N5 practice papers a c with solutions

National 5 Practice Paper A Last updated 04/05/15
M thematics
National 5 Practice Paper A
Paper 1
Duration – 1 hour
Total marks – 40
 You may NOT use a calculator
 Attempt all the questions.
 Use blue or black ink.
 Full credit will only be given to solutions which contain appropriate working.
 State the units for your answer where appropriate.
N5

FORMULAE LIST
The roots of are
Sine rule:
Cosine rule:
Area of a triangle:
Volume of a Sphere:
Volume of a cone:
Volume of a pyramid:
Standard deviation: , where is the sample size.
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0 𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
𝑉 =
4
3
𝜋𝑟3
𝑉 =
1
3
𝜋𝑟2
ℎ
𝑉 =
1
3
𝐴ℎ
𝑎
sin 𝐴
=
𝑏
sin 𝐵
=
𝑐
sin 𝐶
𝑎2
= 𝑏2
+ 𝑐2
− 2𝑏𝑐 cos 𝐴 or cos 𝐴 =
𝑏2
+ 𝑐2
− 𝑎2
2𝑏𝑐
𝐴 =
1
2
𝑎𝑏 sin 𝐶
𝑠 =
𝑥−𝑥 2
𝑛−1
=
𝑥2− 𝑥 2/𝑛
𝑛−1

1. Evaluate 2
2. Factorise 2
+ 2 − 1 . 2
3.
Find the equation of this straight line in the form = + 3
4. Express = 2
+ − in the form = + 2
+ and hence state
the coordinates of the turning point. 3
5. = 3
−
Change the subject of the formula to . 3
3
2
− 1
3
4

6. Two vectors are defined as = (
2
−
) and = (
−4
3
).
(a) Find the resultant vector + 3 . 1
(b) Find | + 3 |. 2
7.
Part of the graph of = cos is shown in the diagram.
State the value of . 1
8. Find the point of intersection of the straight lines with equations
2 + = and − 3 = . 4
9. A parabola has equation = 2
− 3 + .
Using the discriminant, determine the nature of its roots. 3

10. A straight line has the equation 3 − = .
A second line is parallel to this and passes throught the point −3 .
Write down the equation of the second line. 3
11.
The equation of the parabola in the diagram above is = − 2 2
− .
(a) State the coordinates of the minimum turning point of the parabola. 2
(b) Find the coordinates of . 2
(c) is the point −1 0 . State the coordinates of . 1

12. The square and rectangle shown below have the same perimeter.
Show that the length of the rectangle is 3 + 1 centimetres. 2
13. (a) Express as a single fraction in its simplest form. 3
(b) Express 1 − 2 + 2 as a surd in its simplest form. 3
[End of question paper]
3
𝑥
−
𝑥 + 2
𝑥 ≠ 0 𝑥 ≠ 2

M thematics
Paper 2
Duration – 1 hour and 30 minutes
Total marks – 50
 You may use a calculator
 Attempt all the questions.
 Use blue or black ink.
 Full credit will only be given to solutions which contain appropriate working.
 State the units for your answer where appropriate.
N5

1. The population of a city is increasing at a steady rate of 2.4% per annum.
The current population is 528 000.
What is the expected population in 4 years?
Give your answer to the nearest thousand. 3
2. Two groups of 6 students are given the same test.
(a) The marks of Group A are:
73 47 59 71 48 62.
Use an appropriate formula to calculate the mean and the standard
deviation.
Show clearly all your working. 4
(b) In Group B, the mean is 60 and the standard deviation is 29.8.
Compare the results of the two groups. 2
3. Multiply out the brackets and collect like terms.
+ 4 2 2
+ 3 − 1 3

4. Gordon and Brian leave a hostel at the same time.
Gordon walks on a bearing of 045°at a speed of 4.4 kilometres per hour.
Brian walks on a bearing of 100° at a speed of 4.8 kilometres per hour.
If they both walk at stead speeds, how far apart will they be after 2 hours? 5
5. The diagram shows a mirror which has
been designed for a new hotel.
The shape consists of a sector of a
circle and a kite AOCB.
o The circle, centre O, has a
radius of 50 centimetres.
o Angle AOC = 140°
o AB and CB are tangents to the
circle at A and C respectively.
Find the perimeter of the mirror. 5

6. A drinks container is in the shape of a
cylinder with radius 20 centimetres and
height 50 centimetres.
(a) Calculate the volume of the drinks
container.
Give your answer in cubic centimetres,
correct to two significant figures.
3
(b) Liquid from the full container can fill 800 cups, in the shape of cones,
each of radius 3 centimetres.
What will be the height of liquid in each cup? 4
7.
A regular pentagon is drawn in a circle, centre , with radius 10 centimetres.
Calculate the area of the regular pentagon. 5

8. (a) Express 2
(2 −
2 + ) in its simplest form. 2
(b) Use an appropriate formula to solve the quadratic equation
3 2
+ 3 − = 0.
Give your answers correct to 1 decimal place. 4
9. (a) Solve the equation
4 n + = 0, 0 3 0. 3
(b) Show that
n cos = sin . 2

10. A rectangular wall vent is 30 centimetres long and 10 centimetres wide.
It is to be enlarged by increasing both the length and the width by centimetres.
(a) Show that the area, square centimetres, of the new vent is given by
= 2
+ 40 + 300.
The area of the new vent must be at least 75% more than the original area.
(b) Find the minimum dimensions of the new vent. 5

National 5 Practice Paper A - Answers Last updated 04/05/15
Answers
Paper 1
1.
2. ( )( )
3.
4. ( ) , T.P.( )
5. √
6a. ( ) b. 2 29
7.
8. ( )
9. Therefore there are no real roots
10. 3 18y x 
11a. ( ) b. ( ) c. ( )
12. P(rectangle) = P(square)
   2 2 3 4 2 2l x x   
 3 2 2 2l x x   
4 4 3l x x   
3 1l x  as required
13a. ( )
b. √

National 5 Practice Paper A - Answers Last updated 04/05/15
Answers
Paper 2
1. 581 000
2a. ̅ = 60, s = 11.03 (2dp)
2b. On average the marks of both groups are the same since 60 = 60
However, the marks from Group A are much more consistent since 11.03 < 29.8.
3.
4. 8.5 km
5. 466.73 centimetres
6a. 63 000 cm3
b. 8.4 cm (using the answer to part a)
7. 237.76 cm2
8a. b. = 1.1 or -2.1
9a. = 128.66°, 308.66° b. Proof using
sin
tan
cos
x
x
x

10a. Area = length × breadth    2 2
30 10 300 30 10 40 300x x x x x x x          .
10b. length = 35 cm, breadth = 15 cm

National 5 Practice Paper B Last updated 04/05/15
M thematics
National 5 Practice Paper B
Paper 1
Duration – 1 hour
Total marks – 40
o You may NOT use a calculator
o Attempt all the questions.
o Use blue or black ink.
o Full credit will only be given to solutions which contain appropriate working.
o State the units for your answer where appropriate.
N5

FORMULAE LIST
The roots of are
Sine rule:
Cosine rule:
Area of a triangle:
Volume of a Sphere:
Volume of a cone:
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0 𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
𝑉 =
4
3
𝜋𝑟3
𝑉 =
1
3
𝜋𝑟2
ℎ
𝑉 =
1
3
𝐴ℎ
𝑎
sin 𝐴
=
𝑏
sin 𝐵
=
𝑐
sin 𝐶
𝑎2
= 𝑏2
+ 𝑐2
𝑏2
+ 𝑐2
− 𝑎2
2𝑏𝑐
𝐴 =
1
2
𝑎𝑏 sin 𝐶
𝑠 =
𝑥−𝑥 2
𝑛−1
=
𝑥2− 𝑥 2/𝑛
𝑛−1

1. Evaluate
1 − 2 1 3. 2
2. Evaluate
2
3. Solve the inequality − 2 + 1 3
4. Given that = 2
+ , evaluate −3 . 2
5. Vector has components (
3
−2
−1
) and vector has components (
2
−4
1
).
Calculate |4 − 2 |. 2
6. (a) Factorise 2
− 4 2
. 1
(b) Hence simplify .
2
1
1
÷
3
4
𝑝2
− 4𝑞2
3𝑝 + 6𝑞

7.
Find the equation of the straight line shown in the diagram.
Give your answer in the form = + . 3
8.
Part of the graph of = cos is shown above.
If cos 60 = 0 , state two values for for which cos = −0 0 360. 2
9. Multiply out the brackets and collect like terms.
− 3 2
+ 4 − 1 3

10. A sample of students was asked how many times each had visited the cinema
in the last three months.
The results are shown below.
4 5 4 1 4 3 2 2 4 6 2
3 4 4 1 3 1 2 3 1 1
(a) From the above data, find the median, the lower quartile and the
upper quartile. 3
(b) Calculate the semi-interquartile range. 1
(c) The same sample of students was asked how many times each had attended
a football match in the same three months.
The data had a median of 5 and a semi-interquartile range of 3.
Make two appropriate comments comparing students visiting the cinema
and students attending a football match. 2
11. Two functions are given below.
= 2
+ 2 − 1
= + 3
Find the values of for which = . 3
12. Express in its simplest form
2𝑦8
𝑦3 −2

13.
The equation of the parabola in the above diagram is
= − 1 2
− 16.
(a) State the coordinates of the minimum turning point of the parabola. 2
(b) State the equation of the axis of symmetry of the parabola. 1
14. (a) Express 4 − 2 as a surd in its simplest form. 2
(b) Express as a fraction in its simplest form
2
𝑥
𝑦
1
𝑥2
+
1
𝑥
𝑥 ≠ 0

M thematics
Paper 2
Total marks – 50
o You may use a calculator
N5

1. A spider weighs approximately 1 06 10−
kilograms.
A humming bird is 18 times heavier.
Calculate the weight of the humming bird.
Give your answer in scientific notation. 2
2. A microwave oven is sold for £150.
This price includes VAT at 20%.
Calculate the price of the microwave oven without VAT. 3
3. (a) The price, in pence, of a carton of milk in six different supermarkets
is shown below.
66 70 89 75 79 59
Use an appropriate formula to calculate the mean and standard deviation
of these prices.
Show clearly all your working. 4
(b) In six local shops, the mean price of a carton of milk is 73 pence with a
standard deviation of 17.7 pence.
Compare the supermarket prices with those of the local shops. 2

4. A pendulum travels along an arc of a circle, centre C.
The length of the pendulum is 20 centimetres.
The pendulum swings from A to B.
The length of the arc AB is 28.6 centimetres.
Find the angle through which the pendulum swings from A to B. 4
5. A container to hold chocolates is in the shape of part of a cone with
dimensions as shown below.
Calculate the volume of the container.
Give your answer correct to one significant figure. 5

6. Solve the equation
2 2
+ 3 − 1 = 0
Give your answers correct to one decimal place. 4
7. The diagram below shows a circular cross-section of a cylindrical oil tank.
In the figure below,
o O represents the centre of the circle.
o PQ represents the surface of the oil in the tank.
o PQ is 3 metres.
o The radius OP is 2.5 metres.
Find the depth, metres, of oil in the tank. 4

8. The population of Newtown is 50 000.
The population of Newtown is increasing at a steady rate of 5% per annum.
The population of Auldtown is 108 000.
The population of Auldtown is decreasing at a steady rate of 20% per annum.
How many years will it take until the population of Newtown is greater than the
population of Auldtown? 5
9. A TV signal is sent from a transmitter (T) via a satellite (S) to a village (V), as shown
in the diagram. The village is 500 kilometres from the transmitter.
The signal is sent out at an angle of 3 and is received in the village at an
angle of 40 .
Calculate the height of the satellite above the ground. 5
10. Change the subject of the formula to .
= 3 − 2 2

11. Look at the cuboid shown on the coordinate diagram.
The coordinates of point are 3 1
(a) State the coordinates of
(b) State the coordinates of
(c) What is the shortest distance between points and ? 4
12. At the carnival, the height, metres,
of a carriage on the big wheel above
the ground is given by the formula
= 10 + sin ,
seconds after starting to turn.
(a) Find the height of the carriage above the ground after 10 seconds. 2
(b) Find the two times during the first turn of the wheel when the
carriage is 12.5 metres above the ground. 4
𝑧
𝑥
𝑦
𝐴 𝐵
𝐶
𝐷 𝐸
𝐹
𝑂
𝐺

National 5 Practice Paper B - Answers Last updated 04/05/15
Answers
Paper 1
1. 0 88
2.
3.
4. ( )
5. 10
6a. ( )( ) b.
7.
8.
9.
10a. median 1 33, 1 5, 4Q Q    b. 1 25
10c. On average the students went to more football matches than to the cinema since 5 > 3.
The number of times the students visited the cinema was more consistent since 1 25 3  .
11.
12.
13a. Min T.P. ( ) b.
14a. √ 14b.

National 5 Practice Paper B - Answers Last updated 04/05/15
Answers
Paper 2
1. 3.431×10-3
kilograms
2. £125
3a. ̅ = 73, s = 10.5
3b. On average, milk is the same price in both types of shop since 73 = 73.
The price of milk is more consistent in supermarkets since 10 5 17 7   .
4. 81.9°
5. 2000 cm3
6. = 0.3, -1.8
7. = 0.5 metres
8. 3 years
9. 190.8 kilometres
10.
11a. F (5, 3, 0) b. G (0, 3, 0) c. √
12a. 10.9 metres b. 30 seconds and 150 seconds

National 5 Practice Paper C Last updated 04/05/15
M thematics
National 5 Practice Paper C
Paper 1
Duration – 1 hour
Total marks – 40
o You may NOT use a calculator
N5

FORMULAE LIST
The roots of are
Sine rule:
Cosine rule:
Area of a triangle:
Volume of a Sphere:
Volume of a cone:
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0 𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
𝑉 =
4
3
𝜋𝑟3
𝑉 =
1
3
𝜋𝑟2
ℎ
𝑉 =
1
3
𝐴ℎ
𝑎
sin 𝐴
=
𝑏
sin 𝐵
=
𝑐
sin 𝐶
𝑎2
= 𝑏2
+ 𝑐2
𝑏2
+ 𝑐2
− 𝑎2
2𝑏𝑐
𝐴 =
1
2
𝑎𝑏 sin 𝐶
𝑠 =
𝑥−𝑥 2
𝑛−1
=
𝑥2− 𝑥 2/𝑛
𝑛−1

1. Evaluate 5.04 + 8.4 ÷ 7. 2
2. Evaluate 2
3. Simplify 3 2 − 4 − 4 3 + 1 3
4. = − 4
(a) Evaluate −2 1
(b) Given that = , find . 2
5. Solve, by factorising
+ − 2
= 0. 3
2
1
3
4
+
3
8

6. A hotel books taxis from a company called Quick-Cars.
The receptionist notes the waiting time for every taxi ordered over a period
of two weeks. These times, in minutes, are shown below.
12 25 29 37 6 13 26
32 42 7 14 29 35 44
(a) For the given data, calculate:
(i) the median 1
(ii) the lower quartile 1
(iii) the upper quartile 1
(b) Calculate the semi-interquartile range. 1
In another two week period, the hotel books taxis from a company called Fast-Cabs.
The median waiting time for Fast-Cabs is found to be 27.5 minutes and the
semi-interquartile range for Fast-Cabs is found to be 2.5 minutes.
(c) Use this information to compare the two companies. 2
7. Part of the graph of = sin + is shown in the diagram.
State the values of and . 2
𝑦
𝑥𝑂

8. In the diagram below, A is the point −1 − and B is the point 4 3 .
(a) Find the gradient of the line AB. 2
(b) AB cuts the -axis at the point 0 − .
Write down the equation of the line AB. 1
(c) The point 3 lies on AB. Find the value of . 2
9. = 2
+ −
(a) Write in the form + 2
+ . 2
(b) State the coordinates of the turning point of . 1

10. Andrew and Daisy each book in at the Sleepwell Lodge.
(a) Andrew stays for 3 nights and has breakfast on 2 mornings.
His bill is £145.
Write down an algebraic equation to illustrate this information. 1
(b) Daisy stays for 5 nights and has breakfast on 3 mornings.
Her bill is £240.
Write down an algebraic equation to illustrate this information. 1
(c) Find the cost of one breakfast 3
11. (a) Evaluate 2
(b) Simplify 2
(c) Simplify 2
8
2
3
24
2
2𝑥 + 2
𝑥 + 1 2

M thematics
Paper 2
Total marks – 50
o You may use a calculator
N5

1. Bacteria in a test-tube increase at the rate of 4.6% per hour.
At 12 noon, there are 50 000 bacteria.
At 5 pm, how many bacteria will be present?
Give your answer correct to 3 significant figures. 4
2.
The tangent, MN, touches the circle, centre O, at L.
Angle JLN = 47°
Angle KPL = 31°
Find the size of angle JLK. 3
3. Change the subject of the formula
= 3
+ to . 3

4. A mug is in the shape of a cylinder with diameter 10 centimetres and
height 14 centimetres.
(a) Calculate the capacity of the mug. 2
(b) 600 millilitres of coffee are poured in.
Calculate the depth of the coffee in the mug. 3
5. The diagram below shows a big wheel at the fairground.
The wheel has 16 chairs equally spaced on its circumference.
The radius of the wheel is 9 metres.
As the wheel rotates in an anticlockwise direction, find the distance a chair
travels in moving from position T to position P in the diagram. 4

6. Find the roots of the equation
2 2
+ 4 − = 0.
Give your answers correct to one decimal place. 4
7. Two perfume bottles are mathematically similar in shape.
The smaller one is 6 centimetres high and holds 30 millilitres of perfume.
The larger one is 9 centimetres high.
What volume of perfume will the larger one hold? 3
8. Determine the nature of the roots of the equation
− 2 2
− = 0. 4

9. A pony shelter is part of a
cylinder as shown in figure 1.
It is 6 metres wide and 2 metres
high.
The cross-section of the shelter
is a segment of a circle with
centre O, as shown in figure 2.
OB is the radius of the circle.
Calculate the length of OB. 4
10. The diagram shows a parallelogram, PQRS.
(a) Calculate the size of angle PQR. Do not use a scale drawing. 3
(b) Calculate the area of the parallelogram. 3

11. (a) Solve the equation
2 n + = 0 0 3 0 3
(b) Prove that
sin3
+ sin cos2
= sin . 2
12. (a) A driver travels from A to B, a distance of kilometres, at a constant
speed of 75 kilometres per hour.
Find the time taken for this journey in terms of 1
(b) The time taken for the journey from B to A is hours.
Calculate the average speed for the whole journey. 4

National 5 Practice Paper C - Answers Last updated 04/05/15
Answers
Paper 1
Q1. 6 24
Q2.
Q3.
Q4. a 15 b
Q5.
Q6. a (i) 27.5 (ii) Q1 = 13 (iii) Q3 = 35
b 11
c On average, both companies have the same waiting time since 27.5 = 27.5
Waiting times for Fast-Cabs are more consistent since 2.5 < 11
Q7.
Q8. a 2ABm  b c
Q9. a ( ) ( ) b (-3, -16)
Q10. a b
c One breakfast costs £5
Q11. a 4 b √ c

National 5 Practice Paper C - Answers Last updated 04/05/15
Answers
Paper 2
Q1. 62 600
Q2. 102°
Q3. 3
y c
x
a


Q4. a 1100 cm3
b 7.64 cm (2dp)
Q5. 24.74 m
Q6.
Q7. 101.25 ml
Q8. therefore two real and distinct roots
Q9. 3.25 m
Q10. a 78.6° b 92.22 cm2
Q11.
Q11 b  3 2 2 2
sin sin cos sin sin cos sinx x x x x x x    (Since 2 2
sin cos 1x x  )
Q12. a b 60 km/h

N5 practice papers a c with solutions

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