1.1 Question 1 How do you find the equation of a perpendicular bisector of a straight line ?

1.1

Question 1

How do you find the equation of a perpendicular bisector of a straight line ?

Answer to Question 1

(i) find the midpoint of the line(ii) find the gradient of the line(iii) find the gradient

perpendicular to the given line

(iv) Use midpoint and gradient in y-b = m(x-a)

M(a,b)

1.1

Question 2

How do you find the midpoint of a line joining two points ?


Add the coordinates and divide by two

x1+ x

2 , y1+ y

2

2 2( )x

y(x2,y2)

(x1,y1)

1.1

Question 3

How do you find the altitude AN of ΔABC ?


(i) find the gradient of BC(ii) find the gradient of AN, perpendicular to BC

(iii) use y-b=m(x-a), using A as (a,b)

A

NB

C

2.1

Question 4

How do you show that x-1 is a factor of the function f(x)=x3-3x+2 ?


(i) rewrite the function asf(x)=x3+0x2-3x+2

(ii) use synthetic division with 1 on the outside

(iii) show thatremainder = 0

1.2

Question 5

For what values is this function undefined ?f(x) = x

(x+2)(x-3)


-2 and 3

1.2

Question 6

How do you draw the graph of 2f(x) given the graph of f(x) ?


Double the y-coordinates

2.3

Question 7

How do you find theexact value ofsin (α-β),given that sinα =4/5

and cosβ = 12/13 ?


(i) draw triangles for α and β

(ii) work out cosα and sinβ

(iii) expand formula for sin(α-β)

(iv) insert exact values

αα

4

5

12

13

ββ

2.1

Question 8

What is the turning point ofy=2(x-a)2+b ?Max or min ?


(i) (a,b)minimum

(a,b)

1.2

Question 9

How do you draw the graph of f(-x) given the graph of f(x) ?


Reflect the graph in the y-axis

1.2

Question 10

How do you draw a graph of the form y = cos(x+a) or y = sin(x+a) ?


Move the graph of y=cosx or y=sinx

a units to the LEFT

Move the graph of y=cosx or y=sinx

a units to the LEFT

1.3

Question 11

Name the steps you take in order to differentiate functions likef(x) = x2+ 3x + 1

√x


(i) Change roots to powers

(ii) split up into 3 fractions

(iii) simplify each term(iv) differentiate

1.3

Question 12

If f(t) is the distance travelled in a certain time t seconds, then what does f’(t) represent ?


Speed (velocity)

2.2

Question 13

Given f’(x) and a point on the curve, how do you findf(x) ?


(i) integrate(ii) substitute in a

given point to work out value

of C

1.1

Question 14

What do you know about the gradients of two parallel lines?


They are the same

1.1

Question 15

How do you find the equation of a tangent to a curve at the point when x = a ?


(i) Differentiate(ii) fit a into f’(x) to get

the gradient (m) (iii) fit a into f(x) to get

the tangent point (a,b)

(iv) use y-b=m(x-a)

1.3

Question 16

How do you find the rate of change of a function at a particular point ?


(i) differentiate (ii) fit in given x

value

1.3

Question 17

If y is the equation of a curve, what is represented by dy/dx ?


The gradient

1.3

Question 18

How do you find where a curve is increasing ?


(i) differentiate(ii) let f’(x) = 0(iii)solve to find stationary

points(iv) draw nature table(v) read values for which

graph is increasing

1.3

Question 19

How would you find the maximum or minimum value of a function given its equation?


(i) differentiate(ii) let f’(x) = 0(iii) solve to find the

stationary points(iv) draw the nature table(v) read off max or min

1.4

Question 20

Given a rec. relation in the form un+1 = aun + b and 3 consecutive terms, how do you find the values of a and b?


(i) fit 1st term into un and 2nd term into un+1

(ii) fit 2nd term into un and the 3rd term into un+1

(iii) solve simultaneous equations

2.1

Question 21

How do you find the value of a in the polynomial x3+ax2+4x+3 given either a factor of the polynomial, or the remainder when the polynomial is divided by a number ?


(i) do synthetic division(ii)let the expression

= 0 or the remainder(iii) solve the equation

3.2

Question 22

How do you find

∫ (ax + b)n dx ?


(i) increase power by 1(ii) divide by new power(iii) divide by the

derivative ofthe bracket

i.e. (ax+b)n+1

a(n+1)+ C+ C

2.3

Question 23

How do you solve equations of the form

sin2xo = 0.5 ?(0≤x≤360)


(i) decide on the 2 quadrants (sin is +ve)(ii) press INV sin to get

angle(iii) work out your 2 angles

(iv) divide each by 2

2.3

Question 24

How do you solve equations like

cos2xo-5sinxo = 0 ?(0≤x≤360)


(i) fit in 1-2sin2xo for cos2xo

(ii) factorise(iii) solve equation

3.1

Question 25

How do you find a unit vector parallel to a given vector ?


(i) find the length of the given vector

(ii) divide all the components by

this length

2.4

Question 26

How do you prove that a line is a tangent to a circle ?


Rearrange line to makey = or x =

Substitute line into circleProve it has equal roots using b2-4ac = 0 or repeated roots

3.1

Question 27

How do you find the angle between two vectors ?


a.b

a b

a.b

a bcos=

a

b

3.1

Question 28

What is a unit vector ?


A vector of length 1 unitA vector of length 1 unit

3.1

Question 29

What vector is equal to

AB + CD + BC ?


AD

3.1

Question 30

If u = ai+bj+ckthen what is u in component form ?


U =abc

3.2

Question 31

How do you integrate sin ax ?


-1/a cos ax + C

3.2

Question 32

How would you differentiate a function likey = sin3 x ?


(i) write as (sin x)3

(ii) multiply by the power(iii) decrease power by one(iv) multiply by the derivative of

the bracketi.e. 3 sin2x cosx

(i) write as (sin x)3

(ii) multiply by the power(iii) decrease power by one(iv) multiply by the derivative of

the bracketi.e. 3 sin2x cosx

3.3

Question 33

Given experimental data, how do you find an equation in the form y=abx or y=axb ?


(i) take logs of both sides

(ii) rearrange to get a straight line equation

(iii) determine type(iv) Equate and solve for a

and b


(ii) rearrange to get a straight line equation

(iii) determine type(iv) Equate and solve for a

and b

3.2

Question 34

How do you differentiate an expression like

without multiplying it out ?

432 x


(i) multiply by the power

(ii) decrease power by 1

(iii) multiply by derivative of bracket

3.3

Question 35

Given an equation like m = moe-3k and an amount by which it has been decayed, how do you find k ?


(i) fit in m and mo

(ii) rearrange to get e-3k =

(iii) take logs to get -3k =

(iv) solve for k

(i) fit in m and mo

(ii) rearrange to get e-3k =

(iii) take logs to get -3k =

(iv) solve for k

3.3

Question 36

What isloga x + loga y equal to ?


Loga xyLoga xy

3.3

Question 37

How do you solve equations of the form

3x = 0.155 ?



(ii) bring x down to front

(iii) solve the equation


(ii) bring x down to front

(iii) solve the equation

3.3

Question 38

What is loga xn equal to ?


nloga xnloga x

1.3

Question 39

If y =

How should you rewrite y so it is ready to differentiate?

x2

1


1

2

1 x

3.4

Question 40

How do you find the maximum or minimum values ofacosx + bsinx + c ?


(i) change acosx+bsinx into Rcos(x-a)

(ii) max is R+c

1.1 Question 1 How do you find the equation of a perpendicular bisector of a straight line ?

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