Algebra: Sequences and Graphs - Worksheets, Questions and Revision

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KS3 · Algebra

KS3.M-A3 Algebra: Sequences and Graphs

AQA KS3.M-A3 · Calculator allowed · about 100 minutes
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
For each sequence, write down the next two terms and describe the term-to-term rule.
(a)3, 7, 11, 15, ...(2)
(b)800, 400, 200, 100, ...(2)
(c)2, 5, 11, 23, 47, ...(2)
2
Answer the following about special sequences.
(a)Write down the first five triangular numbers.(1)
(b)Write down the first five square numbers, starting from 12.(1)
(c)A Fibonacci-type sequence starts 1, 1, 2, 3, 5, ... Write down the next three terms.(2)
3
Four points are given: A(3, 5), B(-2, 4), C(-4, -3), D(5, -1).
x y -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 O A(3, 5) B(-2, 4) C(-4, -3) D(5, -1)
(a)State which quadrant point A lies in.(1)
(b)State which quadrant point C lies in.(1)
(c)Write down the coordinates of the reflection of point B in the y-axis.(1)
(d)M is the midpoint of A and D. Find the coordinates of M.(2)
4
The nth term of a sequence is given by the expression 3n + 2.
(a)Work out the 1st, 2nd and 3rd terms of the sequence.(1)
(b)Work out the 10th term of the sequence.(1)
(c)A term in the sequence is 47. Work out its position, n, in the sequence.(2)
5
Find an expression, in terms of n, for the nth term of each sequence.
(a)5, 9, 13, 17, ...(2)
(b)2, 8, 14, 20, ...(2)
(c)30, 25, 20, 15, ...(2)
6
A sequence has nth term 4n - 3.
(a)Work out the first three terms of the sequence.(1)
(b)Determine whether 100 is a term in this sequence. You must show your working.(2)
(c)Work out the smallest term in the sequence that is greater than 200.(2)
7
Aisha makes patterns from matchsticks. Pattern 1 uses 4 matchsticks, Pattern 2 uses 7 matchsticks, Pattern 3 uses 10 matchsticks and Pattern 4 uses 13 matchsticks.
(a)Write down the number of matchsticks needed for Pattern 5.(1)
(b)Find an expression, in terms of n, for the number of matchsticks in Pattern n.(2)
(c)Work out the number of matchsticks needed for Pattern 20.(1)
(d)Aisha has 88 matchsticks. Work out the largest pattern number she can build completely, showing that no matchsticks are wasted.(2)
8
y = 2x - 1
x y -3 -2 -1 1 2 3 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 0
(a)Complete a table of values for y = 2x - 1 when x = -2, -1, 0, 1, 2.(2)
(b)State the gradient of the line y = 2x - 1.(1)
(c)State the y-intercept of the line y = 2x - 1.(1)
9
For each equation, write down the gradient and the y-intercept.
(a)y = 5x + 3(1)
(b)y = -2x + 7(1)
(c)y = x - 4(1)
(d)y = 6 - 3x(2)
10
A straight line has gradient 4 and passes through the point (0, -3).
(a)Write down the equation of the line in the form y = mx + c.(1)
(b)A different straight line has gradient -2 and passes through the point (3, 1). Find the equation of this line.(3)
11
A conversion graph is used to change between pounds sterling (GBP) and euros (EUR). The graph is a straight line through the origin, showing that 10 pounds is equivalent to 11.50 euros.
(a)Use the graph information to find how many euros are equivalent to 40 pounds.(2)
(b)Find how many pounds are equivalent to 23 euros.(2)
(c)Write down an equation connecting E (euros) and P (pounds), in the form E = kP.(1)
12
Bilal cycles from his home to the shops, a distance of 12 km. He cycles at a constant speed and covers the 12 km in 40 minutes. He stays at the shops for 15 minutes, then cycles home again, covering the 12 km in 30 minutes.
0 10 20 30 40 50 60 70 80 90 0 2 4 6 8 10 12 Time (minutes) Distance from home (km)
(a)Work out Bilal's average speed, in km/h, on the journey to the shops.(2)
(b)Work out Bilal's average speed, in km/h, on the journey home.(2)
(c)Work out the total time, in minutes, for the whole trip (there and back, including the stop).(1)
13
Priya opens a savings account with 50 pounds. She then pays in 15 pounds at the end of each month (no interest is paid).
(a)Write down the amount in the account at the end of month 1, month 2 and month 3.(1)
(b)Find an expression for the amount, A pounds, in the account at the end of month n.(2)
(c)Work out the amount in the account at the end of month 12.(1)
(d)Work out after how many months the amount in the account will first exceed 300 pounds.(2)
14
Consider the equations x = 3, y = -2, y = x and y = -x.
(a)Which equation gives a vertical line?(1)
(b)Which equation gives a horizontal line?(1)
(c)Which equation gives a line through the origin with positive gradient?(1)
(d)Which equation gives a line through the origin with negative gradient?(1)
15
A pattern of pentagons is made from straws. Pattern 1 uses 5 straws, Pattern 2 uses 9 straws and Pattern 3 uses 13 straws (each new pentagon shares one straw with the previous one).
(a)Write down the number of straws in Pattern 4.(1)
(b)Find an expression, in terms of n, for the number of straws in Pattern n.(2)
(c)Priya has 100 straws. By forming an inequality, find the largest pattern number she can build.(2)
16
Line A has equation y = x + 2. Line B has equation y = -2x + 8.
0 1 2 3 4 2 4 6 8 10 x y
(a)Complete a table of values for Line A and Line B at x = 0, 1, 2, 3.(2)
(b)By comparing your tables, find the coordinates of the point where Line A and Line B intersect.(1)
(c)Check your answer to part (b) algebraically by solving x + 2 = -2x + 8.(2)
17
A sequence begins: 2, 5, 10, 17, 26, ...
(a)Find the first differences between consecutive terms.(1)
(b)Find the second differences.(1)
(c)Using the pattern in the second differences, work out the next two terms of the sequence.(2)
18
A quadratic sequence has nth term of the form an2 + b. The sequence is 3, 6, 11, 18, 27, ...
(a)Show that the second difference of the sequence is 2.(2)
(b)Hence find the value of a.(1)
(c)Find the value of b.(2)
(d)Find an expression for the nth term of the sequence.(1)
(e)Use your expression to work out the 15th term of the sequence.(2)
19
Two graphs are drawn. Graph P has equation y = 3x + 4. Graph Q has equation y = x2 + 4.
(a)State the value of y for each graph when x = 0. What do you notice?(1)
(b)Work out the value of y on each graph when x = 5.(2)
(c)Explain why Graph Q increases more quickly than Graph P for large positive values of x.(1)
(d)State whether Graph Q is a straight line or a curve, giving a reason.(1)
20
A plumber charges a call-out fee plus an hourly rate. The total charge, C pounds, for a job lasting h hours is modelled by C = 35h + 60.
(a)Write down the call-out fee (the fixed charge before any hours are worked).(1)
(b)Write down the hourly rate charged.(1)
(c)Calculate the total charge for a job lasting 3.5 hours.(2)
(d)A customer is charged 235 pounds for a job. Work out how many hours the job lasted.(2)
(e)A second plumber charges according to the sequence of costs for whole numbers of hours: 90, 120, 150, 180, ... (for h = 1, 2, 3, 4 hours). By finding the nth term of this sequence, write down an equation for the second plumber's charge, C pounds, in terms of h hours, and state which plumber is cheaper for a 6-hour job.(4)
Mark scheme · KS3.M-A3 Algebra: Sequences and Graphs

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Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20