Transcript Higher Mathematics - All Saints Secondary School

Higher Mathematics
or
Everything you wanted to
know about Higher Maths
but …
Higher Mathematics: HappySmile
Productions
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Higher Mathematics: HappySmile
Productions
Question 1
How
do you draw
the graph of –f(x)
given the graph of
f(x) ?
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Higher Mathematics: HappySmile
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answer
Answer to Question 1
Reflect
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question
the graph
in the x-axis
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Question 2
For
what values is
the function
f(x) = √(x-a)
undefined?
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Higher Mathematics: HappySmile
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answer
Answer to Question 2
x
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<a
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question
Question 3
If
f(x) =
then what is
’
f (x) ?
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n
x
Higher Mathematics: HappySmile
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answer
Answer to Question 3
n-1
nx
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question
Question 4
What
is the
y
general equation
of a straight
line passing
through (a,b)
with gradient m ?
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(a,b)
x
answer
Answer to Question 4
y
– b = m(x – a)
y
(a,b)
x
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question
Question 5
If
f(t) is the
distance travelled in
a certain time t
seconds, then what
’
does f (t) represent ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 5
Speed
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Return to
question
(velocity)
Higher Mathematics: HappySmile
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Question 6
What
is an
altitude of a
triangle ?
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Higher Mathematics: HappySmile
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answer
Answer to Question 6
A
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question
line from a vertex,
perpendicular to the
opposite side
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Higher Mathematics: HappySmile
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Question 7
What
is logaa
equal to ?
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Higher Mathematics: HappySmile
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answer
Answer to Question 7
1
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Higher Mathematics: HappySmile
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Return to
question
Question 8
Name
the steps you
take in order to
differentiate
functions like
2+ 3x + 1
x
f(x) =
√x
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Higher Mathematics: HappySmile
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answer
Answer to Question 8
(i)
Change roots to
powers
(ii) split up into 3
fractions
(iii) simplify each term
(iv) differentiate
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question
Question 9
How
do you find
where a curve
crosses the y-axis ?
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Higher Mathematics: HappySmile
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answer
Answer to Question 9
Let
x = 0 and
solve
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question
Question 10
How
do you draw
the graph of 2f(x)
given the graph of
f(x) ?
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Higher Mathematics: HappySmile
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answer
Answer to Question 10
Double
the
y-coordinates
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question
Question 11
How
do you find the
gradient of a line,
given the angle the
line makes with the
positive direction of
the x-axis ?
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answer
Answer to Question 11
m
= tan Θ
y
Θ
x
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question
Question 12
What
is the
equation of a
horizontal line
passing through the
point (a,b) ?
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Higher Mathematics: HappySmile
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answer
Answer to Question 12
y
=b
y
(a,b)
x
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question
Question 13
What
do you know
about the gradient
of two perpendicular
lines ?
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answer
Answer to Question 13
multiply
together to
give -1
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question
They
y
i.e. m1xm2 = -1
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m1
x
m2
Question 14
What
is a median of
a triangle ?
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Higher Mathematics: HappySmile
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answer
Answer to Question 14
A
line from a
vertex to the
middle of the
opposite side
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Higher Mathematics: HappySmile
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question
Question 15
What
is the
gradient of a
horizontal line ?
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Higher Mathematics: HappySmile
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answer
Answer to Question 15
0
y
x
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question
Question 16
How
do you find the
rate of change of a
function at a
particular point ?
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answer
Answer to Question 16
(i)
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question
differentiate
(ii) fit in given x
value
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Question 17
How
do you find the
limit of a recurrence
relation from its
formula ?
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answer
Answer to Question 17
(i)
let un+1 = un = L
(ii) solve equation
Remember:
For un+1=aun+b,
a limit exists if -1<a<1
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question
Question 18
How
do you know
when a recurrence
relation of the form
un+1 = aun + b
converges ?
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answer
Answer to Question 18
-1
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<a<1
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question
Question 19
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?
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answer
Answer to Question 19
(i)
Return to
question
fit 1st term into un and
nd
2 term into un+1
(ii) fit 2nd term into un and
rd
the 3 term into un+1
(iii) solve simultaneous
equations
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Question 20
How
do
you show
that a curve
is always decreasing?
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answer
Answer to Question 20
(i)
differentiate
(ii) show that f’(x) is
negative and
a square
2
e.g. –(x+a)
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question
Question 21
How
would you
find the maximum
or minimum value
of a function given
its equation?
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answer
Answer to Question 21
(i)
Return to
question
differentiate
(ii) let f’(x) = 0
(iii) solve to find the
stationary points
(iv) draw the nature table
(v) read off max or min
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Question 22
Which
two points
does the graph of
y = logax always
pass through ?
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answer
Answer to Question 22
(1,0)
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and (a,1)
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question
Question 23
How
do you find
where a curve is
decreasing ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 23
(i)
(ii)
(iii)
(iv)
(v)
Exit
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differentiate
let f’(x) = 0
solve to find st pts
draw nature table
read values where
graph is decreasing
Higher Mathematics: HappySmile
Productions
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question
Question 24
If
y is the equation
of a curve, what is
represented by
dy/dx ?
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answer
Answer to Question 24
The
Exit
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gradient
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question
Question 25
How
do you find
where a curve
crosses the x-axis ?
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Higher Mathematics: HappySmile
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answer
Answer to Question 25
Let
y=0
or f(x) = 0
and solve for x
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question
Question 26
What
is an
equivalent
expression for
x
y = a using logs ?
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answer
Answer to Question 26
x
Exit
= logay
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question
Question 27
How
do you
show that
a curve is
always increasing ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 27
(i)
Return to
question
Differentiate
’
(ii) show that f (x) is a
perfect square
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Higher Mathematics: HappySmile
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Question 28
How
do you find the
equation of a tangent
to a curve at the point
when x = a ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 28
(i)
Return to
question
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)
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Question 29
For
what values of
a function is the
function said to be
undefined ?
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Productions
answer
Answer to Question 29
When
Return to
question
you fit in a value
of x and you cannot get
an answer
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Question 30
How
do you draw
the graph of f(x-1)
given the graph of
f(x) ?
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answer
Answer to Question 30
Move
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question
the graph 1 unit
to the right
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Question 31
How
do you find
f(g(x)) for given
functions f(x) and
g(x) ?
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Productions
answer
Answer to Question 31
Fit
g(x) into f(x)
i.e. each x in f(x) is
replaced by the
function g(x)
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question
Question 32
What
two things do
you require in order
to find the equation
of a straight line ?
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Productions
answer
Answer to Question 32
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question
The
gradient of the
line and a point on the
line
y
1
m
(a,b)
x
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Question 33
How
do you find the
midpoint of a line
joining two points ?
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Productions
answer
Answer to Question 33
the coordinates
and divide by two
x+x , y+y
1
2
1
2
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question
Add
(
2
2
)
(x1,y1)
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y
(x2,y2)
x
Question 34
What
is the
gradient of a
vertical line ?
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answer
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question
Answer to Question 34
undefined
y
x
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Question 35
How
do you find the
median AM of
triangle ABC ?
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Productions
answer
Answer to Question 35
(i)
(ii)
(iii)
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find the
mid point
of BC (M)
C
M
find the
gradient of AM
use y-b = m(x-a)
Higher Mathematics: HappySmile
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question
A
B
Question 36
Which
two points
does the graph
x
y = a always pass
through ?
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answer
Answer to Question 36
(0,1)
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and (1,a)
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question
Question 37
What
is the
perpendicular
bisector of a line ?
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answer
Answer to Question 37
A
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question
line which cuts the
o
given line in half at 90
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Question 38
How
do you draw
the graph of f(x+1)
given the graph of
f(x) ?
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Productions
answer
Answer to Question 38
Move
Return to
question
the graph 1 unit
to the left
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Question 39
How
do you find the
equation of a
perpendicular
bisector of a line ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 39
 (i)
(ii)
(iii)
(iv)
find the midpoint of the line
find the gradient of the line
find the gradient
perpendicular to the given
line
Use midpoint and gradient in
y-b = m(x-a)
M(a,b)
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question
Higher Mathematics: HappySmile
Productions
Question 40
For
what values is
this function
undefined ?
x
f(x) =
(x+2)(x-3)
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Higher Mathematics: HappySmile
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answer
Answer to Question 40
-2
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and 3
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Question 41
What
are the two
formulae used to
find the area of a
triangle ?
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answer
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question
Answer to Question 41
A
= ½base x height
A = ½bcsinA
A
b
C
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B a s e
height
a
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c
B
Question 42
What
three
processes do you go
through in order to
factorise a
quadratic ?
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answer
Answer to Question 42
(i)
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question
common factor
(ii) difference of two
squares
(iii) trinomial
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Question 43
What
is the
equation of a
vertical line passing
through (a,b) ?
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Productions
answer
Answer to Question 43
x
=a
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question
y
(a,b)
x
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Question 44
What
is the
Theorem of
Pythagoras ?
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answer
Answer to Question 44
For
ΔABC,
right-angled at A,
2
2
2
a =b +c
a
B
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C
b
c
Higher Mathematics: HappySmile
Productions
A
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question
Question 45
What
do you know
about the gradients
of two parallel
lines?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 45
They
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are the same
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Return to
question
Question 46
How
do you draw
the graph of f’(x)
given the graph of
f(x) ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 46
 (i)
plot x coords of st. points on
x-axis (SPs become roots)
(ii) look at each part of f(x)
separately:
if rising, graph of f’(x) is
above x-axis
if falling, graph
of f’(x) is
below x-axis
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question
Question 47
How
do you get the
gradient of a line
with an equation like
3x + 2y = 5 ?
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Productions
answer
Answer to Question 47
(i)
Return to
question
Rearrange into the
form
y = mx + c
(ii) read off
gradient = m
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Question 48
What
is loga1
equal to ?
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answer
Answer to Question 48
0
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question
Question 49
How
do you find the
length of a line
joining two points ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 49
√(x2
–
2
x1)
y
+
2
(y2 –y1)
B(x2,y2)
A(x1,y1)
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x
Return to
question
Question 50
What
is the
Converse of
Pythagoras ?
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Productions
answer
Answer to Question 50
If
2
a
2
b
=
+
then ΔABC is
right-angled at A
2
c
a
B
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question
Higher Mathematics: HappySmile
Productions
C
b
c
A
Question 51
How
do you find the
gradient of a line
joining two points ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 51
m
= y2 – y1
x2 – x 1
y
A(x1,y1)
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question
B(x2,y2)
x
Question 52
How
do you find the
altitude AN of
ΔABC ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 52
(i)
(ii)
(iii)
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find the gradient
of BC
find the gradient
A
of AN,
perpendicular
to BC
B
N
use y-b=m(x-a)
Higher Mathematics: HappySmile
Productions
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question
C
Question 53
For
a curve, how do
you find the
stationary points
and their nature ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 53
(i)
differentiate
(ii) let f’(x) = 0
(iii) solve to find
stationary points
(iv) find y-coordinates
(v) draw nature table
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question
Question 54
How
do you draw
the graph of 3+f(x)
given the graph of
f(x) ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 54
move
Exit
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graph up 3
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question
Question 55
How
do you find
where a curve is
increasing ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 55
(i)
(ii)
(iii)
(iv)
(v)
Exit
Exit
Return to
question
differentiate
let f’(x) = 0
solve to find stationary
points
draw nature table
read values for which
graph is increasing
Higher Mathematics: HappySmile
Productions
Question 56
How
do you find
where two lines
intersect ?
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answer
Answer to Question 56
Simultaneous
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Return to
question
equations
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Question 57
How
do you draw
the graph of 3-f(x)
given the graph of
f(x) ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 57
Reflect
the graph in
the x-axis,
then move it up 3
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question
Question 58
How
do you draw
the graph of f(-x)
given the graph of
f(x) ?
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answer
Answer to Question 58
Reflect
the graph in
the y-axis
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question
Question 59
How
do you solve
equations like
100 = 0 ?
42
x
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answer
Answer to Question 59
(i)
Return to
question
multiply by the
denominator of the
2
fraction (here x )
(ii) factorise and solve
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Question 60
How
do you find the
exact values of
sin(A+B), cos(A-B) etc.
given that
3
cosA = /5 and
12
sinB = /13 ?
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Productions
answer
Answer to Question 60
 (i)
 (ii)
 (iii)
 (iv)
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draw
two Δs
find
missing sides
expand
formula
fit in values
from Δs
A
5
3
B
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Productions
13
12
Return to
question
Question 61
How
do you solve
equations like
o
o
Cos2x - 5cosx = 2 ?
(0 ≤ x ≤ 360)
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answer
Answer to Question 61
(i)
o
2
2cos x -1
Return to
question
fit in
for
o
cos2x
(ii) factorise
(iii) solve the equation
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Question 62
What
is
sin x
cos x
equal to ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 62
tan
Exit
Exit
x
Higher Mathematics: HappySmile
Productions
Return to
question
Return to
question
Question 63
How
do you show
that x-1 is a factor
of the function
3
f(x)=x -3x+2 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 63
(i)
Return to
question
rewrite the function as
3
2
f(x)=x +0x -3x+2
(ii) use synthetic division
with 1 on the outside
(iii) show that
remainder = 0
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 64
What
is the
sine rule ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 64

a
b
c
=
=
sinA sinb sinC
A
c
b
B
C
Exit
Exit
a
Higher Mathematics: HappySmile
Productions
Return to
question
Question 65
Given
f’(x) and a
point on the curve,
how do you find
f(x) ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 65
(i)
Return to
question
integrate
(ii) fit in given point
to work out value
of C
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 66
How
do you solve
quadratic
inequations like
2
x - 5x + 6 ≤ 0 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 66
(i)
factorise
(ii) draw graph
(iii) read values
below x-axis
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 67
How
do you change
from radians to
degrees ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 67

Exit
Divide by π and
multiply by 180
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 68
What
is the
condition for real
roots ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 68
2
b
Exit
Exit
– 4ac ≥ 0
Higher Mathematics: HappySmile
Productions
Return to
question
Question 69
How
do you find the value of
a in the polynomial
x3+ax2+4x+3 given a factor
of the polynomial or the
remainder when the
polynomial is divided by a
number ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 69
Return to
question
(i) do synthetic division
(ii) let the expression
= 0 or the remainder
(iii) solve the equation

Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 70
How
do you find f(x) if
f’(x) = 5-3x2 and
the curve passes through
the point (1,9) ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 70
(i)
(ii)
Return to
question
f(x) = ∫f'(x) dx
find C by replacing
point (1,9) into f(x)
(iii) write down completed
formula for f(x)

Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 71
What
is
2
2
sin x + cos x
equal to ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 71

Exit
1
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 72
How
do you find the
equation of the
tangent to a circle at a
particular point on the
circumference ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 72

(i) find the
centre
Return to
question
y
(a,b)
(ii) find gradient
from centre
x
to point
(iii) find perpendicular gradient
(iv) use y-b=m(x-a)
C
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 73
How
do you find
2
x + 1 dx ?
∫ √x
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 73

(i) change root to
power
Return to
question
(ii) split up into fractions
(iii) simplify each term
(iv) integrate each term
(v) REMEMBER +C
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 74
How
do you show
that the root of a
function lies
between two given
values ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 74

fit in two values and
show one is positive
and one is negative
+ve
x
-ve
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 75
How
do you find
exact values of
sin2x and cos2x
3
given cosx = /5 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 75

(i)
(ii)
(iii)
(iv)
Exit
Exit
draw a
right-angled
triangle
find the
missing side
expand the double
angle formula
fit in values from Δ
Higher Mathematics: HappySmile
Productions
Return to
question
5
3
A
Question 76
What
is the turning
point of
2
y=2(x-a) +b ?
Max or min ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 76

(i) (a,b)
minimum
(a,b)
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 77
How
do you
n
integrate x ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 77
n+1
x

n+1
Exit
Exit
+C
Higher Mathematics: HappySmile
Productions
Return to
question
Question 78
How
do you solve
equations like
o
o
cos2x -5sinx = 0 ?
(0≤x≤360)
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 78
Return to
question
o
2
1-2sin x
o
(i) fit in
for cos2x
(ii) factorise
(iii) solve equation

Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 79
How
do you
complete the square
for functions like
2
2x + 12x + 3 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 79
(i) multiply out
2
a(x+p) +q
(ii) compare with
given function
(iii) find a, p and q

Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 80
How
do you solve
equations of the form
o
sin2x = 0.5 ?
(0≤x≤360)
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 80
(i) decide on the
2 quadrants (sin is +ve)
(ii) press INV sin to get
angle
(iii) work out your 2 angles

(iv)
Exit
Exit
divide each by 2
Higher Mathematics: HappySmile
Productions
Return to
question
Question 81
How
do you solve
quadratic inequations
like
2
x +5x-6 ≥ 0 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 81
(i) factorise
(ii) draw graph
(iii) read values
above x-axis

Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 82
What
is the centre
and radius of a
circle with equation
2
2
2
x +y =r ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 82
(i) centre (0,0)
(ii) radius = r

Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 83
How
do you
calculate the area
under a curve ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 83
Return to
question
(i) integrate
(ii) fit in two limits
and subtract to
find area

Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 84
How
do you find the
root of an equation
between two given
values to 1 dp ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 84
iteration
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 85
How
do you solve
equations of the form
o
2
sin x = 0.5 ?
(0≤x≤360)
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 85
(i)
Return to
question
rearrange to get
o
sinx = ± …
(ii) find answers in
all 4 quadrants
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 86
How
do you name the
angle between a line
and a plane ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 86
 (i)
 (ii)
start at end of line (A)
go to where line meets A
the plane (B)
C
 (iii) go to the point B
on the plane
directly under
the start of the line (C)

ABC
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 87
What
is the condition
for equal roots ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 87
2
b –
Exit
Exit
4ac = 0
Higher Mathematics: HappySmile
Productions
Return to
question
Question 88
What
is the turning
point of
2
y = b-3(x-a) ?

max or min ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 88
(a,b)
(a,b)
Maximum
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 89
What
is the quadratic
formula and explain
when it is used ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 89
x
=
2
-b±√(b -4ac)
Return to
question
2a
It is used to find roots
of a quadratic equation
when it is difficult to
factorise.
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 90
How
do you prove that
a line is a tangent to a
circle ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 90
Rearrange
Return to
question
line to make
y = or x =
Fit line into circle
Prove it has equal roots
2
using b -4ac = 0 or
repeated roots
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 91
How
do you find the
exact value of
sin (α-β),
4
given that sinα = /5
12
and cosβ = /13 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 91
 (i)
draw triangles
for α and β
 (ii) work out
cosα and sinβ
expand
formula for sin(α-β)
 (iv) insert exact values
α
5
4
13
 (iii)
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
12
β
Question 92
How
do you solve
equations of the form
o
cosx = - 0.8 ?
(0≤x≤360)
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 92
Return to
question
(i) decide on the
2 quadrants (cos is -ve)
(ii) ignore the sign and
press INV cos to get
angle
(iii) work out your 2 angles

Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 93
How
do you change
from degrees to
radians ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 93
Divide
by 180 and
multiply by π
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 94
How
do you find the
exact values of sin x
or tan x given
cos x = a ?
b
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 94
(i)
draw triangle
(ii)
b
a
use Pythagoras
to fill in missing side
(iii) read values off
triangle using
SOHCAHTOA
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
x
Question 95
How
do you factorise
a cubic expression like
3
2
x -2x -x+2 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 95
Return to
question
Synthetic
division
using factors of last
number
factor
1
-2
-1
2
Remainder=0
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 96
What
is the centre
and radius of a circle
of the form
2
2
x +y +2gx+2fy+c=0 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 96
Centre
(-g,-f)
2
2
Radius √(g +f -c)
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 97
How
do you remember
the exact values of
o
o
o
30 , 45 and 60 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Return to
question
Answer to Question 97
sin30
o
=½
Draw right-angled
triangle
Complete using Pythagoras
Do similar
o
for tan 45 =1
60o
1
2
30o
√3
45o
√2
1
Exit
Exit
Higher Mathematics: HappySmile
Productions
1
45o
Question 98
How
do you calculate
the area between two
curves ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 98
let equations equal
each other
(ii) solve to find limits
(iii) integrate
Return to
question
(i)
(upper - lower)
functions between limits
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 99
How
do you solve an
equation like
3sinx+1 = 0 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 99
(i)
Return to
question
rearrange to sinx =
(ii) decide on 2 quadrants
(iii) ignore any –ve and press
INV sin to get angle
(iv) work out two answers
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 100
What
is the condition
for no real roots ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 100
2
b
Exit
– 4ac < 0
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 101
How
b

Exit
∫
a
Exit
do you find
3
x
dx ?
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 101
3+1
x
[ 3+1


Exit
then
Exit
]
1
Return to
question
b
a
4
/4[(b )
-
Higher Mathematics: HappySmile
Productions
4
(a )]
Question 102
How
do you find
where a line and a
circle intersect ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 102
Rearrange
Return to
question
line to get
x = … or y = …
Fit into circle and solve
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 103
State
the cosine rule
to find an angle
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 103
cos
A=
2
b
2
c
2
a
+ 2bc
A
c
b
B
C
Exit
Exit
a
Higher Mathematics: HappySmile
Productions
Return to
question
Question 104
What
is the centre
and radius of a circle
of the form
2
2
2
(x-a) +(y-b) = r ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Return to
question
Answer to Question 104
Centre
(a,b)
Radius = r
y
r
C (a,b)
x
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 105
State
the cosine rule
to find a missing side
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 105
2
a
=
2
2
b +c -2bccosA
A
c
b
B
C
Exit
Exit
a
Higher Mathematics: HappySmile
Productions
Return to
question
Question 106
How
do you find
n
∫ (ax + b) dx ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 106
 (i)
 (ii)
 (iii)
n+1
 i.e.
Exit
increase power by 1
divide by new power
divide by the
derivative of
the bracket
(ax+b)
a(n+1)
Exit
+ C
Higher Mathematics: HappySmile
Productions
Return to
question
Question 107
How
do you find
the coordinates of a
point which divides a
line in a ratio e.g.
3:2 ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 107

(i)
(ii)
(iii)
(iv)
(v)
Exit
Exit
write in form AB = 3
BC 2
cross-multiply
write AB = (b-a)
solve to find missing
vector
rewrite as point (*,*)
Higher Mathematics: HappySmile
Productions
Return to
question
A
3
B
2
C
Question 108
What
is a
position vector ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 108
A vector
Return to
question
which starts
at the origin
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 109
How
do you express
acosx+bsinx+c
in the form
kcos(x-α) etc?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Return to
question
Answer to Question 109
(i) expand brackets
and equate like terms S A
(ii) find k =√(a2+b2)
T C
(iii) identify quadrant α is in
(iv)
Exit
Exit
find α , tanα = sinα
cosα
Higher Mathematics: HappySmile
Productions
Question 110
How
do you
differentiate a
bracket without
multiplying it out ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 110
(i)
Return to
question
multiply by old power
(ii) decrease power by 1
(iii) multiply by
derivative of bracket
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 111
What
is
Logax – logay
equal to ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 111
x
loga
y

Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 112
What
do you get
when you
differentiate cosx ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 112
-sinx
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 113
How
do you show
that two vectors are
perpendicular ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 113
Show
that a.b=0
a
b
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 114
How
do you
integrate sin ax ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 114
1
- /
a
Exit
Exit
cos ax + C
Higher Mathematics: HappySmile
Productions
Return to
question
Question 115
How
do you draw a
graph of the form
y = acosx
or y = asinx ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 115
Draw
Return to
question
y = cosx
or y = sinx graph
with a maximum of a
and a minimum of -a
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 116
How
do you find the
maximum or
minimum values of
acosx + bsinx + c ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 116
(i)
change
acosx+bsinx into
Rcos(x-a)
(ii) max is R+c
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 117
How
do you find a
unit vector parallel
to a given vector ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 117
(i)
find the length of
the given vector
(ii) divide all the
components by
this length
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 118
How
do you
integrate cos ax ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 118
1
 /a
Exit
Exit
sin ax + C
Higher Mathematics: HappySmile
Productions
Return to
question
Question 119
How
do you draw a
graph of the form

y = cos(x+a)
 or y = sin(x+a) ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 119
Move
the graph of
y=cosx or y=sinx
a units to the LEFT
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 120
What
is a unit
vector ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 120
A vector
of length 1
unit
Exit
Exit
Higher Mathematics: HappySmile
Productions
Return to
question
Question 121
How
do you draw a
graph of the form

y = cos bx
 or y = sin bx ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 121
Draw
Return to
question
the normal graph
but fit in b waves
o
o
between 0 and 360
Exit
Exit
Higher Mathematics: HappySmile
Productions
Question 122
What
is
loga x + loga y equal
to ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 122
Loga
Exit
Exit
xy
Higher Mathematics: HappySmile
Productions
Return to
question
Question 123
What
do you get
when you
differentiate sin x ?
Exit
Exit
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 123
cos
Exit
Exit
x
Higher Mathematics: HappySmile
Productions
Return to
question
Question 124
How
do you find the
angle between two
vectors ?
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Higher Mathematics: HappySmile
Productions
answer
Answer to Question 124

a.b
cosq =
ab
a
q
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b
Return to
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Question 125
Given
an equation
-3k
like m = moe and
an amount by which
it has been decayed,
how do you find k ?
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Productions
answer
Answer to Question 125
(i)
Return to
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fit in m and mo
-3k
(ii) rearrange to get e
=
(iii) take logs
(iv) solve
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Question 126
If
u = ai+bj+ck
then what is u in
component form ?
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answer
Answer to Question 126
U=
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a
b
c
Higher Mathematics: HappySmile
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question
Question 127
What
do you get
when you
differentiate
cosax ?
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Productions
answer
Answer to Question 127
-asinax
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question
Question 128
How
do you solve an
equation of the form
acosx + bsinx + c=0 ?
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answer
Answer to Question 128

Return to
question
Change acosx+bsinx
into Rcos(x- a)
Rearrange and solve
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Question 129
What
is loga
equal to ?
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n
x
Higher Mathematics: HappySmile
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answer
Answer to Question 129

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nloga x
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question
Question 130
How
would you
differentiate a
function like
3
y = sin x ?
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answer
Answer to Question 130
3
x)
(i) write as (sin
(ii) multiply by the power
(iii) decrease power by one
(iv) multiply by the derivative
of the bracket
2
i.e. 3cosx sin x

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Question 131
State
the three
rules of logs ?
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answer
Answer to Question 131

Return to
question
(i) logaxy = logax + logay
x
(ii) loga
=
log
x
–
log
y
a
a
y
(iii)
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n
logax = nlogax
Higher Mathematics: HappySmile
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Question 132
How
do you solve
equations of the
form
x
3 = 0.155 ?
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answer
Answer to Question 132

Return to
question
(i) take logs of both sides
(ii) bring x down to front
(iii) solve the equation
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Question 133
Given
experimental
data, how do you
find an equation in
x
the form y=ab or
b
y=ax ?
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Productions
answer
Answer to Question 133

Return to
question
(i) take logs of both sides
(ii) rearrange to get a
straight line equation
(iii) determine type
(iv) find solution
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Question 134
How
would you
differentiate a
function like
y = sin ax ?
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Higher Mathematics: HappySmile
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answer
Answer to Question 134
dy
 /dx
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= acos ax
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question
Question 135
a
If u = b
c
then what is u ?
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answer
Answer to Question 135
work out length
2
2
2
√(a +b +c )

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question
Question 136
How
do you add or
subtract vectors ?
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Productions
answer
Answer to Question 136
add or subtract
matching
components

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question
Question 137
What
does
a.a equal ?
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answer
Answer to Question 137
2
 a
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Question 138
How
do you prove
that three 3-D
points are
collinear ?
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Productions
answer
Answer to Question 138
Return to
question
Prove they are the
same vector
multiplied by different
or the same numbers

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Question 139
Express
the
n
equation y=kx in the
form of the equation
of a straight line,
Y=nX+c.
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Productions
answer
Answer to Question 139

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question
logy = nlogx + logk
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Question 140
Who
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loves maths ?
Higher Mathematics: HappySmile
Productions
answer
Answer to Question 140

Exit
ME !!!!!!
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Return to
question
Acknowledgements:
 Presentation by
B. McKenna
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