X100/301
NATIONAL
QUALIFICATIONS
2011
WEDNESDAY, 18 MAY
9.00 AM – 10.30 AM
MATHEMATICS
HIGHER
Paper 1
(Non-calculator)
Read carefully
Calculators may NOT be used in this paper.
Section A – Questions 1–20 (40 marks)
Instructions for completion of Section A are given on page two.
For this section of the examination you must use an HB pencil.
Section B (30 marks)
1
Full credit will be given only where the solution contains appropriate working.
2
Answers obtained by readings from scale drawings will not receive any credit.
LI X100/301 6/32010
*X100/301*
©
Read carefully
1 Check that the answer sheet provided is for Mathematics Higher (Section A).
2 For this section of the examination you must use an HB pencil and, where necessary, an
eraser.
3 Check that the answer sheet you have been given has your name, date of birth, SCN
(Scottish Candidate Number) and Centre Name printed on it.
Do not change any of these details.
4 If any of this information is wrong, tell the Invigilator immediately.
5 If this information is correct, print your name and seat number in the boxes provided.
6 The answer to each question is either A, B, C or D. Decide what your answer is, then,
using your pencil, put a horizontal line in the space provided (see sample question
below).
7 There is only one correct answer to each question.
8 Rough working should not be done on your answer sheet.
9 At the end of the exam, put the answer sheet for Section A inside the front cover
of your answer book.
Sample Question
A curve has equation y = x3 – 4x.
What is the gradient at the point where x = 2?
A
8
B
1
C
0
D –4
The correct answer is A—8. The answer A has been clearly marked in pencil with a
horizontal line (see below).
A
B
C
D
Changing an answer
If you decide to change your answer, carefully erase your first answer and, using your pencil,
fill in the answer you want. The answer below has been changed to D.
A
[X100/301]
B
C
D
Page two
FORMULAE LIST
Circle:
The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius
g2 + f 2 − c .
The equation (x – a)2 + (y – b)2 = r2 represents a circle centre (a, b) and radius r.
a.b = | a| |b| cos θ, where θ is the angle between a and b
Scalar Product:
a1
or
a.b = a1b1 + a2b2 + a3b3 where a = a2
a3
Trigonometric formulae:
sin (A ± B)
cos (A ± B)
sin 2A
cos 2A
b1
and b = b2 .
b3
= sin A cos B ± cos A sin B
±
= cos A cos B sin A sin B
= 2sin A cos A
= cos2 A – sin2 A
= 2cos2 A – 1
= 1 – 2sin2 A
Table of standard derivatives:
f (x)
f ′ ( x)
sin ax
a cos ax
cos ax −a sin ax
Table of standard integrals:
f (x)
∫ f (x) dx
sin ax
− a1 cos ax + C
cos ax
1
a sin ax + C
[Turn over
[X100/301]
Page three
SECTION A
ALL questions should be attempted.
1.
2.
⎛ 2⎞
⎛ 1⎞
⎛−4 ⎞
Given that p = ⎜ 5 ⎟ , q = ⎜ 0 ⎟ and r = ⎜ 2 ⎟ , express 2p – q – 1 r in component form.
2
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ −7 ⎠
⎝ −1 ⎠
⎝ 0⎠
⎛ 1⎞
⎜
⎟
A ⎜ 9⎟
⎜ −15 ⎟
⎝
⎠
B
⎛ 1⎞
⎜
⎟
⎜ 11⎟
⎜ −13 ⎟
⎝
⎠
C
⎛ 5⎞
⎜
⎟
⎜ 9⎟
⎜ −13 ⎟
⎝
⎠
D
⎛ 5⎞
⎜
⎟
⎜ 11⎟
⎜ −15 ⎟
⎝
⎠
A line l has equation 3y + 2x = 6.
What is the gradient of any line parallel to l?
A
−2
B
−
C
D
[X100/301]
2
3
3
2
2
Page four
3.
The diagram shows the graph of y = f(x).
y
(1, 2)
•
O
x
•
(–2, –3)
Which of the following shows the graph of y = f(x + 2) – 1?
A
y
(–4, 3)
•
O x
•
(–1, –2)
B
y
3•
x
O
•
(3, –2)
C
y
(3, 1)
•
x
O
•
–4
y
(–1, 1)
D
•
O x
•
(–4, –4)
[Turn over
[X100/301]
Page five
4.
A tangent to the curve with equation y = x3 – 2x is drawn at the point (2, 4).
What is the gradient of this tangent?
5.
6.
A
2
B
3
C
4
D
10
If x2 – 8x + 7 is written in the form (x – p)2 + q, what is the value of q?
A
–9
B
–1
C
7
D
23
The point P(2, –3) lies on the circle
with centre C as shown.
y
The gradient of CP is –2.
What is the equation of the tangent at P?
7.
A
y + 3 = –2(x – 2)
B
y – 3 = –2(x + 2)
C
y+3=
1
(x
2
– 2)
D
y–3=
1
(x
2
+ 2)
C
O
x
P(2, –3)
A function f is defined on the set of real numbers by f(x) = x3 – x2 + x + 3.
What is the remainder when f(x) is divided by (x – 1)?
A
0
B
2
C
3
D
4
[X100/301]
Page six
8.
A line makes an angle of 30° with the positive direction of the x-axis as shown.
y
30°
O
x
What is the gradient of the line?
A
9.
1
3
B
1
2
C
1
2
D
3
2
The discriminant of a quadratic equation is 23.
Here are two statements about this quadratic equation:
(1) the roots are real;
(2) the roots are rational.
Which of the following is true?
A
Neither statement is correct.
B
Only statement (1) is correct.
C
Only statement (2) is correct.
D
Both statements are correct.
[Turn over
[X100/301]
Page seven
10.
11.
Solve 2 cos x = 3 for x, where 0 ≤ x < 2π.
A
π
5π
and
3
3
B
π
2π
and
3
3
C
π
5π
and
6
6
D
π
11π
and
6
6
⌠⎛ 1
⎞
Find ⎮ ⎜ 4x 2 + x −3 ⎟ dx, where x > 0.
⎮⎝
⎠
⌡
−1
2
− 3x− 4 + c
A
2x
B
2x
C
8 2
x − 3x− 4 + c
3
D
8 2 1 −2
x − x +c
3
2
−1
2
1
− x− 2 + c
2
3
3
[X100/301]
Page eight
12.
The diagram shows two right-angled triangles with sides and angles as given.
2
3
5
2
q
p
1
What is the value of sin( p + q )?
13.
A
2 2
+
5 3
B
2
5
+
5 3
C
2
2
+
3 3 5
D
4
1
+
3 5 3
Given that f(x) = 4 sin 3x, find f′ (0).
A
0
B
1
C
12
D
36
[Turn over
[X100/301]
Page nine
14.
An equilateral triangle of side 3 units is shown.
The vectors p and q are as represented in the diagram.
What is the value of p . q?
A
9
B
9
2
C
p
q
9
2
D
15.
16.
0
Given that the points S(–4, 5, 1), T(–16, –4, 16) and U(–24, –10, 26) are collinear,
calculate the ratio in which T divides SU.
A
2:3
B
3:2
C
2:5
D
3:5
1
Find ⌠
⎮ 4 dx, where x ≠ 0.
⌡ 3x
A
−
1
+c
9x3
B
−
1
+c
x3
C
1
+c
x3
D
1
+c
12x3
[X100/301]
Page ten
17.
The diagram shows the graph of a cubic.
y
(1, 2)
•
–1
O
2
x
What is the equation of this cubic?
18.
A
y = –x(x + 1)(x – 2)
B
y = –x(x – 1)(x + 2)
C
y = x(x + 1)(x – 2)
D
y = x(x – 1)(x + 2)
If f(x) = (x – 3)(x + 5), for what values of x is the graph of y = f(x) above the x-axis?
A
–5 < x < 3
B
–3 < x < 5
C
x < –5, x > 3
D
x < –3, x > 5
[Turn over
[X100/301]
Page eleven
19.
Which of the following diagrams represents the graph with equation log3 y = x?
A
y
(1, 3)
x
O
B
y
(1, 1)
O
C
x
y
(1, 3)
1
O
D
x
y
(3, 1)
O 1
[X100/301]
x
Page twelve
20.
On a suitable domain, D, a function g is defined by g(x) = sin2
x −2.
Which of the following gives the real values of x in D and the corresponding values of
g(x)?
A
x≥0
and
–1 ≤ g(x) ≤ 1
B
x≥0
and
0 ≤ g(x) ≤ 1
C
x≥2
and
–1 ≤ g(x) ≤ 1
D
x≥2
and
0 ≤ g(x) ≤ 1
[END OF SECTION A]
[Turn over for SECTION B
[X100/301]
Page thirteen
Marks
SECTION B
ALL questions should be attempted.
21.
A quadrilateral has vertices A(–1, 8), B(7, 12), C(8, 5) and D(2, –3) as shown in the
diagram.
y
B
A
E
C
O
x
D
(a) Find the equation of diagonal BD.
2
(b) The equation of diagonal AC is x + 3y = 23.
Find the coordinates of E, the point of intersection of the diagonals.
(c)
3
(i) Find the equation of the perpendicular bisector of AB.
(ii) Show that this line passes through E.
[X100/301]
Page fourteen
5
Marks
22.
A function f is defined on the set of real numbers by f(x) = (x – 2)(x2 + 1).
(a) Find where the graph of y = f(x) cuts:
(i) the x-axis;
2
(ii) the y-axis.
(b) Find the coordinates of the stationary points on the curve with equation y = f(x)
and determine their nature.
8
(c) On separate diagrams sketch the graphs of:
(i) y = f(x);
(ii) y = –f(x).
23.
3
(a) Solve cos 2x° – 3 cos x° + 2 = 0 for 0 ≤ x < 360.
5
(b) Hence solve cos 4x° – 3 cos 2x° + 2 = 0 for 0 ≤ x < 360.
2
[END OF SECTION B]
[END OF QUESTION PAPER]
[X100/301]
Page fifteen
[BLANK PAGE]
X100/302
NATIONAL
QUALIFICATIONS
2011
WEDNESDAY, 18 MAY
10.50 AM – 12.00 NOON
MATHEMATICS
HIGHER
Paper 2
Read Carefully
1
Calculators may be used in this paper.
2
Full credit will be given only where the solution contains appropriate working.
3
Answers obtained by readings from scale drawings will not receive any credit.
LI X100/302 6/32010
*X100/302*
©
FORMULAE LIST
Circle:
The equation x 2 + y 2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f ) and radius g 2 + f 2 − c .
The equation ( x – a) 2 + ( y – b) 2 = r 2 represents a circle centre (a, b) and radius r.
a.b = | a| |b| cos θ, where θ is the angle between a and b
Scalar Product:
or
a.b = a 1 b 1 + a 2 b 2 + a 3 b 3 where a =
Trigonometric formulae:
sin (A ± B)
cos (A ± B)
sin 2A
cos 2A
= sin A cos B ± cos A sin B
±
= cos A cos B sin A sin B
= 2sin A cos A
= cos2 A – sin2 A
= 2cos2 A – 1
= 1 – 2sin2 A
Table of standard derivatives:
f (x)
f ′ ( x)
sin ax
a cos ax
cos ax
−a sin ax
Table of standard integrals:
[X100/302]
f (x)
∫ f (x) dx
sin ax
− a1 cos ax + C
cos ax
1
a sin ax + C
Page two
Marks
ALL questions should be attempted.
1.
D,OABC is a square based pyramid as shown in the diagram below.
z
D(2, 2, 6)
y
C
B
O
M
A
x
O is the origin, D is the point (2, 2, 6) and OA = 4 units.
M is the mid-point of OA.
2.
(a) State the coordinates of B.
→
→
(b) Express DB and DM in component form.
1
(c) Find the size of angle BDM.
5
3
Functions f, g and h are defined on the set of real numbers by
• f(x) = x3 – 1
• g(x) = 3x + 1
• h(x) = 4x – 5.
(a) Find g( f(x)).
2
(b) Show that g( f(x)) + xh(x) = 3x3 + 4x2 – 5x – 2.
1
(c)
(i) Show that (x – 1) is a factor of 3x3 + 4x2 – 5x – 2.
(ii) Factorise 3x3 + 4x2 – 5x – 2 fully.
5
(d) Hence solve g( f(x)) + xh(x) = 0.
1
[Turn over
[X100/302]
Page three
Marks
3.
(a) A sequence is defined by un + 1
1
= − un
2
with u0 = –16.
1
Write down the values of u1 and u2.
(b) A second sequence is given by 4, 5, 7, 11, . . . .
It is generated by the recurrence relation vn + 1 = pvn + q with v1 = 4.
3
Find the values of p and q.
(c) Either the sequence in (a) or the sequence in (b) has a limit.
(i) Calculate this limit.
3
(ii) Why does the other sequence not have a limit?
4.
The diagram shows the curve with equation y = x3 – x2 – 4x + 4 and
the line with equation y = 2x + 4.
The curve and the line intersect at the points (–2, 0), (0, 4) and (3, 10).
y
y = x3 – x2 – 4x + 4
y = 2x + 4
–2
O
3
10
Calculate the total shaded area.
[X100/302]
x
Page four
Marks
5.
Variables x and y are related by the
equation y = kx n.
The graph of log2 y against log2 x is a
straight line through the points (0, 5)
and (4, 7), as shown in the diagram.
log2 y
(0, 5)
(4, 7)
5
Find the values of k and n.
O
6.
log2 x
(a) The expression 3 sin x – 5 cos x can be written in the form R sin(x + a)
where R > 0 and 0 ≤ a < 2π.
Calculate the values of R and a.
4
(b) Hence find the value of t, where 0 ≤ t ≤ 2, for which
t
∫ (3cos x + 5sin x) dx = 3.
7
0
7.
Circle C1 has equation (x + 1)2 + (y – 1)2 = 121.
A circle C2 with equation x2 + y2 – 4x + 6y + p = 0 is drawn inside C1.
The circles have no points of contact.
What is the range of values of p?
9
[END OF QUESTION PAPER]
[X100/302]
Page five
[BLANK PAGE]
[BLANK PAGE]
[BLANK PAGE]
X100/12/02
NATIONAL
QUALIFICATIONS
2012
MONDAY, 21 MAY
1.00 PM – 2.30 PM
MATHEMATICS
HIGHER
Paper 1
(Non-calculator)
Read carefully
Calculators may NOT be used in this paper.
Section A – Questions 1–20 (40 marks)
Instructions for completion of Section A are given on Page two.
For this section of the examination you must use an HB pencil.
Section B (30 marks)
1
Full credit will be given only where the solution contains appropriate working.
2
Answers obtained by readings from scale drawings will not receive any credit.
LI X100/12/02 6/28110
*X100/12/02*
©
Read carefully
1 Check that the answer sheet provided is for Mathematics Higher (Section A).
2 For this section of the examination you must use an HB pencil and, where necessary,
an eraser.
3 Check that the answer sheet you have been given has your name, date of birth, SCN
(Scottish Candidate Number) and Centre Name printed on it.
Do not change any of these details.
4 If any of this information is wrong, tell the Invigilator immediately.
5 If this information is correct, print your name and seat number in the boxes provided.
6 The answer to each question is either A, B, C or D. Decide what your answer is, then,
using your pencil, put a horizontal line in the space provided (see sample question
below).
7 There is only one correct answer to each question.
8 Rough working should not be done on your answer sheet.
9 At the end of the exam, put the answer sheet for Section A inside the front cover
of your answer book.
Sample Question
A curve has equation y = x3 – 4x.
What is the gradient at the point where x = 2?
A8
B1
C0
D – 4
The correct answer is A—8. The answer A has been clearly marked in pencil with a
horizontal line (see below).
A
B
C
D
Changing an answer
If you decide to change your answer, carefully erase your first answer and, using your
pencil, fill in the answer you want. The answer below has been changed to D.
A
[X100/12/02]
B
C
D
Page two
FORMULAE LIST
Circle:
The equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle centre (–g, –f) and radius
g2 + f 2 − c .
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 q, where q is the angle between a and b
 a1 
 b1 
or
a.b = a1b1 + a2b2 + a3b3 where a =  a2  and b =  b2  .
 
 
 a 
 b 
3
3
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 = 2sin A cos A
cos 2A = cos2 A – sin2 A
= 2cos2 A – 1
= 1 – 2sin2 A
Table of standard derivatives:
f(x)
f ′(x)
sin ax
a cos ax
cos ax
–a sin ax
Table of standard integrals:
f(x)
∫ f (x)dx
sin ax
− a1 cos ax + C
cos ax
− a1 sin ax + C
[Turn over
[X100/12/02]
Page three
SECTION A
ALL questions should be attempted.
1. A sequence is defined by the recurrence relation un + 1 = 3un + 4, with u0 =1.
Find the value of u2.
A7
B10
C25
D35
2. What is the gradient of the tangent to the curve with equation y = x3 – 6x + 1 at the
point where x = –2?
A–24
B3
C5
D6
3. If x2 – 6x + 14 is written in the form (x – p)2 + q, what is the value of q?
A–22
B5
C14
D50
4. What is the gradient of the line
shown in the diagram?
y
A − 3
1
B −
3
C −
1
2
D −
3
2
[X100/12/02]
150 °
O
Page four
x
5. The diagram shows a right-angled triangle with sides and angles as marked.
5
3
a
4
What is the value of cos2a?
A
B
7
25
3
5
C
24
25
D
6
5
3
6.If y = 3x–2 + 2x 2, x > 0, determine
dy
.
dx
4 25
x
5
A
−6x −3 +
B
−3x −1 + 3x 2
C
−6x −3 + 3x 2
D
−3x −1 +
1
1
4 25
x
5
 −3
 1
 1 and v =  t  are perpendicular, what is the value of t?
7.If u =  
 
 2t 
 −1
A–3
B–2
C
2
3
D1
[Turn over
[X100/12/02]
Page five
4 3
pr .
3
What is the rate of change of V with respect to r, at r = 2?
8. The volume of a sphere is given by the formula V =
A
16p
3
B
32p
3
C
16p
D
32p
9. The diagram shows the curve with equation of the form y = cos(x + a) + b
for 0 ≤ x ≤ 2p.
y
O
p
6
7p
6
2p
x
–2
What is the equation of this curve?
A

π
y = cos  x −  − 1
6

B

y = cos  x −

C

π
y = cos  x +  − 1
6

D

y = cos  x +

[X100/12/02]
π
+1
6 
π
+1
6 
Page six
10. The diagram shows a square-based pyramid P,QRST.
TS, TQ and TP represent f, g and h respectively.
P
h
R
S
Q
f
T
g
Express RP in terms of f, g and h.
A–f + g – h
B–f – g + h
C
f – g – h
D
f + g + h
∫
 1 
11. Find ∫  2  dx, x ≠ 0.
 6x 
A–12x–3 + c
B–6x–1 + c
C
– 1 x–3 + c
3
D
– 1 x–1 + c
6
12. Find the maximum value of


2 − 3 sin  x − π 
3

and the value of x where this occurs in the interval 0 ≤ x ≤ 2p.
max value
x
A
−1
11π
6
B
5
11π
6
C
−1
5π
6
D
5
5π
6
[X100/12/02]
Page seven
[Turn over
13. A parabola intersects the axes at x = –2, x = –1 and y = 6, as shown in the diagram.
y
6
–2
–1
O
x
What is the equation of the parabola?
A y = 6(x – 1)(x – 2)
B y = 6(x + 1)(x + 2)
C y = 3(x – 1)(x – 2)
D y = 3(x + 1)(x + 2)
14.Find
1
∫∫ (2x − 1)2 dx where x > 2 .
1
3
(
)
(
)
(
)
A
A
1 2x − 1 2 + c
3
B
B
1 2x − 1 − 2 + c
2
C
C
1 2x − 1 2 + c
2
D
D
1 2x − 1 − 2 + c
3
(
1
3
)
1
 3
15. If u = k  −1 , where k>0 and u is a unit vector, determine the value of k.
 
 0
A
A
1
2
B
B
1
8
C
C
D
D
1
2
1
10
[X100/12/02]
Page eight
16.If y = 3cos4x, find
dy
.
dx
A12cos3x sin x
B12cos3x
C–12cos3x sin x
D–12sin3x
 3
17. Given that a =  4 and a. a + b = 7 , what is the value of a.b?
 
 0
(
A
B
C
D
)
7
25
18
−
5
– 6
–18
18. The graph of y = f(x) shown
has stationary points at (0, p) and (q, r).
y = f(x)
y
Here are two statements about f(x):
(1)f(x) < 0 for s < x < t;
(2)f ′(x) < 0 for x < q.
p
Which of the following is true?
A
Neither statement is correct.
B
Only statement (1) is correct.
C
Only statement (2) is correct.
D
Both statements are correct.
O
t
s
x
(q, r)
[Turn over
[X100/12/02]
Page nine
19. Solve 6 – x – x2 < 0.
A
–3 < x < 2
B
x < –3, x > 2
C
–2 < x < 3
D
x < –2, x > 3
20. Simplify
logb 9a2
logb 3a
, where a > 0 and b > 0.
A2
B3a
Clogb3a
Dlogb(9a2 – 3a)
[END OF SECTION A]
[X100/12/02]
Page ten
SECTION B
Marks
ALL questions should be attempted.
21. (a)
(i) Show that (x – 4) is a factor of x3 – 5x2 + 2x + 8.
(ii) Factorise x3 – 5x2 + 2x + 8 fully.
(iii) Solve x3 – 5x2 + 2x + 8 = 0.
6
(b) The diagram shows the curve with equation y = x3 – 5x2 + 2x + 8.
y
y = x3 – 5x2 + 2x + 8
P
O
Q
R
x
The curve crosses the x-axis at P, Q and R.
6
Determine the shaded area.
(
22.(a) The expression cos x − 3 sin x can be written in the form k cos x + a
where k > 0 and 0 ≤ a < 2p.
)
4
Calculate the values of k and a.
(b) Find the points of intersection of the graph of y = cos x − 3 sin x with
the x and y axes, in the interval 0 ≤ x ≤ 2p.
[Turn over for Question 23 on Page twelve
[X100/12/02]
Page eleven
3
Marks
23.(a)Find the equation of l1,
joining P(3, –3) to Q(–1, 9).
the
perpendicular
bisector
of
the
line
4
(b) Find the equation of l2 which is parallel to PQ and passes through R(1, –2).
2
(c) Find the point of intersection of l1 and l2.
3
(d) Hence find the shortest distance between PQ and l2.
2
[END OF SECTION B]
[END OF QUESTION PAPER]
[X100/12/02]
Page twelve
X100/12/03
NATIONAL
QUALIFICATIONS
2012
MONDAY, 21 MAY
2.50 PM – 4.00 PM
MATHEMATICS
HIGHER
Paper 2
Read Carefully
1
Calculators may be used in this paper.
2
Full credit will be given only where the solution contains appropriate working.
3
Answers obtained by readings from scale drawings will not receive any credit.
LI X100/12/03 6/28110
*X100/12/03*
©
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 q, where q is the angle between a and b
 a1 
 b1 


 
and b = b2 .
a
or
a.b = a1b1 + a2b2 + a3b3 where a = 2
 
 
 a 
 b 
3
3
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 = 2sin A cos A
cos 2A = cos2 A – sin2 A
= 2cos2 A – 1
= 1 – 2sin2 A
Table of standard derivatives:
f(x)
f ′(x)
sin ax
a cos ax
cos ax
–a sin ax
Table of standard integrals:
f(x)
∫ f (x)dx
sin ax
− a1 cos ax + C
cos ax − a1 sin ax + C
[X100/12/03]
Page two
g2 + f 2 − c .
ALL questions should be attempted.
Marks
1.Functions f and g are defined on the set of real numbers by
• f(x) = x2 + 3
• g(x) = x + 4.
(a) Find expressions for:
(i)
f(g(x));
(ii)
g(f(x)).
3
(b) Show that f(g(x)) + g(f(x)) = 0 has no real roots.
3
2.(a) Relative to a suitable set of coordinate axes, Diagram 1 shows the
line 2x – y + 5 = 0 intersecting the circle x2 + y2 – 6x –2y – 30 = 0 at the
points P and Q.
Q
P
Diagram 1
6
Find the coordinates of P and Q.
(b) Diagram 2 shows the circle from (a) and a second congruent circle, which also
passes through P and Q.
Q
P
Diagram 2
6
Determine the equation of this second circle.
[X100/12/03]
Page three
[Turn over
3. A function f is defined on the domain 0 ≤ x ≤ 3 by f(x) = x – 2x – 4x + 6.
3
Marks
2
7
Determine the maximum and minimum values of f.
4. The diagram below shows the graph of a quartic y = h(x), with stationary
points at x = 0 and x = 2.
y
5
O
x
2
y = h(x)
On separate diagrams sketch the graphs of:
(a) y = h'(x);
3
(b) y = 2 – h'(x).
3
5. A is the point (3, –3, 0), B is (2, –3, 1) and C is (4, k, 0).
(a) (i) Express BA and BC in component form.
3
^
(ii) Show that cos ABC = 2
2(k + 6k + 14)
·
(b) If angle ABC = 30 °, find the possible values of k.
[X100/12/03]
Page four
7
5
Marks
π
6. For 0 < x < 2 , sequences can be generated using the recurrence relation
un +1 = (sinx)un + cos 2x, with u0 = 1 .
2
(a) Why do these sequences have a limit?
(b) The limit of one sequence generated by this recurrence relation is
1 sin x .
2
7
Find the value(s) of x.
7. The diagram shows the curves with equations y = 4x and y = 32−x .
y = 4x
y
T
y = 32−x
x
O
The graphs intersect at the point T.
log p
(a) Show that the x – coordinate of T can be written in the form loga q ,
a
for all a > 1.
(b) Calculate the y – coordinate of T.
[END OF QUESTION PAPER]
[X100/12/03]
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2
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