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Flashcards in C4 Deck (54):
1

what is the remainder when p(x) is divided by (ax-b)

p(b/a)

solution of ax-b=0

2

how can (5x+1)/(x-1)(2x+1)(x-5) be expressed in partial fraction form

A/(x-1) + B/(2x+1) + C/(x-5)

3

how can (5x+1)/(x-1)(2x+1)^2 be expressed in partial fractions

A/(x-1) + B/(2x+1) + C/(2x+1)^2

4

how to convert a pair of parametric equations into a single cartesian equation

elimiate the parameter
e.g x=t+4 and y=1-t^2

using x=t+4 t=x-4
so t^2=x^2-8x+16

put into the y equation to make y=1-(x^2-8x+16)

5

curve x=rcosϴ and y=rsinϴ

circle with radius r and centre is the origin

6

curve x=rcosϴ+p and y=rsinϴ+q

circle with radius r and centre (p,q)

7

curve x=acosϴ and y=bsinϴ

ellipse
centre is the origin
width is 2a and height is 2b (because a and b are only half the measurement as to the radius)

8

curve is x=2cosϴ+3 and y=sinϴ -1
find cartesian equation of curve

isolate cos and sin
cosϴ=(x-3)/2
sinϴ=y+1

use the identity sin^2ϴ+cos^2ϴ=1

(y+1)^2 + (x-3/2)^2 = 1

9

translation (a)
(b)

add a to x function and b to y function

10

stretch in x direction

multiply the x function by the required factor

11

stretch in y direction

multiply the y function by the required factor

12

reflection in y axis

multiply the x function by -1

13

reflection in x axis

multiply the y function by -1

14

formula for binomial expansion of (1+ax)^n

1+ nax + n(n-1)/2! (ax)^2 + n(n-1)(n-2)/3! (ax)^3 + ...

15

to do binomial expansion of (x+5)/(3-x)(1+3x)

split expression into partial fractions then expand

16

sin(A+B)=
sin(A-B)=

sinAcosB + sinBcosA
sinAcosB - sinBcosA

17

cos(A+B)=
cos(A-B)=

cosAcosB - sinAsinB
cosAcosB + sinAsinB

18

tan(A+B)=
tan(A-B)=

tanA+tanB/(1-tanAtanB)
tanA-tanB/(1+tanAtanB)

19

sin2A=

2sinAcosA

20

cos2A=

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

21

tan2A=

2tanA/(1-tan^2A)

22

write 3sinx + 4cosx in the form Rcos(x-a)

make them equal each other
3sinx+4cosx = Rcos(x-∝)

sub in ange formula for cos(x-∝)
3sinx+4cosx=R(cosxcos∝+sinxsin∝)

group together the constants
=Rcos∝cosx + Rsin∝sinx
left with 3=Rsin∝ and 4=Rcos∝ as cosx and sinx cancel as on both sides

find R by squaring both equations and adding them together
3^2+4^2=R^2sin^2∝+R^2cos^2∝
25=R^2(sin^2∝+cos^2∝)
25=R^2
R=+or- 5

we will choose R=5 and sub into one of the equations
3=5sin∝
sin∝=3/5
∝=0.643501

put value into other equation
4=5cos0.643501
4=4 so correct value

(if it doesn't work find another value on cos graph)

5cos(x-0.644)

23

find the maximum value of 3sinx+4cosx

y value
5cos(x-0.644)
translation (0.644,0)
stretch s.f 5 in y axis so maximum value 5

24

fidn the smallest value of x for which the maximum value occurs

5cos(x-0.644)=5 (max value)
cos(x-0.644)=1
let A = x-0.644
cosA=1
A=cos^(-1)(1)
A=0

so x-0.644=0
x=0.644

25

solve this differential equation
dy/dx=2x(y+4)

separate the variables so you have all xs on one side and all ys on the other
dy/(y+4)=2xdx

integrate both sides
∫1/(y+4)dy = ∫2xdx
ln(y+4) = x^2 + c

26

what equation represents exponential growth

y=ae^bt

27

what equation represents exponential decay

y=ae^-bt

28

rate of change inversely proportional to cube of time t

dx/dt ∝ 1/t^3

dx/dt = k/t^3

29

rate of decrease directly proportional to square root of x

dx/dt ∝ -(x)^1/2

30

P(t) is population at any time t
Po is intial population at time t=0
population grows exponentially

dP/dt ∝ P
dP/dt = kP

separate the variables
1/P dP = kdt

intergrate
∫1/P dP = ∫k dt
ln|P| = kt + c

|P| = e^(kt+c)
|P|=e^c x e^kt

when t=0, P=Po
|Po|=e^c x e^kx0
Po=e^c

put back into original equation
|P|=Poe^kt

P=Poe^kt

31

differentiate xy^2 with respect to x

using the product rule
d/dx (xy^2) = d/dx(x)y^2+xd/dxy^2

1xy^2 + x2ydy/dx

y^2 + 2xy dy/dx

32

∫(x+9)/(x-3)(x+1) dx

split up into partial fractions

x+9=A(x-3) +B(x+1)
when x+3, 12=4B so B=3
when x=-1, 8=-4A so A = -2

3/(x-3) - 2/(x+1)

integrate separately
=∫3/(x-3)dx - ∫2/(x+1)

then make top equal differential of bottom (in this case both 1) so you can apply the ln rule
3∫1/(x-3)dx-2∫1/(x+1)dx

=3ln|x-3|- 2ln|x+1| +c

33

∫sinxcosxdx

use double angle formula of sin2x=2sinxcosx
so sinxcosx=1/2sin2x

∫1/2sin2x = -1/4cos2x + c

34

the modulus of a vector

is its magnitude
|a|

35

A = 2
B = 3
find magnitude of AB

do pythagorus
AB = 2^2 + 3^2
= (13)^1/2

36

what is the direction of a vector

the angle measured anticlockwise from vector i (x axis)

37

find magnitude and hence direction of
(6)
(5)

magnitude = 6^2 + 5^2 = (61)^1/2

angle we are trying to calculate is between 6 and (61)^1/2
opposite angle = 5
adjacent angle = 6
hypotenuse= (61)^1/2

use Toa so tanϴ=5/6
ϴ=tan^-1(5/6)
ϴ=39.8 degrees

38

coordinates of -2i-3j

(-2,-3)
i is parallel to x axis
j is parallel to y axis
k is parallel to z axis

39

if a vector is parallel to a

it is a multiple of a
e.g a = kb

40

-a and a

same magnitude but opposite direction

41

when are vectors equal

if they have the same magnitude and direction

42

base vector component form

e.g 4i-2j

43

column vector form

e.g (7)
(2)

44

unit vector

vector with magnitude of 1

45

position vector

position of a point in vector form
e.g OP (1)
(2)
(3)

46

displacement vector

how to get from one point to another
e.g P(1,2,3) and Q(-1,2,-3)
so PQ(-2)
(0)
(-6)

47

collinear

if they lie on the same straight line
proven by showing AB = kAC

48

how to find AB when given OA and OB

AB = OB - OA

49

vector equation

r = a + tb
r is position vector of a general point on the line
a is position vector of a particular point on the line
b is a vector in the direction of the line

50

find the vector equation of the line through (5)& (7)
(6) (9)

pick either vector for a
b is the vector between them so
(x) = (5) + t(2)
(y) (6) (3)

51

dot product rule

a.b = |a| |b|cosϴ

52

find angle between a(2) and b (1)
(3) (-1)

|a| is (2^2 + 3^2)^1/2 = (13^1/2)
|b| is (1^2+(-1)^2))^1/2 = (2)^1/2

(2).(1)=13^1/2x2^1/2cosϴ
(3) (-1)

(2).(1) = (2x1) + (3x-1)= -1
(3) (-1)

-1= (13)^1/2 x (2)^1/2 x cosϴ

ϴ = cos^-1(-1/(13^1/2 x 2^1/2)
ϴ = 101.3

53

How to show 2 vector line equations intersect

Combine the line equation to include the t and u values

Make the 2 new equations equal to each other

Find u and find t

Put values back into new equations to find point of intersection

54

How to show 2 vectors are perpendicular

A • b = 0