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maths > PURE YEAR 2 > Flashcards

PURE YEAR 2 Flashcards

1
Q

how can a rational number be expressed

A

a/b
a and b are integers

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2
Q

in an arithmetic sequence what is constant

A

the difference between consecutive terms

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3
Q

formula for nth term of arithmetic sequence

A

un= a + (n-1) d

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4
Q

formula for first n terms of an arithmetic sequence

A

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

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5
Q

a geometric sequence has what

A

a common ratio between consecutive terms

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6
Q

formula for nth term of geometric sequence

A

un = a r^(n-1)

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7
Q

formula for sum of first n terms of geometric sequence

A

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

OR

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

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8
Q

when is a geometric sequence convergent

A

|r|<1

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9
Q

sum to infinity of a geometric sequence

A

a / 1-r

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10
Q

when can you sum to infinity

A

when a geometric sequence is convergent

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11
Q

what is a recurrence relation

A

u (n+1) = f (u(n))
defines each term of a sequence as a function of the previous term

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12
Q

when is a sequence increasing

A

u(n+1) > u(n) for all n

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13
Q

when is a sequence decreasing

A

u(n+1) < u(n) for all n

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14
Q

when is a sequence periodic

A

if the terms repeat in a cycle
for a periodic sequence there is an integer k such that u(n+k) = u(n) for all n
the value k is called the order of the sequence

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15
Q

radian to degrees

A

x 180/pi

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16
Q

degrees to radians

A

x pi/180

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17
Q

arc length equation

A

l= θr
radius x angle (in radians)

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18
Q

sector area equation

A

1/2 r^2 θ
(in radians)

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19
Q

segment area equation

A

1/2 r^2 (θ - sinθ)

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20
Q

when θ is small and measured in radians:
approximation for sinθ

A

θ

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21
Q

when θ is small and measured in radians:
approximation for tanθ

A

θ

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22
Q

when θ is small and measured in radians:
approximation for cosθ

A

1 - θ^2 / 2

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23
Q

sec x =

A

1 / cos x

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24
Q

cosec x =

A

1 / sin x

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25
Q

cot x =

A

1 / tan x
OR
cos x / sin x

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26
Q

equation for binomial expansion applied to negative/fractional values of n to obtain infinite series?

A

(1+x)^n

1 + nx + (n(n-1)x^2)/2! + (n(n-1)(n-2)x^3)/3! + … + (nCr) x^r

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27
Q

when is year 2 binomial expansion equation valid

A

when |x|<1

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28
Q

when is the expansion os (1+bx)^n (where n is negative or a fraction) valid

A

valid for |bx|<1

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29
Q

when is the expansion of (a+bx)^n (where n is negative for a fraction) valid

A

valid for |b/a x|<1

30
Q

if y=sin kx , what is dy/dx

A

k cos kx

31
Q

if y= cos kx , what is dy/dx

A

-k sin kx

32
Q

if y= ln x , what is dy/dx

A

1/x

33
Q

if y= a^(kx), what is dy/dx

A

a^(kx) k ln a

34
Q

chain rule equation

A

dy/dx = dy/du x du/dx

35
Q

product rule equation

A

dy/dx= u’v + uv’

36
Q

quotient rule equation (u/v)

A

dy/dx= u’v - uv’ / v^2

37
Q

how to differentiate if x and y are given as functions of a parameter, t

A

dy/dx = dy/dt / dx/dt

38
Q

when is the function f(x) concave

A

if f’‘(x) <0 for every value of x in the specified interval

39
Q

when is the function f(x) convex

A

if f’‘(x)>0 for every value of x in that interval

40
Q

what is a point of inflection

A

a point at which f’‘(x) changes sign

41
Q

how to solve an equation in the form f(x)=0 by an iterative method

A

rearrange f(x)=0 into the form x=g(x) and use the iterative formula x(n+1)=g(x(n))

42
Q

what is newton raphson formula used for

A

approximating the roots of a function f(x)

43
Q

newton raphson formula

A

x(n+1) = x(n) - f(x(n)) / f’(x(n))

44
Q

sign change to find roots of a function?

A

if the function f(x) is continuous on the interval [a,b] and f(a) and f(b) have opposite signs, then f(x) has at least one root, x, which satisfies a<x<b

45
Q

integration equation for x^n

A

∫x^n dx= (x^(n+1))/ n+ 1 + c

46
Q

integration of e^x

A

e^x + c

47
Q

integration of 1/x

A

ln|x| + c

48
Q

integration of cos x

A

sin x + c

49
Q

integration of sin x

A

-cos x + c

50
Q

integration of f’(ax+b)

A

1/a f(ax+b) + c

51
Q

integration by parts formula

A

∫uv’=uv=∫u’v

52
Q

trapezium rule

A

∫y dx= 1/2 h (y0+2(y1 + y2 + … + y(n-1)) + yn)
where h = b - a /n

53
Q

graph of y=sec x:
symmetry?
period?
vertical asymptotes?
range?

A

symmetry in y-axis
period 360 or 2pi
vertical asymptotes at all the values of x for which cosx=0 (90, 270, 450)
y<-1 or y>1

54
Q

graph of y=cosec x:
period?
vertical asymptotes?
range

A

period 360 or 2pi
vertical asymptotes at values of x for which sin x= 0 (0,180,360)
range y<-1 or y>1

55
Q

graph of y=cot x:
period?
vertical asymptotes?
range?

A

period 180 or pi
vertical asymptotes at all values of x for which tanx=0 (0,180,360)
range y is all real numbers

56
Q

identities derived from sin^2+ cos^2 = 1

A

1 + tan^2 = sec^2
1 + cot^2 = cosec^2

57
Q

y=arcsin x graph:
domain
range

A

domain -1<x<1
range -90<arcsinx<90 (pi/2)

58
Q

y=arccos x graph:
domain
range

A

domain -1<x<1
range 0<arccosx< 180 (pi)

59
Q

y=arctan x graph:
domain
range

A

domain x is all real numbers
range -90<arctan<90 (pi/2)

60
Q

addition formula:
sin (A+B)

A

sinAcosB + cosAsinB

61
Q

addition formula:
sin(A-B)

A

sinAcosB - cosAsinB

62
Q

addition formula:
cos(A+B)

A

cosAcosB - sinAsinB

63
Q

addition formula:
cos(A-B)

A

cosAcosB + sinAsinB

64
Q

addition formula:
tan(A+B)

A

tanA + tanB / 1-tanAtanB

65
Q

addition formula:
tan(A-B)

A

tanA - tanB / 1 + tanAtanB

66
Q

double angle formulae:
sin2A

A

2sinAcosA

67
Q

double angle formula:
cos2A

A

cos^2 A - sin^2 A
2cos^2 A -1
1- 2sin^2 A

68
Q

double angle formula:
tan2A

A

2tanA/1-tan^2 A

69
Q

how can a sin x +- b cos x be expressed

A

Rsin (x +- k)
or
Rcos (x +- k)
(R>0 and 0<k<90)
Rcosk= b, Rsink=a and R= sqrt (a^2 +b^2)

70
Q

how to convert between cartesian and parametric equations

A

using substitution to eliminate parameter t

71
Q

for parametric equations x=p(t) and y=q(t) with cartesian equation y=f(x), what is domain and range of cartesian

A

domain of f(x) is range of p(t)
range of f(x) if range of q(t)

72
Q

when does newton raphson not work

A

stationary point at point x=n
tangent to curve at point would not meet x-axis
tangent to curve at point is horizontal

maths (4 decks)

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