Higher Mathematics
Relationships and Calculus Mixed
[SQA]
1. (a) Express f ( x ) = x2 − 4x + 5 in the form f ( x ) = ( x − a)2 + b.
2
(b) On the same diagram sketch:
(i) the graph of y = f ( x ) ;
(ii) the graph of y = 10 − f ( x ) .
4
(c) Find the range of values of x for which 10 − f ( x ) is positive.
Part
(a)
(b)
(c)
Marks
2
4
1
Level
C
C
C
Calc.
NC
NC
NC
•1 pd: process, e.g.
square
•2 pd: process, e.g.
square
•3
•4
•5
•6
ic:
ic:
ss:
ss:
•7 ic:
[SQA]
Content
A5
A3
A16, A6
completing the
completing the
interpret minimum
interpret y-intercept
reflect in x-axis
translate parallel to y-axis
Answer
a = 2, b = 1
sketch
−1 < x < 5
1
U1 OC2
2002 P1 Q7
•1 a = 2
•2 b = 1
•3 any two from:
parabola; min. t.p. (2, 1); (0, 5)
•4 the remaining one from above list
•5 reflecting in x-axis
•6 translating +10 units, parallel to
y-axis
•7 (−1, 5) i.e. −1 < x < 5
interpret graph
2.
(i) Write down the condition for the equation ax2 + bx + c = 0 to have no real
roots.
1
(ii) Hence or otherwise show that the equation x ( x + 1) = 3x − 2 has no real
roots.
2
Part
Marks
3
hsn.uk.net
Level
C
Calc.
CN
Content
A17
Page 1
Answer
U2 OC1
1999 P1 Q8
c SQA
Questions marked ‘[SQA]’ c
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Higher Mathematics
[SQA]
3. Show that the roots of the equation (k − 2) x2 − (3k − 2) x + 2k = 0 are real.
Part
Marks
1
3
hsn.uk.net
Level
C
A/B
Calc.
CN
CN
Content
A17
A17
Page 2
Answer
U2 OC1
1990 P1 Q18
c SQA
Questions marked ‘[SQA]’ c
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4
Higher Mathematics
[SQA]
4.
Part
(a)
(b)
(b)
(c)
Marks
1
2
4
2
hsn.uk.net
Level
C
C
A/B
A/B
Calc.
CN
CN
CN
CN
Content
A6
C4, CGD
C4, CGD
A17
Page 3
Answer
U2 OC1
1994 P2 Q9
c SQA
Questions marked ‘[SQA]’ c
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Higher Mathematics
[SQA]
5. (a) Given that x + 2 is a factor of 2x3 + x2 + kx + 2, find the value of k.
3
(b) Hence solve the equation 2x3 + x2 + kx + 2 = 0 when k takes this value.
Part
(a)
(b)
Marks
3
2
Level
C
C
Calc.
CN
CN
•1 ss:
use synth
f (evaluation)
•2 pd: process
•3 pd: process
Content
A21
A22
division
Answer
k = −5
x = −2, 21 , 1
or
U2 OC1
2001 P2 Q1
•1 f (−2) = 2(−2)3 + · · ·
•2 2(−2)3 + (−2)2 − 2k + 2
•3 k = −5
•4 2x2 − 3x + 1 or 2x2 + 3x − 2 or
x2 + x − 2
5
• (2x − 1)( x − 1) or (2x − 1)( x + 2) or
( x + 2)( x − 1)
and x = −2, 12 , 1
•4 ss: find a quadratic factor
•5 pd: process
[SQA]
6. Solve the equation 3 cos 2x ◦ + cos x ◦ = −1 in the interval 0 ≤ x ≤ 360.
Part
Marks
5
Level
A/B
Calc.
CR
Content
T10
•1 ss:
know
to
use
cos 2x = 2 cos2 x − 1
•2 pd: process
•3 ss: know to/and factorise quadratic
•4 pd: process
•5 pd: process
Answer
60, 131·8, 228·2, 300
•1
•2
•3
•4
•5
5
U2 OC3
2000 P2 Q5
3(2 cos2 x◦ − 1)
6 cos2 x◦ + cos x◦ − 2 = 0
(2 cos x◦ − 1)(3 cos x◦ + 2)
cos x◦ = 21 , x = 60, 30
cos x◦ = − 23 , x = 132, 228
7. Solve 2 cos 2x − 5 cos x − 4 = 0 for 0 ≤ x < 2π .
Part
Marks
5
Level
B
Calc.
CN
Content
T10, T7
•1 ss:
know to use double angle
formula
•2 ic: express as quadratic in cos x
•3 ss: start to solve
•4 pd: reduce to equations in cos x only
•5 pd: complete solutions to include
only one where cos x = k with
|k| > 1
hsn.uk.net
2
Page 4
5
Answer
x = 2·419, 3·864
•1
•2
•3
•4
•5
U2 OC3
2010 P2 Q4
2 × (2 cos2 x − 1) · · ·
4 cos2 x − 5 cos x − 6 = 0
(4 cos x + 3)(cos x − 2) = 0
cos x = − 34 and cos x = 2
2·419, 3·864 and no solution.
c SQA
Questions marked ‘[SQA]’ c
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Higher Mathematics
[SQA]
8.
Part
(a)
(b)
(c)
[SQA]
Marks
4
4
2
Level
C
C
A/B
Calc.
CR
CR
CR
Content
T13
T16
T16
Answer
U3 OC4
1994 P2 Q5
9. A point moves in a straight line such that its acceleration a is given by
1
a = 2(4 − t) 2 , 0 ≤ t ≤ 4. If it starts at rest, find an expression for the velocity
v where a = dv
dt .
Part
Marks
4
Level
C
Calc.
NC
Content
C18, C22
•1 ss: know to integrate acceleration
•2 pd: integrate
•3 ic: use initial conditions with const.
of int.
4
• pd: process solution
hsn.uk.net
Page 5
Answer
3
V = − 43 (4 − t) 2 +
U3 OC2
32
3
2002 P2 Q8
R
1
•1 V = (2(4 − t) 2 ) dt stated or implied
by •2
3
•2 2 × −13 (4 − t) 2
2
•3 0 = 2 ×
•4 c =
10 23
3
1
( 4 − 0) 2
− 23
+c
c SQA
Questions marked ‘[SQA]’ c
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4
Higher Mathematics
[SQA]
√ dy
= 3 sin(2x ) passes through the point 5π
,
3 .
12
dx
Find y in terms of x .
10. A curve for which
Part
•1
•2
•3
•4
[SQA]
Marks
4
pd:
pd:
ss:
pd:
Level
A/B
Calc.
CN
Content
C18, C23
integrate trig function
integrate composite function
use given point to find “c”
evaluate “c”
Answer
√
y = − 32 cos(2x) + 41 3
4
U3 OC2
2001 P2 Q10
R
•1 3 sin(2x) dx stated or implied by •2
•2 − 32 cos(2x)
√
5
π) + c
•3 3 = − 23 cos(2 × 12
√
1
4
• c = 4 3 (≈ 0·4)
11. Given that f ( x ) = 5(7 − 2x )3 , find the value of f ′ (4) .
Part
Marks
4
hsn.uk.net
Level
A/B
Calc.
NC
Content
C21
Page 6
Answer
4
U3 OC2
1991 P1 Q13
c SQA
Questions marked ‘[SQA]’ c
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Higher Mathematics
[SQA]
12.
(a) By writing sin 3x as sin(2x + x ) , show that sin 3x = 3 sin x − 4 sin3 x .
(b) Hence find
Part
(a)
(a)
(b)
Marks
2
2
4
hsn.uk.net
Z
sin3 x dx .
Level
C
A/B
A/B
Calc.
NC
NC
NC
4
4
Content
T8, T8
T8, T8
C23
Page 7
Answer
U3 OC2
1995 P2 Q9
c SQA
Questions marked ‘[SQA]’ c
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Higher Mathematics
[SQA]
13.
Part
Marks
8
Level
C
Calc.
CN
Content
A21, C8
Answer
U2 OC1
1993 P2 Q1
[END OF QUESTIONS]
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Page 8
c SQA
Questions marked ‘[SQA]’ c Higher Still Notes
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