Higher Mathematics. Functions and Graphs. Paper 1 Section B. 1. [SQA]. The
points A and B have coordinates (a, a2. ) and (2b,4b2. ) respectively. Determine.
PSfrag replacements O x y
Higher Mathematics
Functions and Graphs Paper 1 Section B 1. The points A and B have coordinates (a, a2 ) and (2b, 4b2 ) respectively. Determine the gradient of AB in its simplest form.
[SQA]
frag replacements O x y
2.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 1
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
2
PSfrag replacements O x y
Higher Mathematics
PSfrag replacements y 3. The diagram shows a sketch of part of the graph of y = log2 (x).
[SQA]
y = log2 (x)
(a) State the values of a and b .
(b) Sketch the graph of y = log2 (x + 1) − 3.
Part (a) (b)
Marks 1 3
Level A/B A/B
Calc. CN CN
Content A7 A3
O
(a, 0)
Answer a = 1, b = 3 sketch
(8, b)
1 x
U1 OC2 2001 P1 Q10
•1 pd: use log p q = 0 ⇒ q = 1 and
•1 a = 1 and b = 3
•2 ss: use a translation •3 ic: identify one point •4 ic: identify a second point
•2 a “log-shaped” graph of the same orientation •3 sketch passes through (0, −3) (labelled) 4 • sketch passes through (7, 0) (labelled)
evaluate log p pk
4.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 2
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
3
PSfrag replacements O x y
Higher Mathematics
5.
[SQA]
frag replacements O x y
frag replacements O x y
6.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 3
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
7.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 4
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
8.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 5
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
9.
[SQA]
frag replacements O x y
frag replacements O x y
10.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 6
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
11.
[SQA]
frag replacements O x y
frag replacements O x y
12. On a suitable set of real numbers, functions f and g are defined by f (x) =
[SQA]
1 − 2. x Find f g(x) in its simplest form.
1 x+2
and g(x) =
3
frag replacements O x y
replacements O x y
hsn.uk.net
Page 7
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
13. f (x) = 2x − 1, g(x) = 3 − 2x and h(x) = 41 (5 − x). (a) Find a formula for k(x) where k(x) = f g(x) . (b) Find a formula for h k(x) .
[SQA]
2 2
(c) What is the connection between the functions h and k?
1
frag replacements O x y
x 14. A function f is defined on the set of real numbers by f (x) = , x 6= 1. 1−x Find, in its simplest form, an expression for f f (x) .
[SQA]
frag replacements O x y
replacements O x y
hsn.uk.net
Page 8
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
3
PSfrag replacements O x y
Higher Mathematics
15. The functions f and g, defined on suitable domains, are given by f (x) =
[SQA]
and g(x) = 2x + 1.
1 x2 − 4
(a) Find an expression for h(x) where h(x) = g f (x) . Give your answer as a single fraction.
3
(b) State a suitable domain for h.
1
frag replacements O x y
16. Functions f and g, defined on suitable domains, are given by f (x) = 2x and g(x) = sin x + cos x . Find f g(x) and g f (x) .
[SQA]
frag replacements O x y
replacements O x y
hsn.uk.net
Page 9
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
4
PSfrag replacements O x y
Higher Mathematics
17. Given f (x) = x 2 + 2x − 8, express f (x) in the form (x + a)2 − b .
[SQA]
Part
Marks 2
Level C
Calc. NC
Content A5
Answer (x + 1)2 − 9
•1 ss: e.g. start to complete square •2 pd: complete process
2 U1 OC2 2001 P1 Q4
•1 (x + 1)2 . . . •2 (x + 1)2 − 9 or •1 a = 1 •2 b = 9 or •1 x2 + 2x − 8 ≡ x2 + 2ax + a2 − b •2 a = 1 and b = 9
18.
[SQA]
(a) Express 7 − 2x − x 2 in the form a − (x + b)2 and write down the values of a and b .
2
(b) State the maximum value of 7 − 2x − x 2 and justify your answer.
2
frag replacements O x y
19. Express (2x − 1)(2x + 5) in the form a(x + b) 2 + c.
[SQA]
3
frag replacements replacements O x y
O x y
hsn.uk.net
Page 10
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
20. Express x2 + 6x + 11 in the form (x + a)2 + b and hence state the maximum value 1 of 2 . x + 6x + 11
[SQA]
4
frag replacements O x y
21. Show that x2 + 8x + 18 can be written in the form (x + a)2 + b .
[SQA]
Hence or otherwise find the coordinates of the turning point of the curve with equation y = x2 + 8x + 18.
3
(a) Show that f (x) = 2x2 − 4x + 5 can be written in the form f (x) = a(x + b)2 + c.
3
(b) Hence write down the coordinates of the stationary point of y = f (x) and state its nature.
2
frag replacements O x y
22.
[SQA]
frag replacements replacements O x y
O x y
hsn.uk.net
Page 11
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
23.
[SQA]
frag replacements O x y
frag replacements O x y
24.
[SQA]
(a) Show that the function f (x) = 2x 2 + 8x − 3 can be written in the form f (x) = a(x + b)2 + c where a, b and c are constants.
3
(b) Hence, or otherwise, find the coordinates of the turning point of the function f.
1
frag replacements O x y
replacements O x y
hsn.uk.net
Page 12
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
25. The Water Board of a local authority discovered it was able to represent the approximate amount of water W(t), in millions of gallons, stored in a reservoir t months after the 1st May 1988 by the formula W(t) = 1·1 − sin πt 6 .
[SQA]
The board then predicted that under normal conditions this formula would apply for three years. πt (a) Draw and label sketches of the graphs of y = sin πt 6 and y = − sin 6 , for 0 ≤ t ≤ 36, on the same diagram.
4
(b) On a separate diagram and using the same scale on the t-axis as you used in part (a), draw a sketch of the graph of W(t) = 1·1 − sin πt 6 .
3
(c) On the 1st April 1990 a serious fire required an extra from the reservoir to bring the fire under control.
1 4
million gallons of water
Assuming that the previous trend continues from the new lower level, when will the reservoir run dry if water rationing is not imposed?
frag replacements O x y
replacements O x y
hsn.uk.net
Page 13
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
3
PSfrag replacements O x y
Higher Mathematics
26. (a) Express f (x) = x 2 − 4x + 5 in the form f (x) = (x − a)2 + b .
[SQA]
2
(b) On the same diagram sketch: (i) the graph of y = f (x); 4
(ii) the graph of y = 10 − f (x).
1
(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:
Content A5 A3 A16, A6
completing the completing the
Answer a = 2, b = 1 sketch −1 < x < 5
U1 OC2 2002 P1 Q7
•1 a = 2 •2 b = 1
interpret minimum interpret y-intercept reflect in x-axis translate parallel to y-axis
•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
interpret graph
•7 (−1, 5) i.e. −1 < x < 5
replacements O x y
hsn.uk.net
Page 14
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
27. A sketch of the graph of y = f (x) where f (x) = x 3 − 6x2 + 9x is shown below.
[SQA]
The graph has a maximum at A and a minimum at B(3, 0). y
A
PSfrag replacements O
y = f (x)
B(3, 0)
x
4
(a) Find the coordinates of the turning point at A. (b) Hence sketch the graph of y = g(x) where g(x) = f (x + 2) + 4. Indicate the coordinates of the turning points. There is no need to calculate the coordinates of the points of intersection with the axes. (c) Write down the range of values of k for which g(x) = k has 3 real roots. Part (a) (b)
Marks 4 2
Level C C
Calc. NC NC
(c)
1
A/B
NC
•1 •2 •3 •4
ss: pd: ss: pd:
Content C8 A3 A2
interpret transformation interpret transformation
•7 ic:
interpret sketch
U1 OC3 2000 P1 Q2
dy
know to differentiate differentiate correctly know gradient = 0 process
•5 ic: •6 ic:
Answer A(1, 4) sketch (translate 4 up, 2 left) 4 m g0 > 0
d •1 ss: use dx (quadratic) = linear 2 • ic: interpret stationary point
29.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 16
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
2
PSfrag replacements O x y
Higher Mathematics
PSfrag replacements 30. The graph of a function f intersects the x -axis at (−a, 0) and (e, 0) as shown.
[SQA]
y
There is a point of inflexion at (0, b) and a maximum turning point at (c, d).
(0, b)
Sketch the graph of the derived function f 0 .
Part
Marks 3
•1 ic: •2 ic: •3 ic:
Level C
Calc. CN
(c, d)
(−a, 0)
Content A3, C11
Answer sketch
3 O
(e, 0) x y = f (x) U1 OC3 2002 P1 Q6
•1 roots at 0 and c (accept a statement to this effect) 2 • min. at LH root, max. between roots •3 both ‘tails’ correct
interpret stationary points interpret main body of f interpret tails of f
31. The point P(−2, b) lies on the graph of the function f (x) = 3x 3 − x2 − 7x + 4.
[SQA]
(a) Find the value of b .
1
(b) Prove that this function is increasing at P.
3
frag replacements O x y
replacements O x y
hsn.uk.net
Page 17
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
32. A ball is thrown vertically upwards. The height h metres of the ball t seconds after it is thrown, is given by the formula h = 20t − 5t2 .
[SQA]
(a) Find the speed of the ball when it is thrown (i.e. the rate of change of height with respect to time of the ball when it is thrown).
3
(b) Find the speed of the ball after 2 seconds. Explain your answer in terms of the movement of the ball.
frag replacements O x y
replacements O x y
hsn.uk.net
Page 18
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
2
PSfrag replacements O x y
Higher Mathematics
33. A function f is defined by the formula f (x) = (x − 1) 2 (x + 2) where x ∈ R.
[SQA]
(a) Find the coordinates of the points where the curve with equation y = f (x) crosses the x - and y-axes.
3
(b) Find the stationary points of this curve y = f (x) and determine their nature.
7
(c) Sketch the curve y = f (x).
2
frag replacements O x y
replacements O x y
hsn.uk.net
Page 19
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
34.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 20
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
35. If y = x2 − x , show that
[SQA]
2y dy = 1+ . dx x
3
frag replacements O x y
36. If f (x) = kx3 + 5x − 1 and f 0 (1) = 14, find the value of k.
[SQA]
frag replacements O x y
replacements O x y
hsn.uk.net
Page 21
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
3
PSfrag replacements O x y
Higher Mathematics
37.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 22
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
38.
[SQA]
(a) The function f is defined by f (x) = x 3 − 2x2 − 5x + 6. The function g is defined by g(x) = x − 1. Show that f g(x) = x3 − 5x2 + 2x + 8. (b) Factorise fully f g(x) . (c) The function k is such that k(x) =
4 3
1 . f g(x)
For what values of x is the function k not defined?
frag replacements O x y
replacements O x y
hsn.uk.net
Page 23
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
3
PSfrag replacements O x y
Higher Mathematics
39.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 24
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
40.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 25
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
41.
[SQA]
frag replacements O x y
frag replacements replacements O x y
O x y
hsn.uk.net
Page 26
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
42.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 27
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
43.
[SQA]
frag replacements O x y
frag replacements O x y replacements O x y
hsn.uk.net
Page 28
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
44. Functions f and g are defined on the set of real numbers by f (x) = x − 1 and g(x) = x2 .
[SQA]
(a) Find formulae for (i) f g(x) (ii) g f (x) .
4
(b) The function h is defined by h(x) = f g(x) + g f (x) . Show that h(x) = 2x2 − 2x and sketch the graph of h.
3
(c) Find the area enclosed between this graph and the x -axis.
4
frag replacements O x y
replacements O x y
hsn.uk.net
Page 29
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
45. A function f is defined by the formula f (x) = 4x 2 (x − 3) where x ∈ R.
[SQA]
(a) Write down the coordinates of the points where the curve with equation y = f (x) meets the x - and y-axes.
2
(b) Find the stationary points of y = f (x) and determine the nature of each.
6
(c) Sketch the curve y = f (x).
2
(d) Find the area completely enclosed by the curve y = f (x) and the x -axis.
4
frag replacements O x y
replacements O x y
hsn.uk.net
Page 30
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
46.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 31
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
47.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 32
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
48.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 33
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
49.
[SQA]
frag replacements O x y
frag replacements O x y
50. Functions f (x) = sin x , g(x) = cos x and h(x) = x + set of real numbers.
[SQA]
π 4
are defined on a suitable
(a) Find expressions for: (i) f (h(x)); 2
(ii) g(h(x)). (b) (i) Show that f (h(x)) =
√1 2
sin x +
√1 2
cos x .
(ii) Find a similar expression for g(h(x)) and hence solve the equation f (h(x)) − g(h(x)) = 1 for 0 ≤ x ≤ 2π . Part (a)
Marks 2
Level C
Calc. NC
(b)
5
C
NC
•1 ic: •2 ic: •3 •4 •5 •6 •7
ss: ic: ic: pd: pd:
Content A4 T8, T7
interpret composite functions interpret composite functions
hsn.uk.net
•1 sin(x + π4 ) •2 cos(x + π4 ) •3 sin x cos π4 complete 4 • g(h(x)) =
expand sin(x + π4 ) interpret substitute start solving process process
+
cos x sin π4
U2 OC3 2001 P1 Q7
and
√1 cos x − √1 sin x 2 2 •5 ( √12 sin x + √12 cos x) − ( √12 cos x − √12 •6 √22 sin x •7 x = π4 , 3π accept only radians 4
replacements O x y
Answer (i) sin(x + π4 ), (ii) π cos(x + 4 ) (i) proof, (ii) x = π4 , 3π 4
5
Page 34
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
sin x)
PSfrag replacements O x y
Higher Mathematics
51. Functions f and g are defined on suitable domains by f (x) = sin(x ◦ ) and g(x) = 2x .
[SQA]
(a) Find expressions for: (i) f (g(x)); 2
(ii) g( f (x)).
5
(b) Solve 2 f (g(x)) = g( f (x)) for 0 ≤ x ≤ 360. Part (a) (b)
Marks 2 5
•1 ic: •2 ic:
Level C C
Calc. CN CN
Content A4 T10
Answer (i) sin(2x ◦ ), (ii) 2 sin(x ◦ ) 0◦ , 60◦ , 180◦ , 300◦ , 360◦
U2 OC3 2002 P1 Q3
•1 sin(2x ◦ ) •2 2 sin(x ◦ )
interpret f (g(x)) interpret g( f (x))
•3 •4 •5 •6
ss: equate for intersection ss: substitute for sin 2x pd: extract a common factor pd: solve a ‘common factor’ equation •7 pd: solve a ‘linear’ equation
•3 •4 •5 •6 •7
2 sin(2x ◦ ) = 2 sin(x ◦ ) appearance of 2 sin(x ◦ ) cos(x ◦ ) 2 sin(x ◦ ) (2 cos(x ◦ ) − 1) sin(x ◦ ) = 0 and 0, 180, 360 cos(x ◦ ) = 12 and 60, 300
or •6 sin(x ◦ ) = 0 and cos(x ◦ ) = •7 0, 60, 180, 300, 360
1 2
52.
[SQA]
frag replacements O x y
frag replacements replacements O x y
O x y
hsn.uk.net
Page 35
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
53. (a) Solve the equation sinPSfrag 2x ◦ − cos x ◦ = 0 in the interval 0 ≤ x ≤ 180. replacements y y = sin 2x ◦ (b) The diagram shows parts of two trigonometric graphs, y = sin 2x ◦ and y = cos x ◦ . 180 x 90 O Use your solutions in (a) to write down the coordinates of the point P. y = cos x ◦ P
[SQA]
Part (a) (b) •1 •2 •3 •4
Marks 4 1 ss: pd: pd: pd:
•5 ic:
Level C C
Calc. NC NC
Content T10 T3
Answer 30, 90, 150 √ (150, − 23 )
use double angle formula factorise process process or
interpret graph
PSfrag replacements 54. The diagram shows the graph of a cosine function from 0 to π .
[SQA]
(a) State the equation of the graph. √ (b) The line with equation y = − 3 intersects this graph at point A and B.
Marks 1 3
•1 ic:
replacements O x y
Level C C
Calc. NC NC
Content T4 T7
interpret graph
•2 ss: equate equal parts •3 pd: solve linear trig equation in radians 4 • ic: interpret result
hsn.uk.net
Page 36
1
U2 OC3 2001 P1 Q5
2 sin x ◦ cos x ◦ cos x ◦ (2 sin x ◦ − 1) cos x ◦ = 0, sin x ◦ = 90, 30, 150
1 2
•3 sin x ◦ = 12 and x = 30, 150 •4 cos x ◦ = 0 and x = 90 √ •5 150, − 23
y 2 1 O
π 2
A B
−2
Find the coordinates of B. Part (a) (b)
•1 •2 •3 •4
4
Answer y = 2 cos √2x B( 7π , − 3) 12
x √ y=− 3
π
U2 OC3 2002 P1 Q8
•1 2 cos 2x
√ •1 2 cos 2x = − 3 7π •2 2x = 5π 6 , 6 •3 x = 7π 12 c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
3
PSfrag replacements O x y
Higher Mathematics
4
55. Solve 2 sin 3x ◦ − 1 = 0 for 0 ≤ x ≤ 180.
[SQA]
frag replacements O x y
56. Solve the equation 2 cos2 x = 12 , for 0 ≤ x ≤ π .
[SQA]
3
frag replacements O x y
57. Find the exact solutions of the equation 4 sin2 x = 1, 0 ≤ x < 2π .
[SQA]
frag replacements O x y
replacements O x y
hsn.uk.net
Page 37
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
4
PSfrag replacements O x y
Higher Mathematics
58.
[SQA]
frag replacements O x y
frag replacements O x y
4
59. Solve the equation 2 sin 2x − π6 = 1, 0 ≤ x < 2π .
[SQA]
frag replacements O x y
replacements O x y
hsn.uk.net
Page 38
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
60. frag replacements [SQA]
O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 39
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
61.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 40
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
62.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 41
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
63.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 42
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
64.
[SQA]
Z
π 2
cos 2x dx .
3
(b) Draw a sketch and explain your answer.
2
(a) Evaluate
0
frag replacements O x y
65. Given f (x) = (sin x + 1)2 , find the exact value of f 0 ( π6 ).
[SQA]
frag replacements O x y
replacements O x y
hsn.uk.net
Page 43
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
3
PSfrag replacements O x y
Higher Mathematics
66.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 44
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
67.
[SQA]
frag replacements O x y
frag replacements
replacements O x y
O x y
hsn.uk.net
[END OF PAPER 1 SECTION B] c SQA Questions marked ‘[SQA]’ Page 45
c Higher Still Notes All others
PSfrag replacements O x y
Higher Mathematics
Paper 2 1.
[SQA]
frag replacements O x y
frag replacements O x y
2.
[SQA]
frag replacements O x y
frag replacements O x y replacements O x y
hsn.uk.net
Page 46
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
3.
[SQA]
frag replacements O x y
frag replacements O x y
4.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 47
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
3 , x 6= 0. x (a) Find p(x) where p(x) = f (g(x)).
5. f (x) = 3 − x and g(x) =
[SQA]
(b) If q(x) = Part (a) (b) (b)
Marks 2 2 1
2
3 , x 6= 3, find p(q(x)) in its simplest form. 3−x Level C C A/B
Calc. CN CN CN
Content A4 A4 A4
Answer 3 − 3x x
U1 OC2 2000 P2 Q3
•1 f 3x stated or implied by •2 •2 3 − 3x 3 •3 p 3−x stated or implied by •4 3 •4 3 − 3
•1 ic: interpret composite func. •2 pd: process •3 ic: interpret composite func. •4 pd: process •5 pd: process
•5 x
3−x
6. Functions f and g are defined by f (x) = 2x + 3 and g(x) =
[SQA]
3
x 6= ±5.
x2 + 25 where x ∈ R, x2 − 25
The function h is given by the formula h(x) = g f (x) . For which real values of x is the function h undefined?
frag replacements O x y
replacements O x y
hsn.uk.net
Page 48
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
4
PSfrag replacements O x y
Higher Mathematics
7.
[SQA]
frag replacements O x y
frag replacements O x y
8. The functions f and g are defined on a suitable domain by f (x) = x 2 − 1 and g(x) = x2 + 2. (a) Find an expression for f g(x) . (b) Factorise f g(x) .
[SQA]
frag replacements O x y
replacements O x y
hsn.uk.net
Page 49
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
2 2
PSfrag replacements O x y
Higher Mathematics
9.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 50
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
10.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 51
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
11.
[SQA]
frag replacements O x y
frag replacements O x y
12.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 52
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
13.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 53
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
14.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 54
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
15.
[SQA]
(a) On the same diagram, sketch the graphs of y = log10 x and y = 2 − x where 0 < x < 5. Write down an approximation for the x -coordinate of the point of intersection. (b) Find the value of this x -coordinate, correct to 2 decimal places.
frag replacements O x y
replacements O x y
hsn.uk.net
Page 55
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
3 3
PSfrag replacements O x y
Higher Mathematics
16.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 56
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
y
17. The diagram shows part of the graph of the curve with equation y = 2x3 − 7x2 + 4x + 4. replacements (a) Find the x -coordinate ofPSfrag the maximum turning point.
[SQA]
y = f (x) 5
(b) Factorise 2x3 − 7x2 + 4x + 4.
3 A
(c) State the coordinates of the point A and hence find the values of x for which 2x3 − 7x2 + 4x + 4 < 0. Part (a) (b) (c) •1 •2 •3 •4 •5
Marks 5 3 2 ss: pd: ss: pd: pd:
Level C C C
Calc. NC NC NC
Content C8 A21 A6
know to differentiate differentiate know to set derivative to zero start solving process of equation complete solving process
•6 ss: strategy for cubic, e.g. synth. division 7 • ic: extract quadratic factor •8 pd: complete the cubic factorisation •9 ic: •10 ic:
interpret the factors interpret the diagram
O
(2, 0)
Answer x = 13 (x − 2)(2x + 1)(x − 2) A(− 12 , 0), x < − 12 •1 •2 •3 •4 •5 •6
x 2 U2 OC1 2002 P2 Q3
f 0 (x) = . . . 6x2 − 14x + 4 6x2 − 14x + 4 = 0 (3x − 1)(x − 2) x = 13 ···
2
−7 ··· ···
4 ··· ···
··· •7 2x2 − 3x − 2 •8 (x − 2)(2x + 1)(x − 2)
4 ··· 0
•9 A(− 12 , 0) •10 x < − 12
replacements O x y
hsn.uk.net
Page 57
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
18.
[SQA]
frag replacements O x y
frag replacements O x y replacements O x y
hsn.uk.net
Page 58
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
19.
[SQA]
frag replacements O x y replacements O x y
hsn.uk.net
Page 59
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
20.
[SQA]
(a) Write the equation cos 2θ + 8 cos θ + 9 = 0 in terms of cos θ and show that, for cos θ , it has equal roots.
3
(b) Show that there are no real roots for θ .
1
frag replacements O x y
replacements O x y
hsn.uk.net
Page 60
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
21.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 61
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
22.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 62
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
23.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 63
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
24.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 64
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
25.
[SQA]
frag replacements O x y
frag replacements O x y
26.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 65
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
27.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 66
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
[SQA] 28. frag replacements
O x y
frag replacements O x y
29.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 67
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
30.
[SQA]
frag replacements O x y
frag replacements O x y
31.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 68
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
32.
[SQA]
frag replacements O x y
frag replacements O x y replacements O x y
hsn.uk.net
Page 69
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
33.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 70
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
34. The displacement, √ d units, of a wave after t seconds, is given by the formula d = cos 20t◦ + 3 sin 20t◦ .
[SQA]
(a) Express d in the form k cos(20t ◦ − α◦ ), where k > 0 and 0 ≤ α ≤ 360.
(b) Sketch the graph of d for 0 ≤ t ≤ 18.
(c) Find, correct to one decimal place, the values of t, 0 ≤ t ≤ 18, for which the displacement is 1·5 units.
frag replacements O x y
replacements O x y
hsn.uk.net
Page 71
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
4 4 3
PSfrag replacements O x y
Higher Mathematics
35.
[SQA]
frag replacements O x y
frag replacements O x y
replacements O x y
hsn.uk.net
Page 72
c SQA Questions marked ‘[SQA]’ c All others Higher Still Notes
PSfrag replacements O x y
Higher Mathematics
36.
[SQA]
√ (a) Show that 2 cos(x ◦ + 30◦ ) − sin x ◦ can be written as 3 cos x ◦ − 2 sin x ◦ . √ (b) Express 3 cos x ◦ − 2 sin x ◦ in the form k cos(x ◦ + α◦ ) where k > 0 and 0 ≤ α ≤ 360 and find the values of k and α.
(c) Hence, or otherwise, solve the equation 2 cos(x ◦ + 30◦ ) = sin x ◦ + 1, 0 ≤ x ≤ 360.
frag replacements O x y [END OF PAPER 2]
replacements O x y
hsn.uk.net
Page 73
c SQA Questions marked ‘[SQA]’ c Higher Still Notes All others
3 4 3
![[TOPICS IN PRE-CALCULUS] Functions, Graphs, and Basic ...](https://m.moam.info/img/260x300/topics-in-pre-calculus-functions-graphs-and-basic-_59eee0a21723ddb606484b2b.jpg)
