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Further Maths > Pure > Flashcards

Pure Flashcards

1
Q

Critical damping

A

Discriminant of AE is 0

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2
Q

Underdamping

A

Discriminant is negative

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3
Q

Overdamping

A

Discriminant is positive

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4
Q

Scaler product

A

a.b=|a|.|b|cosx

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5
Q

Vector equation of a plane

A

(r-a).n=0

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6
Q

Cartesian equation of a plane

A

n1x+n2y+n3z+d=0

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7
Q

Find angle between two planes

A

Use scalar product on normals

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8
Q

Vector equation of a line in 2d

A

R=a+¥d

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9
Q

Cartesian equation of line 2d same for 3d

A

¥= x-a/d=y-a1/d1

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10
Q

Special case for Cartesian equation of a line in 3D

A

When d=0 write as x=a , ¥=y-a1/d1

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11
Q

Intersection of two lines in 2d

A

Solve direction simultaneously

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12
Q

How to prove to lines in 2d are skew

A

No unique solution simultaneously

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13
Q

Enlargement scale factor k about origin

A

(K 0)

O k

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14
Q

Stretch scale factor m parallel to x axis

A

(m 0)

0 1

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15
Q

Stretch scale factor n parallel to y axis

A

(1 0)

0 n

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16
Q

Rotation 90 degrees clockwise about origin and any rotation about origin through x degrees

A

(0 1)
(-1 0)

(Cosx -sinx)
(Sinx cosx)

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17
Q

Reflection in x and y axis

A 
X axis (1 0)
           (0 -1)
Y axis (-1 0)
           (0 1)
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18
Q

Reflection y=x

A

(0 1)

1 0

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19
Q

Equation of a shear

A

Shear x axis fixed (1 k)
(0 1)

Y axis fixed (1 0)
(k 1)

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20
Q

How do you rotate 3D shapes

A

Check book if unsure

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21
Q

What is an invariant point

A

A point that doesn’t change when multiplied by a matrix

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22
Q

How to find a line of invariant points

A

Set up a matrix equation will lead to two equations like ax + by = 0, if unique solution then line of invariant points, if not origin only point of invariance

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23
Q

What is an invariant line

A

Any point of a line maps to another point on the line

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24
Q

Work out a invariant line

A 
Suppose for y = mx + c *
Multiply matrix and (x y) = (x’ y’)
Use *
As invariant line y’=mx’ + c
Rearrange for m=0
Once solved quadratic for this
Put back into x’ and y’ equations
Gives invariant lines
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25
Q

Reflection in a plane

A

(-1 0 0)
(0 1 0)
(0 0 1)

Reflection in plane x=0

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26
Q

Inverse of a 2x2 matrix

A

M^-1 = 1/(ad-bc) (d -b)

(-c a)

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27
Q

What is it called if detm = 0

A

Singular

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28
Q

Proof for inverse of a product of matrices

A 
x(mn) = i
xmnn^-1 = in^-1
xm = n^-1
xmm^-1 = n^-1m^-1
x = n^-1m^-1
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29
Q

Inverse of a 3x3 matrix

A

1/det m (inverse)

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30
Q

How can planes are arrange in 3D

A

Can be parallel either two or three parallel
Can have a unique intersection
Can have infinite solutions then sheaf
No solution, only intersection between each plane, prism

31
Q

Sum of a geometric series

A

Sn = a(1-r^n)/1-r

32
Q

Arithmetic sequence

A

Common difference denoted by d

33
Q

Explain two methods of summing series

A

Partial and differences

34
Q

Proof of induction

A 
Prove n = 1
Assume true for n = k
Let n = k+1 for target expression 
Use n=k to find n = k+1
True for all values of n
35
Q

Matrix proof by induction

A

N=1

A^kA= A^k+1

36
Q

When is an improper integral convergent and divergent

A

When a -> infinity , is defined so convergent

37
Q

What does sin and cos integrate and differentiate too

A

Remember circle of them

38
Q

Differentiate y=arcsinx

A

1/(1-x^2)^0.5

39
Q

What does tan differentiate too

A

Sec^2

40
Q

Cartesian too polar form

A

X=rcosa

Y=rsina

41
Q

Area under a polar curve

A

Integral ( 0.5r^2) dx

42
Q

Accuracy of a maclaurin expansion

A

% error = approximate value - exact value / exact value )

43
Q

Complex conjugate

A

Opposite of imaginary

44
Q

Coshx in exponentials

A

(e^x + e^-x)/2

45
Q

Sinhx

A

(e^x - e^-x) / 2

46
Q

tanhx

A

(e^x - e^-x) / (e^x + e^-x)

47
Q

Hyperbolic identities

A

Cosh^2(x) - sinh^2(x) = 1
Cosh2x = cosh^2(x) + sinh^2(x)
sinh2x = 2sinhxcoshx

48
Q

What does coshx and sinhx differentiate too

A

Themselves

49
Q

What does arsinhx = ?

A

Ln(x + (x^2 + 1)^0.5)

50
Q

Artanhx equal too ?

A

0.5ln( (1 + x) / (1 - x) )

51
Q

What does arcoshx differentiate too

A

1/(x^2 - 1)^0.5

52
Q

Area of a volume

A

V = integral ( pi y^2 )

53
Q

Mean of a function f(x)

A

(1/(b-a))integral (f(x))

54
Q

What is a root mean square and what’s the formula

A

How far the function is away from zero on average,,

((1/(b-a)) integral (f(x))^2) ^0.5

55
Q

Describe simple harmonic motion

A

D^2x/dt^2 = -w^2 x
Where period is 2pi / w
General solution is Pcoswt + Qsinwt = x

56
Q

What’s homogenous and non homogeneous

A

Homogeneous is one side is 0

57
Q

Explain how the complementary function works

A

Say dy/dx and d^2/dx^2 are equal too then sub them into differential, divide by Ae^€x to get cf

58
Q

Name two conditions to solve exact equation of a cf with real distinct roots

A

Initial conditions or boundary conditions would

59
Q

Cf for a AE with repeated roots

A

(A + Bx)e^¥x

60
Q

Difference between general and particular solution

A

Particular find constants

61
Q

How do you choose what trig substitution to do for Further integration

A

Match the 1+x^2 to for example sec^2 = 1 + tan^2

62
Q

Modulus argument form

A

R(cosx + sinx)

63
Q

What is a loci in form |z-a| = r

A

Circle

64
Q

What is the locus of an angle

A

Arg(z-a) = x

65
Q

Locus of points on a perpendicular bisector

A

|z-a| = |z-b|

66
Q

Explain why the general solution for a second order differential is what it is

A

Relationship with e^ix

67
Q

Prove how R works as an integrating factor

A

Multiply by R

LHS can be written as d/dx(Ry)

Comparing both sides once differentiating

Only true if dR/dx = RP

So integrate both sides

So R = e^integral p dx

68
Q

Proof for why e^i€ = cos€ + isin€

A

Maclaurin series for sin€ and cos€ is same as series for e^x when added and is x is replaced with i€

69
Q

What does cos€ and sin€ equal in complex numbers

A

Cos€ = 0.5( e^i€ + e^-i€)

Sin€ = 0.5i( e^i€ - e^-i€)

70
Q

What is z^n and z^-n

A

Z^n = Cosnx + isinnx

Z^-n = cosnx - isinnx

71
Q

What does cosnx equal

A

Z^n + z^-n / 2

72
Q

Sinnx equal to?

A

Z^n + z^-n / 2i

73
Q

What is the PI of the linear function, trig function, polynomial of order n, and exponential function

A

ax + b
anx^n + an-1x^n-1….
acospx + bsinpx
ae^px

74
Q

Describe a special case for a non-homogeneous equation

A

If your equation equals e^3x then use PI of axe^3x

Further Maths (4 decks)

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