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Higher Mathematics
GCC Applied Calculus
1.[SQA]
hsn.uk.net Page 1
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
2.[SQA]
hsn.uk.net Page 2
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
3.[SQA]
4.[SQA]
hsn.uk.net Page 3
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
5.[SQA] A goldsmith has built up a solid which consists of a triangularprism of fixed volume with a regular tetrahedron at each end.
The surface area, A , of the solid is given by
A(x) =3√3
2
(
x2 +16
x
)
where x is the length of each edge of the tetrahedron.
Find the value of x which the goldsmith should use tominimise the amount of gold plating required to cover thesolid. 6
x
6.[SQA] The shaded rectangle on this maprepresents the planned extension to thevillage hall. It is hoped to provide thelargest possible area for the extension.
Village hall
Manse Lane
TheVennel
8 m
6 m
The coordinate diagram represents theright angled triangle of ground behindthe hall. The extension has length lmetres and breadth b metres, as shown.One corner of the extension is at the point(a, 0) .
O x
y
lb
(a, 0) (8, 0)
(0, 6)
(a) (i) Show that l = 54a .
(ii) Express b in terms of a and hence deduce that the area, A m2 , of theextension is given by A = 3
4a(8− a) . 3
(b) Find the value of a which produces the largest area of the extension. 4
hsn.uk.net Page 4
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
7. The parabolas with equations y = 10− x2 and y = 25(10− x2) are shown in the
diagram below.
= 10 – 2
R
ST
Q
P
22
5(10 )= −
O
x
x
x y
y
y
A rectangle PQRS is placed between the two parabolas as shown, so that:
• Q and R lie on the upper parabola.
• RQ and SP are parallel to the x-axis.
• T , the turning point of the lower parabola, lies on SP.
(a) (i) If TP = x units, find an expression for the length of PQ.
(ii) Hence show that the area, A , of rectangle PQRS is given by
A(x) = 12x− 2x3· 3
(b) Find the maximum area of this rectangle. 6
8.[SQA]
hsn.uk.net Page 5
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
9.[SQA]
10.[SQA]
11.[SQA]
hsn.uk.net Page 6
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
12.[SQA] The graphs of y = f (x) and y = g(x) areshown in the diagram.
f (x) = −4 cos(2x) + 3 and g(x) is of theform g(x) = m cos(nx) .
(a) Write down the values of m and n . 1
(b) Find, correct to one decimal place,the coordinates of the points ofintersection of the two graphs in theinterval 0 ≤ x ≤ π . 5
(c) Calculate the shaded area. 6
π
= f( )
= g( )
7
3
0
–1
–3x
x
x
y
y
y
[END OF QUESTIONS]
hsn.uk.net Page 7Questions marked ‘[SQA]’ c© SQA
All others c© Higher Still Notes
Higher Mathematics
GCC Applied Calculus
1.[SQA]
Part Marks Level Calc. Content Answer U1 OC3
(a) 1 C CN CGD 1996 P2 Q11
(a) 3 A/B CN CGD
(b) 2 C CN C11
(b) 3 A/B CN C11
hsn.uk.net Page 1
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
2.[SQA]
Part Marks Level Calc. Content Answer U1 OC3
(a) 1 C CN CGD 1997 P2 Q10
(a) 3 A/B CN CGD
(b) 3 C CN C11
(b) 3 A/B CN C11
hsn.uk.net Page 2
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
3.[SQA]
Part Marks Level Calc. Content Answer U1 OC3
(a) 3 A/B CR CGD 1998 P2 Q10
(b) 3 C CR C11
(b) 3 A/B CR C11
hsn.uk.net Page 3
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
4.[SQA]
Part Marks Level Calc. Content Answer U3 OC2
(a) 1 C NC A6 1993 P2 Q11
(a) 2 A/B NC A6
(b) 1 C NC C11, C21
(b) 6 A/B NC C11, C21
hsn.uk.net Page 4
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
5.[SQA] A goldsmith has built up a solid which consists of a triangularprism of fixed volume with a regular tetrahedron at each end.
The surface area, A , of the solid is given by
A(x) =3√3
2
(
x2 +16
x
)
where x is the length of each edge of the tetrahedron.
Find the value of x which the goldsmith should use tominimise the amount of gold plating required to cover thesolid. 6
x
Part Marks Level Calc. Content Answer U1 OC3
6 A/B CN C11 x = 2 2000 P2 Q6
•1 ss: know to differentiate•2 pd: process•3 ss: know to set f ′(x) = 0•4 pd: deal with x−2•5 pd: process•6 ic: check for minimum
•1 A′(x) = . . .
•2 3√32 (2x− 16x−2) or 3
√3x− 24
√3x−2
•3 A′(x) = 0
•4 − 16x2or − 24
√3
x2
•5 x = 2•6 x 2− 2 2+
A′(x) −ve 0 +veso x = 2 is min.
hsn.uk.net Page 5
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
6.[SQA] The shaded rectangle on this maprepresents the planned extension to thevillage hall. It is hoped to provide thelargest possible area for the extension.
Village hall
Manse Lane
TheVennel
8 m
6 m
The coordinate diagram represents theright angled triangle of ground behindthe hall. The extension has length lmetres and breadth b metres, as shown.One corner of the extension is at the point(a, 0) .
O x
y
lb
(a, 0) (8, 0)
(0, 6)
(a) (i) Show that l = 54a .
(ii) Express b in terms of a and hence deduce that the area, A m2 , of theextension is given by A = 3
4a(8− a) . 3
(b) Find the value of a which produces the largest area of the extension. 4
Part Marks Level Calc. Content Answer U1 OC3
(a) 3 A/B CN CGD proof 2002 P2 Q10
(b) 4 A/B CN C11 a = 4
•1 ss: select strategy and carrythrough
•2 ss: select strategy and carrythrough
•3 ic: complete proof
•4 ss: know to set derivative to zero•5 pd: differentiate•6 pd: solve equation•7 ic: justify maximum, e.g. naturetable
•1 proof of l = 54a
•2 b = 35(8− a)
•3 complete proof leading to A = . . .
•4 dAda = . . . = 0
•5 6− 32a
•6 a = 4•7 e.g. nature table, comp. the square
hsn.uk.net Page 6
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
7. The parabolas with equations y = 10− x2 and y = 25(10− x2) are shown in the
diagram below.
= 10 – 2
R
ST
Q
P
22
5(10 )= −
O
x
x
x y
y
y
A rectangle PQRS is placed between the two parabolas as shown, so that:
• Q and R lie on the upper parabola.
• RQ and SP are parallel to the x-axis.
• T , the turning point of the lower parabola, lies on SP.
(a) (i) If TP = x units, find an expression for the length of PQ.
(ii) Hence show that the area, A , of rectangle PQRS is given by
A(x) = 12x− 2x3· 3
(b) Find the maximum area of this rectangle. 6
Part Marks Level Calc. Content Answer U1 OC3
(ai) 2 B CN C11 6− x2 2010 P2 Q5
(aii) 1 B CN C11 2x× (6− x2) = A(x)
(b) 6 C CN C11 max is 8√2
•1 ss: know to and find OT•2 ic: obtain an expression for PQ•3 ic: complete area evaluation
•4 ss: know to and start to differentiate•5 pd: complete differentiation•6 ic: set derivative to zero•7 pd: obtain•8 ss: justify nature of stationary point•9 ic: interpret result and evaluatearea
•1 4•2 10− x2 − 4•3 2x(6− x2) = 12x− 2x3
•4 A′(x) = 12 · · ·•5 12− 6x2•6 12− 6x2 = 0•7
√2
•8 x · · ·√2 · · ·
A′(x) + 0 −•9 Max and 8
√2
hsn.uk.net Page 7
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
8.[SQA]
Part Marks Level Calc. Content Answer U1 OC3
(a) 2 C NC A6 1992 P2 Q5
(a) 2 A/B NC A6
(b) 4 C NC C11
(b) 2 A/B NC C11
hsn.uk.net Page 8
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
9.[SQA]
Part Marks Level Calc. Content Answer U1 OC3
(a) 4 C NC CGD 1994 P2 Q7
(b) 3 C NC C11
(b) 5 A/B NC C11
hsn.uk.net Page 9
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
10.[SQA]
Part Marks Level Calc. Content Answer U2 OC2
(a) 2 C NC A6 1999 P2 Q10
(b) 7 A/B NC C17
hsn.uk.net Page 10
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
11.[SQA]
Part Marks Level Calc. Content Answer U2 OC2
3 C CN C17, CGD 1994 P2 Q10
6 A/B CN C17, CGD
hsn.uk.net Page 11
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
12.[SQA] The graphs of y = f (x) and y = g(x) areshown in the diagram.
f (x) = −4 cos(2x) + 3 and g(x) is of theform g(x) = m cos(nx) .
(a) Write down the values of m and n . 1
(b) Find, correct to one decimal place,the coordinates of the points ofintersection of the two graphs in theinterval 0 ≤ x ≤ π . 5
(c) Calculate the shaded area. 6
π
= f( )
= g( )
7
3
0
–1
–3x
x
x
y
y
y
Part Marks Level Calc. Content Answer U3 OC2
(a) 1 C CN T4 m = 3 and n = 2 2009 P2 Q5
(b) 5 C CR T6 (0·6, 1·3), (2·6, 1·3)(c) 6 B CR C17, C23 12·4
•1 ic: interprets graph
•2 ss: knows how to find intersection•3 pd: starts to solve•4 pd: finds x-coord in 1st quadrant•5 pd: finds x-coord in 2nd quadrant•6 pd: finds y-coordinates
•7 ss: knows how to find area•8 ic: states limits•9 pd: integrate•10 pd: integrate•11 ic: substitute limits•12 pd: evaluate area
•1 m = 3 and n = 2
•2 3 cos 2x = −4 cos 2x+ 3•3 cos 2x = 3
7•4 x = 0·6•5 x = 2·6•6 y = 1·3, 1·3
•7∫ (
−4 cos 2x+ 3− 3 cos 2x)
dx
•8∫ 2·60·6 · · ·
•9 “−7 sin 2x”•10 3x− 7
2 sin 2x
•11 (3× 2·6− 72 sin 5·2)− (3× 0·6− 7
2 sin 1·2)•12 12·4
[END OF QUESTIONS]
hsn.uk.net Page 12Questions marked ‘[SQA]’ c© SQA
All others c© Higher Still Notes
Higher Mathematics
GCC Higher Circles
1.[SQA]
2.[SQA] Find the equation of the circle which has P(−2,−1) and Q(4, 5) as the end pointsof a diameter. 3
3.[SQA] The line y = −1 is a tangent to a circle which passes through (0, 0) and (6, 0) .
Find the equation of this circle. 6
4.[SQA] Find the equation of the tangent at the point (3, 4) on the circlex2 + y2 + 2x− 4y− 15 = 0. 4
5.[SQA]
hsn.uk.net Page 1
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Higher Mathematics
6.[SQA]
7.[SQA]
8.[SQA]
(a) Show that the point P(5, 10) lies on circle C1 with equation(x+ 1)2 + (y− 2)2 = 100. 1
(b) PQ is a diameter of this circle asshown in the diagram. Find theequation of the tangent at Q. 5P(5, 10)
Q
O x
y
(c) Two circles, C2 and C3 , touch circle C1 at Q.
The radius of each of these circles is twice the radius of circle C1 .
Find the equations of circles C2 and C3 . 4
9.[SQA] Explain why the equation x2 + y2 + 2x+ 3y+ 5 = 0 does not represent a circle. 2
10.[SQA] For what range of values of k does the equation x2 + y2 + 4kx − 2ky− k − 2 = 0represent a circle? 5
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Higher Mathematics
11.[SQA] Circle P has equation x2 + y2 − 8x − 10y+ 9 = 0. Circle Q has centre (−2,−1)and radius 2
√2.
(a) (i) Show that the radius of circle P is 4√2.
(ii) Hence show that circles P and Q touch. 4
(b) Find the equation of the tangent to the circle Q at the point (−4, 1) . 3
(c) The tangent in (b) intersects circle P in two points. Find the x -coordinates of
the points of intersection, expressing you answers in the form a± b√3. 3
12.[SQA]
13.[SQA]
[END OF QUESTIONS]
hsn.uk.net Page 3Questions marked ‘[SQA]’ c© SQA
All others c© Higher Still Notes
Higher Mathematics
GCC Higher Circles
1.[SQA]
Part Marks Level Calc. Content Answer U2 OC4
2 C CN G10 1999 P1 Q4
2.[SQA] Find the equation of the circle which has P(−2,−1) and Q(4, 5) as the end pointsof a diameter. 3
Part Marks Level Calc. Content Answer U2 OC4
3 C CN G10 1995 P1 Q9
3.[SQA] The line y = −1 is a tangent to a circle which passes through (0, 0) and (6, 0) .
Find the equation of this circle. 6
Part Marks Level Calc. Content Answer U2 OC4
1 C CN G10 1996 P1 Q20
5 A/B CN G9, G15
hsn.uk.net Page 1
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
4.[SQA] Find the equation of the tangent at the point (3, 4) on the circlex2 + y2 + 2x− 4y− 15 = 0. 4
Part Marks Level Calc. Content Answer U2 OC4
4 C CN G2, G5, G9 1996 P1 Q4
hsn.uk.net Page 2
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
5.[SQA]
Part Marks Level Calc. Content Answer U3 OC1
8 C CN G9, G10, G25 1995 P2 Q8
hsn.uk.net Page 3
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
6.[SQA]
Part Marks Level Calc. Content Answer U3 OC1
5 C CN G9, G25 1997 P1 Q12
7.[SQA]
Part Marks Level Calc. Content Answer U2 OC4
(a) 3 C CN G9 1993 P1 Q5
(b) 1 C CN G10
hsn.uk.net Page 4
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
8.[SQA]
(a) Show that the point P(5, 10) lies on circle C1 with equation(x+ 1)2 + (y− 2)2 = 100. 1
(b) PQ is a diameter of this circle asshown in the diagram. Find theequation of the tangent at Q. 5P(5, 10)
Q
O x
y
(c) Two circles, C2 and C3 , touch circle C1 at Q.
The radius of each of these circles is twice the radius of circle C1 .
Find the equations of circles C2 and C3 . 4
Part Marks Level Calc. Content Answer U2 OC4
(a) 1 C CN A6 proof 2009 P2 Q4
(b) 5 C CN G11 3x+ 4y+ 45 = 0
(c) 4 A NC G15 (x− 5)2 + (y− 10)2 = 400,(x+ 19)2+(y+ 22)2 = 400
•1 pd: substitute
•2 ic: find centre•3 ss: use mid-point result for Q•4 ss: know to, and find gradient ofradius
•5 ic: find gradient of tangent•6 ic: state equation of tangent
•7 ic: state radius•8 ss: know how to find centre•9 ic: state equation of one circle•10 ic: state equation of the other circle
•1 (5+ 1)2 + (10− 2)2 = 100
•2 centre = (−1, 2)•3 Q = (−7,−6)•4 mrad = 8
6
•5 mtgt = − 34•6 y− (−6) = − 34(x− (−7))
•7 radius = 20•8 centre = (5, 10)•9 (x− 5)2 + (y− 10)2 = 400•10 (x+ 19)2 + (y+ 22)2 = 400
hsn.uk.net Page 5
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
9.[SQA] Explain why the equation x2 + y2 + 2x+ 3y+ 5 = 0 does not represent a circle. 2
Part Marks Level Calc. Content Answer U2 OC4
2 C CN G9 1993 P1 Q18
10.[SQA] For what range of values of k does the equation x2 + y2 + 4kx − 2ky− k − 2 = 0represent a circle? 5
Part Marks Level Calc. Content Answer U2 OC4
5 A NC G9, A17 for all k 2000 P1 Q6
•1 ss: know to examine radius•2 pd: process•3 pd: process•4 ic: interpret quadratic inequation•5 ic: interpret quadratic inequation
•1 g = 2k, f = −k, c = −k− 2stated or implied by •2
•2 r2 = 5k2 + k+ 2•3 (real r ⇒) 5k2 + k+ 2 > 0 (accept ≥)•4 use discr. or complete sq. or diff.•5 true for all k
hsn.uk.net Page 6
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Higher Mathematics
11.[SQA] Circle P has equation x2 + y2 − 8x − 10y+ 9 = 0. Circle Q has centre (−2,−1)and radius 2
√2.
(a) (i) Show that the radius of circle P is 4√2.
(ii) Hence show that circles P and Q touch. 4
(b) Find the equation of the tangent to the circle Q at the point (−4, 1) . 3
(c) The tangent in (b) intersects circle P in two points. Find the x -coordinates of
the points of intersection, expressing you answers in the form a± b√3. 3
Part Marks Level Calc. Content Answer U2 OC4
(a) 2 C CN G9 proof 2001 P1 Q11
(a) 2 A/B CN G14
(b) 3 C CN G11 y = x+ 5
(c) 3 C CN G12 x = 2± 2√3
•1 ic: interpret centre of circle (P)•2 ss: find radius of circle (P)•3 ss: find sum of radii•4 pd: compare with distance betweencentres
•5 ss: find gradient of radius•6 ss: use m1m2 = −1•7 ic: state equation of tangent
•8 ss: substitute linear into circle•9 pd: express in standard form•10 pd: solve (quadratic) equation
•1 CP = (4, 5)•2 rP =
√16+ 25− 9 =
√32 = 4
√2
•3 rP + rQ = 4√2+ 2
√2 = 6
√2
•4 CPCQ =√62 + 62 = 6
√2 and “so
touch”
•5 mr = −1•6 mtgt = +1•7 y− 1 = 1(x+ 4)
•8 x2+(x+ 5)2− 8x− 10(x+ 5)+ 9 = 0•9 2x2 − 8x− 16 = 0•10 x = 2± 2
√3
hsn.uk.net Page 7
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
12.[SQA]
Part Marks Level Calc. Content Answer U2 OC4
(a) 4 C CN G10 1990 P2 Q8
(b) 4 C CN G10
hsn.uk.net Page 8
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
13.[SQA]
Part Marks Level Calc. Content Answer U2 OC4
5 C CN G9, G1, G15 1992 P1 Q16
[END OF QUESTIONS]
hsn.uk.net Page 9Questions marked ‘[SQA]’ c© SQA
All others c© Higher Still Notes
Higher Mathematics
GCC Trig Equations and Expressions
1.[SQA] Solve the equation 3 cos 2x◦ + cos x◦ = −1 in the interval 0 ≤ x ≤ 360. 5
2. (a)[SQA] Solve the equation sin 2x◦ − cos x◦ = 0 in the interval 0 ≤ x ≤ 180. 4
(b) The diagram shows parts of twotrigonometric graphs, y = sin 2x◦
and y = cos x◦ .
Use your solutions in (a) to writedown the coordinates of the point P. 1
O x
yy = sin 2x◦
y = cos x◦P
90180
3.[SQA]
(a) Using the fact that 7π12 = π
3 + π
4 , find the exact value of sin(
7π12
)
. 3
(b) Show that sin(A+ B) + sin(A− B) = 2 sin A cos B . 2
(c) (i) Express π
12 in terms ofπ
3 andπ
4 .
(ii) Hence or otherwise find the exact value of sin(
7π12
)
+ sin(
π
12
)
. 4
4.[SQA] Functions f (x) = sin x , g(x) = cos x and h(x) = x+ π
4 are defined on a suitableset of real numbers.
(a) Find expressions for:
(i) f (h(x)) ;
(ii) g(h(x)) . 2
(b) (i) Show that f (h(x)) = 1√2sin x+ 1√
2cos x .
(ii) Find a similar expression for g(h(x)) and hence solve the equationf (h(x)) − g(h(x)) = 1 for 0 ≤ x ≤ 2π . 5
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Higher Mathematics
5.[SQA] On the coordinate diagram shown, A is thepoint (6, 8) and B is the point (12,−5) .Angle AOC = p and angle COB = q .
Find the exact value of sin(p+ q) . 4
O x
yA(6, 8)
C
B(12,−5)
pq
6.[SQA]
7.[SQA]
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
8.[SQA]
9.[SQA]
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
10.[SQA]
[END OF QUESTIONS]
hsn.uk.net Page 4Questions marked ‘[SQA]’ c© SQA
All others c© Higher Still Notes
Higher Mathematics
GCC Trig Equations and Expressions
1.[SQA] Solve the equation 3 cos 2x◦ + cos x◦ = −1 in the interval 0 ≤ x ≤ 360. 5
Part Marks Level Calc. Content Answer U2 OC3
5 A/B CR T10 60, 131·8, 228·2, 300 2000 P2 Q5
•1 ss: know to usecos 2x = 2 cos2 x− 1
•2 pd: process•3 ss: know to/and factorise quadratic•4 pd: process•5 pd: process
•1 3(2 cos2 x◦ − 1)•2 6 cos2 x◦ + cos x◦ − 2 = 0•3 (2 cos x◦ − 1)(3 cos x◦ + 2)•4 cos x◦ = 1
2 , x = 60, 30
•5 cos x◦ = − 23 , x = 132, 228
2. (a)[SQA] Solve the equation sin 2x◦ − cos x◦ = 0 in the interval 0 ≤ x ≤ 180. 4
(b) The diagram shows parts of twotrigonometric graphs, y = sin 2x◦
and y = cos x◦ .
Use your solutions in (a) to writedown the coordinates of the point P. 1
O x
yy = sin 2x◦
y = cos x◦P
90180
Part Marks Level Calc. Content Answer U2 OC3
(a) 4 C NC T10 30, 90, 150 2001 P1 Q5
(b) 1 C NC T3 (150,−√32 )
•1 ss: use double angle formula•2 pd: factorise•3 pd: process•4 pd: process
•5 ic: interpret graph
•1 2 sin x◦ cos x◦•2 cos x◦(2 sin x◦ − 1)•3 cos x◦ = 0, sin x◦ = 1
2•4 90, 30, 150
or
•3 sin x◦ = 12 and x = 30, 150
•4 cos x◦ = 0 and x = 90
•5(
150,−√32
)
hsn.uk.net Page 1
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
3.[SQA]
(a) Using the fact that 7π12 = π
3 + π
4 , find the exact value of sin(
7π12
)
. 3
(b) Show that sin(A+ B) + sin(A− B) = 2 sin A cos B . 2
(c) (i) Express π
12 in terms ofπ
3 andπ
4 .
(ii) Hence or otherwise find the exact value of sin(
7π12
)
+ sin(
π
12
)
. 4
Part Marks Level Calc. Content Answer U2 OC3
(a) 3 C NC T8, T3
√3+ 1
2√2
2009 P1 Q24
(b) 2 C CN T8 proof
(c) 3 B NC T11 π
12 = π
3 − π
4
(c) 1 C NC T11√62 or
√
32
•1 ss: expand compound angle•2 ic: substitute exact values•3 pd: process to a single fraction
•4 ic: start proof•5 ic: complete proof
•6 ss: identify steps•7 ic: start process (identify ‘A’ & ‘B’)•8 ic: substitute•9 pd: process
•1 sin π
3 cosπ
4 + cos π
3 sinπ
4
•2√32 × 1√
2+ 12 × 1√
2
•3√3+ 1
2√2or equivalent
•4 sin A cos B+ cos A sin B+ · · ·•5 · · · + sin A cos B − cos A sin B andcomplete
•6 π
12 = π
3 − π
4
•7 2×√32 × 1√
2
•8√62 or
√
32
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
4.[SQA] Functions f (x) = sin x , g(x) = cos x and h(x) = x+ π
4 are defined on a suitableset of real numbers.
(a) Find expressions for:
(i) f (h(x)) ;
(ii) g(h(x)) . 2
(b) (i) Show that f (h(x)) = 1√2sin x+ 1√
2cos x .
(ii) Find a similar expression for g(h(x)) and hence solve the equationf (h(x)) − g(h(x)) = 1 for 0 ≤ x ≤ 2π . 5
Part Marks Level Calc. Content Answer U2 OC3
(a) 2 C NC A4 (i) sin(x + π
4 ), (ii)cos(x+ π
4 )2001 P1 Q7
(b) 5 C NC T8, T7 (i) proof, (ii) x = π
4 ,3π4
•1 ic: interpret composite functions•2 ic: interpret composite functions
•3 ss: expand sin(x+ π
4 )•4 ic: interpret•5 ic: substitute•6 pd: start solving process•7 pd: process
•1 sin(x+ π
4 )•2 cos(x+ π
4 )
•3 sin x cos π
4 + cos x sin π
4 and
complete
•4 g(h(x)) = 1√2cos x− 1√
2sin x
•5 ( 1√2sin x+ 1√
2cos x)− ( 1√
2cos x− 1√
2sin x)
•6 2√2sin x
•7 x = π
4 ,3π4 accept only radians
5.[SQA] On the coordinate diagram shown, A is thepoint (6, 8) and B is the point (12,−5) .Angle AOC = p and angle COB = q .
Find the exact value of sin(p+ q) . 4
O x
yA(6, 8)
C
B(12,−5)
pq
Part Marks Level Calc. Content Answer U2 OC3
4 C NC T9 6365 2000 P1 Q1
•1 ss: know to use trig expansion•2 pd: process missing sides•3 ic: interpret data•4 pd: process
•1 sin p cos q+ cos p sin q•2 10 and 13•3 8
10 ·1213 + 6
10 ·513
•4 126130
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
6.[SQA]
Part Marks Level Calc. Content Answer U2 OC3
(a-c) 3 C NC CGD 1999 P2 Q8
(d) 4 A/B NC T8, G2
(e) 2 A/B NC G2, G5
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
7.[SQA]
Part Marks Level Calc. Content Answer U2 OC3
1 C NC T9 1996 P1 Q18
4 A/B NC T9
8.[SQA]
Part Marks Level Calc. Content Answer U2 OC3
1 C NC T10 1991 P1 Q20
3 A/B NC T10
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
9.[SQA]
Part Marks Level Calc. Content Answer U2 OC3
(a) 4 C CR T10 1992 P2 Q7
(b) 1 C CR A7
(c) 2 C CR A6
(d) 2 C CR T2
(d) 1 A/B CR T2
hsn.uk.net Page 6
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
10.[SQA]
Part Marks Level Calc. Content Answer U2 OC3
(a) 2 C CN CGD 1991 P2 Q3
(a) 1 A/B CN CGD
(b) 5 C CN T10, T11
[END OF QUESTIONS]
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All others c© Higher Still Notes
Higher Mathematics
GCCWaveFunction
1.[SQA] Express 2 sin x◦ − 5 cos x◦ in the form k sin(x− α)◦ , 0 ≤ α < 360 and k > 0. 4
2.[SQA] Express 8 cos x◦− 6 sin x◦ in the form k cos(x◦ + a◦) where k > 0 and 0 < a < 360. 4
3.[SQA] Find the maximum value of cos x− sin x and the value of x for which it occurs inthe interval 0 ≤ x ≤ 2π . 6
4.[SQA]
5. (a) 12 cos x◦ − 5 sin x◦ can be expressed in the form k cos(x + a)◦ , where k > 0and 0 ≤ a < 360.
Calculate the values of k and a . 4
(b) (i) Hence state the maximum and minimum values of12 cos x◦ − 5 sin x◦ .(ii) Determine the values of x , in the interval 0 ≤ x < 360, at which thesemaximum and minimum values occur. 3
6. (a)[SQA] Write sin(x) − cos(x) in the form k sin(x − a) stating the values of k and awhere k > 0 and 0 ≤ a ≤ 2π 4
(b) Sketch the graph of y = sin(x) − cos(x) for 0 ≤ x ≤ 2π , showing clearly thegraph’s maximum and minimum values and where it cuts the x -axis and they-axis. 3
7.[SQA]
8.[SQA] Solve the equation 2 sin x◦ − 3 cos x◦ = 2·5 in the interval 0 ≤ x < 360. 8
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Higher Mathematics
9.[SQA]
10. (a) The expression 3 sin x− 5 cos x can be written in the form R sin(x+ a) whereR > 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∫
0
(3 cos x+ 5 sin x) dx = 3.
7
11.[SQA] The displacement, d units, of a wave after t seconds, is given by the formula
d = cos 20t◦ +√3 sin 20t◦ .
(a) Express d in the form k cos(20t◦ − α◦) , where k > 0 and 0 ≤ α ≤ 360. 4
(b) Sketch the graph of d for 0 ≤ t ≤ 18. 4
(c) Find, correct to one decimal place, the values of t , 0 ≤ t ≤ 18, for which thedisplacement is 1·5 units. 3
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
12.[SQA]
13.[SQA]
[END OF QUESTIONS]
hsn.uk.net Page 3Questions marked ‘[SQA]’ c© SQA
All others c© Higher Still Notes
Higher Mathematics
GCCWaveFunction
1.[SQA] Express 2 sin x◦ − 5 cos x◦ in the form k sin(x− α)◦ , 0 ≤ α < 360 and k > 0. 4
Part Marks Level Calc. Content Answer U3 OC4
4 C CR T13 k =√29, α = 68·2 1997 P1 Q11
2.[SQA] Express 8 cos x◦− 6 sin x◦ in the form k cos(x◦ + a◦) where k > 0 and 0 < a < 360. 4
Part Marks Level Calc. Content Answer U3 OC4
4 C CR T13 10 cos(x◦ + 36·9◦) 2001 P2 Q5
•1 ss: expand k cos(x◦ + a◦)•2 ic: compare coefficients•3 pd: process•4 pd: process
•1 k cos x cos a − k sin x sin a statedexplicitly
•2 k cos a = 8 and k sin a = 6 statedexplicitly
•3 k = 10•4 a = 36·9
3.[SQA] Find the maximum value of cos x− sin x and the value of x for which it occurs inthe interval 0 ≤ x ≤ 2π . 6
Part Marks Level Calc. Content Answer U3 OC4
6 A/B CN T14 max value√2 when
x = 7π4
2000 P1 Q10
•1 ss: use e.g. k cos(x+ a)•2 ic: expand chosen rule•3 pd: compare coefficients•4 pd: process•5 pd: process•6 ic: interpret trig expression
•1 e.g. use k cos(x+ a)•2 k cos x cos a− k sin x sin a•3 k cos a = 1 and k sin a = 1•4 k =
√2
•5 tan a = 1, a = π
4 (45◦ is bad form)
•6 max. value =√2 when x = 7π
4 (donot accept 45◦)
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
4.[SQA]
Part Marks Level Calc. Content Answer U3 OC4
(a) 4 C CR T13 1992 P1 Q7
(b) 1 C CR T14
(b) 1 A/B CR T14
5. (a) 12 cos x◦ − 5 sin x◦ can be expressed in the form k cos(x + a)◦ , where k > 0and 0 ≤ a < 360.
Calculate the values of k and a . 4
(b) (i) Hence state the maximum and minimum values of12 cos x◦ − 5 sin x◦ .(ii) Determine the values of x , in the interval 0 ≤ x < 360, at which thesemaximum and minimum values occur. 3
Part Marks Level Calc. Content Answer U3 OC4
(a) 4 C CN T13 k = 13, a = 22·6 2010 P2 Q2
(bi) 1 C CN T14 max 13, min −13(bii) 2 C CN T14 max at 337·4, min at 157·4
•1 ss: use addition formula•2 ic: compare coefficients•3 pd: process k•4 pd: process a
•5 ss: state maximum and minimum•6 ic: find x corresponding to max.value
•7 pd: find x corresponding to min.value
•1 k cos x◦ cos a◦ − k sin x◦ sin a◦•2 k cos a◦ = 12 and k sin a◦ = 5•3 13 (do not accept
√169)
•4 22·6 (accept any answer whichrounds to 23)
•5 13, −13•6 maximum at 337·4 and no others•7 minimum at 157·4 and no others
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
6. (a)[SQA] Write sin(x) − cos(x) in the form k sin(x − a) stating the values of k and awhere k > 0 and 0 ≤ a ≤ 2π 4
(b) Sketch the graph of y = sin(x) − cos(x) for 0 ≤ x ≤ 2π , showing clearly thegraph’s maximum and minimum values and where it cuts the x -axis and they-axis. 3
Part Marks Level Calc. Content Answer U3 OC4
(a) 4 C NC T13√2 sin(x− π
4 ) 2002 P1 Q9
(b) 3 C NC T15, T14 sketch
•1 ss: know to expand, and expand•2 ic: compare coefficients•3 pd: write down the value of k•4 pd: process a
•5 ic: sketch a sine curve•6 ic: int/com max. and min. values•7 pd: process intercepts
•1 k sin x cos a − k cos x sin a statedexplicitly
•2 k cos a = 1 and k sin a = 1 statedexplicitly
•3 k =√2
•4 a = π
4 accept in degrees
•5 correct shape of graph (i.e. sin) butnot passing through the origin
•6 graph lies between√2 and −
√2
•7 ( π
4 , 0), (5π4 , 0), (0,−1) accept only
answers in radians
7.[SQA]
Part Marks Level Calc. Content Answer U3 OC4
(a) 4 C CR T13 1996 P2 Q7
(b) 3 C CR T16
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
8.[SQA] Solve the equation 2 sin x◦ − 3 cos x◦ = 2·5 in the interval 0 ≤ x < 360. 8
Part Marks Level Calc. Content Answer U3 OC4
8 A/B CR T16 100·2◦, 192·4◦ 1999 P2 Q9
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
9.[SQA]
Part Marks Level Calc. Content Answer U3 OC4
(a) 4 C CR T13 1994 P2 Q5
(b) 4 C CR T16
(c) 2 A/B CR T16
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
10. (a) The expression 3 sin x− 5 cos x can be written in the form R sin(x+ a) whereR > 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∫
0
(3 cos x+ 5 sin x) dx = 3.
7
Part Marks Level Calc. Content Answer U3 OC4
(a) 4 C CN T13 R =√34, a = 5·253 2011 P2 Q6
(b) 7 B CN C23, T3, T16 t = 0·6
•1 ss: use compound angle formula•2 ic: compare coefficients•3 pd: process R•4 pd: process a
•5 pd: integrate given expression•6 ic: substitute limits•7 pd: process limits•8 ss: know to use wave equation•9 ic: write in standard format•10 ss: start to solve equation•11 pd: complete and state solution
•1 R sin x cos a+ R cos x sin a•2 R cos a = 3 and R sin a = −5•3
√34 (accept 5·8)
•4 5·253 (accept 5·3)
•5 3 sin x− 5 cos x•6 (3 sin t− 5 cos t) − (3 sin 0− 5 cos 0)•7 3 sin t− 5 cos t+ 5•8
√34 sin(t+ 5·3) + 5
•9 sin(t+ 5·3) = − 2√34
•10 t+ 5·3 = 3·5, 5·9•11 t = 0·6
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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
11.[SQA] The displacement, d units, of a wave after t seconds, is given by the formula
d = cos 20t◦ +√3 sin 20t◦ .
(a) Express d in the form k cos(20t◦ − α◦) , where k > 0 and 0 ≤ α ≤ 360. 4
(b) Sketch the graph of d for 0 ≤ t ≤ 18. 4
(c) Find, correct to one decimal place, the values of t , 0 ≤ t ≤ 18, for which thedisplacement is 1·5 units. 3
Part Marks Level Calc. Content Answer U3 OC4
(a) 4 C CR T13 1991 P2 Q8
(b) 2 C CR T1
(b) 2 A/B CR T1
(c) 1 C CR T7
(c) 2 A/B CR T7
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Higher Mathematics
12.[SQA]
Part Marks Level Calc. Content Answer U3 OC4
(a) 1 C CR CGD 1993 P2 Q9
(a) 4 A/B CR CGD
(b) 4 C CR T13
(c) 1 C CR T16
(c) 3 A/B CR T16
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Higher Mathematics
13.[SQA]
Part Marks Level Calc. Content Answer U3 OC4
(a) 4 C CR T13 1989 P2 Q9
(b) 2 C CR T1, A3
(b) 4 A/B CR T1, A3
(c) 1 A/B CR CGD
(d) 2 A/B CR CGD
[END OF QUESTIONS]
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All others c© Higher Still Notes