SCHOLAR
National 5 Mathematics
Revision Session
Thursday 14th 7:30pm
You will need a pencil, paper and a calculator for some of the activities
Margaret Ferguson
SCHOLAR online tutor for Maths
and
Author of the SCHOLAR National 5 & Higher courses
Tonight’s Session will cover
 Hints & Tips for the Exam
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Vectors Journeys
Completing the Square
Algebraic Factions
The Straight Line
Trig Identities
Surds & Indices
Percentages
Simultaneous equations
Functions
Problem solving involving Triangles and Circles
Statistics
Similarity
You will need paper, a pencil and a calculator for some of the activities and I
will invite you to vote for the correct answer during the presentation.
The National 5 Maths Exam
Paper I
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Paper I is the non-calculator paper and lasts for 1 hour
There are usually 13 questions and the paper is worth 40 marks
Try to complete questions 1 - 4 in the first 10 minutes
Allow no more that 5 minutes for each of the remaining questions
This will leave you 5 minutes at the end to go over the paper.
Paper II
 Paper II is the calculator paper and lasts for 1 hour 30 minutes
 There are usually 13 questions and the paper is worth 50 marks
 If you can’t do a question highlight it and come back to any
highlighted questions when you have finished the paper.
Formula List
The roots of ax2 + bx + c = 0 are x =
2
b
( - 4ac)
-b ±
2a
Sine rule:
a
b
c
=
=
sin A sin B sinC
Cosine rule:
a2
Area of a triangle:
A = ½ ab sin C
=
b2
+
c2
2
2
2
b
+
c
a
– 2bc cos A or cos A =
2bc
Volume of a sphere: V = 4 p r 3
3
Volume of a cone:
V = ⅓πr2h
Volume of a pyramid: V = ⅓Ah
Standard deviation:
s=
å( x - x )
n -1
2
å x - (å x )
2
=
n -1
2
/n
, where n is the sample size.
 In the diagram below, ABC is an isosceles triangle and M
is the mid-point of BC.
Express
in terms of u and w.
A
u
w
What is
in terms of u and w? w + u
What is
in terms of u and w? ½(w + u)
= ½w + ½u
What is
B
M
in terms of u and w?
C
= -w + ½w + ½u
= ½u - ½w
Express x2 + 5x + 7 in the form (x + a)2 + b
Vote for the correct answer now
a)
b)
æ
5ö 3
çx + ÷ +
è
2ø 4
2
✓
x2 + 5x + 7 = (x2 + 5x)
c)
d) (x + 2.5)2 + 13.25
+7
æ 5ö æ 5ö
= çx + ÷ -ç ÷
è 2ø è2ø
2
æ
ö
5
25
= çx + ÷ è 2ø
4
2
æ 5ö
1
= çx + ÷ - 6
è 2ø
4
2
+7
+7
2
æ
5ö 3
= çx + ÷ +
è 2ø 4
2
+7
x - 2x - 3
Simplify
2
x -9
2
Factorise x2 – 2x – 3, what do you get? (x – 3)(x + 1)
Factorise x2 – 9, what do you get?
(x – 3)(x + 3)
x 2 - 2x - 3
Replace your answers in
, what do you get?
2
x -9
(x - 3)(x +1) x +1
=
(x - 3)(x + 3) x + 3
What is the gradient of the line with
equation 7x – 2y = 5
Vote for the correct answer now
a) -7
7x – 2y = 5
b) 7
c) d)
7
2
7
2
✓
-2y = -7x + 5
-7
5
y=
x+
-2
-2
7
5
y= x2
2
Prove that tan2 θ - tan2 θsin2 θ = sin2 θ
LHS = tan2 θ - tan2 θsin2 θ
What do you do first?
= tan2 θ(1 – sin2 θ)
What is 1 – sin2 θ the same as?
= tan2 θ x cos2 θ
What is tan2 θ the same as?
sin q
2
=
´
cos
q
2
cos q
What can we do now?
2
= sin2 θ
= RHS
sinq
Remember tanq =
and cos2 q + sin 2 q =1
cosq
8
Rationalise the denominator and simplify fully
3 2
Vote for the correct answer now
a)
b)
c)
d)
3
4
2
3
8
3
4
3
✓
8
8
2
=
´
3 2 3 2
2
16
=
3 4
4 2
4
= =
=
3´ 2 6 3
-½
(2a
-½
3)(a
Expand
+ 2) giving your
answer in its simplest form.
(2a-½ - 3)(a-½ + 2) = 2a-1 + 4a-½ - 3a-½ - 6
= 2a-1 + a-½ - 6
2 1
= +
-6
a
a
In a sale all items are reduced by 20%.
If Graham bought a guitar in the sale for £319.20, how
much was the full price?
Vote for the correct answer now
a) £399.00
✓
b) £383.04
What percentage is £319.20?
c) £255.36
d) £351.12
£319.20 = 80%
What is 1%?
What is 100%?
1% = 319.20 ÷ 80 = 3.99
3.99 x 100 = £399.00
A theatre sells seats in the stalls and the circle.
For opening night of a new play all 640 seats were sold.
Seats in the stalls costs £42 and seats in the circle cost £48.
The total income from tickets was £28632.
How many seats are there in the stalls and the circle
respectively?
Let s be the number of seats in the stalls and c in the circle?
s + c = 640 1
42s + 48c = 28632
What can we do now?
x (-42)
2
-42s - 42c = -26880
+ 42s + 48c = 28632
6c = 1752
c = 292
Let c = 292 in equation 1.
s + 292 = 640
s = 348
There are 348 seats in the stalls and 292 in the circle.
The function f(x) = x2 – 12x + 16
What is the value of x when the function is equal to -20?
Vote for the correct answer now
a) 176
b) -144
c) 18
d) 6
f(x) = -20  x2 – 12x + 16 = -20
x2 – 12x + 36 = 0
How do we find x?
✓
Factorise
(x – 6)(x – 6) = 0
x–6=0
x=6
The shape of the Tyne swing bridge is an arc of a circle.
The length of the span is 86m and the radius of the circle it
was formed from is 84m.
What is the height of the span of the bridge?
What can we add to the diagram?
86m
h
43
What is the length of x?
x = 84 - 43
2
2
x = 72.16
x
84m
How do we find h, the height of the span?
h = radius - x
h = 84 – 72.16 = 11.84m
The mean daytime temperature on the island of
Aruba is 31°C with a standard deviation of 0.7.
The mean daytime temperature in Hawaii is 29°C
with a standard deviation of 1.9.
Make two valid comparisons between the temperature
recordings on the two islands.
What what can you say about the average temperatures?
The average temperature on Aruba is slightly higher than
Hawaii because the mean is 2°C higher.
What what can you say about the standard deviations?
The temperature on Aruba is more consistent than Hawaii
because the standard deviation is lower.
Why might that be important?
The two bottles of perfume are said to be similar.
Calculate the cost of the large bottle.
Do we need a scale factor for
enlargement of reduction?
10.8cm
4.8cm
Enlargement
What is the length scale factor?
£??.??
10.8
What does the price depend on?Volume 3
4.8
æ 10.8 ö
What is the volume scale factor? ç
÷
£19.80
è 4.8 ø
What is the cost of the large bottle?
æ 10.8 ö
ç
÷ ´19.80
è 4.8 ø
3
= £225.53
Question Time
• Make sure you arrive in good time for the exam
• Check the batteries in your calculator, if in doubt get new
ones
• Get plenty of practice over the next few days
• If you are stuck on anything go and see your teacher
• Don’t waste time in the exam if you are stuck
• Carry on and go back to these questions at the end
• Good luck
• I hope to see you next year for Higher
• If you have any questions about the National 5 course or
the exam ask now
• Please give us feedback by following the link provided