1) Find where possible, the value of each of the following expressions, given that a = 6, b = -3

and c = 2

a) 2a + 3b + 4c b) 2a2 c)

d) a2 + b

2 e) f)

g)

h)

i)

j)

k) l)

2) Given that , find the two possible values of z if:

a) and b) and

3) Given that a = 3, b = -4 and c = -1, evaluate:

a) b) c)

4) If find the two possible values of y when

and

5) If

, find when and

6) If

find v when =2, d = 10 and u = 2

7) If , find c in terms of a, b and d

8) If

find in terms of and

9) If , find p when q = -6 and r = -13

10) If

, find when v = 20, u =6 and d = -77

11) If , find p when q = 9 and r = 4

12) If

make d the subject of the formula

13) If , find p in terms of q

14) If a(b + c) = bc, find b in terms of a and c

15) For each of the expression below:

i) Expand the brackets and simplify (where possible)

ii) Then put:

into BOTH the original expression and your

expanded expression and confirm both expressions yield the same result:

a) b) c)

d) e) f)

g)

h) i)

j) k) l)

1) Using a calculator were necessary, approximate each of these to: (i) 3 d.p. and (ii) 3 s.f.

a) b) c)

d) e) f)

g) h)

i)

j) k) 42.839 l) 0.499

2) Approximate the following to (i) 4 s.f. and (ii) 4 d.p.

a) 5.28741 × 10-4

b) 2.9097 × 10

3 c) 4,826.301

d) 0.0022439 e) 42.148 f) 38,995,025

3) Type these into your calculator, ensuring the correct mathematical is obeyed and round

the answer produced by the calculator to 4 s.f.

a) b)

c)

d)

e)

f)

g) h) i)

j)

k)

l)

4) Where necessary, correct numbers to 1 s.f. and hence, without using a calculator, give

approximate answers to each of the following sums:

a) b)

c)

d)

e)

f)

g) h) i)

j)

k)

l)

5) Find the following estimates, without using a calculator, by making suitable approximations:

a) The value of is given by

. Estimate when m =12.6 and u = 9.21

b) Given

, estimate when m = 5.92, a = 21.4 and b = 9.72

c) The value of is given

. Estimate when =41.3, and

6) Find the exact answer to each of these sums, without using a calculator:

Clue: You can factorise each of these expressions first

a) 2.52 + 0.5 × 2.5 b) 5.2

2 + 0.8 × 5.2 c) 1.3 × 3.7 + 3.7

2

d) 2.6 × 3.4 + 3.42

e) 5.92 – 2.9 × 5.9 f) 4.3

2 – 1.3 × 4.3

g) 8.762 - 4.76 x 8.76 h) 16.27

2 – 5.27 x 16.27 i)

1) Mairi weighs a bag of sand using an electronic scale that is only accurate to 1 d.p. It reads 5.6 kg when the bag of sand it placed upon it. The bag of sand is then placed on a scale that is more accurate (to 2 d.p.). Which of these possible readings would be consistent with the reading from the first scale: 5.00 kg, 5.15 kg, 5.49 kg, 5.50 kg, 5.51 kg, 5.53 kg, 5.55 kg, 5.56 kg, 5.57 kg, 5.59 kg, 5.60 kg, 5.62 kg, 5.64 kg, 5.65 kg, 5.66 kg, 5.68 kg, 5.70 kg, 6.00 kg. Note: Write down all of the values from this list that, when rounded to 1 d.p. give 5.6 kg

2) A shopkeeper said he made a profit of £2,500 last month. If figure he gave is correct to the nearest £100, which of these are possible figures for the profit he made last month: £2,000, £2,345, £2,399, £2,425, £2,449, £2,450, £2,480, £2,499, £2,501, £2,535, £2,545, £2,549, £2,550, £2,551, £2,559, £2,580, £2,599, £3,000.

3) Correct to one decimal place, the length of a room ( ) is given as 2.8 m. a) List all the 2 d.p. numbers between 2.70 and 2.90 (there’ll be 21 numbers in your list) b) Highlight all of the numbers in your list that, when rounded to 1 d.p, give 2.8 m c) Write an inequality that expresses your answer to part (b): i.e: ___ ___ Note: The number at the right end of the inequality is the number immediately after the last

one highlighted in your list; but since we use a sign at the right of the inequality, that

number isn’t actually included (rather than sign, which would include the number).

4) On the box for the new brake pads I purchased, it says they are 12.5 mm thick, to 1 d.p. a) List every 2 decimal place number that when rounded to 1 d.p. gives 12.5 mm (for instance 12.48 mm will be in your list, but 12.42 mm will not) b) Write an inequality (like you did in question 3) to express the range of thicknesses, , of brake pads that conform to the label specification.

5) Referring to a football match, a newspaper headline proclaimed 75,000 watched England win. If this figure for the number of spectators is correct to nearest 1,000, find the range in which , the number of people who actually attended the match lies. Clue: Starting from 74,200 and going up by 100 at a time, think of the range of numbers

that will round to 75,000. Keep going until you get to one which doesn’t

6) One of the components parts of a metal hinge is a pin. In order to work properly this pin must have a diameter of 1.25 mm correct to 2 decimal places. Find the range in which the diameter, mm, must lie. Clue: Start from 1.242 mm, then 1.243 mm etc.

7) One hundred people were asked if they ate cornflakes for breakfast that morning. To the nearest 10, 40 people said they did. Of the 100 people interviewed, what is the largest possible number who did NOT eat cornflakes.

8) An advertisement says “For about £50 you can buy this automatic watch.” If this price is correct to the nearest £10, what is the most that you would expect to have to pay for such a beautiful watch?

9) The length of a car is given as 255 cm. If this figure is correct to the nearest 5 cm, find the range in which the actual length lies.

10) Knitting yarn is sold by weight. It is found that 10 g of double knitting pure wool has length of 20 m correct to the nearest metre. What is the minimum length of yarn you would expect in a 50 g ball of double knitting pure wool?

In questions 1 to 9 all values are given correct to nearest whole number. Use the given formula to find the upper and lower bounds of the values for the letter given on the left-hand side:

1) if = 10 and = 6

2) if =5, = 7 and = 6

3)

if = 12

4)

if = 12, = 12 and = 17

5) if = 3, = 4 and =5

6)

if =12 and =5

7) If = 22

8) of = 17

9)

if = 10 and =12

10) Given

, = 1.25 and = 4.93, each correct to 2 d.p. Find the maximum and

minimum values for . Give these values correct to 2 d.p.

11) If and correct to the nearest whole number, find the upper and lower bounds for the value , given that Hint: is largest when is biggest and is smallest

12) If and correct to 1 d.p, find the largest and smallest values of: a) b) c)

13) Two variables and are such that and Calculate the smallest and largest values of:

a) b) c) d)

14) Two variables and are such that and Calculate the smallest and largest values of:

a) b) c)

d)

15) The length of an edge of a cube is 5 cm, correct to the nearest whole number Calculate the smallest and largest values of: a) an edge of the cube b) the surface area of the cube c) the volume of the cube.

16) A boy runs 100 m (nearest m) in 12½ secs (nearest ½ sec). Find his max and min speeds.

17) The opposite side of a triangle can be found by . If it = 4.2m (1 dp) and (nearest whole number) find the maximum and minimum value of .

18) . Find the max and min values of given = 16.5 (1 dp), (2 s.f.)

19) Use

to find max and min value of given: = 48 m (2 s.f.). (2 s.f.)

20) The height of the cylinder is given by

Find the minimum value of if = 800cm3 (1s.f.) and = 40cm (1s.f.)

21) A sphere of gold has radius of 8½ cm (nearest ½ cm). Its volume is given by

Its mass can be found because 1cm3 of gold has a mass of 4.2g (1 d.p.)

Its value can be found because 1g of gold is worth £28 (nearest £) Find the maximum and minimum values of the sphere in gold.