Brainscape's Knowledge GenomeTM


Top Flashcards (Certified)
All Flashcards

A Level Maths > Pure Year 2 > Flashcards

Pure Year 2 Flashcards

(124 cards)

Start Studying
1
Q

What is important to remember about sectors and small angle approximations?

A

Angles in radians NOT degrees

Shift
Setup
2
2

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Arc length formula

A

r{theta}

theta must be in radians

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Sector area formula

A

0.5r²{theta}

theta must be in radians

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Area of a segment formula

A

(0.5r²{theta} - 0.5r²sin{theta}

theta must be in radians

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

When theta is small, 9{theta}²-2{theta}+1 is what?

A

1

theta is small so approximately 0, meaning that the first 2 terms become 0 so just leaves 1

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What is a mapping?

A

Transforms one set of numbers into a different set of numbers
It is a function if every input has a distinct output

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Is a one-to-one mapping a function?

A

Yes

Each input has one distinct output

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

Is a many-to-one mapping a function?

A

Yes

Each input has a distinct output

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Is a one-to-many mapping a function?

A

No

Not every input has a distinct output

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Is y=1/x a function

A

No

No value at 0, doesn’t map anywhere

Can be made a function by restricting domain to x≠0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Domain…

A

The set of X values for which the function is valid

Writen as x=a

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

Range…

A

The set of Y values that the function can take

Written as a<=f(x)<=b

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

How do you deal with partial fractions?

A

Multiply up

Knock out brackets by substituting values in to make some coefficients 0

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

What is a piecewise-defined funtion?

A

A function which consists of more than one part

When drawing:
●=less/more than or equal to
○=less/more than

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

What is an inverse function?

A

The mathematical opposite of the original function

Only exists for one-to-one functions

Y becomes X, X becomes Y, reflected in line Y=X

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

How do you get an inverse function?

A

Replace f(x) with y
Swap y and x
Rearrange to make x the subject
Swap y for f(x)

Check f(x) and not any other function notation e.g g(x)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

When do inverse functions exist

A

One-to-one functions

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

What is the domain of an inverse?

A

The range of the ordinary function

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

What is the range of an inverse function?

A

The same as the domain of the normal function

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

If f(x)=sinx, the inverse function can be written as…

A

f-¹(x)=sin-¹x=arcsinx

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

Sketch the graph of y=arcsinx

State the range and domain

A

X Intercept: 0
Y Intercept: None

Coordinates: (-1,-90) (0,0) (1,90)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

If f(x)=cosx, the inverse function can be written as…

A

f-¹(x)=cos-¹x=arccosx

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

if f(x)=tank, the inverse function can be written as…

A

f-¹(x)=tan-¹x=arctan

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

What is the modulus of a number?

A

It is the non negative numerical value
Also known as the absolute value
Denoted as |x|

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
25
What is the difference between y=f(|x|) and y=|f(x)|?
y=f(|x|) will not have a negative value input, meaning that negative values of x will become the positive values of x. The first quadrant is reflected in the y axis into the second quadrant and the fourth into third y=|f(x)| will not go below the x axis since y cannot be negative. All negative in the third quadrant will reflect up into the second, and all negative in the fourth reflected up into the first
26
Sketch the graph of y=arccosx | State the domain and range
X intercept: 1 Y intercept: 90° Key coordinates: (-1,180) (0,90) (1,180) Asymptotes: ? Reflected in the y=x axis
27
Sketch the graph of y=arctanx | State the range and domain
Online image
28
How do you access modulus feature on calculator?
Shift | Abs (absolute)
29
What's important to remember once you have solutions for a modulus function?
Sub back in to check no phantom solutions (signs not equal once put back in on either side of equation)
30
How do you deal with solving modulus equations?
Draw the graphs look for intersections Make modulus positive, solve, check for phantom solutions Make modulus negative, solve, check for phantom solutions
31
What are reciprocate functions called?
Cosecant=Cosec Secant=Sec Cotangent=Cot
32
What is the rule to remember reciprocal functions?
Look at the third letter for each, will match what is on the bottom
33
Formula for Cosecant?
Cosec(x)=1/sin(x)
34
What is the formula for Secant?
Sec(x)=1/cos(x)
35
Formula for Cotangent?
Cot(x)=1/tan(x) OR cos(x)/sin(x)
36
Sketch the graph of y=cosec(x) in range (range two pie to negative two pie)
Image online
37
Sketch the graph of y=sec(x) in range negative two pi and two pi
Get an image online
38
Sketch the graph of y=cot(x) in the range negative two pi to two pi
Get an image from online
39
Method for solving equation involving reciprocal functions?
Change Flip Solve
40
Solve sec(x)=0
1/cos(x)=0 cos(x)=1/0 1/0 = no solutions
41
Solve cosec(x)=0
1/sin(x)=0 sin(x)=1/0 1/0 = no solutions
42
Solve cot(x)=0 in range 0° to 360°
1/tan(x)=0 tan(x)=1/0 1/0 = asymptotes, meaning 90°, 270°
43
2 trig functions from year 12
tan(x)=sin(x)/cos(x) sin²(x)+cos²(x)=1
44
2 trig identities from year 13, derived from sin²(x)+cos²(x)=1
1+tan²(x)=sec²(x) 1 with a tan is sexy 1+cot²(x)=cosec²(x) 1 in a cot is cosy
45
How can you derive the two trig identities from sin²(x)+cos²(x)
Divide by cos Divide by sin
46
Where do the addition formula appear?
Page 6 of formula booklet
47
Double angle formula for sin2x
2sinxcosx
48
Double angle formulas for cos2x
cos²x-sin²x 2cos²x-1 1-2sin²x
49
Double angle forumla for tan2x
2tanx ______ 1-tan²x
50
When is dy/dx used
Gradient or rate of change Tangent or normal Increasing or decreasing Stationary point or turning point
51
When is d²y/dx² used
Nature of a stationary point Checking if minimum or maximum Finding minimum or maximum
52
Differentiate tan(x)
sec²(x)
53
Differentiate x^n
nx^(x-1)
54
Differentiate e^x
e^x
55
Differentiate 4e^2x
8e^2x
56
Differentiate e^sin(x)
cos(x)e^sin(x)
57
Differentiate ln(x)
1/x
58
Differentiate ln(x²+1)
2x/(x^2+1)
59
Differentiate a^x
a^xln(a)
60
Differentiate 2^3x
3(2^3xln(2))
61
Differentiate sin, cos, -sin, -cos
cos -sin -cos sin
62
Bracket rule for differentiating (f(x))^n
n(f(x))^n-1
63
Steps for product and quotient rule
Identify u and v Differentiate u and v Substitute into formula
64
Product rule formula
u'v+uv'
65
Quotient rule formula
(u'v-uv')/v^2
66
dx/dy ...
reciprocal of dy/dx
67
When is a function concave
If and only if f''(x)<0 for every x-value in given interval
68
When is a function convex
If and only if f''(x)>0 for every x-value in given interval
69
What is a point of inflection
Point at which a curve changes from being concave to convex or vice versa f''(x) changes sign Does not have to be a stationary point
70
How do you prove a point of inflection
d^y/dx^2 Equal to 0 Opposite signs either side very close to point
71
If y=u/v Then dy/dx=...
(u'v-uv')/v²
72
How do you put something in harmonic form
``` Rewrite the RHS using addition formula Compare coefficients Find tan a Use pythagoras to find R Write out your function in full ```
73
How do you find exact values
Draw a triangle Lable OAH Check signs
74
Signs for sin(x) from 0 to 360
P P N N
75
Signs for cos(x) going from 0 to 360
P N N P
76
Signs for tan(x) going from 0 to 360
P N P N
77
Explain a cast diagram
C bottom right (cos) A top right (all) S top left (sin) T bottom left (tan) Starting at A, draw arrow anticlockwise Letter in each quadrant is what is positive for that range
78
Harmonic motion asin(x)+bcos(x)
R sin(x+a)
79
acos(x)+bsin(x)
R cos(x-a)
80
asin(x)-bcos(x)
R sin(x-a)
81
acos(x) - bsin(x)
R cos(x+a)
82
How do you integrate e^x
e^x + c
83
How do you integrate e^4x
0.25 e^4x + c
84
How do you integrate e^0.25x
4e^0.25x + c
85
How do you integrate Sin Cos -Sin -Cos
-Cos Sin Cos -Sin
86
How do you integrate sin(2x)
-0.5cos(2x) + c
87
How do you integrate cos(0.5x)
2sin(0.5x) + c
88
How do you integrate a^x
a^x/ln(a)
89
How do you integrate 2^x
2^x/ln(2) + c
90
Integrate 2^3x
2^3x/3ln(2) + c
91
Reverse Bracket Rule
(f(x))^n = (f(x))^n+1/(n+1)(f'(x))
92
Integrate (2x+1)⁵
(2x+1)⁶/12 + c
93
How can you integrate a complex fraction using Y12 methods
Split the fraction into a polynomial by multiplying up by the denominator ^-1
94
When do you use ln to integrate a fraction
If the numerator is a multiple of the derivative of the denominator
95
Integrate 3/(1+3x)
3ln(1+3x) ÷ 3 = ln(x) + c
96
Integrate 3/x
3ln(x) + c
97
Integrate x/(3x²+1)
xln(3x²+1)/6x + c = ln(3x²+1)/6 + c
98
When do you use algebraic division in integration
If you have an improper fraction with a numerator of same power or higher than the denominator
99
When do you use partial fractions for integration
If the denominator has a higher power than the numerator | AND the denominator can be factories I multiple brackets
100
How do you simplify integrating trig
Use identities Use double angles Check if then in the book
101
Relationship between product rule and integration by parts
Opposites
102
Integration by parts formula
Page 7 uv - ∫ vu' dx
103
Acronym for labelling u then v in integration by parts
Logs Algebra Trig Exponentials
104
Integrate ln(x)
1/x + c
105
Integrate (lnx)^2
u=(lnx)^2 v=x xln(x)^2 - 2xln(x) + 2x + c
106
Integration by substitution method
``` Chose the substitution u Differentiate du/dx Rearrange for dx Substitute, replacing dx and cancelling Sort limits if applicable Simplify Integrate Substitute back out if necessary ```
107
What is the Reimann Sum
Adding together rectangles to estimate the area under a curve More rectangles means more accurate the estimate ALWAYS DO RECTANGLES UNDER CURVE, DO NOT CROSS LINE
108
What happens during the Riemann Sum as the number of rectangles increases
The value for area approaches a limit which is the actual value of the integral
109
How do you prove that increasing number of triangles is more accurate for Riemann Sum
Draw a curve in double positive axis (1st quadrant) Draw one rectangle Repeat with 2 and 3 Conclude with; more triangles means more of the area under the graph is filled by rectangles
110
Definition of a cartesian equation
Give a direct relationship between x and y eg y=(x+1)^2
111
Definition of a parametric equation
x and y are defined in terms of a third variable known as the parameter eg x=t-1 and y=t^2
112
How do you go from a parametric equation to a cartesian equation without trig
Rearrange one to get t | Substitute t into the other equation
113
How do you go from a parametric equation to a cartesian equation with trig
Chose an appropriate identity Rearrange the parametric equation Substitute into identity
114
For a parametric equation x=p(t) and y=q(t) with cartesian equation y=f(x) Domain of f(x) Range of f(x)
``` Domain = Range of p(t) Range = Range of q(t) ```
115
Rule for differentiating parametric equations
(dy/dt)/(dx/dt) = (dy/dt) x (dt/dx) = dy/dx = y'/x'
116
Formula to integrate parametric equations
∫ y dx/dt dt
117
How do you integrate parametric equations
Swap limits from x to t Get dx/dt Plug into formula
118
Segment
Area created by joining two ends of an arc with a straight line
119
Where are small angle formulas for trig found
Page 6 formula booklet
120
Integrating trig
Swap for an identity so easier | Cant have extra x on bottom
121
Integrating trig with sin or cos squared
Use double angle
122
Integrate lnx
by parts u=lnx v=x xlnx-x+c
123
Continous function
No vertical asymptote | Drawn without taking a pen off the page
124
When will Newton raphson fail
Stationary point chosen to start with Derivative will be 0 Can't divide by zero since its invalid Tangent to the x axis means it won't intersect the x axis Start point close to stationary point Gradient of tangent is so small it intercepts the x axis a long way from x0

A Level Maths (6 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2025 Bold Learning Solutions. Terms and Conditions