             8             13           19     …
What is the next term?
....................................................................................................................
Answer .......................................................................
(1)
(Total 4 marks)
Q20.          Billy and Mina are investigating sequences that begin with
1,            2,            4,         ......
(a)     Billy says the fourth term is 8
          What rule could Billy be using?
....................................................................................................................
....................................................................................................................
Answer ........................................
(1)
(b)     Mina says the fourth term is 7
          What rule could Mina be using?
....................................................................................................................
....................................................................................................................
Answer ........................................
(1)
(Total 2 marks)
Q21. In all of the following diagrams numbers are increased by two equal steps.
For example:

       Find a, b and c.

....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer a = ..................., b = …..............., c = ................
(Total 3 marks)

Q22.          Square tiles are used to make patterns on a grid.

(a)     The pattern continues in the same way.
          Draw Pattern 4
(1)
(b)     (i)      Complete this table.
 
Pattern Number	1	2	3	4	5	
Number of tiles	5	8	 	 	 	
(1)
(ii)     How many tiles are in Pattern 9?
...............................................................................................................
Answer .................................................
(1)
(c)     There are 302 tiles in Pattern 100.
          How many tiles are in Pattern 99?
................................................................................................
Answer .................................................
(1)
(Total 4 marks)
 Q23.          (a)     Solve                 a – 2 = 9
....................................................................................................................
Answer a = .........................................
(1)
(b)     Solve                  = 5
....................................................................................................................
Answer b = .........................................
(1)
(c)     Solve                 2c – 3 = 11
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer c = .........................................
(2)
(Total 4 marks)

 

Q24.          In the magic square, the rows, columns and diagonals add to the same number.
 
10	w	x	
5	y	9	
6	11	4	

          Work out the values of w, x and y.
..........................................................................................................
..........................................................................................................
..........................................................................................................
Answer w = ........... x = ........... y = ...........
(Total 3 marks)

 

Solving Equations
Q25. (a)     Solve the equation x + 5 = 20
....................................................................................................................
Answer x = ...........................................
(1)
(b)     Solve the equation 2y + 3y = 20
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer y = ...........................................
(2)
(Total 3 marks)
 Q26.          (a)     Solve the equation            = 7
....................................................................................................................
Answer w = .........................................
(1)
(b)     Solve the equation             7x – 3 = 60
....................................................................................................................
Answer x = ..........................................
(2)
(c) Write down one integer which satisfies the inequality   7y > 63
Answer ................................................
(1)
(Total 4 marks)
 
Q27.          Solve these equations 
(a)     4x – 7 = 5
....................................................................................................................
....................................................................................................................
Answer x = .................................................
(2)
(b)     2(y + 5) = 28
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer y = ..................................................
(3)
(c)     7z + 2 = 9 – 3z
...................................................................................................................
....................................................................................................................
....................................................................................................................
Answer z = .................................................
(3)
(Total 8 marks)

 

Q28.          Solve the equations
(a)     3x = 12
....................................................................................................................
Answer x = .................................................
(1)
(b)     y + 7 = 13
....................................................................................................................
Answer y = .................................................
(1)
(c)     8z – 5 = 11
....................................................................................................................
....................................................................................................................
Answer z = .................................................
(2)
(d)     3(w – 2) = 9
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer w = .................................................
(3)
(Total 7 marks)

Q29.          (a)     You are given that            6x = 48
          Work out the value of 5x
....................................................................................................................
....................................................................................................................
Answer .............................................
(2)
(b)     Solve the equation           2(4y – 1) = 18
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer y = ........................................
(3)
(Total 5 marks)

Trial and Improvement
 Q30. Parveen is using trial and improvement to find a solution to the equation
x3 + 7x = 30
          This table shows her first two trials.
x	x3 + 7x	Comment	
2	22	Too small	
3	48	Too big	

 	 	 	

 	 	 	

          Continue the table to find a solution to the equation.
          Give your answer to 1 decimal place.
Answer .................................................
(Total 3 marks)

Q31.          x is a number such that x(x – 1)(x + 2) = 40
          Use trial and improvement to find a solution for x.
          Give your answer to 1 decimal place.
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer x = ...............................................
(Total 4 marks)
Q32.          Use trial and improvement to find a solution to the equation
x3 – x = 21
          Give your answer to one decimal place.
You must show your working.
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer .................................................
(Total 3 marks)
 Q33.          Kate and Lee are working out this question.
 
A solution of the equation x3 + x = 700 lies between 8 and 9.
Use trial and improvement to find this solution, correct to one decimal place.	
          Kate’s answer is 8.8 
Lee’s answer is 8.9
          Which answer is correct?
You must show all your working.
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer x = .................................................
(Total 3 marks)

 

Inequalities
Q34. (a)     Solve the inequality 2x + 3 ≥ 1
....................................................................................................................
....................................................................................................................
Answer .................................................
(2)
(b)     Write down the inequality shown by the following diagram.

................................................................................................
Answer .................................................
(1)
(c)     Write down all the integers that satisfy both inequalities shown in parts (a) and (b).
................................................................................................
Answer .................................................
(1)
(Total 4 marks)

 

Q35.          (a)     Solve the equation 
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer x = ...............................................
(3)
(b)     Solve the inequality 3x + 8 < 29
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer .................................................
(2)
(Total 5 marks)
 Q36.          (a)     Rearrange the formula to make t the subject.
S = 3t + 40
....................................................................................................................
....................................................................................................................
Answer ..................................................
(2)
(b)     Solve 3t + 40 < 70
....................................................................................................................
Answer ..................................................
(2)
(Total 4 marks)
 Linear & Quadratic Graphs
Q37.          (a)     Complete this table of values for y = 2x – 1
 
x	–1	0	1	2	3	
y	–3	 	1	 	5	
....................................................................................................................
(1)
(b) On the grid draw the graph of y = 2x – 1 for values of x from –1 to +3.

(2)
(Total 3 marks)
 
Q38.          (a)     Complete the table of values for           y = x2 – 2
 
X	–2	–1	0	1	2	
Y	2	 	–2	–1	2	
(1)
(b)     On the grid below, draw the graph of y = x2 – 2 for values of x from –2 to +2

(2)
(c)     Explain how the graph can be used to estimate the solutions of      x2 – 2 = 0
................................................................................................
................................................................................................
(1)
(Total 4 marks)
 
Q39.          (a)     On the grid below draw the graph of y = 2x – 3 for values of x from –1 to 4.
....................................................................................................................
....................................................................................................................
....................................................................................................................
(3)

(b)     The line y = 4.5 crosses the line y = 2x – 3 at P.
Use the graph to work out the coordinates of P.
Answer ( ................. , ............... )
(2)
(Total 5 marks)

Real life graphs
Q40. (a)     The basic monthly charge for a mobile phone contract is £35.
        This includes:
        Option 1:     300 free minutes of calls and 100 free texts
    or Option 2:  100 free minutes of calls and unlimited free texts.
        All other calls are charged at 6p per minute. Extra texts are charged at 10p each.
        On average, each month, Matt makes 500 minutes of calls and sends 250 texts.
      Which option should he choose?
      You must show your working.
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer .................................................
(5)
(b)     Viki has a different mobile phone contract. She never sends texts.
The graph shows how the total monthly charge is calculated for her mobile phone contract for up to 500 minutes of calls.

(i) Write down the basic monthly charge and the number of free minutes of calls.
Basic monthly charge £ ............... Number of free minutes of calls   ...............
(2)
(ii)     Work out the charge per minute for the other calls.
...............................................................................................................
...............................................................................................................
Answer ...................................... pence
(3)
(Total 10 marks)
 Q41. Motorists should drive with a safe gap between their vehicle and the vehicle in front.
         This graph shows the minimum safe gap between vehicles at different speeds.
         Different gaps are recommended for wet roads and dry roads.

(a)     The road is dry.
          A car is travelling at 30 miles per hour behind a lorry.
     What is the minimum safe gap between the car and the lorry?
....................................................................................................................
....................................................................................................................
Answer .................................... metres
(1)

(b)     Tim is driving at 60 miles per hour on a dry road.
          He is driving with the minimum safe gap between his car and the car in front.
          It starts to rain heavily and both cars slow down to 40 miles per hour.
          Should Tim increase the gap between his car and the car in front to continue driving with the minimum safe gap?
          You must show clearly how you obtain your answer.
................................................................................................
................................................................................................
................................................................................................
................................................................................................
................................................................................................
(3)
(Total 4 marks)

 Q42.          Here is a distance-time graph for a train journey.

(a)     For how long does the train stop on the journey?
Answer ..................................... minutes
(1)
(b) (i)      On which part of the journey does the train travel fastest?
      Put a circle around the part of the graph that shows this.
(1)
(ii)     Explain how you know.
................................................................................................................
................................................................................................................
(1)
(Total 3 marks)
Q43. Wayne cycles from Newcastle to Ashington, a distance of 20 miles.
The diagram shows the distance-time graph of his journey.

(a)     How far from Newcastle is Wayne at 11.00?
Answer .......................................... miles
(1)
(b)     Describe what is happening between 12.00 and 13.00
....................................................................................................................
 (1)
(c)  How far does Wayne travel in the first 2 hours of his journey?
Answer .......................................... miles
(1)
(d) What is Wayne’s average speed over the first 2 hours of his journey?
....................................................................................................................
....................................................................................................................
Answer ........................................... mph
(2)
(e) Darren travels from Ashington to Newcastle by bus.
He leaves Ashington at 10.00 and arrives in Newcastle at 11.00
On the diagram draw a possible distance-time graph of Darren’s journey.
(1)
(Total 6 marks)
Q44.          You may use the grid below to help you solve this problem.

At 9 am a man leaves point P and walks along a road at a steady speed of 6 kilometres per hour.
At 12 noon a cyclist leaves P, on the same road in the same direction, at a steady speed of 20 kilometres per hour.
After travelling for an hour the cyclist gets a puncture which delays her for 30 minutes.
She then continues at 20 kilometres per hour until she overtakes the walker.
(a)     At what time did the cyclist overtake the walker?
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer ..............................................
(3)
(b) A motorist leaves P, travelling at a steady speed of 50 kilometres per hour.
The motorist overtakes the walker at the same time as the cyclist.
At what time did the motorist leave P?
....................................................................................................................
....................................................................................................................
....................................................................................................................
Answer ..............................................
(2)
(Total 5 marks)

 

I think I’ve accidentally deleted one of the answers which has then messed up the numbering! RM needs to fix this! 

M1.          (a)     7 (+) 6 (+) 15
M1
          28
A1

(b)     7 × 3 × 5
M1
          105
A1
(c)     0
B1
[5]

 

M2.          (a)     24 × 10 or 240 seen
16 + 24 × 10
M1
          256
A1
(b)     2 (years)
B1

(c)     600 = 120 + 24m or 600 – 120
See 480
M1
          480 = 24m or their 480 ÷ 24

M1 dep
          20
A1
[6]

 

M3.          (5x) = 20
or x = 4 seen
B1
          (5 – 8) ÷ 3
Allow 5 – 8 ÷ 3 or 3y = –3
M1
          (y =) –1
A1
          3 × (their) 4× – (their) –1
M1
          13
ft (their) x or x = 4 and (their) y
A1ft
[5]
 
M4.          (a)     3x + 7y
B1 for 1 term correct
B2
(b)     5 × 4 (+) 2 × –7 or sight of 20 and –14
M1
          6
A1
(c)     25 (–) 9
M1
          16
A1
[6]

 

M5.          (a)     5
B1
(b)     5 × 100 ( = £5)
Or 5 + 5
M1
          10
A1

(c)     £2.50 ÷ 5
250 ÷ 5; their (7.50 – 5) ÷ 5 if shown
M1
          50
A1
[5]

 

M6.          (a)     2x + 3y + 4x + 2y + 3x + 5x + y
14x or 6y seen
M1
          14x + 6y
A1

(b)     ‘Their 14’ × 4 + ‘their 6’ × y = 68
M1
          ‘Their 6y’ = ‘their 12’
M1
          (y = ) 2
ft Their answer for (a) with 2 terms
A1ft
[5]

 

 

 

M7.          (a)     7c
B1
(b)     4x – 3y
B1 For 4x or –3y
B2
[3]

 

M8.          (a)     12a – 3b
B1 For one correct term seen, 12a or – 3b
B2
(b)     x2 + 7x
B1
[3]

 

M9.          (a)     3a
Allow 3 × a and a × 3; not a3
B1
(b)     6b + 10
B1 for 6b or + 10
fw eg, 16b; deduct 1 mark
B2
(c)     12
B1
[4]
M10.          (a)     8x – 4 + 3x + 18
Allow one error
M1
11x + 14
fw that does not contradict is not penalised but fw
such as = 25x do not award A1
A1

(b)     4x2 – 2x3
B1 each term fw such as = 2x5 only give B1
4x2 – 2x2 = 2x2 is B1, 4x2– 2x2 = 6x4is BO
B2
[4]

 

M11.          (a)     6x + 5
B1 6x or + 5  eg 6x – 5 scores B1
B2

M12.          (a)     a6
B1
(b)     b6
B1
(c)     c11
B1
[3]

M13.          (a)     x5
B1
(b)     y3
B1
(c)     x4 – 3x
B1 x4 or 3x
Penalise fw
B2
[4]

 

M14.          (a)     (i)      y9
B1
(ii)     y5
B1
(iii)     y14
B1

(b)     (i)      y14  or  (a)(iii)
ft their answers from Part (a)
B1 ft
(ii)     y5  or  a(ii)
ft their answers from Part (a)
B1 ft
[5]
 M15.          
 	4	
7	9	

B1 For two correct
B1
[1]
  M16.          (a)     16
B1
(b)     Add 3
oe Allow n + 3
B1
[2]

M17.          (a)     14
B1
          41
ft their first answer × 3 – 1
B1 ft

(b)     10, 15, 20
B1
(c)     3n
M1
          3n + 4
A1
[5]

M18.          (a)     64
B1
(b)     (i)

B1
          (ii)     13
B1
(c)     26
B1
[4]
 
M19.          (a)     Term to term rule → × 2
oe eg, double each time
or one more than the sum of the previous terms
B1

(b)     Term to term rule → + consecutive integers
oe eg, add 1 more each time
or sum of previous 3 terms
B1
[2]
M20.          1a = 17;  b = 47; c = 15
ft c from their b
B1, B1, B1 ft
[3]
 M21.          (a)     Correct diagram
B1
(b)     (i)      3 → 11, 4 → 14 and 5 → 17
B1
          (ii)     29
B1
(c)     299
B1
[4]

 

M22.          (a)     11
B1
(b)     15
B1
(c)     (2c = 11 + 3 = 14)
Their 14 ÷ 2
M1
          7
A1
[4]

M23.          w = 3         x = 8           y = 7
B1 each
B3
[3]
 
M24.          (a)     15
Allow embedded answer
B1
(b)     5y = 20
Allow embedded answer with contradiction
M1
          4
Allow embedded answer with no contradiction
A1
[3]
 
M25.          (a)     (w =) 63
B1
(b)     7x = 63
63 ÷ 7
9 embedded    M1
M1
          (x = ) 9
A1

(c)     One integer > 9
Any whole number greater than 9
B1
[4]

 M26.          (a)     4x = 12
M1
          3
A1
(b)     y + 5 = 28 ÷ 2
or 2y + 10 = 28
M1
          y = 14 – 5
or y = 
M1 dep
          9
A1

(c)     7z + 3z or 9 – 2
M1
          10z = 7
A1
          
oe
A1
[8]
 M27.          (a)     4
B1
(b)     6
B1

(c)     8z = 16
8z = 11 + 5
M1
          2
A1

(d)     3w = 6 = 9
w – 2 = 9/3
M1
          3w = 15
w – 2 = 3
DM1
          5
SC1 for 3w – 2 = 9, 3w = 11, w = 11/3
A1
[7]
 
M28.          (a)     40
B1 (x =) 8
B2

(b)     8y – 2 (= 18)
(4y – 1 =) 18 ÷ 2
M1
          8y = 18 + 2
4y = 9 + 1
M1
          2.5 or  or 
oe
A1
[5]

 

M29.          All values can be rounded or truncated to nearest whole number
          Testing integers until solution between x = 3 and x = 4 found
x = 3 gives 30, x = 4 gives 72
M1
          Testing 1 dp values until a solution between x = 3.2 and x = 3.3 found
x = 3.2 gives 36.608, x = 3.3 gives 40.227
M1

          Testing a value between 3.25 and 3.3 to confirm answer
x = 3.25 gives 38.39  3.26 gives 38.75
      3.27 gives 39.12  3.28 gives 38.46
M1
          x = 3.3
3.29 ⇒ 39.9, 3.3 ⇒ 40.3
so x = 3.3 is 4 marks
A1
[4]

 

M30.          Trial between 2 and 3
B1
          Trial between 2.3 and 2.4 inclusive that “bracket” the answer
B1
          Trial at 2.35 or 2.36 or 2.37 and 2.4 stated as answer
DB1
In this question final answer on its own will not get any marks
Working must be seen.
All trials must be correctly evaluated either rounded or truncated to a degree of accuracy that allows comparison.
[3]

 

M30.          Trial above 2.8796
2 gives 6, 3 gives 24
M1
Trial below 2.8796
2.9 gives 21.489, all values to at least l dp
rounded or truncated
M1
Testing a value that justifies 2.9 as answer
2.5 →13.125, 2.6 → 14.976, 2.7 → 16.983
2.8 → 19.152, 2.85 gives 20.299
DM1
x = 2.9
Dep on any M mark
A1
[4]

 

M31.         x + y = 15 and x – y = 3 
or at least 2 valid trials
eg 8 – 7 ≠ 3, 8 + 7 = 15 one valid trial 
10 + 5 = 15 2nd valid trial
M1
          (x =) 9 and (y =) 6
Note: x and y may not be seen
A1
          54
54 on its own scores SC2
A1
[3]

 

M32.          Trial at 8.8 or 8.9 (or between)
690(.272) or 713(.869)
B1
          Trials at 8.85 (702) and 8.8
or any 2 trials for 8.8 ≤ x ≤ 8.85 
which bracket 700 
Note: These 2 trials score first B1 
also
B1
          8.8 or Kate
Dependent on second B1
Alternative method:
Trial at 8.85        M1 
702            A1 
8.8            A1
B1 dep
[3]

 

M33.          (a)     2x –2, 2x > 1 – 3
Allow > but not = unless recovered in answer
M1
          x ≥ –1
Do not allow > in final answer.
A1
(b)     x < 2, 2 > x   < x <2
Condone alternative letters
B1
          x = –1, 0, 1
Follow through their inequalities for (a) and (b) unless part (b) is an ”invented inequality‘ such as x – 3 < 2.
B1 ft
[4]

 

M34.          (a)     23 – 2x = 15
4.6 – 0.4x = 3 gets M1 allow one error
M1
23 – 15 = 2x
1.6 = 0.4x A1
A1
4
f.t. if M1 awarded.
A1 ft

(b)     3x < 21
3x = 21 gets M1 iff recovered
M1
x < 7
Must have inequality in answer.
Accept ≤..
A1
[5]

 

M35.          (a)     S – 40

M1
          
oe

A1

(b)     3t < 70 – 40
M1
          t < 10
A1
[4]

 

M36.          (a)     –1 and 3
B1
(b)     Line drawn from (–1, –3) to (3,5)
B1 for 3 points plotted correctly
B2
[3]

 

M37.          (a)     –1
B1
(b)     Correct curve from x = –2 to 2 ± 2 mm from integer points
B1   Four points plotted correctly from (–2, 2)
       (their) (–1, –1), (0, –2), (1, –1) and (2, 2)
B2
(c)     (x values at points where) curve crosses x-axis
oe
B1
[3]

 

M38.          (a)     Two points calculated or plotted
B1 For each point or             (–1, –5)
B1 Line through (0, –3)         (0, –3)
B1 Line gradient 2                 (1, –1)
                                              (2, 1)
                                              (3, 3)
                                              (4, 5)
B2
          Straight line drawn
B1

(b)     Attempt to read off at y = 4.5 or 2x = 7.5
or 4.5 as y coordinate
B1
          3.75
ft Their graph
± 1 mm ( square)
B1 ft
[5]

 

M39.          (a)     (500 – 300) × 6
or 200 × 6
or (£) 12
or 1200
(250 – 100) × 10
or 150 × 10
or (£) 15
or 1500
Option 1
M1

          (£) 27 or 62     or      24 or 59
Accept pence
A1
          (500 – 100) × 6 or 400 × 6
Option 2
M1
          (£) 27 and 24    or    62 and 59
Accept pence
A1
          Option 2 (cheaper)
B1 ft

(b)     (i)      £ 25
B1
                   150
B1
          (ii)     Attempt to read one coordinate between 150 and 500
± £1
eg,   250, 30
       (500, £41.50 - 43.50 inclusive)
Allow 350, 42.5 (± 1)
Not (150, 25)
M1

                   Correct statement
eg,   £1 for 20 mins
       £2 for 40 mins
       £5 for 100 mins
       £10 for 200 mins
Their £17.50 (± 1) for 350 mins
(Their change –25) and (their mins – 150)
0.05
M1 dep
                   5
£0.05
A1
[10]
 
M40.          (a)     13 to 14 inclusive
B1
(b)     26 to 28 seen
B1
          35 to 37 seen
B1
Allow incorrect written values if they are correctly
and unambiguously indicated on the graph

          Yes or increase gap (by 9 m)
oe
This mark can only be awarded if justified by
(their) values from the graph.
B1
[4]

 

M41.          (a)     10 minutes
B1
(b)     (i)      Last section indicated
B1
          (ii)     Steepest line
oe
Do not accept 20 km in 20 mins and 25 km in 50 mins
Do not accept greater distance in shorter time
B1dep
[3]

 

M42.          (a)     8
B1
(b)     Not moving
Same distance from Newcastle oe
B1
(c)     16
B1

(d)     Their 16/2
M1
8
A1 ft
(e)     Line or curve from (10.00, 20) to (11.00,0)
B1
[6]

 

M43.          (a)     Line from (9 am, 0) to (2 pm, 30)
B1
          Line from (12, 0) to (1, 20) plus 30 minute break, then line from
(1:30, 20) to (2 pm, 30)
B1
          2 (pm)
B1

          Alt 1
          Table or list of values showing steady speed of 6 km/h for at least
3 values
eg, (10, 6), (11, 12), (12, 18), (1, 24), (2, 30)
B1
          Table or list of values showing steady speed of 20 km/h with break
to at least (1:30, 20)
eg, (1, 20), (1:30, 20), (2, 30) [(2:30, 40)]
B1
          2 (pm)
B1

          Alt 2
          6x = 20(x – 3 – 0.5)
M1
          x = 5 hours
A1
          2 (pm)
A1

(b)     Line from their ‘(2 pm, 30)’ with gradient 50 km/h
30 ÷ 50 (= 36 minutes)
M1
          1:23 – 1:25 (pm)
Their (a) – 36 minutes
A1 ft

          Alt 1
          Table of values back from their ‘(2, 30)’ for at least 1 hour
eg, (1, 5)
B1
          1:23 – 1:25 (pm)
Their (a) – 36 mins
B1 ft

          Alt 2
          30 ÷ 50 (= 0.6h = 36m)
M1
          1:24 (pm)
Their (a) – 36 mins
A1ft
[5]