Maths

Question 0

Solve the inequality x^2 - 2x - 3 >= 0

Answer 0

x^2 - 2x - 3 = (x + 1)(x - 3)

Line crosses x axis at x values of -1 and 3

The line is above or equal to the x axis when x <= -1 or x >= 3 so this is the inequality

Question 1

Complete the square of x^2 - 8x + 3 = 0

Answer 1

x^2 - 8x = -3

(8 / 2 = 4 and 4^2 = 16)

x^2 - 8x + 16 = -3 + 16

(x - 4)^2 = 13

x - 4 = +- sqr(13)

Calculate answers from here

Question 2

How do you complete the square?

Answer 2

In terms of ax^2 + bx + c

Move c to the other side of the equation

Take b, halve it, square the answer and add the result to both sides of the equation

Factorise the left side

Square root both sides and put a plus / minus sign before the right side

Calculate x from here

Question 3

How do you solve quadratic inequalities?

Answer 3

For x^2 - 2x - 3 < 0

(x + 1)(x - 3) < 0

So the curve crosses the x axis when x = -1 and x = 3

Because we are looking for <0, the inequality is -1 < x < 3 because the line is under the x axis between x values of -1 and 3

Question 4

What else is completing the square useful for?

Answer 4

Identifying the line of symmetry and the vertex

For example,

y = x^2 - 8x + 3 can be written as y = (x - 4)^2 - 13

From this, the line of symmetry is x = 4 and the vertex is (4,-13)

Question 5

How do you complete the square?

Answer 5

Revise from book

Question 6

What is the bowtie theorem?

Answer 6

If every point of a bowtie shape touches the circumference, the two angles are equal and the two angles at the bottom are equal

Question 7

What is the theorem that tells us about tangents meeting at an external point?

Answer 7

2 tangents meeting at an external point are equal in length

Question 8

What is a cyclic quadrilateral?

Answer 8

A quadrilateral where every point is on the circumference of the circle

Question 9

Using the same shape as the last question what relates POQ and PXQ?

Answer 9

POQ = 180 - PXQ

Question 10

What is the theorem linking a tangent and a radius?

Answer 10

The angle between a tangent and a radius is 90 degrees

Question 11

What are supplementary angles?

Answer 11

C angles

The 2 angles equal 180 degrees

Question 12

What is the arrowhead theorem?

Answer 12

The angle at the centre is twice the angle at the circumference

Question 13

What is the theorem that tells us when a line will bisect a chord?

Answer 13

The perpendicular from the centre to a chord bisects the chord

Question 14

What is the cyclic quadrilateral theorem?

Answer 14

Opposite angles in a cyclic quadrilateral add up to 180 degrees

Question 15

How do you calculate the interior angle size of a regular polygon?

Answer 15

(180 * (number of sides - 2)) / number of sides

Question 16

What are corresponding angles?

Answer 16

F angles

The two angles equal each other

Question 17

What are alternate angles?

Answer 17

Z angles

The two angles equal each other

Question 18

How do you calculate the external angle size of a regular polygon?

Answer 18

360 / number of sides

Question 19

What is alternate segment theorem?

Answer 19

If a triangle always touching the circumference is labelled ABC and there is a tangent (T) at point B, the angle between T and BC = BAC and the angle between T and BA = ACB

Question 20

If the centre is O, and there are 2 tangents called P and Q that meet at point X and a line is drawn from X to O, which angles are the same?

Answer 20

OPX and OQX

PXO and QXO

Question 21

When drawing a graph that has negative and positive values, what do you have to remember?

Answer 21

Draw the graph with a cross axis

Question 22

What do you have to remember in probability tree questions?

Answer 22

Make a section of the tree for each named condition

Question 23

In an inequality graph, what do dotted and solid lines mean?

Answer 23

Dotted line is <>

Solid line is <= and >=

Question 24

In an inequality diagram, what do hollow and filled dots mean?

Answer 24

Hollow is <>

Full is <= and >=