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Algebra

1) Factorise completely 120x2 - 10x - 5

(120x + p)(120x + q) / 120

p and q must add to -10 and multiply to 120*-5

Write in p and q...       

(120x + [-30])(120x + [20]) / 120 

...and cancel to get (12x - 3)(120x + 20) / 12

and then 1/4 (4x - 1)(120x + 20)

 

Geometry

2) A hemispherical wine glass of radius 6 cm is filled with wine to the brim. The wine is then poured into a cylindrical bottle of radius 6 cm and height 20 cm. How high up the bottle will the wine go?

The volume of the wine does not change regardless of the container!

volume = 1/2 (1/3 π * 33)

so there is 18π cm3 of wine in the glass

It is now poured into the bottle

formula for cylinder = πr2h

so volume of wine in this cylinder is π*32*h and also we know the volume of the "cylinder of wine" is 18π

so 18π = π*32*h

so h = 18π / 32π

so h = 2cm

 

Indices

3) Evaluate 80.5 * 16-0.5

8 is 23

16 is 24

so (23)0.5 * (24)-0.5

= 21.5 * 2-2

= 2-3

= 1/23

= 1/8

 

Irrational numbers

4) Write 0.123123123123123......  as a fraction

x = 0.123123123123123.......

1000x = 123.123123123123123.......

999x = 123

x = 123/999

x = 41/333

 

Circle theorems

5) Prove that the alternate segment theorem is always true

see notes on Circle Theorems

 

Simultaneous equations

6) Solve by substitution:

2x2 + 3y = 7

5x + y - 2 = 3

 

y = 5 - 5x

2x2 + 3(5 - 5x) = 7

2x2 - 15x + 8 = 0

now just solve like a normal quadratic, then go back to find y

 

Equations of circles

7) Write down the formula for a circle of radius 9cm centred on the origin

x2 + y2 = 81

 

Proportion

8) The gravitational field strength of a planet is inversely proportional to the square of distance from the centre. At a distance of 100Km the field strength is 5 units. What will be the field strength at a distance of 215Km ?

g = k 1/d2

5 = k/1000

k = 5000

now put in the new value!

g = 5000 1/2152

g = 5000/2152

g = 0.108 units (3 s.f.)

 

Rearranging equations

9) Make r the subject of the following equation:

F-r = mv2r + mr + m

F - m = mv2r + mr + r

F - m = r(mv2 + m + 1)

r = (F - m)/(mv2 + m + 1)

 

Sampling

10) State one advantage and one disadvantage of systematic sampling

Advantage) Very quick & efficient

Disadvantage) The subdivisions could seriously affect the results e.g.) getting the third best person on every football team

 

Probability

11) In the card game POKER, the best hand you can possibly get is called a "royal flush"

Calculate the probability that I will get an Ace of Hearts, followed by a King of Hearts, followed by a Queen of Hearts, followed by a Jack of Hearts, followed by a Ten of Hearts... IN THAT ORDER!

1/52 x 1/51 x 1/50 x 1/49 x 1/48 = 1/312,000,000  (one in threehundred million, which is why you shouldn't gamble!!!)

 

Proof

12) Prove that the sum of any three consecutive integers is always a multiple of three

(n) + (n + 1) + (n + 2)

= 3n + 3

= 3(n + 1)

"n + 1" is an integer therefore 3(n + 1) is divisible by 3

 

Accuracy and Error

13) I measured a table and wrote down that it is 1.62 m long. What are the upper and lower bounds of this measurement?

Upper = 1.625

Lower = 1.615

 

Trigonometric functions

14) Write down the cosine rule for finding side lengths

a2 = b2 + c2 - (2bc cosA)

 

Transformations

15) y = f(x) is transformed into y = f(0.5*x). State the type of transformation and describe what happens to the appearance of the line

It is a stretch...

...scale factor 2, in the direction of the x axis

 

EXTRA!

16) Show how to construct an angle of exactly 60o using just a ruler, compass and pencil!

 

 

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