1.
Points A, B and C are on the circumference of a circle, ce ntre O and radius r. ˆ A trapezium OABC is formed such that AB is parallel to OC, and the angle AOC π
is θ, 2 ≤ θ < π.
diagram not to scale
(a )
ˆ Show that angle BOC is π – θ. (3)
(b)
Sho Show that the area, T , of the trapezium can be expressed as
1 T = T = 2
r 2 sin θ −
1 2
r 2 sin 2θ . (3)
IB Questionbank Mathematics Higher Level 3rd edition
1
(c )
Show that when the area is maximum, the value of θ satisfies
(i)
cos θ = 2 cos 2 θ. (ii)
Hence determine the maximum area of the trapezium when r = r = 1. (Note: It is not required to prove that it is a maximum.) (5) (Total 11 marks)
2.
André wants to get from point A located in the sea to point Y located on a straight stretch of beach. P is the point on the beach nearest nearest to A such that AP = 2 km and PY = 2 km. He does this by swimming in a straight line to a point Q located on the beach and then running to Y.
When André swims he covers 1 km in 5 5 minutes. When he runs he covers 1 km in 5 minutes.
(a )
T minutes taken by André to If PQ = x km, 0 ≤ x ≤ 2, find an expression for the time T minutes reach point Y. (4)
dT (b)
Show that
d x
=
5 5 x x
2
+4
− 5. (3)
dT (c )
(i)
Solve d x
= 0.
(ii)
Use the value of x x found in part (c) (i) to determine the time, T minutes, T minutes, taken for André to reach point Y.
2
d T d x 2
(iii)
Show that minimum.
=
20 5
( x + 4) 2
3 2
and hence show that the time found in part (c) (ii) is a (11) (Total 18 marks)
3.
A packaging company makes boxes for chocolates. An example of a box is shown below. This box is closed and the top and bottom bottom of the box are identical regular hexagons hexagons of side x cm.
diagram not to scale
3 3 x 2
(a) (a)
Show Show that that the the area area of each each hexa hexago gon n is is
2
2
cm . (1)
(b) (b)
3 3 Give Given n tha thatt the the volu volume me of the the box box is 90 cm , show that when x when x = 20 the total surface area of the box is a minimum, justifying that this value gives a minimum.
(7) (Total 8 marks)
4.
2
The curve y curve y = x – 5 is shown below.
IB Questionbank Mathematics Higher Level 3rd edition
3
A point P on the curve has x has x-coordinate -coordinate equal to a.
(a) (a)
Sho Show tha thatt th the dis disttance ance OP is
a4
− 9a 2 + 25 . (2)
(b)
Find th the values of of a a for which the curve is closest to the origin. (5) (Total 7 marks)
5.
A skydiver jumps from a stationary balloon at a height of 2000 m above the ground. –1
–0.2t –0.2t
Her velocity, v m s , t seconds t seconds after jumping, is given by v = 50(1 – e (a) (a)
).
Find Find her her acce accele lera ratio tion n 10 sec secon onds ds aft after er jum jumpi ping ng.. (3)
(b)
How far above above the the grou ground nd is is she she 10 secon seconds ds afte afterr jumpi jumping? ng? (3) (Total 6 marks)
6.
–1
A rocket is rising vertically at a speed of 300 m s when it is 800 m directly above the launch site. Calculate the rate of change of the distance between the rocket and an observer, who is 600 m from the launch site and on the same horizontal level as the launch site.
diagram not to scale (Total 6 marks)
IB Questionbank Mathematics Higher Level 3rd edition
5
7.
A lighthouse L is located offshore, 500 m etres from the nearest point P on a long
straight shoreline. The narrow beam of light from the lighthouse rotates at a constant rate of 8π radians per minute, producing an illuminated spot S that moves along the shoreline. You may assume that the height of the lighthouse can be ignored and that the beam of light lies in the the horizontal plane defined by sea level.
When S is 2000 metres from P, (a)
show show that the the speed speed of S, corr correct ect to thre threee signific significant ant figu figure res, s, is 214 214 000 metre metress per minute; (5)
(b) (b)
fin find the the acce accele lera rati tion on of S. S. (3) (Total 8 marks)
8.
The diagram below shows a circle with centre at the origin O and radius r > r > 0.
A point P( x, x, y), y), ( x > 0, y 0, y > 0) is moving round the circumference of the circle.
tan arcsin
Let m =
y
r .
d y (a )
Given that dt = 0.001r 0.001r , show that
dm = dt 10
3
r 2
r
− y 2
. (6)
dm (b) (b)
Stat Statee the the geo geome metr tric ical al mean meanin ing g of of dt . (1) (Total 7 marks)
IB Questionbank Mathematics Higher Level 3rd edition
7
9.
–1
A helicopter H is moving vertically upwards with a speed of 10 m s . The helicopter is h m directly above the point Q, which is situated on level ground. The helicopter is observed from the point P, which is also at ground level, and PQ = 40 m. This information is represented in the diagram below.
diagram not to scale
When h = 30, (a) (a)
show show that that the the rat ratee of of cha chang ngee of of
ˆQ HP
is 0.16 radians per second; (3)
(b) (b)
fin find the the rate ate of of ch change ange of PH. PH. (4) (Total 7 marks)
10.
2
The quadratic function function f f ( x) x) = p = p + qx – x x has a maximum value of 5 when x when x = 3. (a )
Find th the value of of p p and the value of q. of q. (4)
(b)
The graph of f f ( x) x) is translated 3 units in the positive direction parallel to the x-axis. x-axis. Determine the equation of the new graph. (2) (Total 6 marks)
− 5 x + 4 2 Consider f f ( x) x) = x + 5 x + 4 . x 2
11.
(a) (a)
Find Find the the equa equati tion onss of all all asy asymp mpto tote tess of the the gra graph ph of of f. f. (4)
(b)
Find Find the the coor coordin dinate atess of the point pointss where where the graph graph of of f meets f meets the x the x and y and y axes. (2)
(c) (c)
Fin Find the the coor coordi din nate ates of (i) (i)
the the max maxim imum um poin pointt and and just justif ify y you yourr ans answe wer; r;
(ii) (ii)
the the mini minimu mum m poin pointt and and just justify ify your your ans answe wer. r. (10)
(d)
Ske Sketch the graph of f f , clearly showing all the features found above. (3)
(e)
of f. Hence, write down the number of points of inflexion of the graph of f. (1) (Total 20 marks)
12.
2 x
The function f function f is is defined by f by f (( x) x) = x = x e . It can be shown that f that f derivative of f ( f ( x). x). (a )
(n)
n
( x) x) = (2 x + n 2
n−1
)e
2 x
for all n∈
+
(n)
, where f where f
th
( x) x) represents the n
(n)
By considering f ( x) x) for n for n =1 and n = 2, show that there is one minimum point P on the graph of f f , and find the coordinates of P. (7)
(b)
Show that f has f has a point of inflexion Q at x at x = −1. (5)
IB Questionbank Mathematics Higher Level 3rd edition
9
(c) (c)
Dete Determ rmine ine the the int inter erval valss on on the the doma domain in of f where f where f f is is (i)
concave up;
(ii)
concave down. (2)
(d)
Sketch f , clearly showing any intercepts, asymptotes and the points P and Q. (4)
(e) (e)
Use Use mat mathe hema matic tical al indu induct ctio ion n to to pro prove ve that that f f f
(n)
(n)
n
( x) x) = (2 x + n2
n−1
)e
2 x
for all n∈
+
, where
th
( x) x) represents the n derivative of f ( f ( x). x). (9) (Total 27 marks)
13.
2 3
2
A family of cubic functions is defined as f k ( x) x) = k x − kx + x, x, k ∈ (a )
+
.
Express in in te terms of of k k (i)
f ′ f ′ k ( x) x) and f and f ′ ′ ′k ( x); x);
(ii) (ii)
the the coor coordi dina nate tess of the the poi point ntss of inf infle lexi xion on P P k on the graphs of f f k . (6)
(b)
Show that all P k lie on a straight line and state its equation. (2)
(c) (c)
Sho Show tha thatt for for all all val valu ues of k of k , the tangents to the graphs of f f k at P at P k are parallel, and find the equation of the tangent lines. (5) (Total 13 marks)
(a) (a)
14.
Find Find the the roo roott of of the the equa equati tion on e
2–2 x
– x
= 2e
giving the answer as a logarithm. (4)
(b)
2–2 x
The curve y = e
– x
– 2e – 2e
has a minimum point. Find the coordinates of this minimum.
(7)
(c )
2–2 x
The curve y = e
– x
– 2e – 2e
is shown below.
Write down the coordinates of the points A, B and C. (3)
(d) (d)
Henc Hencee sta state te the the set set of valu values es of k of k for for which the equation e has two distinct positive roots.
2–2 x
– x
– 2e – 2e
= k (2) (Total 16 marks)
15.
( x x + y) y) Show that the points (0, 0) and ( 2π , − 2π ) on the curve e = cos ( xy) xy) have a common tangent.
(Total 7 marks)
1 16.
The curve C has C has equation y equation y = 8
(9 + 8 x 2
− x 4 )
.
d y (a) (a)
Find Find the the coo coord rdin inat ates es of of the the poin points ts on on C at C at which d x = 0. (4)
IB Questionbank Mathematics Higher Level 3rd edition
11
(b)
The tangent to C at C at the point P(1, 2) cuts the x the x-axis -axis at the point T. Determine the coordinates of T. (4)
(c )
The normal to C at C at the point P cuts the y-axis y-axis at the point N. Find the area of triangle PTN. (7) (Total 15 marks)
17.
x 2 − 2 x −1.5 e x) = The function f function f is is defined by f by f ( x) .
(a )
Find f ′( ′( x). x). (2)
f ( x) (b)
You are gi given th that y = x − 1 has a local minimum at x at x = a, a > 1. Find the value of a. of a. (6) (Total 8 marks)
18.
Below is a sketch of a Ferris wheel, an amusement park device carrying passengers around the rim of the wheel.
(a)
The circ circula ularr Ferris Ferris wheel wheel has a radi radius us of 10 10 metres metres and and is revol revolvin ving g at a rate of 3 radia radians ns per minute. Determine how fast a passenger passenger on the wheel is going vertically upwards when the passenger is at point A, 6 metres higher than the centre of the wheel, and is rising. (7)
(b)
The operat operator or of the Ferris Ferris wheel stands stands directl directly y below below the the centre centre such such that the bottom bottom of the Ferris wheel is level with his eyeline. As he watches the passenger his line of sight makes an angle α with the horizontal. Find the rate of change of α at point A. (3) (Total 10 marks)
19.
1 d y − ln (1 + e 2 x ) , show that d x If y y = 3
=
2 3 (e – y – 3). (Total 7 marks)
5π 20.
– x
Consider the graphs y graphs y = e
– x
and y and y = e
sin 4 x, x, for 0 ≤ x ≤ x ≤ 4 .
5π (a)
On the the same same set of axe axess draw, draw, on gra graph ph pape paper, r, the the grap graphs, hs, for 0 ≤ x ≤ 4 . π Use a scale of 1 cm to 8 on your x-axis x-axis and 5 cm to 1 unit on your y your y-axis. -axis. (3)
nπ (b)
– x
Show that the x-intercepts x-intercepts of the graph y graph y = e
sin 4 x are 4 , n = 0, 1, 2, 3, 4, 5. (3)
(c )
x –
Find the x-coordinates x-coordinates of the points at which the graph of y of y = e
sin 4 x meets the graph
– x
of y y = e . Give your answers in terms of π. (3)
(d)
– x
(i) Show that when the graph of y = e their gradients are equal.
IB Questionbank Mathematics Higher Level 3rd edition
– x
y = e , sin 4 x meets the graph of y
13
(ii) (ii)
Hence Hence explai explain n why thes thesee three three meetin meeting g points points are are not not local local maxim maximaa of the the – x
graph y graph y = e
sin 4 x. x. (6)
(e )
(i)
y-coordinates, y y1, y2 and y Determine the y-coordinates, and y3, where y where y1 > y2 > y3, of the 5π
x – local maxima of y y = e sin 4 x for 0 ≤ x ≤ x ≤ 4 . You do not need to show that they are maximum values, but the values should be simplified.
(ii)
Show that y1, y2 and y and y3 form a geometric sequence and determine the common ratio r. (7) (Total 22 marks)
