Module 12 & 13 revision

Question 1

What is the rule for angles in a semi-circle?

Answer 1

they equal 90°

Question 2

What is the rule for angles in the same segment?

Answer 2

they are equal

Question 3

What is the rule for angles at a centre?

Answer 3

they are twice the angle on the edge

Question 4

What is a cyclic quadrilateral?

What is the rule?

Answer 4

all four corners are on the edge of the circle

diagonally opposite angles add to 180°

Question 5

What is the rule for alternate segments?

Answer 5

diagonally opposite angles are equal

Question 6

What is the rule for the radius and a tangent?

Answer 6

the radius and tangent make a right angle

Question 7

What is the rule for a triangle formed by two radii?

Answer 7

it is isosceles

Question 8

How do you calculate the upper and lower bounds?

Answer 8

0.5 more and 0.5 less

e.g. 2.5

UB = 2.55
LB = 2.45

Question 9

How do you calculate the HCF and LCM?

Answer 9

1) draw a frequency tree for both numbers
2) write down all of the numbers circled multiplied together

HCF is the numbers that appear in both (use lowest index)
LCM is all the numbers (use highest index)

e.g. 150 and 250

150 = 5² x 2 x 3
250 = 5³ x 2

HCF = 5² x2 = 50
LCM = 5³ x 2 x 3 = 750

Question 10

What are the three rules of indicies?

Answer 10

when x, + the indicies
when /, - the indicies
when there are (), multiply indicies
e.g. (2³)³ = 2 to the power of 9

Question 11

What is the rule of indicies when it is to the power of 0?

Answer 11

it equals 1

Question 12

What is the rule of indicies when it is to a negative power?

Answer 12

it becomes 1/x to the power of x
e.g. 2-³ = 1/2³

Question 13

What is the rule of indicies when the power is a fraction? (fractional indicies)

Answer 13

a number to the power of a fraction is the square root of that number
the denominator is the root
the numerator is the power

e.g. 27 to the 2/3 = (the cube root of 27)² = 3² = 9

Question 14

What is the rule of indicies when there is a fraction to the power of -1?

Answer 14

you just flip the fraction

e.g. (2/3) to the -1 = 3/2

Question 15

What is the rule of indicies when there is a fraction to the power of x?

Answer 15

you add the power to both numbers in the fraction

e.g. (2/5)² = 2²/5² = 4/25

remember to simplify if possible!!

Question 16

What is the rule for negative fractional indicies? (a number to the power of a negative fraction)

Answer 16

the same as fractional indicies but you just add 1/x

e.g. 27 to the -1/3 = 1/ the cube root of 27 = 1/3

Question 17

How do you simplify surds?

Answer 17

find two factors of the number - one must be a square number

e.g. √75 = √25√3 = 5√3

Question 18

How do you add or subtract surds?

Answer 18

the surds must be the same
it’s like collecting ‘like terms’

e.g. 3√2 + 5√2 = 8√2

2√27 - 3√3 = 6√3 - 3√3 = 3√3
simplify = 2 x√9√3 = 2x3√3 = 6√3

Question 19

What is the rule for multiplying surds that are the same?

Answer 19

they equal the normal number

e.g. √3 x √3 = √9 = √

√5 x √5 = 5
√2 x √2 = 2 etc

Question 20

How do you expand double brackets with surds?

Answer 20

expand brackets like normal except :

multiplying the same surd equals the normal number
if you are multiplying two different surds you just multiply the two numbers and add the √
e.g. √3 x √3 = 3

√3 x √2 = √6

you then collect the like terms

e.g. (3 + √2) (5 - √2) = 15 - 3√2 + 5√2 - 2 = 13 + 2√2

(3 - √5) (4 - √2) = 12 - 3√2 - 4√5 + √10

Question 21

How do you rationalise surds?

Answer 21

it is just multiplying out the surd in the denominator - so multiple the numerator and denominator by the denominator

e.g. 7/√2 = 7/√2 x √2/√2 = 7√2/2

Question 22

How do you find a percentage of a number without a calculator?

Answer 22

• find 10% of the number (divide by 10)
• find 1% of the number (divide by 100)
• multiply the 10% by the amount of tens in the number
• multiple the 1% by the amount of ones in the number

e.g. find 56% of 8200

10% = 820
1% = 82

820 x 5 = 4100
82 x 6 = 492

4100 + 492 = 4592

Question 23

How do you find a percentage of a number with a calculator?

Answer 23

% means out of 100 so…

• divide percentage by 100 to get a decimal
• multiply the decimal by the number

e.g. find 56% of 8200

0.56 x 8200 = 4592

Question 24

How do you calculate percentage increase?

Answer 24

• the original number is 100%
• add the percentage increase onto 100%
• divide by 100 to get a decimal
• multiply the original number by the decimal

e.g. £1552 increasing by 20%

100% + 20% = 120% = 1.2

1552 x 1.2 = £1862.4

Question 25

How do you calculate percentage decrease?

Answer 25

• the original number is 100%
• minus the percentage decrease from 100%
• divide by 100 to get a decimal
• multiply the original number by the decimal

e.g. £250 decreasing by 15%

100% - 15% = 85% = 0.85

250 x 0.85 = 212.5

Question 26

How do you write a number out of a number as a percentage?

Answer 26

smaller number/ larger number x 100

e.g. 94 out of 200

94/200 x 100 = 47%

Question 27

How do you calculate percentage change? (same for increase and decrease)

Answer 27

percentage change = actual change/ original number x 100

e.g. 40 increases to 56, what is the percentage increase

actual increase = 56 - 40 = 16

percentage increase = 16/ 56 x 100 = 40%

Question 28

How do you calculate compound interest?

Answer 28

total = original amount x multiplier (percentage (as decimal) inc/dec) to the power of the number of years

e.g. £3000 in a bank for 3 years, earns 2% compound interest

total = 3000 x 1.02³ = £3183.62

Question 29

How do you calculate reverse percentages?

Answer 29

(they are to find the original amount and are usually in sale questions, asks for the cost before)

• write down the answer
• write a ? to represent the original amount
• next to ? x by the percentage multiplier
• rearrange so that you have the answer over the multiplier
• solve

e.g. on sale for £22.05 reduced by 10%, what was the price before?

? x 0.9 = 22.05

22.05/0.9 = £24.50

Question 30

How do you calculate reverse percentages?

Answer 30

(they are to find the original amount and are usually in sale questions, asks for the cost before)

• write down the answer
• write a ? to represent the original amount
• next to ? x by the percentage multiplier
• rearrange so that you have the answer over the multiplier
• solve

e.g. on sale for £22.05 reduced by 10%, what was the price before?

? x 0.9 = 22.05

22.05/0.9 = £24.50