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Maths- A-level > Pure Year 1 > Flashcards

Pure Year 1 Flashcards

1
Q

What is the cosine formula- both variations?

A 
a^2 = b^2 + c^2 - 2bc x cos(A)
Cos(A) = b^2 + c^2 - a^2 / 2bc
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2
Q

What is the sine formula- both variations?

A

sin (A) / a = sin (B) / b - angles

a / sin (A) = b / sin (B) - sides

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3
Q

What is pythagoras’ theorem?

A

a^2 + b^2 = c^2

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4
Q

What is a rational fraction?

A

A fraction with a surd in the denominator

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5
Q

What is 8^2/3?

A

4
Bottom number outside
Top number inside

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6
Q

How do you rationalise a surd?

A

Multiply by either the denominator itself if it is on its own or by itself & the opposite to either + or -

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7
Q

What is the quadratic formula?

A

x = -b +- root (b^2 - 4ac) / 2a

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8
Q

What is the discriminant and when does it show the number of roots a quadratic has?

A

b^2 - 4ac = 0 - 1 root (tangent if root repeated)
b^2 - 4ac > 0 - 2 roots
b2 - 4ac < 0 - no roots

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9
Q

How do you find points of intersection?

A

Equate the 2 equations

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10
Q

How do you write an inequality when the equation is greater than (>) ?

A

2 separate inequalities with the smaller one as less than ()
Because above the graph- opposite directions of solutions

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11
Q

How do you write an inequality when the equation is less than (

A

As one inequality because below the graph

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12
Q

What does the reciprocal graph y = 1 / x look like and where are it’s asymptotes?

A

Line in top right quadrant & bottom left quadrant with asymptotes at x = 0 & y = 0

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13
Q

What does the reciprocal graph y = -2 / x look like and where are it’s asymptotes?

A

Line in top left quadrant & bottom right quadrant with asymptotes at x = 0 & y = 0

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14
Q

What does the graph y = 2 / x^2 look like & where are it’s asymptotes?

A

Line in top left quadrant & top right quadrant with asymptotes at x = 0 & y = 0

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15
Q

What does the graph y = -5 / x^2 look like & where are it’s asymptotes?

A

Line in bottom left quadrant & bottom right quadrant with asymptotes at x = 0 & y = 0

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16
Q

In a function such as (a / b ) what direction does a & b control?

A

a controls left/right movements

b controls up/down movements

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17
Q

When a graph is translated, are its asymptotes translated with it?

A

When a graph is translated its asymptotes are translated with it

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18
Q

What does the translation y = -f(x) mean?

A

Reflection in the x-axis

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19
Q

What does the translation y = f(-x) mean?

A

Reflection in the y-axis

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20
Q

How do you calculate the gradient of the line between 2 points?

A 
gradient = change in y / change in x
gradient = y2 - y1 / x2 - x1
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21
Q

What is the product of the gradients of 2 perpendicular lines?

A

-1

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22
Q

What is unique about the gradients of 2 lines which are perpendicular to one another?

A

Their gradients are negative reciprocals of one another

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23
Q

What is unique about the gradients of 2 lines which are parallel to one another?

A

Their gradients are the same

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24
Q

How do you find the distance between 2 points?

A

Find the net coordinates by minusing them from one another and then use pythagoras’ theorem
Root {(x2 - x1)^2 + (y2 - y1)^2}

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25
Q

How do you calculate the mid point of 2 points?

A

Add each of the x & y coordinates separately and divide each one by 2 separately to form the midpoints coordinates

(x1 + x2/2, y1 + y2/2)

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26
Q

What is a perpendicular bisector and what is its gradient?

A

A line that passes directly in the middle of another line or 2 points and forms a 90 degree angle with that line or the line the 2 points form
Its the gradient of the negative reciprocal of the line it cuts or the line formed by the 2 points

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27
Q

What is the general equation of a circle?

A

x2 + y2 = r2

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28
Q

How do you find the gradient and radius of a circle when its equation is given in its expanded form? (x2 + y2 + 2fx + 2gy + c = 0)

A

Complete the square

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29
Q

What is unique about the tangent of a circle and its radius?

A

They touch only once and when they do they form a 90 degree angle

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30
Q

What is unique about the perpendicular bisector of a chord of a circle?

A

It will go through the centre of the circle

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31
Q

Define what the circumcircle of a triangle is

A

A circumcircle is a circle drawn using the 3 vertices of any triangle

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32
Q

Define what the circumcentre of a triangle is

A

It is the centre of the circumcircle and is the point where the perpendicular bisector of each side of the triangle intersect

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33
Q

What is different about the circumcircle of a right angled triangle?

A

The hypotenuse of the right angled triangle is the diameter of the circle because the angle in a semi circle is always a right angle

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34
Q

How do you find the centre of a circle when you have 3 points on its circumference

A

Join the points together to form 3 lines and find the perpendicular bisector of 2 of the lines. Find the midpoints of of the chords and plug into the perpendicular bisector equation to find C. Equate the 2 perpendicular bisector equations and the intersection found is the midpoint.

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35
Q

Define a theorem

A

Statement that has been proven

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36
Q

Define a conjecture

A

Statement that has yet to be proven

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37
Q

What is proof by deduction?

A

Use of algebra to find proof

38
Q

What is proof by exhaustion?

A

Work out using all possibilities- several examples

39
Q

What is disproof by counter example?

A

When you prove a statement wrong with one example

40
Q

What is 0 factorial (0!) equal to?

A

1

41
Q

How do you find the rth entry in the nth row of Pascal’s triangle?

A

n-1 C r-1

42
Q

What is a natural number (like x € N)?

A

All positive integers- therefore x is a positive integer (positive whole number)

43
Q

What is a real number (like x € R)?

A

All numbers positive or negative, decimal or integer- therefore x is any number

44
Q

How do you find the area of a triangle?

A
  1. 5 x a x b x Sin (C)

0. 5 x base x height

45
Q

What is a unit circle?

A

A circle with a radius of 1 unit

46
Q

How do you find the value of a sin, cos, tan 960?

A

Use the cast diagram but first take away 360 until you are left with a value which is between 0 and 360 so that it minimises time takes and chances of error of going around and around the CAST diagram. Therefore you would use the CAST diagram to find the value of 240 degrees which is equal to the value of 960 degrees.

47
Q

What is the CAST diagram and how do you use it?

A

Four quadrants are split into CAST starting with C in the bottom right
C- cos only positive
A- all positive (sin, cos, tan)
S- sin only positive
T- tan only positive
You always you go anti-clockwise for positive angles and clockwise for negative angles

48
Q

What are the trigonometric identities?

A

sin2x + cos2x = 1

tan x = sin x / cos x

49
Q

How do your prove trigonometric identities?

A

Draw triangle from positive x-axis of a circle and label adjacent side as X, opposite side as Y and hypotenuse as 1. Then use the SohCahToa to find values in terms of X & Y SO sin x = y and cos x = x and tan x = y/x. As the equation of a circle is x2 + y2 = r2, sin2x + cos2x = 1 & tan = y/x = tan x = sin x / cos x

50
Q

What are the inverse functions of sin, cos & tan called?

A

arcsin, arcos, arctan

51
Q

What is the resultant vector?

A

The sum of 2 or more vectors

52
Q

What is a unit vector?

A

Vector of length 1

53
Q

How do you find the magnitude of a vector?

A

Add the i and j and square root the answer- pythagoras

54
Q

How can you write a vector?

A

Column/position vector form (x/y)

xi + yj

55
Q

How do you find the gradient of any point on a curve?

A

Find the gradient of the tangent to that point on the curve

56
Q

What is 1^-368

A

1

1^x where x is any number = 1

57
Q

Find the turning point of the equation πŸ“ βˆ’ πŸ‘(𝒙 βˆ’ πŸ’)^𝟐 = 𝟎

A

The turning point (maximum) will be at (4, 5)

Notice that the turning point is a maximum because the π‘₯^2 term is βž–

58
Q

Solve the equation 𝒙^4 + πŸ‘π’™^2 + 𝟐 = 𝟎

A

The equation could be rewritten as (𝒙^2)^𝟐 + πŸ‘π’™^2 + 𝟐 = 𝟎
So you can let y=π‘₯^2 and you have: 𝑦^2 +3𝑦+2=0
Now you can solve by factorising

59
Q

Solve the equation 𝒙^6 + πŸ“π’™^3 + πŸ” = 𝟎

A

The equation could be rewritten as (𝒙^3)^2+ πŸ“π’™^3 + 𝟐 = 𝟎
So you can let y=π‘₯^3 and you have: 𝑦^2 + 5𝑦 + 6=0
Now you can solve by factorising

60
Q

What MUST you remember with linear inequalities?

A

If you βœ–οΈ or βž— by a βž– number you need to reverse the inequality sign

61
Q

What MUST you remember with logs and inequalities?

A

Logs between 0 and 1 with any base number are βž– and therefore when βž— or βœ–οΈ by them you MUST reverse the sign

62
Q

Sketch the inequality: π’š < 𝒙 βˆ’ πŸ’

A

In this case you want the area below the line because 𝑦 is less than (

63
Q

What is the equation of a straight line/tangent?

A

𝑦 βˆ’ 𝑦1 = π‘š(π‘₯ βˆ’ π‘₯1)

64
Q

What are 3 properties of circles that you need to remember?

A

1) The angle in a semi-circle is a right angle
2) If you draw a line from the centre of the circle, perpendicular to a chord, then the line will bisect the chord (halfway point of the chord)
3) Any tangent to a circle is perpendicular to the radius at the point where it touches the circle

65
Q

What is the binomial probability equation and what does it mean?

A 
𝑷(𝑿) = 𝒏π‘ͺ𝒙𝒑^(x)q^(nβˆ’x)
Where: 
𝑛 = number of trials
π‘₯ = number of successes
𝑝 = probability of success
π‘ž = probability of failure (i.e. (1 βˆ’ 𝑝))
66
Q

How would you solve the following:

If you throw a die 8 times what is the probability of throwing three 6s?

A 
𝑛 = 8
π‘₯ = 3 (we want 3 6s)
𝑝 = 1/6 (probability of throwing a 6)
π‘ž = 5/6 (probability of NOT throwing a 6)
𝑃(3 6𝑠) = 8𝐢3 (1/6)^3 (5/6)^5 = 0.104
67
Q

Differentiate π’š = 𝒙^2 + πŸ’π’™ from first principles

A

𝑦 = π‘₯^2 + 4π‘₯
Lim as h->0 = [(π‘₯+h)^2 +4(π‘₯+h)βˆ’π‘₯^2 - 4π‘₯] / [h]
… in first principles formula f(x+h) = (π‘₯+h)^2 +4(π‘₯+h)
AND f(x) = π‘₯^2 - 4π‘₯
… as h β†’ 0 we have
= [2xh + (h)^2 + 4h] / [h]
= 2π‘₯ + h + 4 = 2x + 4

68
Q

What does the second derivative tell you?

A

Tells you where the stationary point(s) is/are
If d𝑦 > 0 and it’s a minimum (positive is minimum)
If d𝑦 < 0 and it’s a maximum (negative is maximum)

69
Q

What MUST you remember when integrating?

A

INCLUDE C

70
Q

What is the 1 exception to including the C when integrating?

A

When definite integrals with limits

71
Q

What must you remember with integration with limits?

A

βž• answer = area above π‘₯-axis
βž– answer = area below π‘₯-axis
IGNORE βž– when considering area
If there is mixture (above and below)- find each area separately and add areas (ignoring βž– sign)

72
Q

What does |π‘Ž| mean?

A

Magnitude of a- note ALSO means modulus- ALWAYS POSITIVE- magnitude scalar quantity … ALWAYS positive too

73
Q

How can you write vectors?

A

1) Component form

2) Magnitude, direction form

74
Q

What does component form look like?

A

1) Use of i and j

75
Q

What does magnitude direction form look like?

A

(Magnitude of vector, angle of vector made with horizontal)

E.g. (4, 40) … = Turn 40Β° from horizontal and draw line with length of 4

76
Q

How do you find the resultant vector?

A

Add the vectors involved

77
Q

What must you remember with position vectors?

A

βƒ—βƒ—βƒ—βƒ—βƒ—βƒ— βƒ—βƒ—βƒ—βƒ—βƒ—βƒ— 𝑨𝑢 = βˆ’π‘Άπ‘¨
βƒ—βƒ—βƒ—βƒ—βƒ—βƒ— βƒ—βƒ—βƒ—βƒ—βƒ—βƒ— ⃗⃗⃗⃗⃗⃗𝑨𝑩 = 𝑢𝑩 βˆ’ 𝑢𝑨
βƒ—βƒ—βƒ—βƒ—βƒ—βƒ—βƒ— βƒ—βƒ—βƒ—βƒ—βƒ—βƒ— 𝑢𝑴 (midpoint) = 𝑢𝑨 + 1/𝟐 𝑨𝑩

(IMAGINE ARROWS ABOVE ALL)

78
Q

How do you find the distance between 2 points?

A

Pythagoras

|𝑨𝑩|=√(π’ŠπŸ βˆ’π’ŠπŸ)^𝟐 +(π’‹πŸ βˆ’π’‹πŸ)^𝟐

79
Q

Find the distance between the points 𝑨 and 𝑩 given the vectors |𝑂𝐴| = 3π’Š βˆ’ 𝒋 and |𝑂𝐡| = 5π’Š + 2𝒋

A 
|𝐴𝐡|=√(𝑖2 βˆ’π‘–1)^2 +(𝑗2 βˆ’π‘—1)^2
βƒ—βƒ—βƒ—βƒ—βƒ—
|𝐴𝐡|=√(5βˆ’3)^2 +(2βˆ’βˆ’1)^2
βƒ—βƒ—βƒ—βƒ—βƒ—
|𝐴𝐡| = √(2)^2 + (3)^2
βƒ—βƒ—βƒ—βƒ—βƒ—
|𝐴𝐡| = √13
80
Q

What does the a^x graph look like?

A
  • upwards sloping graph- same shape as e^x
  • always crosses the 𝑦-axis at1(π‘Ž^0 =1)
  • the π‘₯-axis is an asymptote as you can never get a y- value of 0 (π‘Ž^x β‰  0)
    SEE MATHS WORD DOC CALLED GRAPHS TO KNOW
81
Q

What does the e^x graph look like?

A
  • upwards sloping graph- same shape as a^x as you have just replaced π‘Ž with e
    SEE MATHS WORD DOC CALLED GRAPHS TO KNOW
83
Q

What do you do if you can’t remember what a graph looks like or what its shape is?

A
  • Sub in values into its equation to plot a few points
  • NOTE- math error typically means asymptote
  • SEE MATHS WORD DOC CALLED GRAPHS TO KNOW
84
Q

What is the answer to: 2^[log(base 2) (5)]and why?

A

5- because 2 and 2^log(base 2) cancel each other out

85
Q

What is the answer to: log(base 2) (2^5) and why?

A

5- because log(base 2) and 2 cancel out

86
Q

What is the answer to: e^ln(7) and why?

A

7- because e and ln cancel out

87
Q

What is the answer to: ln(𝑒^7) and why?

A

7- because e and ln cancel out

88
Q

What does the graph of ln(x) look like?

A
  • increasing graph but a decreasing gradient
  • always crosses the π‘₯-axis at 1 (ln1 = 0)
  • the 𝑦-axis is an asymptote as you cannot get an answers for ln(0) (try it on your calculator, you will get an error - you can’t raise e to any power and get the answer 0)
    SEE MATHS WORD DOC CALLED GRAPHS TO KNOW
89
Q

What are the laws of logs?

A

1) π₯𝐨𝐠(𝒙) + π₯𝐨𝐠(π’š) = π₯𝐨𝐠(π’™π’š)
2) π₯𝐨𝐠(𝒙) βˆ’ π₯𝐨𝐠(𝐲) = π₯𝐨𝐠 (𝒙/π’š)
3) π₯𝐨𝐠(π’™π’Œ) = π’Œπ₯𝐨𝐠(𝒙)
4) π₯𝐨𝐠(𝟏) = 𝟎
5) log(less than 1) = negative βž–

90
Q

What MUST you remember about logs that are less than 1?

A

They are negative βž–

… with inequalities MAKES SURE YOU SWITCH THE SIGH WHEN DIVIDING βž— OR MULTIPLYING βœ–οΈ BY THEM

91
Q

Solve the equation: πŸ‘^(π’™βˆ’πŸ“) = 𝟐

A 
log 3^(π‘₯βˆ’5) = log 2
(π‘₯ βˆ’ 5)log3 = log 2
(π‘₯ βˆ’ 5) = (log2)/(log3)
(π‘₯ βˆ’ 5) = 0.6309
π‘₯ = 0.6309 + 5 
𝒙 = πŸ“. πŸ”πŸ‘πŸŽπŸ—
92
Q

How do you plot π’š = π’Œπ’™^𝒏 and … find k and n?

A 
log(𝑦) = log(π‘˜π‘₯^𝑛)
log(𝑦) = log(π‘˜) + log(π‘₯^𝑛)
log(𝑦) = log(π‘˜) + 𝑛log(π‘₯)
log(𝑦) = 𝑛 log(π‘₯) + log(π‘˜)
Y = MX + C
... gradient = 𝑛 
... intercept = log k
93
Q

How do you plot π’š=𝒂𝒃^𝒙 and … find a and b?

A 
log(𝑦) = log(π‘Žπ‘^π‘₯)
log(𝑦) = log(π‘Ž) + log(𝑏^π‘₯) 
log(𝑦) = log(π‘Ž) + π‘₯log(𝑏) 
log(𝑦) = π‘₯log(𝑏) + log(π‘Ž)
Y = MX + C
... gradient = log𝑏 
... intercept = logπ‘Ž

Maths- A-level (6 decks)

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