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National Curriculum: Algebra - KS3 - Exemplification


Created on 25 October 2013 by ncetm_administrator
Updated on 04 June 2014 by ncetm_administrator
 

Exemplification

 

Example A

  1. (i) The first five terms of an arithmetic sequence are
      4 11 18 25 32  
    Find, in terms of n, an expression for the nth term of the sequence.
    Jane says that 697 is a term in the arithmetic sequence,

    (ii) Is Jane correct?
    You must justify your answer.
  2. The nth term of a different number sequence is 4n + 5
    Work out the first three terms of this number sequence.
  3. Here are some patterns made from matchsticks:
    four match sticks four match sticks four match sticks
    Pattern number 1 Pattern number 2 Pattern number 3

    (i) Draw Pattern number 4.

    (ii) Complete this table for the pattern sequence

    Pattern number 1 2 3 4 5   100
    Number of matchsticks 4 7 10          

    (iii) Find, in terms of n, an expression for the nth term of the sequence

Example B

  1. Eggs are sold in boxes.
    A small box holds 6 eggs.

    Hina buys x small boxes of eggs.

    Write down, in terms of x, the total number of eggs in these small boxes.
     
  2. Mr Smith owns minibuses and coaches.

    Each minibus has 12 seats.

    (i) Write an expression, in terms of m, for the number of seats in m minibuses.

    Each coach has 48 seats

    (ii) Write an expression, in terms of
    m and c, for the number of seats in m minibuses and c coaches.

Example C

Simplify these expressions:

  1. 3a + 2b + 2a – b
  2. 4x + 7 + 3x – 3 – x
  3. 3(x + 5)
  4. 12 – (n – 3)
  5. m(n – p)
  6. 4(a + 2b) – 2 (2a + b)
  7. 3(x – 2) – 2(4 – 3x)
  8. (n + 1) 2 – (n + 1) + 1

Factorise:

  1. 4a + 10b
  2. p2 + 6p
  3. 6x2 - 9xy

Example D

  1. Solve the equation 5x = 30
  2. Solve x + 2x = 12
  3. Solve 2y – 1 = 13
  4. Solve 4(x + 3) = 6

Example E

  1. P = 3a + 5b

    a = 5.8
    b = –3.4

    Work out the value of P.
  2. S = 2p + 3q

    p = – 4
    q = 5

    (i) Work out the value of S.

    T = 2m + 30
    T = 40

    (ii) Work out the value of m.
  3. P = x2 - 7x

    Work out the value of P when x = - 5
  4. P = 𝜋r + 2r + 2a
    P = 84
    r = 6.7

    Work out the value of a.
    Give your answer correct to 3 significant figures.

Example F

  1. Expand and simplify (x + 7)(x – 4)
  2. Expand y(y3 + 2y)
  3. Expand p(qp2)
  4. Expand and simplify 5(3p + 2) – 2(5p – 3)

Example G

  1. Rearrange the equation y = 4x - 5 to find x in terms of y
  2. Make t the subject of the formula v = 5t + u
  3. Make c the subject of the formula a = 3c – 4

Example H

Harry left school at 3 30 pm.
He walked home.
On the way home, he stopped to talk to a friend.

His sister, Sophie, left the same school at 3 45 pm.
She cycled home using the same route as Harry.

Here are the distance-time graphs for Harry’s and Sophie’s complete journeys.

distance time graph

  1. Find the distance Harry walked during the first 10 minutes of his journey.
  2. Find the total time that Harry stopped to talk to his friend.
  3. Write down the distance that Harry had walked when Sophie cycled past him.

Example I

Here are five graphs labelled A, B, C, D and E.

Graph A
Graph B
Graph C
Graph D
Graph E

 

Each of the equations in the table represents one of the graphs A to E.

Write the letter of each graph in the correct place in the table.

Equation Graph
x + y = 5  
y = x - 5  
y = -5 - x  
y = -5  
x = -5  

Example J

  1. diagram showing two straight lines intersecting at a point

    Diagram NOT accurately drawn

    The diagram shows two straight lines intersecting at point A.
    The equations of the lines are

    y = 4x – 8
    y = 2x + 3

    Work out the coordinates of A.
  2. Solve the simultaneous equations

    5a + 3b = 9
    2a – 3b = 12
  3. Solve the simultaneous equations

    6x – 2y = 33
    4x + 3y = 9
  4. The graphs of the straight lines with equations 3y + 2x = 12 and y = x – 1 have been drawn on the grid.

    graph to solve the simultaneous equations

    Use the graphs to solve the simultaneous equations
    3y + 2x = 12
    y = x –1

Example K

  1. A straight line passes through the points (0, 5) and (3, 17).
    Find the equation of the straight line.
  2. Find the gradient of the straight line with equation 5y = 3 – 2x.

 


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