N5 practice papers d f with solutions

National 5 Practice Paper D Last updated 04/05/15
M thematics
National 5 Practice Paper D
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:
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. Find the gradient of the straight line joining points A(3, -7) and B(-5, 3) 2
2. Evaluate 3
3.
Find the equation of the straight line AB. 3
2
5
of 3
1
2
+
4
5

4. The marks scored by a group of students in their October test are listed below.
41 56 68 59 43 37 70 58 61 47 75 66
(a) Calculate:
(i) The median mark;
(ii) The semi-interquartile range for the data. 2
The teacher arranges extra revision classes for the students before their next
test in December.
In this test the median mark is 67 and the semi-interquartile range is 7.
(b) Make two appropriate comments comparing the marks in the October
and December tests. 2
5. Marmalade is on special offer.
Each jar on special offer contains 12.5% more than the standard jar.
A jar on special offer contains 450 g of marmalade.
How much does the standard jar contain? 3

6. The diagram shows a square based pyramid .
Express ⃗⃗⃗⃗⃗ in terms of and . 3
7. William Watson Fast Foods use a logo based on parts of three identical parabolas.
The logo is represented on the diagram below.
The first parabola has turning point P and equation = + 2 2
− 1
(a) State the coordinates of P. 2
(b) If R is the point (2,0), find the coordinates of Q, the minimum turning
point of the second parabola. 1
(c) Find the equation of the parabola with turning point S. 2

8. Write as a single fraction in its simplest form. 3
9. Part of the graph of = cos is shown in the diagram.
State the values of and . 2
10. Evaluate 2 + 3−1
. 2
11. Express 12 + 5 3 − 2 as a surd in its simplest form. 3
3
𝑚
+
4
𝑚 + 1

12. A piece of gold wire 10 centimetres long is made into a circle.
The circumference of the circle is equal to the length of the wire.
Show that the area of the circle is exactly
2
square centimetres. 4
13. Two variables and are connected by the relationship = + .
Sketch a possible graph of against to illustrate this relationship when
and are both less than zero. 3
[End of question paper]

M thematics
Paper 2
Duration – 1 hour and 30 minutes
Total marks – 50
o You may use a calculator
N5

Q1. The average Scottish house price is £153 100.
The average price is expected to rise by 2.5% per month.
What will the average Scottish house price be in 3 months?
Give your answer correct to three significant figures. 3
Q2. The heights, in millimetres, of six seedlings are given below.
15 18 14 17 16 19
(a) Calculate:
(i) the mean; 1
(ii) the standard deviation; 3
of these heights.
Show clearly all your working.
(b) Later the same six seedlings are measured again.
Each has grown by 4 millimetres.
State:
(i) the mean; 1
(ii) the standard deviation; 1
of these new heights.
Q3. Solve algebraically the equation
4 sin + 1 = −2 0 3 0. 3

Q4. A sports centre charges different entrance fees for adults and children.
(a) One evening 14 adults and 4 children visited the sports centre.
The total collected in entrance fees was £55.00.
Let pounds be the adult entrance fee and pounds be the child’s
entrance fee.
Write down an equation in and to describe the above information. 1
(b) The following evening 13 adults and 6 children visited the sports centre.
The total collected in entrance fees was £54.50.
Write down a second equation in and . 1
(c) Calculate the entrance fee for an adult and the entrance fee for a child. 4
Q5. The diagram below shows part of a parabola with equation of the form
= + 2
+ .
(a) Write down the equation of the axis of symmetry of the graph. 1
(b) Write down the equation of the parabola. 2
(c) Find the coordinates of C. 2

Q6. A garden, in the shape of a quadrilateral, is represented in the diagram.
Calculate:
(a) the length of the diagonal BD;
Do not use a scale drawing. 3
(b) the area of the garden.
Q7. The curved part of a doorway is an arc of a circle with radius 500 millimetres
and centre C.
The height of the doorway to the top of the arc is 2000 millimetres.
The vertical edge of the doorway is 1800 millimetres.
Calculate the width of the doorway. 5

Q8. Find the values of such that the equation 2 2
− + 2 = 0 has equal roots. 4
Q9. A gift box, 8 centimetres high, is prism shaped.
The uniform cross-section is a regular pentagon.
Each vertex of the pentagon is 10 centimetres from the centre O.
Calculate the volume of the box. 5
Q10. Express
4 3
, 3, 0,
3
x x
x x
   

as a fraction in its simplest form. 3

Q11. A rectangular lawn has a path, 1 metre wide, on 3 sides as shown.
The breadth of the lawn is metres.
The length of the lawn is three times its breadth.
The area of the lawn equals the area of the path.
(a) Show that 3 2
− 5 − 2 = 0. 3
(b) Hence find the length of the lawn. 4

National 5 Practice Paper D - Answers Last updated 04/05/15
Answers
Paper 1
Q1.
5
4
ABm  
Q2. or
Q3.
Q4. a(i) Median 58 5  (ii) SIQR = 11
b On average, after the extra revision classes, there was an improvement in
the students’ marks since 67 58 5  .
After the extra classes the marks were more consistent since 7 < 11.
Q5. 400 grams
Q6. or equivalent
Q7. a P(-2, -16) b Q(6, -16) c ( )
Q8. ( )
Q9. a = 4, b = 3 Q10. Q11. √
Q12. Circumference 2 10r  ,
10 5
2
r
 
  . Area
2
2
2
5 25 25
r  
  
   
      
   
Q13. 𝑦
𝑥

National 5 Practice Paper D - Answers Last updated 04/05/15
Answers
Paper 2
Q1. £165 000
Q2. a (i) 16.5 mm (ii) 1.87 mm (2dp)
Q2. b (i) 20.5 mm (ii) 1.87 mm (2dp)
Q3.
Q4. a b
c Adult entrance fee £3.50. Child entrance fee £1.50.
Q5. a b ( ) c C(0,5)
Q6. a 15.597… m b 111.6 m2
(1dp)
Q7. 800 mm
Q8.
Q9. 1902.1 cm3
(1dp)
Q10. ( )
Q11. a Area lawn = Area path
Area of lawn = Total area of garden – Area of lawn
  3 1 3 2 3x x x x x x     
2 2 2
3 3 5 2 3x x x x   
2
3 5 2 0x x  
b The lawn is 6 metres in length.

National 5 Practice Paper E Last updated 04/05/15
M thematics
National 5 Practice Paper E
Paper 1
Duration – 1 hour
Total marks – 40
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
of 3
2. Multiply out the brackets and collect like terms.
4 + 2 − + 3 3
3. In an experiment involving two variables, the following values for and
were recorded.
1 2 3 4
4 2 0 -2
The results were plotted and a straight line was drawn through the points.
Find the gradient of the line and write down its equation. 3
4. Solve the equation
3
5. Given 2 2
− 2 − 1 = 0 show that
4
2
𝑥
+ 9 = 16
𝑥 =
1 ± 3
2
1
2
2
1
3
+
6

6. The diagram below shows part of the graph of = 36 − − 2 2
(a) State the coordinates of the maximum turning point. 2
(b) State the equation of the axis of symmetry. 1
The line = 20 is drawn.
It cuts the graph of = 36 − − 2 2
at R and S as shown below.
(c) S is the point (6, 20). Find the coordinates of R. 2

7. A badge is made from a circle of radius 5 centimetres.
Segments are taken off the top and bottom of the circle as shown.
The straight edges are parallel.
The badge measures 7 centimetres from the top to the bottom.
The top is 8 centimetres wide.
Calculate the width of the base. 5
8. Sketch the graph of = sin 2 0 360 3
9. = 4 + 2
(a) Find the value of 2 as a surd in its simplest form. 3
(b) Find the value of given that = 3 2. 3

10. The height of a triangle is 2 − centimetres and the base is 2 centimetres.
The area of the triangle is 7 square centimetres.
Calculate the value of 5

M thematics
Paper 2
Total marks – 50
N5

1. = 2
.
Find the value of when = 3 6 10−2
and = 3 10
Give your answer in scientific notation. 3
2.
Calculate the area of triangle PQR. 4
3. In the evening, the temperature in a greenhouse drops by 10.4% per hour.
At 8 p.m. the temperature was 28° Celsius.
Find the temperature at 11 p.m. 3

4. Relative to coordinate axes, the point A
has coordinates (2, 4, 6).
(a) Find the coordinates of C and D. 2
(b) Write down the coordinates of B. 1
5. Shampoo is available in travel size and salon size bottles.
The bottles are mathematically similar.
The travel size contains 200 millilitres and is 12 centimetres in height.
The salon size contains 1600 millilitres.
Calculate the height of the salon size bottle. 3

6. A jeweller uses two different arrangements of bead and pearls.
The first arrangement consists of 2 beads and 5 pearls and has an overall
length of 5.2 centimetres.
The second arrangement consists of 3 beads and 2 pearls and has an overall
length of 5.6 centimetres.
Find the length of one bead and the length of one pearl. 6
7. A pharmaceutical company makes vitamin pills
in the shape of spheres of radius 0.5 centimetres.
(a) Calculate the volume of one pill. 3
Give your answer to 3 significant figures.
The company decides to change the shape of each pill to a cylinder.
The new pill has the same volume as the original and its diameter is 1.4 centimetres.
(b) Calculate the height of the new pill. 3

8. David walks on a bearing of 050° from hostel A to viewpoint V, 5 kilometres away.
Hostel B is due east of hostel A.
Susie walks on a bearing of 294° from hostel B to the same viewpoint.
Calculate the length of AB, the distance between the two hostels. 5
9. The chain of a demolition ball is 12.5 metres long.
When vertical, the end of the chain is 1.5 metres from the ground.
It swings to a maximum height of 2.5 metres above the ground on both sides.
(a) At this maximum height, show that the angle which the chain makes
with the vertical, is approximately 23°. 4
(b) Calculate the maximum length of the arc through which the end of the
chain swings. Give your answer to 3 significant figures. 4
A B
VN
5 km
50°

10. Find the range of values of such that the equation 2
− 4 + 2 = 0 0
has real roots. 4
11. (a) Solve algebraically the equation
3 sin − 1 = 0 0 360 3
(b) Simplify
n cos 2

National 5 Practice Paper E - Answers Last updated 04/05/15
Answers
Paper 1
Q1.
Q2.
Q3.
Q4.
Q5. Proof (using quadratic formula)
Q6. (a) (2,36) (b) (c) R(-2,20)
Q7. 6 centimetres
Q8.
Q9. (a) √ (b)
Q10.
1
-1
O 90 180 270 360

National 5 Practice Paper E - Answers Last updated 04/05/15
Answers
Paper 2
Q1. 3.24 × 1015
Q2. 187.5 cm2
(1 decimal place)
Q3. 20°C (nearest degree)
Q4. (a) C(4,3,4), D(6,2,2) (b) B(6,4,2)
Q5. 24 centimetres
Q6. Bead 1.6 centimetres, pearl 0.4 centimetres
Q7. (a) 0.524 cm3
(3 SF) (b) 0.34 cm (2 decimal places)
Q8. AB = 11.05 kilometres (2 decimal places)
Q9. (a) proof (b) 10.0 metres (3 SF)
Q10.
Q11. a (1 decimal place)
b Proof using the fact that

National 5 Practice Paper F Last updated 04/05/15
M thematics
National 5 Practice Paper F
Paper 1
Duration – 1 hour
Total marks – 40
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
. 2
2. (a) Factorise
4 2
− 2
. 1
(b) Hence simplify
. 2
3. A group of people attended a course to help them stop smoking.
The following table shows the statistics before and after the course.
Mean number of
cigarettes smoked per
person per day
Standard deviation
Before 20.8 8.5
After 9.6 12.0
Make two valid comments about these results. 2
4𝑥2
− 𝑦2
6𝑥 + 3𝑦
1
3
5
+ 2
4
7

4. Teams in a quiz answer questions on film and sport.
This scatter graph shows the scores of some of the teams.
A line of best fit is drawn as shown above.
(a) Find the equation of this straight line. 3
(b) Use this equation to estimate the sport score for a team with
a film score of 20. 1
5. Given that ⃗⃗⃗⃗⃗ = (
3
0
−3
) calculate |⃗⃗⃗⃗⃗ |.
Give your answer as a surd in its simplest form. 3

6. Solve the inequation
13 + 4 1 − 7 2 − . 3
7. The graph of = 2
has been moved to the position shown in the diagram.
Write down the equation of the graph shown. 2
8. A straight line is represented by the equation 2 + = 6.
(a) Find the gradient of this line. 2
(b) This line crosses the -axis at 0 . Find the value of . 1

9. The tangent SV touches the circle,
centre O, at T.
o Angle PTQ is 37°.
o Angle VTR is 68°.
Calculate the size of angle PQR. 3
10. The graph shown below has an equation of the form = cos − .
Write down the value of . 1

11. Cleano washing powder is on special offer.
Each box on special offer contains 20% more powder than the standard box.
A box on special offer contains 900 grams of powder.
How many grams of powder does the standard box contain? 3
12. A parabola has equation = 2
− 3 + 5.
(a) Show that the parabola has no real roots. 2
(b) Write the equation in the form = − 2
+ . 2
(c) Sketch the graph of = 2
− 3 + 5, showing the coordinates of
the turning point and the point of intersection with the -axis. 3

13. Triangle ABC is right-angled at B.
The dimensions are shown.
(a) Calculate the area of triangle ABC. 1
BD, the height of triangle ACB is drawn as shown.
(b) Use your answer to part (a) to calculate the height BD. 3

M thematics
Paper 2
Total marks – 50
N5

1. The orbit of a planet around a star is circular.
The radius of the orbit is 4.96 × 107
kilometres.
Calculate the circumference of the orbit.
Given your answer in scientific notation. 3
2. A boat was bought for £35 000. Its value decreases by 8% each year.
How much will the boat be worth after 4 years? 3
3. Change the subject of the formula below to .
+ =
2
4. Solve algebraically the system of equations
4 + 2 = 13
5 + 3 = 17. 3

5. A child’s toy is in the shape of a hemisphere
with a cone on top, as shown in the diagram.
The toy is 10 centimetres wide
and 16 centimetres high.
Calculate the volume of the toy.
Give your answer correct to two significant figures. 5
6. The diagram shows the base of a
loudspeaker stand which has the shape
of part of a circle.
 The centre of the circle is O.
 EF is a chord of the circle.
 EF is 18 centimetres.
 The radius, OF, of the circle is 15
centimetres.
Find the width of the stand. 4
16 cm

7. There are three mooring points on Lake Sorling.
o From A, the bearing of B is 074°.
o From C, the bearing of B is 310°.
(a) Calculate the size of angle ABC. 2
B is 230 metres from A and 110 metres from C.
(b) Calculate the direct distance from A to C.
Give your answer to 3 significant figures. 4
8. Express
3
+ 1
−
1
− 2
−1 −2
as a single fraction in its simplest form. 3

9. A set of scales has a circular dial.
The pointer is 9 centimetres long.
The tip of the pointer moves through an arc of 2 centimetres for each
100 grams of weight on the scales.
A parcel, placed on the scales, moves the pointer through an angle of 284°.
Calculate the weight of the parcel. 4
10. The number of diagonals, , in a polygon of sides is given by the formula
=
1
2
− 3
(a) How many diagonals does a polygon of 7 sides have? 2
(b) A polygon has 65 diagonals.
Show that for this polygon, 2
− 3 − 130 = 0. 2
(c) Hence find the number of sides in this polygon. 3

11. Emma goes on the “Big Eye”.
Her height, ℎ metres, above the ground is given by the formula
ℎ = −31 cos + 33
where is the number of seconds since the start.
(a) Calculate Emma’s height above the ground 20 seconds after the start. 2
(b) When will Emma first reach a height of 60 metres above the ground? 3
(c) When will she next be at a height of 60 metres above the ground? 1

12. In triangle ABC,
o BC = 8 centimetres
o AC = 6 centimetres
o PQ is parallel to BC
o M is the midpoint of AC.
o Q lies on AC, centimetres from M, as shown in the diagram.
(a) Write down an expression for the length of AQ . 1
(b) Show that PQ = (4 +
3
) centimetres. 3

National 5 Practice Paper F - Answers Last updated 04/05/15
Answers
Paper 1
Q1.
Q2. (a) ( )( ) (b)
Q3. On average the number of cigarettes smoked per person went down after the course
since 9 6 20 8.  
However, the number of cigarettes smoked per person before the course was more
consistent since 8 5 12 0.  
Q4. (a) 2 3y x  (b) 43
Q5. √
Q6.
Q7. ( )
Q8. (a) (b)
Q9. 121°
Q10.
Q11. 750 grams
Q12. (a) therefore no real roots
(b) ( ) (c)
Q13. (a) 150 m2
( )
(b) 12 metres
5
,
x
y

National 5 Practice Paper F - Answers Last updated 04/05/15
Answers
Paper 2
Q1. 3.12 × 108
kilometres
Q2. £25 073.75
Q3. ( ) (or equivalent)
Q4. , (or , )
Q5. 550 cubic centimetres (to 2 SF)
Q6. 27 centimetres
Q7. (a) 124° (b) 305 metres (to 3 SF)
Q8. ( )( )
Q9. 2230.5 grams (to 1 decimal place)
Q10. (a) 14 diagonals (b) proof (c) 13 sides
Q11. (a) 3.87 metres (1 decimal place)
(b) 150.6 seconds (c) 209.4 seconds
Q12. (a) AQ = + 3
(b) PQ =
( )
as required.

N5 practice papers d f with solutions

More Related Content

Similar to N5 practice papers d f with solutions (20)

More from sjamaths (20)

Recently uploaded (20)

N5 practice papers d f with solutions