__Algebra__

**1) Factorise completely
120x**^{2} - 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 * 3^{3})

so there is 18π cm^{3
}of wine in the glass

*It is now poured into the
bottle*

formula for cylinder = πr^{2}h

so volume of wine in this
cylinder is π*3^{2}*h and also we know the volume of the
"cylinder of wine" is 18π

so 18π = π*3^{2}*h

so h = 18π / 3^{2}π

so h = 2cm

#

__
Indices__

**3) Evaluate 8**^{0.5}
* 16^{-0.5}

#
8 is 2^{3}

16 is 2^{4}

so
(2^{3})^{0.5
}*
(2^{4})^{-0.5}

= 2^{1.5} * 2^{-2}

= 2^{-3}

= 1/2^{3}

= 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:**

**2x**^{2} + 3y = 7

**5x + y - 2 = 3**

y = 5 - 5x

2x^{2} + 3(5 - 5x) =
7

2x^{2} - 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**

# x^{2
}+ y^{2} = 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}/_{d}**2**

5 = k/1000

k = 5000

*now put in the new value!*

g = 5000 ^{1}/_{215}**2**

g = 5000/215^{2}

g = 0.108 units (3 s.f.)

#

__
Rearranging equations__

**9) Make r the subject of
the following equation:**

**F-r = mv**^{2}r + mr + m

F - m = mv^{2}r + mr + r

F - m = r(mv^{2} + 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**

# a^{2}
= b^{2} + c^{2} - (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

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__
EXTRA!__

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