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Final Maths Flashcards

1
Q

What is the formula for nCr?

A

r!-(n-r)!

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2
Q

What is (X+Y)^n ?

A

(n) (n-r) r

(r) x y

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3
Q

What is
*
———————–
(x+a)(x+b)^2

In partial fractions?

A

A B C
—– + ——— + ———
(x+a) (x+b) (x+b)^2

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4
Q

What is
*
———————–
(x+a)(x^2 + bx +c)

In partial fractions?

A

A Bx + C
—– + ——————–
(x+a) (x^2 + bx + c)

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5
Q

Sin2x=

A

2sinxcosx

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6
Q

Cos2x=

A

cos^2x - sin^2x
1 - 2sin^2x
2cos^2x - 1

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7
Q

Cos(-x)=

A

Cos(x)

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8
Q

Sin(x)=

A

-Sin(x)

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9
Q

What is it when f’‘(0) is less than 0?

A

Maximum

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10
Q

What is S’(T) and S’‘(T) ?

A

Speed and acceleration

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11
Q

Sin^2X=

A

1/2 (1-cos2x)

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12
Q

Cos2x=

A

-1/2 (1+cos2x)

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13
Q

What is an odd function?

A

When f(-x) = -f(x) or when the function has half turn symmetry.

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14
Q

What is an even function?

A

When f(x)=f(-x) or when the function is symmetrical across the y axis

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15
Q

What is the function used to describe the position of an asymptote?

A

X—> n +/- : {let x = (n +/- 0.1)} y —> +/- infinity

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16
Q

What is used to describe the domain?

A

{ X: XER, X () n}

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17
Q

How can you determine the range of a function?

A

The domain of the inverse

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18
Q

What are natural numbers (N)?

A

1,2,3,4,5

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19
Q

What are whole numbers (W)?

A

0,1,2,3,4

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20
Q

What are integer numbers (Z)?

A

-2,-1,0,1,2,3

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21
Q

What are rational numbers (Q)?

A

Numbers produced by a fraction

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22
Q

What are irrational numbers ?

A

Numbers which cant be written as a fraction

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23
Q

What are Real numbers (R)?

A

Amy number that isn’t complex

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24
Q

What are complex numbers (C)?

A

The combination of a real and an imaginary number

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25
Q

What is ill-conditioning?

A

When a small change in the equation leads to a significant change in the solution.

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26
Q

When is a matrix redundant?

A

000:0

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27
Q

When is a matrix inconsistent?

A

000:3

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28
Q

Derivative of sin(ax)

A

acos(ax)

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29
Q

Derivative of cos(ax)

A

-asin(ax)

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30
Q

Derivative of tan(ax)

A

asec^2(ax)

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31
Q

Derivative of sec(ax)

A

asec(ax)tan(ax)

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32
Q

Derivative of cosec(ax)

A

-acosec(ax)cot(ax)

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33
Q

Derivative of cot(ax)

A

-acosec^2(ax)

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34
Q

Derivative of e^(f(x))

A

f’(x) x e(x)

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35
Q

Derivative of Ln(f(x))

A

f’(x)/f(x)

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36
Q

Derivative of sin^-1 (x/a)

A

1
———— x(Chain rule factor)
(a^2 - x^2)^1/2

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37
Q

Derivative of cos^-1 (x/a)

A
  • 1
    ———— x(Chain rule factor)
    (a^2 - x^2)^1/2
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38
Q

Derivative of tan^-1 (x/a)

A

a
———— x(Chain rule factor)
(a^2 + x^2)

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39
Q

Derivative of lny

A

1/y dy/dx

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40
Q

Derivative of Log10x

A

1/xln10

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41
Q

Integral of ax^n

A

1
———- ax^(n+1) +C
n+1

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42
Q

Integral of (ax +b)^n

A

1
———- (ax+b)^(n+1) +C
a (n+1)

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43
Q

Integral of cos(ax + b)

A

1
—– sin(ax +b) +C
a

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44
Q

Integral of sin(ax + b)

A

-1
—– cos(ax +b) +C
a

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45
Q

Integral of Tan(x)?

A

lnsecx

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46
Q

Integral of e^(ax + b)

A

1/a x e^(ax + b)

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47
Q

Integral of sec^2(ax+b)

A

1/a tan(ax+b)

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48
Q

Integral of 1/(ax+b)

A

1/a ln(ax + b)

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49
Q

Integrate 1
————
(1 + x^2)

A

Tan^-1 (x) + C

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50
Q

Integrate 1
————
(1 - x^2)^(1/2)

A

Sin^-1(x) + C

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51
Q

Integrate 1
————
(a^2+ x^2)

A

1/a Tan^-1(x/a) +C

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52
Q

Integrate 1
————
(a^2 - x^2)^(1/2)

A

1/a Sin^-1(x/a) + C

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53
Q

Integrate fg’

A

fg- int of f’g

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54
Q

What are the two ways you can calculate the derivative of the inverse?

A
  1. Swap x and y values, rearrange so y is the subject and differentiate.
  2. Swap x and y values, differentiate by dx/dy. find the reciprocal of dx/dy = dy/dx. sunstitute a value for y
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55
Q

How do you find the area between two curves and the y axis?

A

int of f(y) by dy with the limits as y values

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56
Q

How do you find the volume of revolution around the x axis?

A

int of pie times (f(x))^2 dx with limits set as x values

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57
Q

What is the domain and range of sin^-1 x?

A

Domain: -90

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58
Q

What is the domain and range of cos^-1 x?

A

Domain: 0

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59
Q

What is the domain and range of tan^-1 x?

A

Domain: -90

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60
Q

How do you implicitly differentiate something?

A

d/dx = d/dy x dy/dy

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61
Q

What is a point of inflection?

A

When f’‘(x) = 0

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62
Q

How do you differentiate a parametric equation?

A

dy/dy = dy/dt x dt/dx

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63
Q

How do you find the second derivative of a parametric equation?

A

d/dt dy/dx x dt/dx

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64
Q

What is the conjugate of z= a + bi

A

a-bi

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65
Q

What does a complex number multiplied by the conjugate produce?

A

Whole number

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66
Q

How do you find the square root of a + bi?

A

Make it equal to a+bi and equate real and imaginary parts

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67
Q

How do you find a (argz)

A

Tan^-1 (b/a)

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68
Q

How do you find r the modulus of Z?

A

square root of a^2 + b^2

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69
Q

What are the quadrants of the cartesian diagram?

A

pie - theta 1 Theta 1
1 Re
————————————————-
-pie + theta 1 -Theta
1
1 im

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70
Q

How do you write Z in polar form?

A

r(cos (x) + isin (x) )

71
Q

How do you find z1 x z2 ?

A

r1 x r2(cos (x1 + x2) + isin (x1 + x2))

72
Q

How do you find z1 / z2 ?

A

r1 / r2(cos (x1 - x2) + isin (x1 - x2))

73
Q

How do you find z^n ?

A

r^n(cos (nx) + isin (nx) )

74
Q

How do you find sin 5x in sines?

A

Binomial expansion of (cos (x) + isin (x) )^5
Equate isin(x) with imaginary part of expansion
Factor out i
Sub in (1-sin^2 x) as cos^2 x

75
Q

How do you find the geometrical interpretation of [z+5]=4

A

Sub in z = x + iy
Group real number with real part
Let the square root of x + 5+ iy = 4
Square both sides

76
Q

What is the centre of the circle and radius with the equation x^2 + y^2 + 2gx + 2fy + C = 0

A 
Centre = (-g,-f)
Radius = The root of g^2 + f^2 - C
77
Q

What is the centre of the circle and the radius of a circle with the equarion (x-a)^2 + (y-b)^2 = r^2

A 
Centre = (a,b) 
Radius = r
78
Q

What is 1 + tan^2 x ?

A

sec^x

79
Q

What is the compound angle formula for sin(a+b)?

A

sinacosb + cosasinb

80
Q

What is the compound angle formula for cos(a+b)?

A

cosacosb-sinasinb

81
Q

What is the compound angle formula for sin(a-b)?

A

sinacosb - cosasinb

82
Q

What is the compound angle formula for cos(a-b)?

A

cosacosb+sinasinb

83
Q

How do you find the roots of a polar equation?

A

n root of r (cos((2pie x k)/n) + isin((2pie x k)/n))

84
Q

What is the formula for a geometric sequence?

A

Un=ar^(n-1)

85
Q

What is the sum of a geometric series?

A

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

86
Q

What’s the sum to infinity of a geometric series?

A

a/(1-r)

87
Q

Whats the formula for an arithmetic sequence?

A

un= a + (n-1)d

88
Q

What’s the sum of an arithmetic sequence?

A

n/2 (2a + (n-1)d)

89
Q

What’s the sum of an arithmetic series when you know the last term?

A

n/2 (a+l)

90
Q

What’s the sum of sequential numbers?

A

n(n+1)/2

91
Q

What’s the sum of sequential squared numbers?

A

n(n+1)(2n+1)/6

92
Q

What’s the sum of sequential cubed numbers?

A

n^2(n+1)^2 / 4

93
Q

What is the order of a matrix?

A

row x column

94
Q

What is the transpose?

A

Replacing rows with columns

95
Q

When is a matrix symmetrical?

A

When A = transpose

96
Q

When is a matrix skew-symmetric?

A

When -A = transpose (main diagonal will consist of zero’s)

97
Q

What is matrix A - Matrix B equal to ?

A

A + -(B)

98
Q

What can A be re-written as with respect to transpose properties?

A

(A’)’

99
Q

What can (A+B)’ be re-written as with respect to transpose properties?

A

A’ + B’

100
Q

What can (AB)C be re-written as with respect to matrixes

A

A(BC)

101
Q

What can A(B+C) be re-written as with respect to matrixes

A

AB + AC

102
Q

What can (AI) be re-written as with respect to matrixes

A

A

103
Q

How do you find A^3 with respect to matrixes?

A

A.A^2

104
Q

What does it mean if AB=BA=I with respect to matrixes?

A

A is the inverse of B

105
Q

What is the determinant of a 2x2 matrix?

A

AD-BC

106
Q

What does it mean in the determinant of a 2x2 matrix is a)0 or b) not zero

A

a) Singular - matrix / no inverse

b) Non-singular - matrix / Inverse exists

107
Q

How do you find the inverse of a 2x2 matrix?

A

1/ad-bc (d -c)

(-b a)

108
Q

How can you find det (AB) ?

A

Det A x Det B

109
Q

How do you find the determinant of a 3x3 matrix?

A

a(ei-fh) - b(ai-gf) + c(dh-ge)

110
Q

How do you find the inverse of a 3x3 matrix?

A

1/det A x transpose of co-factors

111
Q

How do you find solutions to equations using the inverse?

A

AX=B so X=A-1 x B

where x is the column matrix

112
Q

What are the direction cosines?

A

cos (alpha) = x/|a|
cos (beta) = y/|a|
cos (y) = z/|a|

113
Q

What is a unit vector?

A

A vector with the length of 1

114
Q

How do you tell if two vectors are perpendicular?

A

a.b = 0

115
Q

How do you find the angle between two vectors?

A

a.b / |a| x |b| = cos x

116
Q

What is the cross product / vector product?

A

The determinant, sub top line as i j k

117
Q

What is the area of a triangle?

A

1/2 absinc

118
Q

What is the substitution of an area for a triangle?

A

|a x b| = |a| x |b| sinx

so area = 1/2 |a x b|

119
Q

What is the area of a parallelogram?

A

|a x b|

120
Q

What can a x b be re-written as with respect to vectors?

A

-b x a

121
Q

What can a(b+c) be re-written as with respect to vectors?

A

ab + ac

122
Q

What can (a+b)c be re-written as?

A

ac+bc

123
Q

What is the result of a x a with respect to vectors?

A

0

124
Q

What is a(bc) not equal to ?

A

(ab)c

125
Q

How do you write in vector form?

A

r= i, j, k + lambda(i, j, k)

126
Q

How do you write in parametric form?

A 
x= a1 + lambda d1
y= a2 + lambda d2
z= a3 + lambda d3
127
Q

How do you write in symmetric form? or cartesian form?

A

x-a1/d1 = y-a2/d2 = z-a3/d3 = lambda

128
Q

How do you find the vectors perpendicular to both u and v ?

A

+/- (v x u)

129
Q

What two types of vectors exist?

A

Identical
Parallel
Intersecting
Skew

130
Q

How do you find if a vector intersects?

A

Put into parametric form
Simultaneous equations to work out the unknowns
Check for consistency with the third equations

131
Q

How do you find if two vectors are skew?

A

Same method as intersecting but show that there is inconsistency with the solutions

132
Q

What is the angle between two lines?

A

d1 . d2 / |d1| x |d2| = cos x

133
Q

What is the vector equation of a plane?

A

r.n = a.n

134
Q

What is the parametric equation of a plane?

A

r = a + lambda b + ewe c
Where a is a positive vector from the origin
b and c are non-parallel vectors, parallel to the plane
Lambda and ewe are scalars

135
Q

How do you find the normal of a parametric equation of a plane?

A

bxc

136
Q

What are the different ways two planes can exist together?

A

Parallel
Coinsident
Intersect on a line

137
Q

How do you calculate where two planes intersect at a line?

A

we need a and d
a= a point on the line, sub x, y, or z as equal to zero then solve simultaneously for the other two values
d= n1 x n2

138
Q

What is the angle between two planes?

A

n1 . n2 / |n1| x |n2| = cos x

139
Q

What ways can three planes exist together?

A

Intersect at a point , intersect at a line (sub in parameter) and not intersect. All use gaussian elimination. Point will have a set of solutions. The line will have a a redundant matrix no intersection will have an inconsistent matrix

140
Q

What ways can a plane and a line exist together?

A

Intersect (one solution), lie on the plane(many solutions 4=4) or parallel ( no solutions, n1.d1 = 0)

141
Q

How do you find the angle between a line and a plane?

A

d1 . n1 / |d1| x |n1| = cos x (90-x)

142
Q

How do you find out the integrating factor?

A

e^int of p(x) dx

143
Q

What do you do once you have found the integrating factor?

A

Multiply both sides by it, giving

d/dx(I(x)y) = Int of I(x) x f(x)

144
Q

What is the general solution of a homogenous equation when the values of n are real and not equal?

A

Y= Ae^mx + Be^mx

145
Q

What is the general solution of a homogenous equation when the values of n are real and equal?

A

Y= Ae^mx + Bxe^mx

146
Q

What is the general solution of a homogenous equation when the values of n are not real?

A

Y= e^p(x) (Acosqx + Bsinqx)

147
Q

What is the particular integral when f(x) is linear?

A

ax+b

148
Q

What is the particular integral when f(x) is quadratic?

A

ax^2 +bx +c

149
Q

What is the particular integral when f(x) is a sine or cosine function?

A

psin(nx) + qcos(nx)

150
Q

What is the particular integral when f(x) is exponential?

A

Ke^rx

151
Q

What is the particular integral when f(x) is exponential and r is equal to a root of the equation?

A

Kxe^rx

152
Q

What is the particular integral when f(x) is exponential and r is equal to both roots of the equation?

A

Kx^2e^rx

153
Q

What is the particular integral when f(x) is a sum of two functions?

A

find each one separately then add to find overall yp

154
Q

What is a power series?

A

The sum of anx^n where n starts at 0

155
Q

What is the McClaurin series sum

A

f(r) (0) x^r / r! where r=0 and n=infinity

156
Q

What is the expansion for e^x ?

A

1 + x + x^2/2! + x^3/3! + x^4/4! +….

157
Q

What is the expansion for sinx?

A

x - x^3/3! + x^5/5! - x^7/7! +…..

158
Q

What is the expansion for cosx ?

A

1 - x^2/2! + x^4/4! - x^6/6! +…..

159
Q

When does ln(1+x) converge?

A

-1

160
Q

When does 1/1+x converge?

A

|X|

161
Q

When does than^-1 converge?

A

|X|

162
Q

What functions do not have a M series?

A

Ln (x) + x^1/2

163
Q

How do you find the series for two functions?

A

Expand both and multiply together

164
Q

What is the expansion for tan^-1 x ?

A

x - x^3/3 + x^5/5 - x^7/7 +…..

165
Q

What is the expansion for ln(x+1)

A

x - x^2/2 + x^3/3 - x^4/4 +….

166
Q

How do you know if an iterative equation will converge?

A

If G’(x0)

167
Q

What is the euclidean algorithm?

A

a=qb+r

168
Q

What is the anti-clockwise rotational matrix?

A

cosx -sinx

sinx cosx

169
Q

What is the clockwise rotational matrix?

A

cosx sinx

-sinx cosx

170
Q

What is the reflection matrix?

A

cos2x sin2x

sinsx -sin2x

171
Q

What is the scaling matrix?

A

(x 0)

0 x

172
Q

How do you convert numbers to a different base?

A

Use division algorithm where the quotient is always the desired number. keep going until the divisor is 0

173
Q

What is the equation for speed?

A

ds/dt = ((dx/dt)^2 + (dy/dt)^2)^1/2

Maths (4 decks)

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