Pure Chapters 1-7

Question 1

What is the domain?

Answer 1

The set of possible inputs for a function

Question 2

What is the range?

Answer 2

The set of possible outputs for a function

Question 3

Where are the asymptotes of the graph y=k/x or y=k/x^2?

Answer 3

x = 0
y = 0

Question 4

Transformations: y = f(x) + a

Answer 4

Translation by the vector (0 a)

Question 5

Transformations: y = f(x + a)

Answer 5

Translation of (-a 0)

Question 6

Transformations: y = af(x)

Answer 6

Stretch by scale factor a in the vertical direction

Question 7

Transformations: y = f(ax)

Answer 7

Stretch by scale factor 1/a in the horizontal direction

Question 8

Transformations: y = -f(x)

Answer 8

Reflection in the x-axis

Question 9

Transformations: y = f(-x)

Answer 9

Reflection of the graph in the y-axis

Question 10

What happens to asymptotes in translations?

Answer 10

They are translated in the same way as the graph

Question 11

What forms can the equation of a line take? (2)

Answer 11

y = mx + c

0 = ax + by + c

Question 12

What is similar about parallel lines?

Answer 12

They have the same gradient

Question 13

What are the gradients of perpendicular lines? (2)

Answer 13

A perpendicular to a line of gradient m has a gradient of -1/m
The product of the gradients of two perpendicular lines is -1

Question 14

What does a graph of direct proportion look like? (2)

Answer 14

The two quantities increase at the same rate meaning the graph is a straight line
The graph passes through the origin

Question 15

How is an algebraic fraction simplified?

Answer 15

Factorise the numerator and denominator wheee possible and cancel

Question 16

What is the factor theorem?

Answer 16

If f(x) is a polynomial then:
If f(p)=0, then (x-p) is a factor of f(x)
If (x-p) is a factor of f(x), then f(p)=0

Question 17

What is proof by deduction?

Answer 17

Start from known facts or definitions and use logical steps to reach the correct conclusion

Question 18

What must a mathematical proof contain? (4)

Answer 18

Any information or assumptions being used
Every step shown clearly
All possible cases covered
A statement of proof at the end of the working

Question 19

How do you prove an identity? (3)

Answer 19

Start with the expression on one side of the identity
Manipulate that expression until it matches the other side
Leave the other expression unchanged

Question 20

What is proof by exhaustion?

Answer 20

Break the statement into each case and prove them all separately

Question 21

How do you find a remainder in polynomial division? (2)

Answer 21

Use the grid method and then work out what is needed to reach the desired constant
Substitute the result of the dividing term into the initial equation to find the remainder

Question 22

What is a counter-example? (2)

Answer 22

One example that does not work for the statement

Only one example is needed to disprove a statement completely

Question 23

What is a theorem?

Answer 23

A statement that has already been proven

Question 24

What is a conjecture?

Answer 24

A statement that has yet to be proven

Question 25

What is the quotient?

Answer 25

The result of polynomial division

Question 26

What is a polynomial?

Answer 26

A finite expression with positive whole number indices