Pure Maths Flashcards
Pythagorean triples
4,3. 5
12,5. 13
24,7. 25
15,8. 17
Sin(A)=
Cos(90-A)
Tan θ
sin θ / cos θ
Sec x
1/cos x
Cosec x
1/sin x
Cot x
1 / tan x = cos x / sin x = cosec x / sec x
Y = sec θ (graph)
- symmetry in y-axis
- has period 360
- Asymptotes of -90, 90, 270
Y = sec θ (graph domain and range)
- domain x∈R, x≠90, 270, 450
- range y ≤-1 or y≥1
Y = cosec θ (graph)
- has period 360
- vertical asymptotes for which sin x = 0 (180, 360)
Y = cosec θ (graph domain and range)
- domain x∈R, x ≠ 0, 180, 360
- range y ≤-1 or y≥1
Y = cot θ (graph)
- has period 180
- vertical asymptotes for which tan x = 0 (0, 180, 360
Y = cot θ (graph domain and range)
- domain x∈R, x ≠ 0, 180, 360
- range y∈R,
Sec^2x
1 + tan^2 x
Cosec^2x
1 + cot^2x
Arcsin x (domain and range)
Domain -1≤ x ≤ 1
Range -90 ≤ arcsinx ≤ 90
Arccos x (domain and range)
Domain -1≤ x ≤ 1
Range 0 ≤ arccos x ≤ 180
Arctan x (domain and range)
Domain x∈R
Range -90 ≤ arctan x ≤ 90
Un (arithmetic)
A + (n-1) d
Sn (arithmetic)
N/2 (2a + (n-1) d)
Un (geometric)
Ar^(n-1)
Sn (geometric)
(A (r^(n-1)))/r-1
Sum to infinity ∞
A/1-r
|r| < 1
V=ab^t
A= initial value
B= the annual proportional decrease (-ve)/increase (+ve) in the value of the car
Y=log 9 (x+a)
(0, log 9 (x+a)) (-a+1, 0)
Discriminant
b^2-4ac >0 2
b^2-4ac = 0 1
Proof for sum of arithmetic series
Proof for sum of a geometric series
Proof for sum to infinity
Sigma notation proof
Sin (2A)
2SinACosA
Cos(2A)
Cos^2A-sin^2A
2cos^2A-1
1-2sin^2A
Tan2A
2tanA/1-tan^2A
Sin4X
2Sin2Acos2A
1-cos^4A
(1+cos^2A)(1-cos^2A)
E = km + b
K is the increase in E when m increase by 1
b is E at the beginning or the fixed charged for E
Y = f(x) + a
X, y+a
Y = f(x + a)
X-a, y
Y = af(x)
X, ay
Y = f(ax)
X/a , y
Y = -f(x)
X, -y
Y = f( - x)
-x, y
Gradient for parallel and perpendicular
Parallel = same gradient
Perpendicular = neg reciprocal
Y=kx
If x increases by 1 unit, y increases by k units
Ncr formula
N!/r! (N-r)!
Cosine rule
A^2 = b^2 + c^2 -2bc cos A
Area of triangle
1/2 ab sinC
Y = sinx
- repeat’s every 360 and crosses the x axis at (-180, 0, 180, 360)
- maximum of 1 and minimum value of -1
Y= cosx
- repeat’s every 360 and crosses the x axis at (-90, 90, 270, 450)
- maximum of 1 and minimum value of -1
Y=tanx
- repeat’s every 180 and crosses the x axis at (-180, 0, 180, 360)
- has no maximum or minimum
Unit of a vector
- size of vector is 1
- a (2
3). |a| = √3
Unit = (2/√3)
3/√3）
Collinear
Lie of the same line
Find the gradient of the tangent to the curve
- Find f’(x) = dy/dx
- Sub the given point into f’(x) = gradient
- Form equation
Find the gradient of the normal to the curve
- Find f’(x) = dy/dx
- Sub the given point into f’(x) = gradient
- negative reciprocal of gradient
- Form equation
Increasing function
If f’(x) ≥ 0
Decreasing function
If f’(x) ≤ 0
Interval of which a function is decreasing
F(x) = x^3+3x^2-9x
F’(x) = 3x^2+6x-9 ≤ 0
3(x+3)(x-1)≤0
-3≤x≤1
Find a stationary point
- Find dy/dx and set to 0 to find an x value
- sub the x value into the original equation f(x) to find y value
Finding out whether a stationary point is a local minimum or maximum or point of inflection method 1
Stationary point (a,b)
F’’(a) < 0 max
F’’(a) = 0
F’’(a) > 0
Finding out whether a stationary point is a local minimum or maximum or point of inflection method 2
Stationary point (a,b)
Consider points just below and above x and sub into f’(x)
Finding the constant of integration (c)
- integrate the function
- sub the point on curve into integrated fuction
- solve to find c
Sketching gradient functions
y=f(X) y=f’(X)
Maximum or minimum cuts the a xis
Point of inflection touches the x axis
Positive gradient above the x axis
Negative gradient below the x axis
Vertical asymptote same
Horizontal asymptote y=0
Definite integral (area)
Modulus function graphs
Y = |f(x)|
Reflected on x axis
Arc length
R theta
Area of sector
1/2 r^2 theta
Natural numbers
N
only the positive integers i.e. 1, 2, 3, 4,5,6, ………. excluding zero, fractions, decimals and negative numbers
Z
Integers
A whole number
-2, -1, 0, 1, 2, 3
Q/R
Rational numbers
Anything that can be written asa fraction of 2 integers
R
Real numbers
Anything
Null set
Nothing in set, empty
Element of a set
E
Included in set
Not an element of a set
E with strike
P
Irrational numbers
Cannot be written as a fraction of integers
Pi, sqrt (5)
Z+
Positive integers
State one limitation of the model
It is unlikely that the car’s value with decrease with age
The lorry won’t drive at the same pace throughout the whole journey
Tangent
Only touches once
Congruent
Same length
Scalene
Has 3 different lengths
What’s the shortest distance between two parallel lines
The perpendicular distance
Sin(x)cos(x)
0.5sin(2x)
Explain why the function has no inverse?
The function is many to one
8sinxcosx
4(2sinxcosx)
= 4sin2x
How to find gradient of curve?
Differentiate and sub in point
Find the exact range of values of x for which the curve is increasing.
Differentiate and set > 0
Explain relationship between gradients of points on curve
As h —> 0, 12+ 3h —> the gradient of the chord tends to the gradient of the tangent to the curve
Show that the curve has no stationary points
Differentiate
- use the Discriminant to show ≠ 0 hence the curve has no turning points
Verify the curve has a stationary point when x=4
Differentiate and sub in 4 which should = 0
Determine the nature of this stationary point
Differentiate twice and sub in x >0 is a minimum
How to find turning point of cubic.
Differentiate and factorise the quadratics
When to add c for integration?
First of all remember that you only need to include the +c if you are integrating without limits (ie you don’t have the two numbers at the top and bottom of the integration symbol). We differentiated A by differentiating each small part of it separately and adding them together at the end.
Find the minimum perimeter of the pool
Differentiate and set to 0 to find value for x and sub into equation
Increasing Interest rates
Starting value is 20,000
8% interest increase
(20,000x1,08) year 2
(20,000x1.08^2)
Value of r = 1,08
Decrease in value of car
Starting value is 20,000
8% decrease
(20,000x0.08) year 2
(20,000x0.08^2)
Value of r = 0.08
Condition for an infinite geometric series with common ratio r to be convergent
|r| < 1
Surface are of cyclinder
H= height
R= radius
2πrh+2πr2
Volume of sphere
4/3πr^3
Volume of cyclinder
πr2h
Area of rhombus
diagonal lines times together and divide by 2
Area of parrellogram
base x perpendicular height
Concave function
If the second derivative is ≤ 0
f’’(x) ≤ 0
Convex function
If the second derivative is ≥ 0
F’’(x) ≥ 0
Two odd numbers
2n+1, 2m+1
Difference of two squares
A^+b^+2ab
Cos^2+sin^2+2coxsinx (cosx+sinx)
Increase by percentage (sequence)
2100 is starting, increase by 1.2%
Find from 2017-2030 (include the 2017)
1.2/100 +1 = 1.012
2100((1.012)^14-1/1.012-1
Why can’t you use turning point, stationary point on curve (newton raphson)
The gradient is 0 = tangent never touches the x axis which is required by x axis
differentiate 2xy with respect to x
2y + 2x(dy/dx)
Rate of change of depth
Dh/dt
Vertical line gradient
M = infinity
Dy/dx = infinity
Dx/dy = 0
Horizontal line
Dy/dx = 0
Differentiate cos, sin from first principles
Rate of flow of water
Dv/dt
Parametric equation domain and range
The domain of f(x) is the range of x
The range of 𝑓(𝑥) is the range of y
Determine whether or not this iteration formula can be used to find an approximation for a, justifying your answer
- annotate the graph by drawing cobweb diagram
- “the iteration formula can be used to find an approximation for a because the cobweb spirals inwards for the cobweb diagram”
Prove by counter example: prove for all positive real values of x….
Let x = 1/8
Let x = 6
Rocket is described with equation. Show that the rocket returns to the ground between 19.3 and 19.4 seconds after launch
Sub 19.3 and 19.4 show there is a change of sign
Figure 1 is a graph of the price of a stock during a 12-hour trading window. The equation of the curve is given above. Show that the price reaches a local maximum in the interval .
Differentiate equation and then sub
Explain why (for certain equations) can the Newton Raphson method cannot be used?
Accept any reasons why the Newton-Raphson method cannot be used with x1 = 0 which refer or allude to either the stationary point or the
tangent. E.g.
- There is a stationary point at x = 0
- Tangent to the curve (or y = 2x3 x2 =) would not meet the x-axis
For binomial limits (modulus) in fractions, which value do you use?
The smaller value
P(AnB) for independent
P(A) x P(B)
P(A|B)
P(AnB)/P(B8
Non mutually exclusive/independent P(AUB)
P(A) + P(B) - P(AnB)
Mutually exclusive P(AuB)
P(A) + P(B)
Mutually exclusive P(AnB)
P(AnB) = 0
Which value is more accurate when subbing in x for a binomial expansion?
The smaller the x value, the more accurate
Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of S.
Increase the number of strips
Explain why, for this question, the Newton-Raphson method cannot be used with x1 = 0
There is a stationary point at x=0
- the tangent to the curve would not meet the x axis
Find turning point
Differentiate and set to 0
Find the minimum perimeter of the pool, giving your answer to 3 significant figures.
Differentiate and set to 0 then sub back into the equation
Surface area of a shape
Area of each face and add together
Surface area of a cylinder
2πrh+2πr^2
Volume of sphere
4/3πr^3
