Past Papers 2016 - 2018

1. *X7477611*
©
National
Qualications
2016H
Total marks — 60
Attempt ALL questions.
You may NOT use a calculator.
Full credit will be given only to solutions which contain appropriate working.
State the units for your answer where appropriate.
Answers obtained by readings from scale drawings will not receive any credit.
Write your answers clearly in the spaces provided in the answer booklet. The size of the space
provided for an answer should not be taken as an indication of how much to write. It is not
necessary to use all the space.
Additional space for answers is provided at the end of the answer booklet. If you use this space
you must clearly identify the question number you are attempting.
Use blue or black ink.
Before leaving the examination room you must give your answer booklet to the Invigilator; if you
do not, you may lose all the marks for this paper.
X747/76/11 Mathematics
Paper 1
(Non-Calculator)
THURSDAY, 12 MAY
9:00 AM – 10:10 AM
A/PB

2. Page 02
FORMULAE LIST
Circle:
The equation x2
+ y2
+ 2gx + 2fy + c = 0 represents a circle centre (−g, −f ) and radius
The equation (x − a)2
+ (y − b)2
= r2
represents a circle centre (a, b) and radius r.
Scalar Product: a.b = |a||b| cos θ, where θ is the angle between a and b
or a.b = a1b1 + a2b2 + a3b3 where a =
1 1
2 2
3 3
=and
a b
a b
a b
b
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠
.
Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B
cos (A ± B) = cos A cos B
±
sin A sin B
sin 2A = 2 sin A cos A
cos 2A = cos2
A − sin2
A
= 2 cos2
A − 1
= 1 − 2 sin2
A
Table of standard derivatives:
Table of standard integrals:
f (x) f ′(x)
sin ax
cos ax
a cos ax
– a sin ax
f (x)
∫ f (x)dx
sin ax
cos ax
cos ax + c
sin ax + c
1–
a
1
a
2 2
+ –g f c .

3. Page 03
MARKS
Attempt ALL questions
Total marks – 60
1. Find the equation of the line passing through the point ( )2, 3− which is parallel to
the line with equation 4 7+ =y x .
2. Given that
3
12 8= +y x x, where 0>x , find
dy
dx
.
3. A sequence is defined by the recurrence relation
1
10
3
+ = +n nu u1
with 3 6=u .
(a) Find the value of u4.
(b) Explain why this sequence approaches a limit as → ∞n .
(c) Calculate this limit.
4. A and B are the points −7, 3( ) and ( )1, 5 .
AB is a diameter of a circle.
y
x
B
A
O
Find the equation of this circle.
2
3
1
1
2
3
[Turn over

4. Page 04
MARKS
5. Find
∫ ( )8cos 4 1+x dx.
6. Functions f and g are defined on  , the set of real numbers.
The inverse functions f −1
and g−1
both exist.
(a) Given ( ) 3 5= +f x x , find ( )1−
f x .
(b) If g ( )2 = 7, write down the value of g−1
( )7 .
7. Three vectors can be expressed as follows:
FG = −2i −6j + 3k
GH = 3i + 9j −7k
EH = 2i + 3j + k
(a) Find FH.
(b) Hence, or otherwise, find FE.
8. Show that the line with equation =3 5−y x is a tangent to the circle with equation
2 2
2 4 5 0+ + − − =x y x y and find the coordinates of the point of contact.
2
3
1
2
2
5

5. Page 05
MARKS
9. (a) Find the x-coordinates of the stationary points on the graph with equation
( )=y f x , where 3 2
( ) 3 24= + −f x x x x.
(b) Hence determine the range of values of x for which the function f is strictly
increasing.
10. The diagram below shows the graph of the function ( ) 4log=f x x, where 0>x .
y
xO (1, 0)
(4, 1)
f x( )=log4 x
The inverse function, f −1
, exists.
On the diagram in your answer booklet, sketch the graph of the inverse function.
11. (a) A and C are the points ( )1, 3, 2− and ( )4, , 4− 3 respectively.
Point B divides AC in the ratio 1 : 2.
Find the coordinates of B.
(b) kAC is a vector of magnitude 1, where 0>k .
Determine the value of k.
4
2
2
2
3
[Turn over

6. Page 06
MARKS
12. The functions f and g are defined on  , the set of real numbers by
( ) 2
2 4 5= − +f x x x and ( ) 3= −g x x.
(a) Given ( ) ( )( )=h x f g x , show that ( ) 2
2 8 11= − +h x x x .
(b) Express ( )h x in the form p x q r+( ) +
2
.
13. Triangle ABD is right-angled at B with angles BAC = p and BAD = q and lengths as
shown in the diagram below.
D
C
B
2
1
4 A
q
p
Show that the exact value of ( )cos −q p is
19 17
85
.
2
3
5

7. Page 07
MARKS
14. (a) Evaluate log5 25.
(b) Hence solve ( )4 4 5log log 6 log 25x x+ − = , where 6>x .
15. The diagram below shows the graph with equation ( )=y f x , where
( ) ( )( )2
= − −f x k x a x b .
y
(1, 9)
−5 O 4
x
(a) Find the values of a, b and k.
(b) For the function ( ) ( )= −g x f x d , where d is positive, determine the range of
values of d for which ( )g x has exactly one real root.
[END OF QUESTION PAPER]
1
5
3
1

8. Page 08
[BLANK PAGE]
do not write on this page

9. Write your answers clearly in the spaces provided in this booklet. The size of the space provided
for an answer should not be taken as an indication of how much to write. It is not necessary to
use all the space.
Additional space for answers is provided at the end of this booklet. If you use this space you must
clearly identify the question number you are attempting.
Use blue or black ink.
Before leaving the examination room you must give this booklet to the Invigilator; if you do not
you may lose all the marks for this paper.
FOR OFFICIAL USE
X747/76/01 Mathematics Paper 1 (Non-Calculator)
Answer Booklet
Fill in these boxes and read what is printed below.
Full name of centre Town
Forename(s) Surname Number of seat
Day Month Year Scottish candidate number
*X7477601*
*X747760101*
©
Mark
Date of birth
H National
Qualications
2016
THURSDAY, 12 MAY
9:00 AM – 10:10 AM
A/PB

10. *X747760102*
Page 02
DO NOT
WRITE IN
THIS
MARGIN
QUESTION
NUMBER
2.
1.

11. *X747760103*
Page 03
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QUESTION
NUMBER
3.(a)
3.(b)
3.(c)

12. *X747760104*
Page 04
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QUESTION
NUMBER
4.

13. *X747760105*
Page 05
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QUESTION
NUMBER
5.
6.(a)
6.(b)

14. *X747760106*
Page 06
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QUESTION
NUMBER
7.(b)
7.(a)

15. *X747760107*
Page 07
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QUESTION
NUMBER
8.

16. *X747760108*
Page 08
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QUESTION
NUMBER
9.(b)
9.(a)

17. *X747760109*
Page 09
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QUESTION
NUMBER
10.
y
(1, 0)O
(4, 1)
x
f(x) = log4 x

18. *X747760110*
Page 10
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WRITE IN
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MARGIN
QUESTION
NUMBER
11.(a)

19. *X747760111*
Page 11
DO NOT
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THIS
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QUESTION
NUMBER
11.(b)

20. *X747760112*
Page 12
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THIS
MARGIN
QUESTION
NUMBER
12.(a)

21. *X747760113*
Page 13
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THIS
MARGIN
QUESTION
NUMBER
12.(b)

22. *X747760114*
Page 14
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THIS
MARGIN
QUESTION
NUMBER
13.

23. *X747760115*
Page 15
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QUESTION
NUMBER
14.(a)
14.(b)

24. *X747760116*
Page 16
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QUESTION
NUMBER
15.(a)
15.(b)

25. *X747760117*
Page 17
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QUESTION
NUMBER ADDITIONAL SPACE FOR ANSWERS

26. *X747760118*
Page 18
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QUESTION
NUMBER ADDITIONAL SPACE FOR ANSWERS

27. *X747760119*
Page 19
ADDITIONAL SPACE FOR ANSWERS
DO NOT
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THIS
MARGIN
QUESTION
NUMBER

28. *X747760120*
Page 20
For Marker’s Use
Question No Marks/Grades

29. *X7477612*
©
National
Qualications
2016H
Total marks — 70
Attempt ALL questions.
You may use a calculator.
Full credit will be given only to solutions which contain appropriate working.
State the units for your answer where appropriate.
Answers obtained by readings from scale drawings will not receive any credit.
Write your answers clearly in the spaces provided in the answer booklet. The size of the space
provided for an answer should not be taken as an indication of how much to write. It is not
necessary to use all the space.
Additional space for answers is provided at the end of the answer booklet. If you use this space
you must clearly identify the question number you are attempting.
Use blue or black ink.
Before leaving the examination room you must give your answer booklet to the Invigilator; if you
do not, you may lose all the marks for this paper.
X747/76/12 Mathematics
Paper 2
THURSDAY, 12 MAY
10:30 AM – 12:00 NOON
A/PB

30. Page 02
FORMULAE LIST
Circle:
The equation x2
+ y2
+ 2gx + 2fy + c = 0 represents a circle centre (−g, −f ) and radius
The equation (x − a)2
+ (y − b)2
= r2
represents a circle centre (a, b) and radius r.
Scalar Product: a.b = |a||b| cos θ, where θ is the angle between a and b
or a.b = a1b1 + a2b2 + a3b3 where a =
1 1
2 2
3 3
=and
a b
a b
a b
b
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠
.
Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B
cos (A ± B) = cos A cos B
±
sin A sin B
sin 2A = 2 sin A cos A
cos 2A = cos2
A − sin2
A
= 2 cos2
A − 1
= 1 − 2 sin2
A
Table of standard derivatives:
Table of standard integrals:
f (x) f ′(x)
sin ax
cos ax
a cos ax
– a sin ax
f (x)
∫ f (x)dx
sin ax
cos ax
cos ax + c
sin ax + c
1–
a
1
a
2 2
+ –g f c .

31. Page 03
MARKS
Attempt ALL questions
Total marks — 70
1. PQR is a triangle with vertices P( )0, 4− , Q( )6,2− and R( )10,6 .
y
x
Q
O
P
R
M
(a) (i) State the coordinates of M, the midpoint of QR.
(ii) Hence find the equation of PM, the median through P.
(b) Find the equation of the line, L, passing through M and perpendicular to PR.
(c) Show that line L passes through the midpoint of PR.
2. Find the range of values for p such that x2
− 2x + 3 − p = 0 has no real roots.
[Turn over
1
2
3
3
3

32. Page 04
MARKS
3. (a) (i) Show that ( )1x + is a factor of 2x3
− 9x2
+ 3x + 14.
(ii) Hence solve the equation 2x3
− 9x2
+ 3x + 14 = 0.
(b) The diagram below shows the graph with equation y = 2x3
− 9x2
+ 3x + 14.
The curve cuts the x-axis at A, B and C.
y = 2x3
− 9x2
+ 3x + 14
y
x
A B C
O
(i) Write down the coordinates of the points A and B.
(ii) Hence calculate the shaded area in the diagram.
4. Circles C1 and C2 have equations ( ) ( )2 2
5 6 9x y+ + − =
and x2
+ y2
− 6x −16 = 0 respectively.
(a) Write down the centres and radii of C1 and C2.
(b) Show that C1 and C2 do not intersect.
2
3
1
4
4
3

33. Page 05
MARKS
5. The picture shows a model of a water molecule.
H H
O
Relative to suitable coordinate axes, the oxygen atom is positioned at
point ( )2 2 5−A , , .
The two hydrogen atoms are positioned at points ( )10 18 7−B , , and ( )4 6 21− −C , ,
as shown in the diagram below.
( )10 18 7−B , ,
( )2 2 5−A , ,
( )4 6 21− −C , ,
(a) Express AB and AC in component form.
(b) Hence, or otherwise, find the size of angle BAC.
6. Scientists are studying the growth of a strain of bacteria. The number of bacteria
present is given by the formula
( ) 0 107
200 t
B t e= 
,
where t represents the number of hours since the study began.
(a) State the number of bacteria present at the start of the study.
(b) Calculate the time taken for the number of bacteria to double.
[Turn over
2
4
1
4

34. Page 06
MARKS
7. A council is setting aside an area of land to create six fenced plots where local
residents can grow their own food.
Each plot will be a rectangle measuring x metres by y metres as shown in the
diagram.
y
x
(a) The area of land being set aside is 108 m2
.
Show that the total length of fencing, L metres, is given by
( )
144
9L x x
x
= + .
(b) Find the value of x that minimises the length of fencing required.
3
6

35. Page 07
MARKS
8. (a) Express 5cos x − 2sin x in the form k cos (x + a),
where k > 0 and 0 < a < 2π.
(b) The diagram shows a sketch of part of the graph of y = 10 + 5cos x − 2sin x
and the line with equation 12y = .
The line cuts the curve at the points P and Q.
y
x
P Q
O
y = 10 + 5cos x − 2sin x
y = 12
Find the x-coordinates of P and Q.
9. For a function f , defined on a suitable domain, it is known that:
• ( )
2 1x
f x
x
+
=′
• ( )9 40f =
Express ( )f x in terms of x.
[Turn over for next question
4
4
4

36. Page 08
MARKS
10. (a) Given that ( )
1
2 27y x= + , find
dy
dx
.
(b) Hence find
2
4
7
x
dx
x +
⌠
⎮
⌡
.
11. (a) Show that sin 2x tan x = 1 − cos 2x, where
3
2 2
x
π π
< < .
(b) Given that ( ) sin 2 tanf x x x= , find ( )f x′ .
[END OF QUESTION PAPER]
2
1
4
2

37. Write your answers clearly in the spaces provided in this booklet. The size of the space provided
for an answer should not be taken as an indication of how much to write. It is not necessary to
use all the space.
Additional space for answers is provided at the end of this booklet. If you use this space you must
clearly identify the question number you are attempting.
Use blue or black ink.
Before leaving the examination room you must give this booklet to the Invigilator; if you do not
you may lose all the marks for this paper.
FOR OFFICIAL USE
X747/76/02 Mathematics Paper 2
Answer Booklet
Fill in these boxes and read what is printed below.
Full name of centre Town
Forename(s) Surname Number of seat
Day Month Year Scottish candidate number
*X7477602*
*X747760201*
©
Mark
Date of birth
H National
Qualications
2016
THURSDAY, 12 MAY
10:30 AM – 12:00 NOON
A/PB

38. *X747760202*
Page 02
1.(a)
(i)
1.(b)
1.(a)
(ii)
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MARGIN
QUESTION
NUMBER

39. *X747760203*
Page 03
1. (c)
2.
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MARGIN
QUESTION
NUMBER

40. *X747760204*
Page 04
3.(a)
(i)
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QUESTION
NUMBER
3.(a)
(ii)

41. *X747760205*
Page 05
3.(b)
(i)
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QUESTION
NUMBER
3.(b)
(ii)

42. *X747760206*
Page 06
4.(a)
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NUMBER

43. *X747760207*
Page 07
4.(b)
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NUMBER

44. *X747760208*
Page 08
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QUESTION
NUMBER
5.(b)
5.(a)

45. *X747760209*
Page 09
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QUESTION
NUMBER
6.(a)
6.(b)

46. *X747760210*
Page 10
7.(a)
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QUESTION
NUMBER

47. *X747760211*
Page 11
7.(b)
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NUMBER

48. *X747760212*
Page 12
8.(a)
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NUMBER

49. *X747760213*
Page 13
8.(b)
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50. *X747760214*
Page 14
9.
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NUMBER

51. *X747760215*
Page 15
10.(a)
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QUESTION
NUMBER
10.(b)

52. *X747760216*
Page 16
11.(a)
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NUMBER

53. *X747760217*
Page 17
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11.(b)
QUESTION
NUMBER

54. *X747760218*
Page 18
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ENTER
NUMBER
OF
QUESTION
ADDITIONAL SPACE FOR ANSWERS

55. *X747760219*
Page 19
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ENTER
NUMBER
OF
QUESTION
ADDITIONAL SPACE FOR ANSWERS

56. *X747760220*
Page 20
For Marker’s Use
Question No Marks/Grades