Pure Maths Flashcards
(114 cards)
1
Q
How would you find the midpoint of 2 coordinates?
A
xm = (x1 + x2) / 2
ym = (y1 + y2) / 2
2
Q
How would you find the length between 2 coordinates?
A
the root of (x2 - x1)^2 + (y2 - y1)^2
3
Q
What is the formula for the equation of a line?
A
y - y1 = m(x - x1)
4
Q
How do you know if 2 lines are perpendicular?
A
Their gradients multiply to equal -1
5
Q
What is a perpendicular bisector?
A
A line passing through the midpoint of AB at a right angle to AB
6
Q
What is the discriminant?
What does it mean?
A
b^2 - 4ac of quadratic formula
If discriminant < 0, there are no real roots.
If discriminant = 0, there is one repeated real root
If discriminant > 0, there are two distinct real roots
7
Q
How would you find x in the equation:
b^x = a?
A
x = logb(a)
8
Q
Give the laws of logarithms
A
- logb(b^x) = x
- logb(1) = 0
- logb(b) = 1
- logc(a) + logc(b) = logc(ab)
- logc(a) - logc(b) = logc(a/b)
- nlog(a) = log(a^n)
- loga(b) = logc(b) / logc(a)
9
Q
Where would you draw the asymptote and intercepts on the graphs:
- y = 3^x ?
- y = 2^x +2 ?
A
- asymptote at y = 0, intercept at (0,1)
- asymptote at y = 2, intercept at (0,3)
10
Q
Where would you draw the asymptote and intercepts on the graph:
- y = ln(x) ?
A
- asymptote at x = 0, intercept at (1,0)
11
Q
Where is the intercept of the graph:
- y = root ?
Where does it exist on the axes?
A
Intercept at (0,0).
Only exists in the positive quadrant
12
Q
What does it mean if dy/dx is > 0? < 0?
A
- When dy/dx > 0, graph is increasing
- When dy/dx < 0, graph is decreasing
13
Q
How would you draw the gradient function of a curve?
A
Draw a line with the polynomial order -1 of that of the original line.
The line crosses the x axis at the x values of the original turning points
14
Q
What is the order/degree of a polynomial?
A
The highest power in the expression
15
Q
Complete:
(x^3 + 2x^2 + 3x +2) / (x+1)
by inspection
A
x^2 + x + 2
16
Q
What is the factor theorem?
A
If (x-a) is a factor of f(x), f(a) = 0
If (bx-a) is a factor of f(x), f(a/b) = 0
17
Q
What is the remainder theorem?
A
When f(x) is divided by (bx-a), the remainder is f(a/b)
18
Q
What does a triple root mean for the shape of the graph?
A
The gradient of the line at that point is 0
19
Q
Where would you draw the asymptotes on the graphs:
- y = 1/x ?
- y = 1/x^2 ?
What is the difference in their shapes?
A
Both at y = 0, x = 0
Both have one part of the curve in the positive quadrant, but y = 1/x is reflected in the line y = -x, where y = 1/x^2 is reflected in the line x = 0
20
Q
How would you describe the translations:
- y = f(x-a) ?
- y = f(x+a) ?
And how would you find the new equation?
A
- Translation by vector (a 0), replace x with (x-a)
- Translation by vector (-a 0), replace x with (x+a)
21
Q
How would you describe the translations:
- y = f(x) + a ?
- y = f(x) - a ?
And how would you find the new equation?
A
- Translation by vector (0 a), replace y with (y-a)
- Translation by vector (0 -a), replace y with (y+a)
22
Q
How would you describe the translations:
- y = -f(x)
- y = f(-x)
And how would you find the new equation?
A
- Reflection in x axis, replace y with -y
- Reflection in y axis, replace x with -x
23
Q
How would you describe the translations:
- y = f(1/a x)
- y = f(ax)
And how would you find the new equation?
A
- Stretch parallel to x axis by scale factor a, replace x with (1/a)x
- Stretch parallel to x axis by scale factor 1/a, replace x with ax
24
Q
How would you describe the translations:
- y = af(x)
- y = (1/a)f(x)
And how would you find the new equation?
A
- Stretch parallel to y axis by scale factor a, replace y with (1/a)y
- Stretch parallel to y axis by scale factor 1/a, replace y with ay
25
What is the equation of a circle?
(x-a)^2 + (y-b)^2 = r^2
where the centre of the circle is (a,b), and the radius is r
26
How would you determine if a point is on the outside, or inside a circle?
Substitute the coordinates into the circle equation:
- if result > r, the point sits outside
- if result = r, the point lies on the circle
- if result < r, the points sits inside
27
How would you find the centre of the circle from 3 points lying on its edge?
The intersection of the perpendicular bisectors of 3 points lies on the centre of the circle
28
What are the circle theorems you need to know?
- The perpendicular bisector of a chord will go through the centre of the circle
- The radius meets a tangent at 90 degrees
- Chords on the ends of the diameter meet at 90 degrees
29
How would you determine how many times a line touches a circle?
Let the equations equal each other and simplify to a quadratic.
- If b^2 - 4ac < 0, they don't intersect
- if b^2 - 4ac = 0, they meet once
- if b^2 - 4ac > 0, they meet twice
30
What are the sets of numbers you need to know?
- N = natural numbers = positive integers
- Z = integers
- Q = rational numbers
- R = real numbers
- C = complex numbers
- I = imaginary numbers = multiples of i
31
How would you write 2 < x <= 4 in set notation?
{x : xeR, 2 < x <= 4}
32
How would you write 2 < x <= 4 in interval notation?
xe (2,4]
such that () = not inclusive, [] = inclusive
33
How would you write x <= -1, x > 3 in set notation?
{x: xeR, x <= -1} u {x: xeR, x > 3}
34
How would you write x <= -1, x > 3 in interval notation?
xe(-infinity, -1] u (3, infinity)
35
How would you divide or times by x in an inequality and why?
Must divide or multiply by an even power of x, as x could be negative and flip the sign
36
How are the first and second derivatives written?
first = f'(x), dy/dx
second = f''(x), d^2y/dx^2
37
How would you find the second derivative?
Differentiate the first derivative
38
What does the second derivative tell you?
The rotation of the curve:
- Positive = anticlockwise, convex
- Negative = clockwise, concave
- 0 = point of inflection
The acceleration of the curve:
- If at a turning point, positive means a minimum point, negative means a local maximum
39
What is a stationary point and what are the different types?
A point where the gradient = 0.
- Local maximum
- Local minimum
- Stationary point of inflection
40
What is a stationary point of inflection?
A point on the curve where rotation has changed, or concavity is changing. Not necessarily a turning point
41
How would you generally use optimisation?
- Find an equation for the problem
- Differentiate and solve dy/dx = 0 to find the turning points
- Find the max/min depending on the question using f''(x) - this is the answer
42
How would you find where the curve is concave / convex?
- Solve f''(x) > 0 for convex
- Solve f''(x) < 0 for concave
43
How would you find where the curve is increasing / decreasing?
- Solve f'(x) > 0 for increasing
- Solve f'(x) < 0 for decreasing
44
How would you find the non-stationary points of inflection of a curve?
- Solve f''(x) = 0 for the possible points of inflection
- Substitute values left and right of each x coordinate into f''(x). If the signs are the same, it is a point of inflection
45
How would you find and classify the stationary points of a curve?
- Solve f'(x) = 0
- Find f''(x) and substitute in the x coordinates. If it is <0, the point is a local maximum, >0 means a minimum
- If it =0, substitute the left and right values of the x coordinate into f'(x).
- -ve to +ve is a minimum, +ve to -ve is a maximum, same sign is a stationary point of inflection
46
How would you find the area between a curve and the x axis?
Integrate the equation. Substitute in the upper limit, and then the lower limit. Do the upper limit - lower limit
47
What is the trapezium rule and what is it used for?
Area = 0.5h [first + last + 2(rest)]
where h = distance between x coordinates (must be constant), first/last/rest refer to the heights of the bars (y coordinates)
Used for approximating the area under curves
48
When would the trapezium rule be an overestimate?
Underestimate?
If the curve is convex (f''(x) > 0), the trapezium rule is an overestimate.
If the curve is concave (f''(x) < 0), the trapezium rule is an underestimate
49
What does a=> b mean?
What does a <=> b mean?
- If a is true, then b is true
- a is true if and only if b is true
50
How would you prove something by counter-example?
Find an example that doesn't fit
51
How would you prove something by deduction?
Use n and m to represent any two integers, form an equation and solve
52
How would you prove something by exhaustion?
Check that every possible option fits the rule
53
How do you convert from degrees to radians?
Radians to degrees?
- Divide by 180 then multiply by pi
- Multiply by 180 and divide by pi
54
What is the arc length of a sector of a circle, angle x degrees in radians?
arc length = xr
55
What is the area of a sector of a circle, angle x degrees in radians?
area = 0.5r^2x
56
How would you find the angle x between two vectors, a and b?
cos(x) = (a.b) / (|a||b|)
Where a.b is is each the sum of the values of a and b multiplied together, and |a| is the length of a found using pythagoras
57
How would you find the next angles from:
- sin?
- cos?
- tan?
- sin: ans, 180/pi - ans, then + 360/2pi to each
- cos: ans, 360/2pi - ans, then + 360/2pi to each
- tan: ans, 180/pi + ans
58
What is:
- cosec?
- sec?
- cot?
- cosec = 1 / sin
- sec = 1 / cos
- cot = 1 / tan
59
What are the two types of sequences (not arithmetic / geometric)?
- Position-to-term (deductive) - where you have a nth formula
- Term-to-term (inductive) - where you have a rule of how to get from one term to the next
60
Give an example of an inductive definition of a sequence
u1 = 3 (first term)
u(n+1) = 2u(n) (recurrence relation - how you get from one term to the next)
61
How would you find the limit of a sequence, if it exists?
If there is a limit, as n tends to infinity, u(n+1) = u(n), which tends to L
To find L, replace both u(n+1) and u(n) with L in the definition, and solve for L
62
What is a converging sequence?
The sequence approaches some limit
63
What is a diverging sequence?
The sequence does not converge
64
What is a periodic sequence?
A sequence where the same terms repeat over and over. The number of repeated terms is the period
65
What is a decreasing sequence?
A sequence where u(n+1) < u(n) for all n
66
What is an increasing sequence?
A sequence where u(n+1) > u(n) for all n
67
What is an arithmetic sequence?
Where the same amount is added to each term each time
68
What is the formula for the nth term of an arithmetic sequence?
u(n) = a + (n-1)d
where a is the first term and d is the common difference
69
What are the two formulae for the sum of the first n terms in an arithmetic series?
Sn = 0.5n(a+l)
where a is the first term and l is the last term
Sn = 0.5n[2a + (n-1)d]
where a is the first term and d is the common difference
70
What is the difference between a sequence and a series?
A sequence is just the list of terms, where a series is the sum of the terms
71
What is a geometric sequence?
A sequence where you get from one term to the next by multiplying by a constant called the common ratio. Causes exponential growth / decay
72
What is the formula for the nth term of a geometric sequence?
u(n) = ar^(n-1)
73
What is the formula for the sum of the first n terms in a geometric series?
Sn = [a(1-r^n)] / (1-r)
74
What is the condition for a geometric sequence to sum to infinity?
What is the formula for the sum to infinity of a geometric sequence?
Only sums to infinity if r^n converges to 0, which happens when -1
75
What is a function?
A bridge between sets where for any input there can only be one output
76
What are the different types of functions?
- One-to-one - one y value gives one x value. Function has no turning points
- Many-to-one - one y value gives many x values. Is a function
- One-to-many - one x value gives many y values. Not a function
- Many-to-many - many x values give many y values. Not a function
77
What is the domain of a function?
The set of all possible x values that can be inputted
78
What is the range of a function?
All the possible values of f(x) that can be output
79
What are the different ways you can write composite functions?
f(g(x)) is:
- fg(x)
- fog(x)
ff(x) is f^2(x)
80
How would you find the domain of a composite function?
Must first consider the domain of the first (inner) function, then find the domain of the composite function, and combine the two
81
What are even functions?
What are odd functions?
Functions symmetrical along the y axis. f(-x) = f(x)
Functions which have a rotational symmetry of 180 degrees. f(-x) = -f(x)
82
How would you determine if a function is odd or even?
Find f(-x), if it equals f(x), it is even, if it equals -f(x), it is odd
83
What is a periodic function?
A function that repeats its values at regular intervals
84
What is the order of graph transformations for:
y = f(x) => y = af(bx+c)+d
cbad.
If one transformation is inside the function and one is outside, the order doesn't matter
85
What type of function can have inverse functions?
One-to-one functions, any others need to restrict the domain first
86
How would you translate f(x) to f^-1(x)?
Reflect in the line y = x
87
How would you find the inverse function?
Rearrange the equation to make x the subject, then call x f^-1(x) and call f(x) x
88
How would you find the domain and range of an inverse function?
The domain is the range of the original, and the range is the domain of the original
89
What is a self-inverse function?
ff(x) = x
e.g, f(x) = 1/x, so ff(x) = f^-1(x)
90
What is a modulus function?
|x|
For any real number input, you always get the positive value as the output
Transformations work in the same way as if || were () of a normal function
91
How would you go about solving a modulus equation?
Must sketch the graph, then find which portions of the modulus function intersects with the other expression. Can split the modulus into positive and negative gradient portions to solve each equation separately
92
How would you write -n < x < n as a modulus inequality?
|x| < n
93
How would you write a < x < b as a modulus inequality?
Find the average of a and b, then take this number from each term in the inequality.
You will get n < x < n, which can be written as |x| < n
The same works for x < a, x > b
94
What is the inverse of:
- sin?
- cos?
- tan?
- arcsin
- arccos
- arctan
95
What would you restrict the domain of sin to to find arcsin?
-pi/2 < x < pi/2
96
What would you restrict the domain of cos to to find arccos?
0 < x < pi/2
97
What would you restrict the domain of tan to to find arctan?
-pi/2 < x < pi/2
98
Where would you draw the asymptotes on a cosec graph?
0, pi, 2pi
99
Where would you draw the asymptotes on a sec graph?
pi/2, 3pi/2
100
Where would you draw the asymptotes on a cot graph?
pi/2, 3pi/2
101
What are the trig identities you need to know?
- sin^2x cos^2x =1
- tanx = sinx / cosx
- 1 + cot^2x = cosec^2x
- tan^2x + 1 = sec^2x
102
How do you differentiate:
- sinkx?
- coskx?
- tankx?
- kcoskx
- -ksinkx
- ksec^2kx
103
How do you differentiate:
- e^kx?
- lnkx?
- ke^kx
- 1/x
104
How would you differentiate y = a^x?
lna x a^x
105
What is the chain rule and what is it used for?
Used to differentiate composite functions.
For the function fg(x):
dy/dx = g'(x)f'(g(x)
106
How would you differentiate ln(f(x)?
dy/dx = f'(x) / f(x)
107
What does sin(x +- y) equal?
sinxcosy +- cosxsiny
108
What does cos(x +- y) equal?
cosxcosy -+ sinxsiny
109
What does tan(x +- y) equal?
(tanx +- tany) / (1 -+ tanxtany)
110
What is the connected rates of change equation?
dA/dB = dA/dC x dC/dB
111
What does sin2x equal?
2sinxcosx
112
What does cos2x equal?
cos^2x - sin^2x
So:
- 1 - 2sin^2x
- 2cos^2x - 1
113
What does tan2x equal?
2tanx / (1 - tan^2x)
114
What is the product rule?
y = f(x)g(x)
dy/dx = f'(x)g(x) + g'(x)f(x)
