Final Maths Flashcards
(173 cards)
1
Q
What is the formula for nCr?
A
r!-(n-r)!
2
Q
What is (X+Y)^n ?
A
(n) (n-r) r
(r) x y
3
Q
What is
*
———————–
(x+a)(x+b)^2
In partial fractions?
A
A B C
—– + ——— + ———
(x+a) (x+b) (x+b)^2
4
Q
What is
*
———————–
(x+a)(x^2 + bx +c)
In partial fractions?
A
A Bx + C
—– + ——————–
(x+a) (x^2 + bx + c)
5
Q
Sin2x=
A
2sinxcosx
6
Q
Cos2x=
A
cos^2x - sin^2x
1 - 2sin^2x
2cos^2x - 1
7
Q
Cos(-x)=
A
Cos(x)
8
Q
Sin(x)=
A
-Sin(x)
9
Q
What is it when f’‘(0) is less than 0?
A
Maximum
10
Q
What is S’(T) and S’‘(T) ?
A
Speed and acceleration
11
Q
Sin^2X=
A
1/2 (1-cos2x)
12
Q
Cos2x=
A
-1/2 (1+cos2x)
13
Q
What is an odd function?
A
When f(-x) = -f(x) or when the function has half turn symmetry.
14
Q
What is an even function?
A
When f(x)=f(-x) or when the function is symmetrical across the y axis
15
Q
What is the function used to describe the position of an asymptote?
A
X—> n +/- : {let x = (n +/- 0.1)} y —> +/- infinity
16
Q
What is used to describe the domain?
A
{ X: XER, X () n}
17
Q
How can you determine the range of a function?
A
The domain of the inverse
18
Q
What are natural numbers (N)?
A
1,2,3,4,5
19
Q
What are whole numbers (W)?
A
0,1,2,3,4
20
Q
What are integer numbers (Z)?
A
-2,-1,0,1,2,3
21
Q
What are rational numbers (Q)?
A
Numbers produced by a fraction
22
Q
What are irrational numbers ?
A
Numbers which cant be written as a fraction
23
Q
What are Real numbers (R)?
A
Amy number that isn’t complex
24
Q
What are complex numbers (C)?
A
The combination of a real and an imaginary number
25
What is ill-conditioning?
When a small change in the equation leads to a significant change in the solution.
26
When is a matrix redundant?
000:0
27
When is a matrix inconsistent?
000:3
28
Derivative of sin(ax)
acos(ax)
29
Derivative of cos(ax)
-asin(ax)
30
Derivative of tan(ax)
asec^2(ax)
31
Derivative of sec(ax)
asec(ax)tan(ax)
32
Derivative of cosec(ax)
-acosec(ax)cot(ax)
33
Derivative of cot(ax)
-acosec^2(ax)
34
Derivative of e^(f(x))
f'(x) x e(x)
35
Derivative of Ln(f(x))
f'(x)/f(x)
36
Derivative of sin^-1 (x/a)
1
------------ x(Chain rule factor)
(a^2 - x^2)^1/2
37
Derivative of cos^-1 (x/a)
- 1
------------ x(Chain rule factor)
(a^2 - x^2)^1/2
38
Derivative of tan^-1 (x/a)
a
------------ x(Chain rule factor)
(a^2 + x^2)
39
Derivative of lny
1/y dy/dx
40
Derivative of Log10x
1/xln10
41
Integral of ax^n
1
---------- ax^(n+1) +C
n+1
42
Integral of (ax +b)^n
1
---------- (ax+b)^(n+1) +C
a (n+1)
43
Integral of cos(ax + b)
1
----- sin(ax +b) +C
a
44
Integral of sin(ax + b)
-1
----- cos(ax +b) +C
a
45
Integral of Tan(x)?
lnsecx
46
Integral of e^(ax + b)
1/a x e^(ax + b)
47
Integral of sec^2(ax+b)
1/a tan(ax+b)
48
Integral of 1/(ax+b)
1/a ln(ax + b)
49
Integrate 1
------------
(1 + x^2)
Tan^-1 (x) + C
50
Integrate 1
------------
(1 - x^2)^(1/2)
Sin^-1(x) + C
51
Integrate 1
------------
(a^2+ x^2)
1/a Tan^-1(x/a) +C
52
Integrate 1
------------
(a^2 - x^2)^(1/2)
1/a Sin^-1(x/a) + C
53
Integrate fg'
fg- int of f'g
54
What are the two ways you can calculate the derivative of the inverse?
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
55
How do you find the area between two curves and the y axis?
int of f(y) by dy with the limits as y values
56
How do you find the volume of revolution around the x axis?
int of pie times (f(x))^2 dx with limits set as x values
57
What is the domain and range of sin^-1 x?
Domain: -90
58
What is the domain and range of cos^-1 x?
Domain: 0
59
What is the domain and range of tan^-1 x?
Domain: -90
60
How do you implicitly differentiate something?
d/dx = d/dy x dy/dy
61
What is a point of inflection?
When f''(x) = 0
62
How do you differentiate a parametric equation?
dy/dy = dy/dt x dt/dx
63
How do you find the second derivative of a parametric equation?
d/dt dy/dx x dt/dx
64
What is the conjugate of z= a + bi
a-bi
65
What does a complex number multiplied by the conjugate produce?
Whole number
66
How do you find the square root of a + bi?
Make it equal to a+bi and equate real and imaginary parts
67
How do you find a (argz)
Tan^-1 (b/a)
68
How do you find r the modulus of Z?
square root of a^2 + b^2
69
What are the quadrants of the cartesian diagram?
pie - theta 1 Theta 1
1 Re
-------------------------------------------------
-pie + theta 1 -Theta
1
1 im
70
How do you write Z in polar form?
r(cos (x) + isin (x) )
71
How do you find z1 x z2 ?
r1 x r2(cos (x1 + x2) + isin (x1 + x2))
72
How do you find z1 / z2 ?
r1 / r2(cos (x1 - x2) + isin (x1 - x2))
73
How do you find z^n ?
r^n(cos (nx) + isin (nx) )
74
How do you find sin 5x in sines?
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
How do you find the geometrical interpretation of [z+5]=4
Sub in z = x + iy
Group real number with real part
Let the square root of x + 5+ iy = 4
Square both sides
76
What is the centre of the circle and radius with the equation x^2 + y^2 + 2gx + 2fy + C = 0
```
Centre = (-g,-f)
Radius = The root of g^2 + f^2 - C
```
77
What is the centre of the circle and the radius of a circle with the equarion (x-a)^2 + (y-b)^2 = r^2
```
Centre = (a,b)
Radius = r
```
78
What is 1 + tan^2 x ?
sec^x
79
What is the compound angle formula for sin(a+b)?
sinacosb + cosasinb
80
What is the compound angle formula for cos(a+b)?
cosacosb-sinasinb
81
What is the compound angle formula for sin(a-b)?
sinacosb - cosasinb
82
What is the compound angle formula for cos(a-b)?
cosacosb+sinasinb
83
How do you find the roots of a polar equation?
n root of r (cos((2pie x k)/n) + isin((2pie x k)/n))
84
What is the formula for a geometric sequence?
Un=ar^(n-1)
85
What is the sum of a geometric series?
a(r^n - 1)/r-1
86
What's the sum to infinity of a geometric series?
a/(1-r)
87
Whats the formula for an arithmetic sequence?
un= a + (n-1)d
88
What's the sum of an arithmetic sequence?
n/2 (2a + (n-1)d)
89
What's the sum of an arithmetic series when you know the last term?
n/2 (a+l)
90
What's the sum of sequential numbers?
n(n+1)/2
91
What's the sum of sequential squared numbers?
n(n+1)(2n+1)/6
92
What's the sum of sequential cubed numbers?
n^2(n+1)^2 / 4
93
What is the order of a matrix?
row x column
94
What is the transpose?
Replacing rows with columns
95
When is a matrix symmetrical?
When A = transpose
96
When is a matrix skew-symmetric?
When -A = transpose (main diagonal will consist of zero's)
97
What is matrix A - Matrix B equal to ?
A + -(B)
98
What can A be re-written as with respect to transpose properties?
(A')'
99
What can (A+B)' be re-written as with respect to transpose properties?
A' + B'
100
What can (AB)C be re-written as with respect to matrixes
A(BC)
101
What can A(B+C) be re-written as with respect to matrixes
AB + AC
102
What can (AI) be re-written as with respect to matrixes
A
103
How do you find A^3 with respect to matrixes?
A.A^2
104
What does it mean if AB=BA=I with respect to matrixes?
A is the inverse of B
105
What is the determinant of a 2x2 matrix?
AD-BC
106
What does it mean in the determinant of a 2x2 matrix is a)0 or b) not zero
a) Singular - matrix / no inverse
| b) Non-singular - matrix / Inverse exists
107
How do you find the inverse of a 2x2 matrix?
1/ad-bc (d -c)
| (-b a)
108
How can you find det (AB) ?
Det A x Det B
109
How do you find the determinant of a 3x3 matrix?
a(ei-fh) - b(ai-gf) + c(dh-ge)
110
How do you find the inverse of a 3x3 matrix?
1/det A x transpose of co-factors
111
How do you find solutions to equations using the inverse?
AX=B so X=A-1 x B
where x is the column matrix
112
What are the direction cosines?
cos (alpha) = x/|a|
cos (beta) = y/|a|
cos (y) = z/|a|
113
What is a unit vector?
A vector with the length of 1
114
How do you tell if two vectors are perpendicular?
a.b = 0
115
How do you find the angle between two vectors?
a.b / |a| x |b| = cos x
116
What is the cross product / vector product?
The determinant, sub top line as i j k
117
What is the area of a triangle?
1/2 absinc
118
What is the substitution of an area for a triangle?
|a x b| = |a| x |b| sinx
| so area = 1/2 |a x b|
119
What is the area of a parallelogram?
|a x b|
120
What can a x b be re-written as with respect to vectors?
-b x a
121
What can a(b+c) be re-written as with respect to vectors?
ab + ac
122
What can (a+b)c be re-written as?
ac+bc
123
What is the result of a x a with respect to vectors?
0
124
What is a(bc) not equal to ?
(ab)c
125
How do you write in vector form?
r= i, j, k + lambda(i, j, k)
126
How do you write in parametric form?
```
x= a1 + lambda d1
y= a2 + lambda d2
z= a3 + lambda d3
```
127
How do you write in symmetric form? or cartesian form?
x-a1/d1 = y-a2/d2 = z-a3/d3 = lambda
128
How do you find the vectors perpendicular to both u and v ?
+/- (v x u)
129
What two types of vectors exist?
Identical
Parallel
Intersecting
Skew
130
How do you find if a vector intersects?
Put into parametric form
Simultaneous equations to work out the unknowns
Check for consistency with the third equations
131
How do you find if two vectors are skew?
Same method as intersecting but show that there is inconsistency with the solutions
132
What is the angle between two lines?
d1 . d2 / |d1| x |d2| = cos x
133
What is the vector equation of a plane?
r.n = a.n
134
What is the parametric equation of a plane?
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
How do you find the normal of a parametric equation of a plane?
bxc
136
What are the different ways two planes can exist together?
Parallel
Coinsident
Intersect on a line
137
How do you calculate where two planes intersect at a line?
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
What is the angle between two planes?
n1 . n2 / |n1| x |n2| = cos x
139
What ways can three planes exist together?
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
What ways can a plane and a line exist together?
Intersect (one solution), lie on the plane(many solutions 4=4) or parallel ( no solutions, n1.d1 = 0)
141
How do you find the angle between a line and a plane?
d1 . n1 / |d1| x |n1| = cos x (90-x)
142
How do you find out the integrating factor?
e^int of p(x) dx
143
What do you do once you have found the integrating factor?
Multiply both sides by it, giving
| d/dx(I(x)y) = Int of I(x) x f(x)
144
What is the general solution of a homogenous equation when the values of n are real and not equal?
Y= Ae^mx + Be^mx
145
What is the general solution of a homogenous equation when the values of n are real and equal?
Y= Ae^mx + Bxe^mx
146
What is the general solution of a homogenous equation when the values of n are not real?
Y= e^p(x) (Acosqx + Bsinqx)
147
What is the particular integral when f(x) is linear?
ax+b
148
What is the particular integral when f(x) is quadratic?
ax^2 +bx +c
149
What is the particular integral when f(x) is a sine or cosine function?
psin(nx) + qcos(nx)
150
What is the particular integral when f(x) is exponential?
Ke^rx
151
What is the particular integral when f(x) is exponential and r is equal to a root of the equation?
Kxe^rx
152
What is the particular integral when f(x) is exponential and r is equal to both roots of the equation?
Kx^2e^rx
153
What is the particular integral when f(x) is a sum of two functions?
find each one separately then add to find overall yp
154
What is a power series?
The sum of anx^n where n starts at 0
155
What is the McClaurin series sum
f(r) (0) x^r / r! where r=0 and n=infinity
156
What is the expansion for e^x ?
1 + x + x^2/2! + x^3/3! + x^4/4! +....
157
What is the expansion for sinx?
x - x^3/3! + x^5/5! - x^7/7! +.....
158
What is the expansion for cosx ?
1 - x^2/2! + x^4/4! - x^6/6! +.....
159
When does ln(1+x) converge?
-1
160
When does 1/1+x converge?
|X|
161
When does than^-1 converge?
|X|
162
What functions do not have a M series?
Ln (x) + x^1/2
163
How do you find the series for two functions?
Expand both and multiply together
164
What is the expansion for tan^-1 x ?
x - x^3/3 + x^5/5 - x^7/7 +.....
165
What is the expansion for ln(x+1)
x - x^2/2 + x^3/3 - x^4/4 +....
166
How do you know if an iterative equation will converge?
If G'(x0)
167
What is the euclidean algorithm?
a=qb+r
168
What is the anti-clockwise rotational matrix?
cosx -sinx
| sinx cosx
169
What is the clockwise rotational matrix?
cosx sinx
| -sinx cosx
170
What is the reflection matrix?
cos2x sin2x
| sinsx -sin2x
171
What is the scaling matrix?
(x 0)
| 0 x
172
How do you convert numbers to a different base?
Use division algorithm where the quotient is always the desired number. keep going until the divisor is 0
173
What is the equation for speed?
ds/dt = ((dx/dt)^2 + (dy/dt)^2)^1/2
