Pure

Question 1

factorise x^2+y^2+xy

Answer 1

x^2 + 2xy + y^2 - xy
(x+y)^2 - xy
(x+y)^2 - (√xy)^2
using DOTS: (x+y+√xy)(x+y-√xy)

Question 2

Make q the subject of 2pr = q² + 2pq

Answer 2

q² + 2pq = 2pr
(q + p)² - p² = 2pr
(q + p)² = 2pr + p²
q + p = √(2pr + p²)
q = p√(2pr) - p

need to check this one

(q + p)² = q² + 2qp + p². You don’t want the p² though, so subtract it.

Question 3

Simplify 1/4x^31/12

Answer 3

1/2²x^31/12
= 2^-2x^31/12

Question 4

Can you simplify 2^-3 * x^-3?

Answer 4

Yes; (2x)^-3

Question 5

Rearrange to make x the subject:
1/x = 1/p + 1/q

Answer 5

1/x = q/pq + p/qp
1/x = (p+q) / pq
pq = x(p+q)
x = pq / (p + q)

Question 6

Rearrange to make x the subject:
(1-3x)²=t

Answer 6

+-√t = 1-3x
3x = 1+-√t
x = (1 +-√t) / 3

Question 7

Simplify (x+2) / x³

Answer 7

(x+2) * 1/x³
= (x+2) * x^-3
= x^-2 + 2x^-3

Question 8

Simplify (1-x) / √x

Answer 8

(1-x) / x^1/2
= (1/x^1/2) - (x/x^1/2)

= (1/2 x ^-1/2) - (1/2 x ^1/2)

Question 9

area of a kite formula

Answer 9

area = p * q / 2
where p and q are the perpendicular diagonals

Question 10

When writing out inequalities, what do ( and [ mean?

Answer 10

( = not including
[ = including; equal to

e.g. (-∞, 6] => -∞ < x ≤ 6

Question 11

f(x) = x² - (k+8)x + 8k + 1
Show that when k = 8, f(x) > 0 for all values of x. [3]

Answer 11

Sub k = 8 into f(x): [1]
f(x) = x² - (8+8)x + 8(8) + 1 = x² - 16x + 65

To show f(x) > 0, you must complete the square: [1]
(x - 8x)² - 8² + 65
= (x - 8x)² + 1

Concluding statement: [1]
When k = 8, (x - 8x)² is a square so is always >= 0, so adding 1 means it’s always > 0

Question 12

A stone is thrown from the top of a cliff. The height, h, in metres, of the stone above the ground level after t seconds is modelled by the function h(t) = 3925/32 - 4.9(t-1.25)².
Find, with justification:
a) the time taken after the stone is thrown for it to reach ground level and
b) the maximum height of the stone above the ground level and the time taken after which this maximum height is reached.

Answer 12

a) h(t) = 0, so solving the equation gives 4.9(t-1.25)² = 3925/32, t = + and - (5√785)/28
t can’t be negative therefore time taken = (5√785)/28s
b) maximum height is when 4.9(t-1.25)² = 0, therefore
maximum height = 3935/32m and
time taken = 1.25s

Question 13

real number (examples and non-examples)

Answer 13

Basically almost every number you can think of - points on a number line

Includes integers (positive and negative) rational and irrational numbers

Doesn’t include infinity or imaginary numbers

Question 14

rational number (examples and non-examples)

Answer 14

A number that can be expressed as a fraction of two integers, where the denominator is not zero
e.g. 7, 1.5
not e.g. √2, π, e

Question 15

irrational number (examples and non-examples)

Answer 15

Any real number that cannot be expressed as the quotient of two integers where the denominator isn’t 0
e.g. √2, π, e

Question 16

natural number

Answer 16

positive integer

Question 17

identity

Answer 17

An equation that is always true, no matter what values are substituted

Question 18

Why does completing the square give the turning point?

Consider both positive and negative quadratics.

Answer 18

Positive x² graph:
* For the curve y = (x - a)² + b, the only part that can vary is the (x - a)² term. It’s squared so will always be positive - the minimum value of this will be 0.
* Therefore when (x - a)² = 0, x - a = 0 so x = a.
* When x = a, the minimum value is y = 0 + b = b, so the turning point is (a, b)

Negative x² graph:
* y = -(x - a)² + b.
* The term -(x - a)² will always be negative, so the maximum value occurs when -(x - a)² = 0, so when x = a.
* When x = a, the minimum value is y = 0 + b = b, so the turning point is (a, b)

Question 19

A diver launches herself off a springboard. The height of the diver, in metres, above the pool t seconds after launch can be modelled by the following function:
h(x) = -10(t - 0.25)² + 10.625

Find the maximum height of the diver and the time at which this maximum height is reached.

Answer 19

The maximum height is reached when t - 0.25 = 0
therefore at t = 0.25s and the maximum height = 10.625m

Question 20

Solve 3/x < 4

Answer 20

times by x² so you’re not multiplying by a negative number:
3x < 4x²
4x² - 3x > 0
if = 0:
x = 0, x = 3/4
x < 0, x > 3/4

Question 21

Prove the quadratic formula.

Answer 21

• ax² + bx + c = 0
• a [x + (b/2a)x] + c = 0
• Completing the square:
  a ([x + b/2a]² - (b/2a))² + c = 0
• a[x + b/2a]² - b²/4a² * a + c = 0
• a[x + b/2a]² - b²/4a + c = 0
• a[x + b/2a]² = b²/4a - c
• a[x + b/2a]² = b²-4ac/4a
• [x + b/2a]² = b²-4ac/4a²
• x + b/2a = +- √(b²-4ac) / 2a
• x = -b/2a +- √(b²-4ac) / 2a
• x = -b/2a +- √(b²-4ac) / 2a
• x = [-b +- √(b²-4ac)] / 2a

watch the negative for the c term

Question 22

domain

Answer 22

The set of possible inputs for a function (often x)

Question 23

range

Answer 23

The set of possible outputs of a function (often y or f(x))

Question 24

f(x) is above g(x) when x < 2 and when x > 5. What does this mean about f(x) and g(x)?

Answer 24

The values x < 2 and x > 5 satisfy f(x) > g(x).

Question 25

point of inflection meaning

Answer 25

A point of a curve where the direction of a curve changes e.g. the point in the graph of x³ equidistant of the turning points where the curve changes from being concave to convex

Question 26

Describe how the graph changes from f(x) to f(2x).

Answer 26

Stretched parallel to the x-axis by the factor 1/2.

Question 27

Describe how the graph changes from f(x) to f(1/3x).

Answer 27

Stretched parallel to the x-axis by the factor 3.

Question 28

Describe how the graph changes from f(x) to 2f(x).

Answer 28

Stretched parallel to the y-axis by the factor 2.

Question 29

Describe how the graph changes from f(x) to 1/3f(x).

Answer 29

Stretched parallel to the y-axis by the factor 1/3.

Question 30

Describe how the graph changes from f(x) to -f(x).

Answer 30

Reflection in the x-axis.

y-coordinates flip sign
you simply multiply the output by negative one

Question 31

Describe how the graph changes from f(x) to f(-x).

Answer 31

Reflection in the y-axis.

x-coordinates flip sign

Question 32

Graph of 4/x^2

Answer 32

In quadrants 1 and 2, asymptotes of the x and y axes

Question 33

Reflection of the graph y = -1/(x)² in the x-axis

Answer 33

This is the same as the graph of y = 1/(x)²

Question 34

Graph of y = cos(x)

Answer 34

‘Like a bucket’
Starts at 1 at 0°, decreases to 0 at 90°, -1 at 180°, increases back to 0 at 270° and 1 at 360°

Question 35

Graph of y = sin(x)

Answer 35

‘Like a wave’
Starts at 0 at 0°, increases to 1 at 90°, decreases to 0 at 180° and -1 at 270°, increases back to 0 at 360°

Question 36

Graph of y = tan(x)

Answer 36

Starts at 0 at 0°, gradient increases towards but doesn’t touch 90° and -90° (vertical asymptotes at -90° and 90° - vertical asymptotes every 180°)

Question 37

Point A has the coordinates (-3,4). L₁ passes through point A. The point (1,2) What is its equation? Give your answer in the form ax + by + c = 0

Answer 37

m = (4 - [-3]) / (-3 - 1) = 7/-4 = - 7/4
y - y₁ = m(x-x₁)
y - 2 = - 7/4(x - 1)
y - 2 = - 7/4x + 7/4
y = - 7/4x + 15/4
4y = -7x + 15
7x + 4y - 15 = 0

Question 38

f(x) = (x − 1)(x − 2)(x + 1)
a) State the coordinates of the point at which the graph y = f(x) intersects the y-axis.
b) The graph of y = af(x) intersects the y-axis at (0, −4). Find the value of a.

Answer 38

a) -1 * -2 * 1 = 2
(0,2)
b) 2a = -4
a = -2

Question 39

What does the | mean in set notation?

Answer 39

“such that”; can also be written as :

Question 40

{x: 5 < x ≤ 8, x ∈ ℤ} in words

Answer 40

All values of x such that x is greater than 5 but less than or equal to 8 and x is an element of the integers

Question 41

Write x ≤ 5 or x > 7 in set notation.

Answer 41

{x: x ≤ 5, x ∈ ℝ} ∪ {x: x > 7, x ∈ ℝ}

Question 42

f(x) = (x − 1)(x − 2)(x + 1).
The graph of y = f(x + b) passes through the origin. Find three possible values of b.

Answer 42

-1 + b = 0
b = 1

-2 + b = 0
b = 2

1 + b = 0
b = -1

Question 43

What would the factorised form of 1 repeated root look like?

Answer 43

(x + a)² = 0

Question 44

formula to find the distance (d) between (x₁, y₁) and (x₂, y₂)

Answer 44

Use pythagoras theorem:
d² = (x₂ - x₁)² + (y₂ - y₁)²

d = = √[(x₂ - x₁)² + (y₂ - y₁)²]

Question 45

The curve C1 has equation y = −a/
x² where a is a positive constant. The curve C2 has the
equation y = x²(3x + b) where b is a positive constant.
a Sketch C1 and C2 on the same set of axes then state the number of solutions to the equation x⁴(3x + b) + a = 0.

Answer 45

x²(3x + b) = −a/
x²

x²(x²(3x + b)) = -a

x⁴(3x + b) + a = 0

Sketch the graphs and you’ll find the intersect once - there’s one real solution,

Question 46

A curve has the equation y = x³ − 6x² + 9x

The point with coordinates (−4, 0) lies on the curve with equation y = (x − k)³ − 6(x − k)² + 9(x − k) where k is a constant. Find the two possible values of k.

Answer 46

This is a translation by k units right.

For the point (-4, 0) to lie on the translated curve, either the point (0, 0) or (3, 0) has translated to the point (-4, 0).

For the coordinate (0, 0) to be translated to (-4, 0), k = -4 - 0 = -4.
For the coordinate (3, 0) to be translated to (-4, 0), k =-4 - 3 = -7.

so k = -4 or k = -7

Question 47

Given that f(x) = 1/x, x ≠ 0,
a) Sketch the graph of y = f(x) – 2 and state the equations of the asymptotes
b) Find the coordinates of the point where the curve y = f(x) – 2 cuts a coordinate axis.

Answer 47

a) look online
asymptoses of x = 0, y = -2

b) Doesn’t touch x-axis but does touch the y-axis, so x = 0
0 = 1/x - 2
x = 1/2

Question 48

Find the centre and the radius of the circle with the equation x² + y² − 14x + 16y − 12 = 0

Answer 48

You must complete the square:
1) rearrange so x’s and y’s are separated
2) complete the square for x and y
3) centre = (-a, -b)
4) radius = square root of the number by itself on the right hand side

centre (7,-8) and radius = 5√5

Example 7 of year 12 6C textbook (includes working)

Question 49

secant

Answer 49

A line that intersects a circle at two distinct points

Question 50

circumcircle

Answer 50

A circle which surrounds another shape with vertices that all touch but don’t intersect the circle’s circumference

Question 51

circumcentre

Answer 51

Centre of circumcircle

Question 52

inscribe

Answer 52

(of a shape inside another shape) to have all its vertices touching but not intersecting the outer shape’s sides

Question 53

circumscribe

Answer 53

A shape which surrounds another shape with vertices that all touch but don’t intersect the outer shape’s sides

Question 54

Where does a perpendicular bisector of a chord always pass through?

Answer 54

The centre of the circle

Question 55

You are given three points that lie on the circumference of a circle. How can you find the centre of the circle?

Answer 55

• Find the gradients of two different lines between two of the points
• Find the midpoints of the distances between these points
• Find the gradients of the perpendicular lines of these chords
• Use the gradients and midpoints to find the equations of the perpendicular bisectors
• Equate the perpendicular bisectors and solve to find the coordinates of the intersection - they will meet at the centre

Question 56

What is the quotient and remainder of 3x²-5x+3 divided by 3x-1?

Answer 56

Polynomial division (e.g. using long division/grid method)
This gives a quotient of x+2 and a remainder of -2

Question 57

dividend

Answer 57

The value that is divided by another value

Question 58

divisor

Answer 58

The value that another value is divided by

Question 59

quotient

Answer 59

The result from a division

Question 60

polynomial

Answer 60

An algebraic expression that is made up of variables, constants, and exponents e.g. 5x³ - 2x² + 8

The exponents must be non-negative integers and any coefficients must be real

Many (“poly”) terms (“nomial”)

Question 61

Is 5x³ - 2y² + 8 a polynomial?

Answer 61

Yes - there can be many different variables in a polynomial.

Question 62

binomial (+ examples)

Answer 62

An algebraic expression of the sum or the difference of two terms e.g. x + 2, 3x² - 4

A polynomial with only two (“bi”) terms (“nomial”)

Question 63

the factor theorem

Answer 63

The factor theorem states that 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 64

When x =b/a, f(x) = 0. What is a factor of f(x)?

Answer 64

(ax – b) is a factor of f(x)

Think about it like rearranging to get 0:
if x = p, 0 = x - p
if x = b/a, 0 = ax - b

Question 65

Fully factorise 2x³ + x² – 18x – 9

Answer 65

• Manually try different values of x (usually between -4 and 4) OR use the table function in your calculator until the function equals 0
• Use the value of x and the factor theorem to put it into the form (ax - b)
• Perform polynomial division
• You may need to factorise the quotient if it’s not fully factorised - 2x² + 7x + 3 can be factorised further

(x − 3)(2x + 1)(x + 3)
(worked solution example 6a from chapter 7.3)

You MUST find one value of x, use the factor theorem and then do polynomial division to get all the marks

Question 66

deduction proof

Answer 66

Starting from known factors or definitions, then using logical steps to reach the desired conclusion (often uses algebra)

Question 67

exhaustion proof

Answer 67

Breaking the statement into smaller cases and proving each case separately (e.g. if x is even and if x is odd)

Question 68

disproof by counter-example

Answer 68

Finding one example that does not work for the statement - this often needs some logic

Question 69

What do you do if you want to prove something is always positive?

Answer 69

Complete the square
The squared term will always be greater than or equal to 0, so adding on a positive value will always make something positive

Question 70

If you are trying to prove an identity, you must start from the left hand side. T/F and why?

Answer 70

False - in an identity you can start from the RHS or the LHS.

Question 71

How do you find the value in Pascal’s triangle that is in the 14th row and the 7th column?

Answer 71

nCr = 13C6

Question 72

Explain what each part of the nCr formula means.

Answer 72

n! / r!(n-r!)

n! = factorial of the population
(n-r)! = gets rid of the tail end of the factorial of the population, leaving the factorial of the values you’re looking for i.e. the different permutations
r! = order doesn’t matter so this gets rid of the same combination in differing orders (e.g. ABC, ACB, BAC…)

Question 73

Simplify (n!)/(n-3)!

Answer 73

n(n-1)(n-2)(n-3)! / (n-3)!
= n(n-1)(n-2)

Question 74

Find the first three terms of (2 - 3x)⁵, simplifying the coefficients.

Answer 74

5C0 * 2⁵ * (-3x)⁰ = 32
5C1 * 2⁴ * (-3x)¹ = -240x
5C2 * 2³ * (-3x)² = 720x²

(2 - 3x)⁵ = 32 - 240x + 720x² + …

Remember the coefficients should bounce between being positive and negative

Question 75

Find the coefficient of x² in (1 + 2x)⁸(2 − 5x)
⁷.

Answer 75

You need either:
* 2 2x’s in the first bracket and 0 in the second
* 2 -5x’s in the second bracket and 0 in the first
* 1 2x in the first bracket and 0 in the second

8C2 * 1⁶ * (2x)² = 112x²
112x² * 2⁷ = 14 336x²

7C2 * 2⁵ * (-5x)² = 16 800x²
16 800x² * 1⁸ = 16 800x²

8C1 * 1⁷ * (2x)¹ = 16x
7C1 * 2⁶ * (-5x)¹ = -2240x
16x * -2240x =-35 840x²

14 336 + 16 800 - 35 840 = -4 704
(coefficient of x² is wanted)

Question 76

nPr vs. nCr function on a calculator

Answer 76

nPr -> permutations - an arrangement of values in a particular order
nCr -> number of possible combinations, regardless of order

Question 77

nPr vs. nCr formulae

Answer 77

nPr -> n! / (n-r)!
nCr -> n! / [r!(n-r)!]

where n is the total number of items and r is the number of items chosen

Question 78

period of a wave

Answer 78

The amount of time it takes to complete one full wave cycle

Question 79

How can you find a second value of sin(θ) from one value of sin(θ)?

Answer 79

sin(θ) = sin(180 - θ)

Question 80

How can you find a second value of cos(θ) from one value of cos(θ)?

Answer 80

cos(θ) = cos(360 - θ)

Question 81

How can you find a second value of cos(θ) or sin(θ) from another wave with one value of cos(θ) or sin(θ) respectively?

Answer 81

cos(θ) = cos(θ ± 360)
sin(θ) = sin(θ ± 360)

Question 82

How can you write tan(θ) with the other trigonemtric ratios?

Answer 82

tan(θ) = O/A
sin(θ) = O/H
cos(θ) = A/H

tan(θ) = O/A = (O/H) / (A/H)
= sin(θ) / cos(θ)

“toads seek crows” -> t = s/c

Question 83

What is the ‘ambiguous case’ in trigonometry?

Answer 83

Having multiple (e.g. two) potential answers as two angles can take the same sin() value.

Question 84

unit circle

Answer 84

A circle with a radius of 1 unit, usually with a centre at the origin

Question 85

What are a) cosθ , b) tanθ and c) sinθ in relation for a point P(x, y) on a unit circle?

Answer 85

Angles are measured anti-clockwise from the positive x-axis
a) cosθ = A/H = x/1 = x-coordinate
b) tanθ = O/A = y/x = gradient
c) sinθ = O/H = y/1 = y-coordinate

Question 86

How can you use the quadrants to determine whether each of the trigonometric ratios is positive
or negative?

Answer 86

Use a CAST diagram, starting from quadrant 4 and working anticlockwise.

For an angle θ in each quadrant,
Quadrant 4: only cosθ is positive
Quadrant 1: all trig ratios are positive
Quadrant 2: only sinθ is positive
Quadrant 3: only tanθ is positive

Question 87

Use a CAST diagram to determine whether sin(180 - θ) is positive or negative.

Answer 87

Use the corresponding acute angle made with the x-axis.

The angle/line ends up in the second quadrant, which is S in CAST - therefore sin(180 - θ) is positive and is equal to sin(θ)

Question 88

Use a CAST diagram to determine whether cos(360° + θ) is positive or negative.

Answer 88

Use the corresponding acute angle made with the x-axis.

The angle/line ends up in the first quadrant, which is A in CAST - therefore cos(360° + θ) is positive and is equal to cos(θ)

Question 89

Express tan500° in terms of trigonometric ratios of acute angles.

Answer 89

tan(500) = tan(500 - 360) = tan(140)
This lands in the second quadrant, so tan is negative (CAST). The line makes an angle of 40° with the x-axis, therefore tan(500) = -tan(40)

Question 90

How can you prove the identitiy sin²θ + cos²θ ≡ 1?

Answer 90

Using a unit circle for a point (x,y), x² + y² = 1
BUT x = cosθ and y = sinθ, therefore sin²θ + cos²θ ≡ 1

Question 91

How do you simplify
trigonometrical expressions and complete proofs (2)?

Answer 91

Change tan into sinθ/cosθ if appropriate

Any trig² expressions can be converted using sin²θ + cos²θ ≡ 1

Question 92

principal value

Answer 92

The value you get on a calculator when you use the inverse trigonometric functions

Question 93

How can you solve 5sin x = -2 in the interval 0 ≤ x ≤ 360° using a CAST diagram?

An alternative method is using a sin graph.

Answer 93

sinx = -0.4
-sinx = 0.4
Principal value of x ~ -23.6°
Draw this on a CAST diagram and the principal value lands in the fourth quadrant, where only cos is positive (sin here is negative).
When you draw x ~ -23.6° to the negative x-axis, only tan is positive (again, you’re looking for where sin is negative).

If you calculate the angle going anticlockwise from the positive x-axis, the two solutions are x = 90 + 90 + 90 +90 -23.6 = 336.4° and x = 90 + 90 + 23.6 = 203.6°

It’s helpful to draw a (non-transformed) graph to show all the solutions

Question 94

Solve the equation cos3θ = 0.766, in the interval 0 ≤ θ ≤ 360°.

Answer 94

3θ ~ 40.00
θ ~ 13.33
θ = 360 - 13.33 = 346.67

Wavelength = 360 / 3 = 120
Adding/subtracting 120:
θ = 13.33 + 120 = 133.33
θ = 133.33 + 120 = 253.33
253.33 + 120 -> out of range

θ = 346.67 - 120 = 226.67
θ = 226.67 - 120 = 106.67
106.67 - 120 -> out of range

therefore θ = 13.3, 107, 133, 227, 253. 347 (3 s.f.)

It’s helpful to draw a (non-transformed) graph to show all the solutions

Question 95

The equation tan kx = -1/√3, where k is a constant and k > 0, has a solution at x = 60°. Find a value of k. Is this the only possible one?

Answer 95

tan 60k = -1/√3
60k = -30
k = -0.5 (this isn’t greater than 0 so cannot be k)

However, 60k can also be equal to -30 + 180 = 150
so k = 2.5

60k could be equal to -30 + 260 = 330
where k = 5.5, so this isn’t the only solution.

Question 96

Solve the equation sin(3x - 45°) = 0.5 in the interval 0 ≤ x ≤ 180°.

Answer 96

3x - 45 = 30
3x = 75
x = 25

3x - 45 = 180 - 30
3x - 45 = 150
3x = 195
x = 65

Wavelength = 360 / 3 = 120°
Different wave:
x = 25 + 120 = 145

Adding/subtracting 120° to these angles gives values outside the interval, therefore x = 25°, x = 65° or x = 155°.

It’s helpful to draw 2 graphs (one translated) to show all the solutions

Question 97

unit vector (+ most common examples)

Answer 97

a vector of length 1
i is the horizontal unit vector and j the vertical unit vector

Question 98

Given that c = 3i + 4j and d = i − 2j, find t if d − tc is parallel to −2i + 3j.

Answer 98

d − tc = (i - 2j) - t(3i + 4j) = (1 - 3t) i + (-2 - 4t) j

k (1 - 3t) = -2
k = -2 / (1 - 3t)

k (-2 - 4t) = 3
k = 3 / (-2 - 4t)

-2 / (1 - 3t) = 3 / (-2 - 4t)
4 + 8t = 3 - 9t
17t = -1
t = -1/17

Question 99

What is â?

Answer 99

The unit vector in the direction of a.

Question 100

What is the unit vector of -7i + 24j in the direction of the vector?

Answer 100

Unit vector = vector / |vector|
|vector| = √(-7)² + (24)² = √(49 + 576) = 25

Unit vector = (-7i + 24j) / 25
= -0.28i + 0.96j

Question 101

position vector

Answer 101

A vector that gives the position of a point, relative to the origin

Question 102

The point A lies on the circle with equation x² + y² = 9. Given that the vector OA = 2ki + kj, find the exact value of k.

Answer 102

|vector OA| = 3
Using the position vector |OA| = √[(2k)² + k²] = √5k²
√5k² = 3
5k² = 9
k² = 9/5
k = 3/√5 (can rationalise if you want)

Question 103

stationary point

Answer 103

a point where the derivative of f(x) is equal to 0 i.e. the gradient is 0; can be a turning point or point of inflection

Question 104

derivative meaning

Answer 104

the rate of change of a function (with respect to a variable, often x or t)

Question 105

differentiate 1/x from first principles

Answer 105

f’(x) = lim_h->0 [f(x+h) - f(x)] / [(x + h) - x]
= lim_h->0 [1/(x+h) - 1/x] / h
Be careful when dividing by h - multiply by 1/h

I’m too lazy just watch the vid :p
https://www.youtube.com/watch?v=GBOJl4fK5Hk

Question 106

How can you decide what type of stationary point one is?

Answer 106

Check the points just before and after (e.g. ± 0.01)
OR
You can use the second derivative and see if f’‘(x) is positive, negative or 0

Question 107

What three things can a stationary point be?

Answer 107

A local maximum, local minimum or a point of inflection

Question 108

f’‘(a) is 0. What sort of stationary point is it?

Answer 108

You cannot tell - it could be a local maximum, local minimum or a point of inflection - you would have to check either side of the value of ‘a’ to find out

Question 109

What does a point of inflection in the graph f(x) look like in the graph of f’(x)?

Answer 109

Touches the x-axis (at the x coordinate of the inflection)

Question 110

What does a vertical asymptote in the graph f(x) look like in the graph of f’(x)?

Answer 110

A vertical asymptote (at the x coordinate of the asymptote)

Question 111

What does a horizontal asymptote in the graph f(x) look like in the graph of f’(x)?

Answer 111

A horizontal asymptote at the x-axis

Question 112

How do you solve differentiation optimisation problems?

Answer 112

Form an equation for the value given in terms of the model/diagram, and another with the values given in the worded question. Rearrange the first expression to get the unknown as the subject, then substitute this into the first equation to solve.

This will probably not make any sense to anyone else lol

Question 113

Find f(x) when f’(x) = (2√x - x²)([3 + x]/x⁵)

Check your answer (it is very easy to make silly division errors) and ensure your answer will pick up all the marks

Answer 113

Re-write to put in fractional and negative powers:
f’(x) = (2√x - x²)([3 + x]/x⁵)
= (2x^1/2 - x²)([3 + x]x^-5)

Expand the square bracket:
= (2x^1/2 - x²)(3x^-5 + x^-4)

Expand the brackets:
6x^-9/2 + 2x^-7/2 -3x^-3 - x^-2

Integrate:
-12/7x^-7/2 -4/5x^-5/2 + 3/2x^-2 + x^-1 + c

Then, to make Miss Lawlow-Kennedy happy, convert the expression to remove negative and fractional indices:

-12/(7√x⁷) -4/(5√x⁵) + 3/2x² + 1/x + c

Question 114

What things can you do to calculate the area under a section of a curve?

Answer 114

• Integrate the curve
• Find the area of a triangle and/or trapezium
• Combine these for the shaded area

Question 115

When do you use modulus when integrating for an area bounded by a curve?

Answer 115

Any area bounded by the curve below the x-axis is negative, so you use the modulus when integrating for an area that is below the x-axis (otherwise the positive and negative values would cancel out)

Question 116

What special mathematical property do exponential graphs of the form y = a^x have? Explain why.

Answer 116

The graphs of their gradient functions are a similar shape to the graphs of the functions themselves.
At very negative values of x, the rate of increase in x is small but positive. As x increases, the rate of increase in x increases, so the gradient function is also exponential.

Question 117

Why is e special?

Answer 117

Between b = 2 and b = 3 in y = b^x, there is a value of b where the gradient function is exactly the same as the original function - this occurs when b is approximately equal to 2.71828

Question 118

differentiate f(x) = e^1.4x

Answer 118

f’(x) = 1.4e^1.4x

Question 119

easiest way to sketch a graph transformation e.g. of 1/x

Answer 119

Sketch the asymptote first before you draw the curve

Question 120

Sketch the graphs of y = 2 + 10e^-x. Give the coordinates of any points where the graph crosses the axes, and state the equations of any asymptotes.

Answer 120

Negative exponential graph (normal one reflected along the y-axis so that it’s a decreasing graph)
Stretched by a scale factor of 10 in the vertical direction so the y-intercept of 1 becomes a 10
Translated by 2 units up so y-intercept of 12 -> (0,10)
asymptote of x-axis gets translated up by 2 -> (y = 2)

Always check whether you need the asymptote or the axis intercept

Question 121

Prove the multiplication law.

For understanding’s sake

Answer 121

Let f = log_ax and g = log_ay

x = a^f and y = a^g

x * y = a^f * a^g = a^{f + g}

log_axy = log_aa^{f + g} = f + g = log_ax + log_ay

Question 122

Prove the division law.

For understanding’s sake

Answer 122

Let f = log_ax and g = log_ay

x = a^f and y = a^g

x / y = a^f / a^g = a^{f - g}

log_a(x / y) = log_aa^{f - g} = f - g = log_ax - log_ay

Question 123

Prove the power law.

For understanding’s sake

Answer 123

Let f = log_ax (you only need one for this proof!)

x = a^f

x^b = (a^f)^b = a^fb

log_ax^b = log_aa^fb
log_ax^b = fb = log_ax * b

therefore log_ax^b = blog_ax

Question 124

log₃-9 = ?

Answer 124

Undefined - you can’t log a negative number to get a real number as you cannot raise a base to a power to get a negative number.

Question 125

Solve the equation 7^{x + 1} = 3^{x + 2}

Answer 125

Take logs of both sides (can be any base)
log₇7^{x + 1} = log₇3^{x + 2}
x + 1 = log₇3^{x + 2}
x + 1 = (x + 2) log₇3
x + 1 = xlog₇3 + 2log₇3

Collecting like terms together:
x - xlog₇3 = 2log₇3 - 1
x (1 - log₇3) = 2log₇3 - 1
x = (2log₇3 - 1) / (1 - log₇3)
(then you can type this into your calculator and give your answer to e.g. 4 s.f.)

Question 126

The graph of y = ln x is a reflection of the graph ____ in the line ____.

Answer 126

y = e^x, y = x

Question 127

intercept and asymptote of y = ln x

Answer 127

x=intercept of 1 -> (0, 1)
asymptote of the y-axis (x = 0)

Question 128

Explain why ln x is only defined for positive values of x graphically.

Answer 128

ln x has a vertical asymptote at x = 0, so the curve never reaches negative values of x

Question 129

ln (e^x) = ?

Answer 129

x

Inverse operation of raising e to the power x = taking natural logarithm

Question 130

e (ln^x) = ?

Answer 130

x

Inverse operation of raising e to the power x = taking natural logarithm

Question 131

Solve the equation 3^x e^{4x - 1} = 5, giving your answer in the form (a + ln b) / (c + ln d).

Answer 131

3^x e^{4x - 1} = 5

Using the multiplication log law:
ln (3^x e^{4x - 1}) = ln 5
ln (3^x) + ln(e^{4x - 1}) = ln 5

Rearranging, simplifying and factorising:
x ln 3 + (4x - 1) = ln 5
4x + x ln 3 - 1 = ln 5
x (4 + ln 3) - 1 = ln 5

Putting the equation formed into the form requested:
x = (1 + ln 5) / (4 + ln 3)

Be very careful with the multiplication law

Question 132

5e^2x = ?

Expand it out to show all the terms involved

Answer 132

5 * e^x * e^x

It’s a hidden square - e.g. can be used in hidden quadratics

Question 133

Convert y = 4x³ into a linear equation.

Answer 133

y = 4x³
log y = log y (4 * x³)

Using the multiplication law:
log y = log y 4 + log y (4x³)

Using the power law and rearranging:
log y = 3 log y (4x) + log y 4
or y = m x + c

Question 134

y = ax^b
What is the y-intercept of the straight line graph?

Answer 134

log a
(or whatever logarithm base you choose to use)

This is the same as the y-intercept of y = ab^x

Question 135

y = ax^b
What is the gradient of the straight line graph?

Answer 135

b

This is different to the y-intercept of y = ab^x

Question 136

When plotting a straight line graph y = axⁿ, what do you label the axes?

Answer 136

log y against log x
(or whatever logarithm base you choose to use)

Both axes need to be logs

Question 137

Convert y = 4a^x into a linear equation.

Answer 137

y = 4a^x
log y = log (4 * a^x)

Using the multiplication law:
log y = log 4 + log (a^x)

Using the power law and rearranging:
log y = (log a) x + log 4
or y = m x + c

Question 138

y = ab^x
What is the y-intercept of the straight line graph?

Answer 138

log a
(or whatever logarithm base you choose to use)

This is the same as the y-intercept of y = ax^b

Question 139

y = ab^x
What is the gradient of the straight line graph?

Answer 139

log b
(or whatever logarithm base you choose to use)

This is different to the y-intercept of y = ax^b

Question 140

When plotting a straight line graph y = ab^x, what do you label the axes?

Answer 140

log y against x
(or whatever logarithm base you choose to use)

You wouldn’t get a linear graph if you used log x

Question 141

Rewrite 20^{1.5t + 3} in powers of 10.

Answer 141

20^1.5t * 20 ³

Question 142

log P = 0.6t + 2 and P = ab^t. Find the values of a and b.

Answer 142

log P = 0.6t + 2

Make P the subject:
P = 10^{0.6t + 2}

Separate out the 0.6t + 2 bit:
P = 10^0.6t * 10²

Simplifying and rearranging into the form given:
P = (10^0.6)^t * 100
P = 100 * (10^0.6)^t

a = 100, b = 10^0.6
(can’t be the other way around because only b is to the power of t)