(a) (a)
1.
(b)
2.
2
Fact Facto orise rise x – 3x – 10. 2
Solve Solve the equatio equation n x – 3x – 10 = 0.
The diagram represents the graph of the t he function :x f
(x – p) (x – q).
y
–
1 2
x
2 C
(a) (b)
3.
Write Write down down the values values of p and q. The function function has a minimum minimum value at the the point point C . Find the x-coordinate of . C
The graph represents the function f: x
p cos x, p ∈
.
y 3
π
x
2
–
Find: (a) 4.
3
the value of (b) p; (b)
the the area area of the the shad shaded ed regi region on..
Two functions f and g are defined as follows: f (x) = cos x,
0 = ≤ x ≤ 2π ;
(x) = 2x + 1, g
x∈
.
o f Solve the equation ( g ) (x) = 0.
5.
2x + 1 The function f is given by F (x) = x − 3 , x ∈ (a)
(i)
, x ≠ 3.
y = f Show that y = 2 is an asymptote of the graph of (x).
1
(ii)
Find the vertical vertical asymptote asymptote of the graph. graph.
(iii)
Write down the coordinate coordinatess of of the point point P at which the asymptotes intersect.
(b)
Find the points points of of interse intersection ction of the the graph graph and the axes. axes.
(c)
y = f Hence Hence sketch sketch the graph graph of (x), showing the asymptotes by dotted lines. l ines.
−7 ( x − 3) 2
(d) (d)
Sho Show that hat f (x) = ′ ′ the point S where x = 4.
(e)
The tangen tangentt at the point point T on the graph is parallel to the tangent at S .
and hence find the equation of the tangent at
Find the coordinates of . T f) 6.
Show that P is the midpoint of [ ST ]. ].
The diagram shows the parabola y = (7 – are the x-intercepts and the point B is x)(l + x). The points A and C the maximum point. y Find the coordinates of . A, B and C
7.
Three of the following diagrams I, II, III, IV represent the graphs of (a) y = 3 + cos 2 x (b) y = 3 cos(x + 2) Identify which diagram represents which graph.
(c)
y = 2 cos x + 3.
y
I
y
I I 4
2 1 2
x –
1
– π
π
2
1
–
x –
1
– π
π
2
1
–
2
–π
3
π
2
–π
y
I I I
–
1
–
2
2
–π
π
3 2
–π
y
I V 3
5
2
4
1
3 x
–
1
– π
π
2
1
–
2
–π
3
π
2
2
–π
1 x –
–
3
1n ( x − 2)
1
– π
π
2
1
–
2
–π
π
3 2
–π
8.
The function f is given by f (x) =
9.
A population of bacteria is growing at the rate of 2.3 % per minute. How long will it take for the size of the population to double? Give your answer to the nearest minute.
10.
Let f (x) = √x, and g (x) = 2 . Solve the equation ( f o g )( )(x) = 0.25.
11.
Two functions f , g are defined as follows: f : x → 3x + 5 andg : x → 2(1 – g x).
x
(a) 12.
–1
f (2);
. Find the domain of the function.
–1
Find
(b) (g o f )(–4). )(–4). 2
The quadratic equation 4x + 4kx + 9 = 0, k > 0 has exactly one solution for x. Find the value of k. B y
A
C x
13.
The diagram shows three graphs. 3
x
A is part of the graph of y=x
Write ite down: (a) cuts the x-axis. C 14.
is the reflection of graph B in line A. B is part of the graph of y = 2 . C
the equation of in the form y =f (x); C
(b) (b)
the the coor coordi dina nate tess of the the poin pointt wher wheree
2
The diagrams show how the graph of (x) in three steps. y = x is transformed to the graph of y = f For each diagram give the equation of the curve. y
y
( a ) 1 0 y =2
x
0
x
1
x
y ( b
)
( c
y
) 7
4 3
0
1
x
0
x
1
15.
3x + π 2 (x) = 4 sin . For what what value valuess of will the equation f (x) = k have no solutions? f k
16.
The diagram shows the graph of the function y = ax + bx + c.
2
y x
Complete belowpto is positive, zeronegative or zero. Expresthe siotable n osshow itive whether each negexpression ative a c 2
b – 4ac
b 17.
Initially a tank contains 10 000 litres of liquid. At the time t = 0 minutes a tap is opened, and liquid then t flows out of the tank. The volume of liquid, V litres, which remains in the tank after minutes is given by t
= 10 000 (0.933 ). V (a) (a)
Find Find the the valu valuee of after 5 minutes. V
(b)
Find how long, to the nearest nearest second, it takes for half of the initial amount amount of liquid to flow out of the tank.
(c)
The tank is regarded as effectively empty when 95% of the liquid has flowed out. Show that it takes almost three-quarters of an hour for this to happen.
(d)
18.
(b)
19.
(i)
Find the value of 10 000 – when t = 0.001 minutes. V
(ii)
Hence or otherw otherwise, ise, estimate estimate the the initial initial flow rate of the the liquid. liquid. Give your answer in litres per minute, correct to two significant figures.
(a)
Expr Expres esss f (x) = x – 6x + 14 in the form f (x) = (x – , where h and k are to be determined. h) + k
2
2
2
Hence, or otherwise, otherwise, write down down the coordinates of the the vertex vertex of the parabola parabola with with equation equation y – x – 6x + 14.
A group of ten leopards is introduced into a game park. After years the number of leopards, N , is modelled t by N = 10 e (a) (a) (b)
0.4t
.
How How many many leop leopar ards ds are are ther theree afte afterr 2 year years? s? How long will it take for the number number of leopards to reach 100? Give your answers to an appropriate degree of accuracy.
Give your answers to an appropriate degree of accuracy. 20.
Consider the function (a) (a) (b)
21.
:x f
1 x + 1, x ≥ – –1
Dete Deterrmin mine the the inv invers erse funct unctio ion n f . –1
What What is the domain domain of f ?
The diagram shows the graph of (x), with the x-axis as an asymptote. y = f
5
y
B
( 5
,
4
)
x
A
(a) (b)
22.
( –
5
,
– –
4
)
On the same axes, axes, draw the graph of (x + 2) – 3, indicating the coordinates of the images of the y =f points A and B. Write down the equation equation of the the asymp asymptote tote to the the graph graph of (x + 2) – 3. y = f
(a)
2
Sketch, Sketch, on the given axes, axes, the graphs graphs of y = x and y – sin x for –1 ≤ x ≤ 2. y
–
1
–
0
. 5
0
0
. 5
1
1
. 5
2
x
(b)
23.
2
Find the positive positive solution solution of the equation equation x = sin x, giving your answer correct to 6 significant figures.
A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the ground after t 2
seconds is given by h = 2 + 20t – 5t , t ≥ 0 (a) (a)
Find Find the the initial height above the ground of the ball (that is, its height at the instant when it is released).
(b)
Show that the height height of the the ball ball after after one one second second is is 17 metres. metres.
(c)
At a later later time time the ball ball is again at a height of 17 metres. (i)
Write Write down down an equat equation ion that that t must satisfy when the ball is at a height of 17 metres.
(ii) (ii)
Solve Solve the equati equation on algebraically. dh
(d)
24.
Find dt .
(i) (ii) (ii)
Find Find the the initial velocity of the ball (that is, its velocity at the instant when it is released).
(iii (iii))
Find Find when the ball reaches its maximum height.
(iv)
Find the maximum maximum height height of of the the ball. 2
The diagram shows part of the graph with equation y = x + px + q. The graph cuts the x-axis at –2 and 3. y 6 4 2 –
25.
3
–
2
–
1 0
1
–
2
–
4
–
6
2
3
4
x
Find the value of (a) (b) p; q. Each year for the past five years the population of a certain country has increased at a steady rate of 2.7% per annum. The present population is 15.2 million. (a)
What was the population population one year ago?
(b)
What was the population population five years ago? (a)
26.
(b)
x
On the diagram, diagram, sketch sketch the graphs graphs of y = e and y = cos x for –2 ≤ x ≤ 1. x
The equati equation on e = cos x has a soluti solution on betwee between n –2 and –1.
Find Find this this solut solution ion..
7
:xa f
x≤
3 – 2x ,
3 2
.
–1
27.
The function f is defined by
28.
The following diagram shows the graph of (x). It has minimum and maximum points at y = f 1, (0, 0) and (
Evaluate f (5) .
1 2
). y 3
. 5 3
2
. 5 2
1
. 5 1
0
–
2
–
. 5
1
0 –
1 0
– –
(a) (b)
29.
30.
3
x
. 5 1
1 –
–
2
. 5 2
2
. 5
3 ( x – 1) + y = f 2. On the same diagram, diagram, draw the graph of What are the coordinate coordinatess of the minimum minimum and maximu maximum m points points of 3 ( x – 1) + y = f 2?
Michele invested 1500 francs at an annual rate of interest of 5.25 percent, compounded annually. (a)
Find the the value of Michele’s investment after 3 years. years. Give Give your your answer answer to the nearest nearest franc.
(b)
How many many complete complete years years will it take for Michele’s initial investment investment to double double in value?
(c)
What should should the interest interest rate be if Michele’s initial investment investment were to double in value value in 10 years?
Note: Radians are used throughou t hroughoutt this question.
Let f (x) = sin (1 + sin x).
(a)
(i)
y = f Sketch the graph of (x), for 0
≤ x≤
6.
(ii) (ii)
(b)
(c) (c) 31.
f Write Write down down the the x-coordinates of all minimum and maximum points of , for 0 ≤ x ≤ 6 . Give your answers correct to four significant figures.
Let S be the region in the first quadrant completely enclosed by the graph of and both coordinate f axes. (i)
Shade S on your diagram.
(ii)
S Write down the integral integral which represen represents ts the the area area of .
(iii)
Evaluate Evaluate the area of to four significant figures. S
x. Give Give reas reason onss why why f (x) ≥ 0 for all values of
ˆ In the diagram below, the points O(0 , 0) and A(8 , 6) are fixed. The angle OPA varies as the point P( x , 10) moves along the horizontal line y = 10. y
P x(
,
1 1
A
O
( 0
,
0
0
)
( 8
,
y=
6
1
)
)
x diagram to scale
(a)
(i) (ii)
(b)
Show that
AP =
x
2
– 16 x + 80.
x. Write down a simila similarr expres expression sion for OP in terms terms of
Hence, Hence, show show that that cos OPˆ A =
8 x + 40 x 2 –
√ {( x 2
(c)
ˆ Find, Find, in degree degrees, s, the angle angle OPA when x = 8.
(d)
Find Find the positiv positivee value value of x such that
– 16 x
ˆ A = 60° OP
+ 80) ( x 2 + 100)}
,
.
Let the function f be defined by ( x) = cos OPˆ A = f (e)
32.
8x x 2 –
+
40
√ {( x 2 – 16 x + 80) ( x 2 + 100)}
, 0
≤ x ≤ 15.
Consid Consider er the equati equation on f (x) = 1. (i)
Explain, in terms terms of the position position of of the points O, A, and P, why this this equation equation has a solution.
(ii) (ii)
Find Find the the exact solution to the equation. 2
2
y = x and y = 5 – 3( x – 4) . The diagram shows parts of the graphs of
9
y y
2
=
x
8 6
y =
2
5
–
3
(
4 2
–
2
0
2
4
x
6
2
2
The graph of transformations. y = x may be transformed into the graph of y = 5 – 3(x – 4) by these transformations. A reflection in the line y = 0 a vertical stretch with scale factor k p units a horizontal translation of q units. a vertical translation of Write down the value of(a) 33.
; k
(b)
p; x
followed by followed by followed by
(c)
q.
y = x sin 3 , for 0 ≤ x < m, and 0 ≤ y < n, where x is in radians and The diagram below shows the graph of m and n are integers. y n
n–
1
0
Find the value of
m –
(a) 3x
m;
(b)
1
m
x
n.
–1
34.
Given that f (x) = 2e , find the inverse function f (x).
35.
:x The diagram below shows part of the graph of the function f
– x3
+ 2 x 2 + 15 x .
y 4 3 3 2 2 1 1
A
0 5 0 5 0 5 0
Q
5 –
3
–
2 –
P
B
1– 5 – 1 0 – 1 5 – 2 0
1
2
3
4
5
x
The graph intercepts the x-axis at A(–3,0), B(5,0) and the origin, O. There is a minimum point at P and a maximum point at Q. (a) (b)
x( x – a ) ( x – b), : x – The function function may also be written written in the form f where a < b . Write down the value of (i) (ii) a; b. Find (i) f ′ (x); (ii) the exact values of '(x) = 0; (iii) (iii) x at which f function at Q.
(c)
(i) (ii)
(d)
the the valu valuee of the the
f Find the equation of the tangent to the graph of at O.
This tangent tangent cuts the graph of at another point. Give the x-coordinate of this point. f
Determine Determine the area of the shaded shaded region. region. x –1
36.
x 2 , (x ≠ 2). Find: Let f (x) = 2 , and g (x) = x –
37.
y = a (x – h) + k The diagram shows part of the graph of . The graph has its vertex at P, and passes through the point A with coordinates (1, 0).
(a)
(g ) (3); ° f
(b)
g (5).
2
y P
2
1
A –
(a)
Write Write down down the value value of(i)
(b)
Calcul Calculate ate the value value of a.
h;
(ii)
1
0
1
x
k .
11
1
38.
–kx
Consider functions of the form y = e (b) Let k = 0.5
(a) Show that
–0.5x
1
e dx ∫ = k (1 – e – kx
0
– k k
).
, for –1 ≤ x ≤ 3, indicating the coordinates of the
(i) (i)
Sket Sketch ch the the grap graph h of y=e y-intercept.
(ii)
Shade the region region enclosed enclosed by this graph, graph, the x-axis, y-axis and the line x = 1.
(iii)
Find the area of this region. region. dy
(c)
(i)
Find
dx
–kx
in terms of , where y = e k
. –kx
The point P(1, 0.8) lies on the graph of the function y = e
39.
40.
(ii) (ii)
Find Find the the valu valuee of in this case. k
(iii)
Find the the gradient of the tangent tangent to the the curve curve at P. P.
.
2
Consider the function f (x) = 2x – 8x + 5. 2
(a) (a)
Expr Expres esss f (x) in the form a (x – p) + q, where a, p, q ∈
(b)
Find Find the minimu minimum m value value of (x). f
.
x
Solve the equation e = 5 – 2 x, giving your answer correct to four significant figures. 6 – x
41.
Consider the functions f :x
4 (x – 1) and g :x
2
.
–1
(a)
Find g .
(b)
Solve Solve the equatio equation n (f ° g ) (x) = 4.
–1
42.
$1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum number of months required for the value of the investment to exceed $3000.
43.
The sketch shows part of the graph of (x) which passes through the points A(–1, 3), B(0, 2), C(l, 0), D(2, y = f 1) and E(3, 5).
8 7 6 E 5 4 A 3 B 2 D 1 C –
4
–
3
–
2
–
1
1
0 –
1
–
2
3
2
5
4
A second function is defined by g (x) = 2f (x – 1).
44.
(a) (a)
Calc Calcul ulat atee g (0), (0), g (1), (1), g (2) (2) and g (3). (3).
(b)
On the same axes, axes, sketch sketch the graph of the function function g (x). x
The diagram below shows a sketch of the graph of the function y = sin(e ) where –1 ≤ x ≤ 2, and x is in radians. The graph cuts the y-axis at A, and the x-axis at C and D. It has a maximum point at B. y B A
–
1
0
1
C
D
2
x
(a)
Find Find the coordi coordinat nates es of A.
(b)
k The coordinate coordinatess of C may be written written as (ln k , 0). Find the exact value of . dy
(c)
(i)
Write down the y-coordina -coordinate te of of B. B. (ii) Find
dx
.
(iii) (iii) Hence, Hence, show show that that at B, x =
13
π ln 2 . (d)
(i) integral.
Write down the integral which ich represents the shaded area. (ii)
Evalua luate this
(e) (e)
(i) (i)
Copy Copy the the abov abovee diag diagra ram m into into your your answ answer er book bookle let. t. (The (There re is no need need to copy copy the the 3
y=x . shading.) On your diagram, sketch the graph of
(ii) 45.
The two graphs graphs intersect intersect at the the point point P. Find the x-coordinate of P. 4
3
2
Let g (x) = x – 2x + x – 2. 2x 3 Let f (x) =
( x) g
(a)
Solve g (x) = 0.
+1 . A part of the graph of (x) is shown below. f y
C A
0
x
B
(b)
The graph has vertical vertical asymptote asymptotess with equations equations x = a and x = b where a < b. Write down the values of (i) a; (ii) b.
(c)
The graph has a horizontal horizontal asymptote asymptote with equation equation y = l. Explain why the value of (x) approaches 1 f as x becomes very large.
(d)
The graph graph inters intersect ectss the x-axis at the points A and B. Write down the exact value of the x-coordinate at (i) A; (ii) B.
(e)
The curve curve inters intersects ects the y-axis at C. Use the graph to explain why the values of (x) and f f ′ ′ ′ ′ (x) are zero at C. x – x,
and g (x) = 1 + x , x ≠ –1. –1. Find Find::
46.
Let f (x) = e
47.
A family of functions is given by
(a) (a)
–1
f (x)
(b)
(g )( )(x). ° f
2
f (x) = x + 3x + k , where k ∈ {1, 2, 3, 4, 5, 6, 7}.
One of these functions is chosen at random. Calculate the probability that the curve of this function crosses the x-axis. 48.
Consider the following relations between two variables x and y.
A.
y = sin x
B.
y is directly proportional to x
C.
y = 1 + tan x
D.
speed y as a function of time x, constant acceleration
E.
y=2
F.
distance y as a function of time x, velocity decreasing
x
Each sketch below could represent exactly two of the above relations on a certain interval. (i) y
( i i )
( i i i )
y
y
x
x
x
Complete the table below, by writing the letter for the two relations that each sketch could represent. sketch
relation letters
(i) (ii) (iii) 49.
2
The diagram shows part of the graph of the curve y = a (x – , where a, h, k ∈ h) + k
.
y
2
0
1
5
1
0 5 0
1
2
3
4
5
x
6
.
(a)
h and of k The vertex vertex is at the the point point (3, 1). Write down the value of .
b)
The point point P(5, 9) is on the graph. graph. Show that a = 2.
(c)
Hence show that the equation equation of the the curve curve can be written written as y = 2x – 12x + 19.
2
15
dy
(d)
(i)
Find
dx
.
A tangent is drawn to the curve at P (5, 9). (ii)
Calculate Calculate the gradient gradient of this tangent, tangent,
(iii)
Find the equation equation of this tangent. tangent.
50.
2
The equation kx + 3x +1 = 0 has exactly one solution. Find the value of . k q
(a) The diagram diagram shows shows part part of the graph of the function function f (x) = point A (3, 10). The line (CD) is an asymptote.
51.
y 1
5
1
0
.
x – p
The curve passes through the
C
A
5
–
1
5
–
1
0
–
5
0
5
–
(b) (b)
1
0
1
5x
5
–
1
0
–
1
5
D
Find the value of (i) (ii) p; q. f The The grap graph h of (x) is transformed as shown in the following diagram. The point A is t ransformed to A ′ (3, –10).
17
y 1
5
1
0
C
5
–
1
5
–
1
0
–
5
0
5
–
1
0
1
5x
5
–
1
0
–
1
5
A
D
Give a full geometric description of the transformation. 52.
53.
The mass m kg of a radio-active substance at time t hours is given by (a)
Write Write down down the initia initiall mass. mass.
(b)
The mass is reduce reduced d to 1.5 kg. How long does this take?
–0.2t
m = 4e
.
2
The function f is given by f (x) = x – 6x + 13, for x ≥ 3. 2
(a)
Write f (x) in the form ( x – a) + b.
(b)
Find Find the invers inversee functi function on f .
c)
Stat Statee the the doma domain in of f .
54.
Let f (x) = 2x + 1.
(b)
Let g (x) = f (x +3) –2. On the grid below draw the graph of (x) for –3 g
–1
–1
(a)
f On the grid below draw the the graph of (x) for 0
≤ x≤
≤ x≤
–1.
2.
y 6 5 4 3 2 1
–
55.
56.
6
–
5
–
4
–
3
–
2
–
0
1
1
–
1
–
2
–
3
–
4
–
5
–
6
2
3
4
5
6
x
1 3 7 Let A and B be events such that P( A) = 2 , P(B) = 4 and P(A ∪ B) = 8 . (a) (a)
Calc Calcul ulat atee P(A ∩ B).
(b) (b)
Calc Calcul ulat atee P(A B).
(c) (c)
Are Are the the even events ts A and B independent? Give a reason for your answer.
Let f (x) = sin(2 x + 1), 0 ≤ x ≤ (a) (a)
.
π
Sket Sketch ch the the curv curvee of (x) on the grid below. y = f
19
y 2 1
. 5 1
0
. 5 0
–
0 –
–
(b) (b)
0
. 5
1
1
. 5
2
2
. 5
3
3
. x5
. 5 1
1
. 5
– 2 Find Find the the x-coordinates of the maximum and minimum points of (x), giving your answers correct to f one decimal place.
2
57.
The equation x – 2kx + 1 = 0 has two distinct real roots. Find the set of all possible values of . k
58.
There were 1420 doctors working in a city on 1 January 1994. After n years the number of doctors, D, working in the city is given by D = 1420 + 100 n. (a) (a)
(i) (i)
How How many any doctor ctorss were ere the there work workin ing g in the city ity at the the star startt of 2004 004?
(ii) In what year year were there first more than 2000 doctors working in the city? At the beginning of 1994 the city had a population of 1.2 million. After , of the city n years, the population, P n
is given by P = 1 200 000 (1.025) . (b)
(i) (ii)
Calculate the percentage percentage growth in population population between 1 January January 1994 and 1 January January 2004. 2004.
(iii)
In what what year will the population first become become greater greater than than 2 million?
(c) (c)
(i) (i) (ii) (ii)
59.
What What was the avera verage ge numb umber of peo people per docto ctor at the the beg beginni innin ng of 1994 994?
Afte Afterr how how many many complete years will the number of people per doctor first fall below 600? 2
Let f (x) = 2x + 1 and g (x) = 3x – 4. (c)
60.
Find the population P at the beginning of 2004.
Find:
(a)
–1
f (x);
(b)
(g )(–2); ° f
(f )( )(x). ° g
Let f (x) = 2 + cos (2 x) – 2sin (0.5 x) for 0 ≤ x ≤ 3, where x is in radians. (a)
y = f On the grid below, below, sketch sketch the curve curve of (x), indicating clearly the point P on the curve where the
derivative is zero. y 4 3 2 1 0
(b)
0
–
1
–
2
–
3
–
4
. 5
1
1
. 5
2
2
3 x
. 5
Write Write down down the solut solution ionss of (x) = 0. f 0.05t
61.
The population p of bacteria at time t is given by p = 100e (b) the rate of incre increase ase of the the popula population tion when t = 10.
62.
Part of the graph of the periodic function f is shown below. The domain of is 0 ≤ x ≤ 15 and the period is f 3. f(
.
Calc Calcul ulate ate:: (a) (a)
the the valu valuee of = 0; p when t
x ) 4 3 2 1 0 0
63.
1
2
(i) f (2);
(ii)
3
4
′ (6.5); (iii) f
5
6
7
8
9
1
0
x
′ (14). f
(a)
Fi Find:
(b)
How many solutions solutions are there there to the equation equation f (x) = 1 over the given domain? 2
The function f (x) is defined as f (x) = –(x – . The diagram below shows part of the graph of (x). The h) + k f maximum point on the curve is P(3, 2).
21
y 4
P
( 3
, 2
3
4
)
2 –
1
1 –
2
–
4
–
6
–
8
–
1
0
–
1
2
2
(a)
Write down the value of:
(i)
h;
(ii)
(b) (b)
x + 6x – 7. Sho Show that hat f (x) can be written as f (x) = –
(c)
Find f ′ (x).
5
6
x
. k
2
The point Q lies on the curve and has coordinates (4, 1). A straight line L, through Q, is perpendicular to the tangent at Q. (d)
(i)
L. Calculate the gradient of
(ii) (ii)
Find Find the equati equation on of L.
(iii (iii))
The The line line L intersects the curve again at R. Find the x-coordinate of R.
Let f (x) = 1 + 3 cos(2x) for 0 ≤ x ≤
64.
(a)
(i) (ii) (ii)
, and x is in radians.
π
Find f ′ (x).
Find Find the values values for ′ (x) = 0, giving your answers in terms of π . x for which f π
The function g (x) is defined as g (x) = f (2 (2x) – 1, 0 ≤ x ≤ 2 . (b)
f f g (i) The graph of may be transformed to the graph of by a stretch 1in the x-direction with scale factor followed by another transformation. Describe fully this other t ransformation. ransformation.
(ii) 65.
Find the solution solution to the equation equation g (x) = f ( x)
h(x) is shown below. There is a minimum point at R Let h(x) = (x – 2)sin(x – 1) for –5 ≤ x ≤ 5. The curve of and a maximum point at S. The curve intersects the x-axis at the points ( a, 0) (1, 0) (2, 0) and ( b, 0).
y 4 3 2
(a –
(a)
5 –
4 –
Find Find the exact exact value value of (i)
, 3 –
1
0 0 2 –
a;
S )
(b
1
1
–
1
–
2
–
3
–
4
–
5
–
6
–
7
(ii)
R
2
3
4
, 5
x
b.
The regions between the curve and the x-axis are shaded for a ≤ x ≤ 2 as shown. (b)
(i) (ii) (ii)
(c)
Hence or otherwise otherwise,, find find the range of values values of of for which the equation ( x – 2)sin(x – 1) = k has k four distinct solutions.
(b) (b)
68.
The y-coordinate of R is –0.240. Find the y-coordinate of S.
: 3x, g :x The functions f and g are defined by f (a)
67.
Calcul Calculate ate this this total total area. area. (i)
(ii)
66.
Write down an expression which represents the total area of the shaded regions.
x
+2.
Find Find an expres expressio sion n for ( f ° g) (x). –l
–l
Sho Show that hat f (18) + g (18) = 22. 3 ( x) = , f 2 x 9 – The function f is defined by for –3 < x < 3. (a)
On the grid, grid, sketch sketch the graph of . f
(b)
Write down the equati equation on of each vertical vertical asymp asymptote. tote.
(c)
Write down the range range of the function function f . 2
The quadratic function f is defined by f (x) = 3x – 12x + 11. 2
(a)
Write f in the form f (x) = 3(x – . h) – k
(b) (b)
The The grap graph h of f is translated 3 units in the positive x-direction and 5 units in the positive y-direction. 2
Find the function g for the translated graph, giving your answer in the form g (x) = 3(x – p) + q. 69.
2x
The diagram below shows the graphs of (x) = 1 + e , g (x) = 10x + 2, 0 ≤ x ≤ 1.5. f y
f
23
(a)
(i) (ii) (ii)
Write down an expression ion for the vertic tical dista istan nce p between the graphs of and g . f
Give Given n that that p has a maximum value for 0
≤ x≤
1.5, find the value of x at which this occurs.
The graph of (x) only is shown in the diagram below. When x = a, y = 5. y = f y
1
6
1
2
8 5 4
0
(b)
1
1
. 5
x
–1
(i) (ii)
. a5
Find f (x).
Hence show that a = ln 2.
(c)
70.
71.
72.
The region region shaded shaded in the the diagram diagram is rotate rotated d through through 360° about about the x-axis. Write down an expression for the volume obtained. Consider the line L with equation y + 2x = 3. The line L1 is parallel to L and passes through the point (6, –4). (a)
L1. Find Find the gradie gradient nt of
(b)
Find Find the equati equation on of L1 in the form y = + mx + b.
(c) (c)
Find Find the the x-coordinate of the point where line L1 crosses the x-axis. (x–11)
The function f is given by f (x) = e
–8.
–1
(a)
Find f (x).
(b)
Write Write down down the domai domain n of f (x).
–l
The graph of (x) is shown in the diagram. y = f y 2 1
–
2
–
1 0
1
–
1
–
2
2
3
4
5
6
7
8
x
(a)
On each of the following following diagrams diagrams draw the required required graph, graph, (i)
(x); y = 2f y 2 1
–
(ii)
2
–
1 0 –
1
–
2
1
2
3
4
5
6
7
8
x
1
2
3
4
5
6
7
8
x
y = f (x – 3). y 2 1
–
2
–
1 0 –
1
–
2
25
(b)
73.
f y = – The point point A (3, –1) is on the graph of . The point A′ is the corresponding point on the graph of f (x) + 1. Find the coordinates of A′ .
p)(x – q). The curve intersects the x-axis at A(– The equation of a curve may be written in the form y = a(x – 2, 0) and B(4, 0). The curve of (x) is shown in the diagram below. y = f y 4 2 A –
(a)
(i)
4
–
B 0
2
2
–
2
–
4
–
6
4
6 x
p and of q. Write down the value of
(ii)
Given that the point point (6, 8) is on the the curve, curve, find find the value value of a.
(iii)
Write the equati equation on of of the curve curve in the form form y = ax + bx + c.
2
dy
(b)
(i) (ii)
(c) (c)
Find
dx
.
A tangent is drawn drawn to the curve curve at a point point P. The gradient gradient of this tangent tangent is 7. Find the coordinates of P.
The The line line L passes through B(4, 0), and is perpendicular to the tangent to the curve at point B. (i) (i)
Find Find the the equa equati tion on of L.
(ii)
Find the x-coordin x-coordinate ate of the point point where where L intersects the curve again.
