Higher

Question 1

How do you find the equation for a perpendicular bisector

Answer 1

• find gradient of line from co ordinates
• m1m2=-1 (perpendicular)
• find midpoint
• sub points into y=m(x-a)

Question 2

How do you find the circumcentre

Answer 2

Find the equation for both lines and do simultaneous equations

Question 3

How do you find the equation of an altitude of a line?

Answer 3

find the gradient of the line that the altitude is perpendicular to
m1m2=-1
sub the point and the gradient into y-b=m(x-a)

Question 4

How do you find the equation of the median?

Answer 4

• find the midpoint of the side it bisects
• calculate the gradient of the median
• substitute the gradient and either point on the line into y-b=m(x-a)

Question 5

how do you find the vector journey PQ?

Answer 5

q-p

Question 6

How do you find the magnitude of a vector AB

Answer 6

square root of a squared + b squared

Question 7

how do you prove that a vector journey is collinear?

Answer 7

find vector journey A to B
then B to C
if AB=2BC(or anything like that) it means AB is parallel to BC
B is a common point to both AB and BC so A, B and C are collinear

Question 8

if P is the position vector of the point p that divides AB into the ratio m:n then what is the formula to fine the point p??

Answer 8

p= (x1+x2)/m+n , (y1+y2)/m+n

Question 9

how do you know if 2 vectors are perpendicular?

Answer 9

a.b=0

Question 10

how do you find the angle between 2 vectors?

Answer 10

cosx=a.b/(magn a x magn b) = a1b1+a2b2+a3b3/(mag axmag b)

Question 11

what are the steps for sketching a quadratic function?

Answer 11

1. find the shape = the coefficient of x^2
2. find y intercept
3. find the roots of the equation (factorise and set=0)
4. find midway between the roots
5. find the coordinates of the TP (complete the square and use a and b)

Question 12

how do you find the coordinates of the TP of a quadratic

Answer 12

complete the square

y=ax^2+bx+c can be written in the form y=a(x+p)^2+q. so the TP is (-p,q)

Question 13

how can you solve a quadratic equation?

Answer 13

quadratic equations may be solved by: 
-the graph
-factorising 
-completing the square
- using quadratic formula 
(find the x's)

Question 14

how can solve a quadratic inequation?

Answer 14

by sketching the quadratic function

• factorise equation
• find roots by setting factorised equation to 0
• solve for x’s

Question 15

how do you find the number’ if there are roots

Answer 15

use the discriminant (b^2-4ac)
if b^2-4ac=0 then the roots are real and equal
if b^2-4ac<0 there are no real roots
if b^2-4ac>0 the roots are real and unequal

Question 16

how can you find unknown coefficients in a quadratic equation?

Answer 16

use the discriminant and solve for the unknown coefficient (e.g p)

Question 17

how can you determine if a line is a tangent to a curve?

Answer 17

use the discriminant
if the b^2-4ac=0 then there is only one point of intersection
-if b^2-4ac>0 there are 2 distinct points of intersection
-if b^2-4ac<0 the line does not intersect with the curve

Question 18

how do you find the equation of a tangent

Answer 18

1. set the 2 equation equal to eachother and bring both to one side so it =0
2. use discriminant to find missing coefficent
3. sub number into y=mx+c

Question 19

how do you find the point of intersection between a curve and a tangent

Answer 19

1. set both equations equal to eachother
2. use discriminant to find out how many points of intersection
3. complete the square/factorise and find coordinates

Question 20

how do you find the degree of the polynomial?

Answer 20

its the number of the highest power in the equation

Question 21

how do you know if something is a factor of a polynomial?

Answer 21

if the remainder is 0

Question 22

how can you factorise a polynomial equation fully

Answer 22

• use synthetic division to fine the factor(s) of the equation
• use the new numbers you found from the synthetic division and put it into brackets along with your factor
• factorise the new equation

Question 23

how can you find a polynomials coefficient?

Answer 23

• use synthetic division and solve through
• since you know the factor you know the remainder must be 0
• set your last number to 0 and solve for your coefficient (e.g q)
• if you are given more 2 factors or have 2 unknown coefficient then solve through and use simultaneous equations

Question 24

how can you find the roots of a polynomial equation?

Answer 24

• synthetic division
• factorise fully
• set equal to 0
• solve for x

Question 25

how can you find an expression for f(x) in a polynomial equation

Answer 25

- find k by making f(x) the y coordinate | - sub value for k into original equation and simplify

Question 26

how do sketch a curve using a polynomial equation?

Answer 26

1. find the y intercept 2. use synthetic division to find the x intercepts 3. find the stationary points by differentiating the original equation (find x and sub into og equation to find y (the TPs)) 4. do nature table using factorised differentiated equation to find max and min TPs 5. sketch

Question 27

how do you draw a trigonometric graph?

Answer 27

1. start with a simple cos or sin graph 2. put in the scale to match the constant,c. 3. slide the graph vertically to match the constant a

Question 28

how do you convert from degrees to radians?

Answer 28

multiply by pi/180

Question 29

how do you convert from radians to degrees?

Answer 29

multiply by 180?

Question 30

how do you know if you need to differentiate in a question?

Answer 30

when the question says: - the rate off change of the function - the gradient of the tangent to the graph of function x

Question 31

how can you find the equations of the tangent in terms of differentation

Answer 31

1. find the point of intersection 2. differentiate to find the gradient 3. sub all these into y-b=m(x-a)

Question 32

how can you find the stationary points.

Answer 32

1. derive the equation and st equal to 0 2. find the x values and sub into the ORIGINAL equation and solve for the y values 3. you now have the TP(s) 4. do a nature table to determine the nature of the SPs

Question 33

what are the steps for curve sketching in terms of differentiating

Answer 33

1. Intercepts a) where does the curve cut the x axis when y=0 b) where does the curve cut the y axis when x=0 2. Stationary points a) where is the curve stationary (f'(x)=0 b) what is the nature of the stationary points 3. behaviour a) what happens for large +/- values of x

Question 34

what happens to a graph when y=f(x)+a and y=f(x)-a

Answer 34

all points move up by 'a' units if a is positive and down if a is negative

Question 35

what happens to a graph when y=f(x+a) and y=f(x-a)

Answer 35

all points are moved to the left by 'a' units if a is positive and to the right if a is negative

Question 36

what happens to a graph when y=-f(x) and y=f(-x)

Answer 36

for y=-f(x) the y coordinates become negative. the graph is reflected in the x axis for y=f(-x) the x coordinates become negative. the graph is reflected in the y axis

Question 37

what happens to the graph y=ksinx

Answer 37

the graph is stretched vertically by a factor of k when k>1. | when k<1 the graph is compressed vertically by a factor of k

Question 38

what happens to the graph y=fsinkx

Answer 38

the graph is compressed horizontally by a factor of k when k<1 the graph is stretched horizontally by a factor of k when k>1

Question 39

what happens to a graph of an inverse function

Answer 39

the graph of an inverse function is found by reflecting the in the line y=x

Question 40

how do you find the inverse of a function algebraically

Answer 40

1. rewrite the function, but this time replace the term 'f(x)' with 'y' 2. rearrange the formula using the algebriac manipulation to make x the subject (i.e obtain x by itself) 3. interchange the letters x and y (swap x for y and y for x) 4. replace y with f^-1(x)

Question 41

what is the sin addition fomula

Answer 41

sin(a+b)=sinacosb+cosasinb

Question 42

what is the cos addition formula

Answer 42

cos(a+b)=cosacosb-sinasinb

Question 43

what does sin2a equal

Answer 43

2sinacosa

Question 44

what does cos2a equal

Answer 44

cos^2a-sin^2a cos^2a-1 (if cosa appears in eqation) sin^2a-1 (if sina appears in the equation)

Question 45

what can cos^2a-1 be rearranged to be

Answer 45

1/2(1+cos2a)

Question 46

what can sin^2a-1 be rearranged to be

Answer 46

1/2(1+sin2a)

Question 47

if y=a^x then...

Answer 47

x=loga(y)

Question 48

what are the properties of the Log function?

Answer 48

1. since every exponential function f(x)=a^x passes theough (0,1), then the corresponding log function must pass through the point (1,0) 2. since every 'basic' exponential function tends towards 0 as x-->-infinity, then every basic log function will tend towards -infinity as x tends towards 0

Question 49

what is the formula for intergration?

Answer 49

ax^n dx =( ax^n+1)/(n+1) +c

Question 50

area between 2 curves

Answer 50

upper - lower

Question 51

formula for distance between 2 points

Answer 51

square root((x2-x1)+(y2-y1))

Question 52

what is the general equation of a circle?

Answer 52

x^2+y^2+2gx+2fy+c=0

Question 53

how can you find the circles centre and the radius

Answer 53

centre(-g,-f ) | r= root(g^2+f^2-c)

Question 54

how can you prove that something is a tangent to a circle?

Answer 54

you can either solve the equation or use the discriminant

Question 55

whats the equation for finding a limit for a recurrence relation

Answer 55

L=b/(1-a)

Question 56

what happens when you differentiate sinx

Answer 56

it turns into cosx

Question 57

differentiate cosx

Answer 57

-sinx

Question 58

what is the formula for the chain rule if y=(ax+b)^n

Answer 58

= an(ax+b)^n-1

Question 59

what is the formula for the chain rule if y=(x+b)^n

Answer 59

=n(x+b)^n-1`

Question 60

integrate (ax+b)^n

Answer 60

=(ax+b)^n/a(n+1)

Question 61

integrate cos(ax+b)

Answer 61

=1/a sin(ax+b)+c

Question 62

integrate sin(ax+b)

Answer 62

1/a -cos(ax+b)+c