Department of Economics | University of Warwick
IMPORTANT: Please do NOT use a calculator in any of these revision exercises. The purpose of these exercises is to increase your understanding, not merely to get the answers right. Therefore, to use a calculator would destroy most of the value of the exercise. But, by all means CHECK your answers with a calculator, but only AFTER you have completed the exercise.
Revision Exercise R1
Topics covered in this exercise:
(1) Rules of arithmetic
(2) Powers and roots
(3) Decimal-fraction conversion, percentages, ratios
-----------------------------------------------
(1) Rules of arithmetic
1. (Letts Revise GCSE Mathematics, 2001 edition for 2002 exams, ch. 1.3, 2.2)
Calculate:
(a)
(c)
6 + (-4) - 8
(-3) X (-4) X 2
(b)
(d)
(-3) + (-7) - (-9)
(-1) X (-1) X (-1)
(e) (-60)

(-3) (f)

40
8
(g)
40

8
Simplify:
(i)
126
56
2. (Letts ch 2.2)
Calculate:
(j)
24
54
(h)

40

8
(k)
21
61
(a) 6(4 - 6)(3 + 2) + 5 (b) -48

(4 X 3) + 5
(c)
3
7

2
9
(d)
3
8
 
4
7
16
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(e)
(g)
2
3
7
X
8
1
5



1
3

1
2


(f)



(5 X 3 )

30

X 5
(h)
3
8

4
9
(i)
1
11

2
9
(k)
1
3
5
of
8
-------------------------------------
(2) Powers and roots (Letts ch 2.1, 4.4)
(j)
(l)
1
3
3
of
8
1
11
1
of
9
3. (a) Calculate: (i) 9 2 (ii) 9 3 (iii) 2
5
(iv) 3
-2
(v) 2
-5
(b) Give the two square roots of 81.
(c) Explain briefly why the square root of -9 cannot be found.
(d) Calculate:
(i)
(iv)
( )
2
-2
-0.
5
(ii)
( )
3
-3
(iii)
(v)
--------------------------------------------------
(3) Decimal-fraction conversion, percentages, ratios (Letts 1.1, 2.4)
4. Convert the following fractions into decimals (remember, without using a calculator!
):
1
(a)
2
5.
; (b)
3
8
; (c)
7
11
Calculate ( without using a calculator!
):
(a) 4% of 30 (b) 5% of 3.5 (c) 96 as a percentage of 72
6. Arrange the following numbers in order of size, starting with the smallest ( without using a calculator!
):
(i)
3
8
,
8
25
,
1
3
, 0.35
(ii)
------------------------------------------------------
(4) Approximations
3
2
, 1
1
10
, 1.065, 1.014, 1.12
7. Taking 1 kilometre to equal 0.62 miles;
(i) how many kilometres are there in 44 miles?
(ii) how many miles are there in 7 kilometres?
(iii) if I assume that one kilometre = 2
3 mile, what percentage error in my calculations will result? (You may use your calculator for this)
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(iii) if I assume that one kilometre = 0.6 mile, what percentage error in my calculations will result? (You may use your calculator for this)
Does the greater simplicity of these calculations justify the loss of accuracy?
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Revision Exercise R2
Topics: Brackets, signs etc.
Adding, subtracting, multiplying, dividing algebraic expressions
Powers
1. (Letts ch 6.2, 6.3)
Simplify (i.e. write in more compact form):
(a) 4a + (1 + 2a)
(d) 4z - z - 2z
(b) 4a - (+2a)
(e) -(-3a) + (-2a)
(c) 4a - (-4a)
(f) 2w X (-6w)
(g) 4z - (z + 2z)
(i) (a + 2ab)

a
(h) 3(a + b -c) - 2(a - b - c)
3.
2. (Letts 2.2 again)
Find the lowest common denominator, and hence express each of the following as a single algebraic fraction:
(b) a b
 c d
(c)
4

5
3 a 2 b
(a)
3

4 a b
(d) a
 a b
(g) b
 a
2 b ab
(e)
(h)
4

5
3 a 2 a a a
1
 b )

(
1
 b )
(f) a a
 b
 b b
 c
 c c
 d
Express each of the following as a single algebraic fraction:
(a) a b
X c d
(b) a bc b
X ac
(c) ab
 b c
(d) r
 pr
100 400
(e) Given x
 y x ( x
 y ) explain why no cancelling of x or y between the numerator and denominator is possible.
4. (Letts 6.4, page 63 only)
Factorise (i.e. take common terms outside a bracket):
(a) 4x + 4y (b) 5x - 10 (c) -4x - 6xy (d) ax - ay
(e) 5ab + 15b (f) 3y - 9y
2
(g) 3ba
2
- 9a (h) mn
2
- 2m
2 n
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5. Letts 2.1 (negative powers); 6.5 (powers greater than 1, and powers of powers); 7.6
(negative powers)
Remove the brackets from the following:
(i) ( xy )
2
(iii) ( 100 x )
1
2
(v)
(vii)

 x
 y
1
2


2
 
 1
2
6. Simplify:
(i) a
3
X a
0
(iii)
A 1
2 x
1
2 y
-
1
2
A 1
2 x
-
1
2 y
1
2
(ii)
(iv)
( x
3 y )
1
3

 x y

 n
(vi)
(viii)
 
1
2

 x
 y
1
2


2
(ii) a
0
X a
-1
(iv)
A
 x
 
1 y
1
 
A(1 -

) x
1
  y
 
1
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Revision Exercise R3
Topics: Linear equations; algebraic and graphical treatment.
Simultaneous linear equations; algebraic and graphical treatment.
1. (Letts ch. 6.6)
For each of the following equations, find an expression for x in terms of a, b, and c. (In other words, isolate x on one side of the equation):
(a) ax + b = 0 (b) 3x + 2b = 2(x + 3b)
(c) b(x - a) = c (d) a
(f) x
 x
3 a x
8

1 b


3 x ab
4
 x
(e) a(x - a) = b(x + b)
2. (Strangely, not included in Letts; nearest is ch. 6.9)
Solve the following linear equations:
(a) 7x = 84 (b) x
2
= 24
(c)
x
3
= 8
(e) x
+
1
=
10
5
7
(g)
3 x
2
1
4 x
3
8
(d) 3x + 4 = 9
(f)
(h) x
8
7 x
8
-
4
=
5
0
- (x - 2) = 12
3.
(j) 3(x + 1) - 2(x - 1) =
4 x2
7
(Letts ch. 5.5)
Draw the graphs (preferably on graph paper) of the following linear equations:
(b) y = 3x - 2 (a) y = 3x + 1
(c) y = -2x + 4 (d) x = 2y + 12
(e) (y + 1)/9x = 2; (f) 3x -5y = 7.
Note: since these are all linear (how do we know this?) you need to find only two points which lie on them in order to draw their graphs. Great accuracy is not necessary.
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4. For each of the equations in the previous question, check algebraically that each graph shows the x and y intercepts correctly (i.e. find x when y = 0 and find y when x = 0).
5. Given the linear relationship between x and y: ax + by + c = 0 (where a, b and c are constants)
Prove algebraically that the graph of this function has intercepts -c/a and -c/b on the x and y axes respectively.
6. (Letts ch.6.10)
Solve algebraically the following pairs of simultaneous equations. In each case, check your answers by substitution.
(a) y = 2x
3x + 2y = 21
(c)
(e) x
 y x
3

1

2

4 y y a
2
 b

1
3 a
 b
3

31
2
(b) 3x - 2(y + 3) = 2
2(x - 3) + 4 = 3y - 5
(d)
(f) x

1
 y
3
1

2
 x
2


6
3 y x
2

3 y
4 y

1
2 x

4
2

1
2
7. (Letts ch.6.10)
Solve the following pairs of simultaneous equations algebraically. Then draw the graphs of their functions and use the graphs to check your solutions.
(a) y = 3x + 1, y = 10 - x/2;
(b) 4y = 6 + x, y = 9(x - 1/9);
(c) 3x + 4y + 2 = 0, 10x = y - 1
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2.
Revision Exercise R4
Topics covered in this exercise:
(i) Expansion of binomial expressions, factorisation of quadratic expressions, solution of quadratic equations by factorisation and by use of formula.
(ii) Graphs of quadratic functions and graphical solution of quadratic equations.
1. (Letts ch. 6.3)
Expand the following binomial expressions:
(i) (x - 1) (x + 2) (ii) (x + a)2
(iii) (x + a) (x - a) (iv) (x + y)2
(v) (x - y)2 (vi) (x + y)(x - x)
(vii) Explain briefly why (x + a)
(Letts ch. 6.4)
2  x 2 + a 2
By "trial and error", factorise the following quadratic expressions (where possible):
Hint: note the pattern (x +  )(x  ) = x 2 + (  )x + 
(i) x2 + 4x + 3 (ii) x2 + 6x + 8
(iii) x2 - x - 12 (iv) x2 - 9
(v)
(vii) x2 + a2 x2 + x - 12
(vi) 100x2 - 9a2
(viii) x =
2 x
1
3. Using your answers to the previous question, find the factors (where possible) and hence the roots (= solutions) of the following quadratic equations.
(i) x2 + 4x + 3 = 0 (ii) x2 + 6x + 8 = 0
(iv) (iii) x2 - x - 12 = 0
(v) x2 + a2 = 0
(vii) x = 12 - x
2
(vi)
(viii) x2 - 9 = 0
100x2 - 9a2 = 0 x
 2 x

1

0
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4.
5.
6.
(Letts ch. 7.2)
The formula for the solution of the quadratic equation ax2 + bx + c = 0 (where a, b and c are constants) is: x

 b
 b
2 
4 ac
2 a
Use this formula to solve the following: (You may use your calculator for this).
(i) 12x2 + 4x - 3 = 0
(iii)
5 x
2
2

3 x
2

1
8

0
(ii)
(iv)
0.5x
2 + 2x - 6 = 0
2 x
2 
6 x
 27
8

0
By examining the formula for the solution of the quadratic equation ax2 + bx + c = 0, state the relationship between the values of a, b and c which must hold if the equation is to have:
(i) two distinct real roots
(ii) a single (repeated) real root
(iii) no real roots.
(Letts ch. 5.4, 6.12, 7.3)
(a) Sketch the graphs of the following functions, identifying clearly the intercepts (and turning points):
(i) y = x
2
+ 10x + 4 (ii) y = x2 - 4
(iii) y = 1 -x2 (iv) y = x2 + 10x + 9.
(v) y = x2 + 6x + 9.
(Hints: Take values of x between plus and minus 5. Find the x-intercepts by solving the corresponding quadratic equations. Great accuracy is not important.)
(b) Hence give a general diagrammatic explanation why a quadratic equation may have two distinct real roots, a single (repeated) root, or no real roots
(depending on parameter values).
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Revision Exercise R5
Topics: (1) The cubic equation y = ax3 + bx2 + cx + d
(2) The rectangular hyperbola y = x
 a b
+ c
(3) Inequalities
1. (Letts ch. 5.4)
Sketch the following graphs with sufficient accuracy to bring out clearly the important features of their shapes
(i) y = x 3 (for values of x between -4 and +4)
(ii) y = x
3
+ 2x
2
- 5 (for x = -4, , +2)
(iii) y = x
3
+ 4x
2
- 1 (for x = -4, , +2)
(iv) y = x
3
+ 5x
2
+ 3x - 9 (for x = -4, … , +2) Examine carefully the behaviour of y at x = 1 and x = -3.
2. Using what you have learned from question 1, sketch some possible shapes for the graph of y = ax3 + bx2 + cx + d (where a, b, c, d are constants). Use your sketches to explain why the equation ax3 + bx2 + cx + d = 0 may have 1 or 3 distinct real solutions (roots), but cannot have 2 such solutions. (Hint: it is not necessary to assign specific numerical values to the parameters. Just give a general diagrammatic indication of possible shapes)
3. (i) Expand: (x - 1)(x + 1)(x + 2). (Hint: do it in two stages)
(ii) Hence draw a sketch graph of y = x3 + 2x2 - x - 2, paying particular attention to the x-intercepts.
(iii) Hence write down the roots of the cubic equation x3 + 2x2 - x - 2 = 0
4. (Letts ch.5.4)
(i) Sketch the graph of y =
1 x
(Note: in the Letts book this is called a reciprocal expression; it is also known as a rectangular hyperbola)
Hint: Take the following values for x:
-1,000; -100; -10; -1; - , - , 0 , , , 1 , 10 , 100 , 1000
10
1
100
1
100
1
10
(ii) If x is allowed to take increasingly large (positive) values, we say that "x approaches (plus) infinity". What happens to the value of y in this case? How does this affect the graph?
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8.
Similarly if x is allowed to take increasingly large (negative) values, we say that "x approaches (minus) infinity". What happens to the value of y in this case? How does this affect the graph?
(iii) Suppose x is allowed to become smaller and smaller (whether positive or negative). What happens to the value of y in this case? How does this affect the graph?
5. Using what you have learned from the previous question, roughly sketch the graphs of
(a) y =
10 x
and (b) y = -
1 x
6 Sketch the graph of y =
10
+ 5 x - 1
How do the three constants (in this case, 10, -1 and 5) affect the shape and position of the curves?
(Letts 6.7 and 6.8)
7. (i) For what values of a and b is it true that
1 a
 1 b
?
(ii) For each of the following expressions, find the values of x for which y > 0. In each case, sketch the graph and indicate your solution on it.
(a) y = 3x - 15
(b) y = x2 - 4
Given that 0 < x < 2, derive the corresponding inequalities (i.e. lower and upper bounds) for y, given
(i) y = 3x + 5
(ii) y = (x - 1)2
9. Demonstrate what restriction on the values of the constants a, b, j and h must hold if a b
 a b

 h j
(Assume a, b, h and j are all positive)
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