HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
Applications of Differentiation
(V106A006)
M2V1 Chapter 6
Find the equation of the tangent to the curve y 
Applications of Differentiation
8
at the point (ln 2, 2) .
e 2
x
(3 marks)
(V106A007)
Level 1 Questions
x
§6.1
Find the equation of the tangent to the curve y  e 2 at the point (2, e).
Tangents to Curves
(3 marks)
(V106A001)
dy
 1  x . Find the equation of the tangent to
The slope at any point (x, y) of a curve C is given by
dx
(V106A008)
Find the equation of the tangent to the curve y  2 x  1 at the point (5, 3).
C at the point (3, 4).
(3 marks)
(2 marks)
(V106A009)
Find the equation of the tangent to the curve y  x 2 ln x at the point (e, e2 ) .
(V106A002)
It is given that the curve C: y  f ( x) passes through the point P(0, –1) and f ( x)  e x . Find the
(3 marks)
1
equation of the tangent to C at P.
(2 marks)
(V106A010)
Find the equation of the tangent to the curve y = sin x + cos x at the point (0, 1).
(3 marks)
(V106A003)
Find the equation of the tangent to the curve y = 2x – 3x2 at the point (1, –1).
(3 marks)
(V106A011)
Find the equation of the tangent to the curve y 2  5 x  1 at the point (2, 3).
© Pearson Education Asia Limited 2019
(V106A004)
Find the equation of the tangent to the curve y = exln x at the point (1, 0).
(3 marks)
(3 marks)
(V106A012)
Find the equation of the tangent to the curve x  y 3  3 y 2 at the point (–2, 1).
(V106A005)
 1
Find the equation of the tangent to the curve y = cos2x at the point  ,  .
 4 2
(3 marks)
(3 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106A013)
Find the equation of the tangent to the curve xy = x + y at the point (2, 2).
Applications of Differentiation
(V106A019)
(3 marks)
(V106A014)
 6
Find the equation of the tangent to the curve ye x  4 x  2 at the point  1,  .
 e
(3 marks)
(V106A015)


Find the stationary points of the graph of y  sin  x   , where 0  x  2 . State whether each of
3

them is a maximum point, a minimum point or neither.
(5 marks)
(V106A020)
Find the stationary points of the graph of y = (x – 2)2(x + 3). State whether each of them is a
maximum point, a minimum point or neither.
 
Find the equation of the tangent to the curve x2 – 2sin y = 3 at the point  2,  .
 6
(5 marks)
(3 marks)
§6.2
(V106A021)
Find the stationary points of the graph of y  2 x 4  4 x3  3 . State whether each of them is a
Local Extrema and First Derivative Test
maximum point, a minimum point or neither.
(V106A016)
Let f(x) = 4x2 – 2x. Find the range of x for which f(x) is
(5 marks)
2
(a) increasing,
(b) decreasing.
(V106A022)
(4 marks)
(V106A017)
Let g(x) = x3 + 2x2. Find the range of values of x for which g(x) is
2x
.
x 4
(a) Find the range of values of x such that f ( x) is
It is given that f ( x) 
2
(i) increasing,
(ii) decreasing.
(b) Hence, find the local extrema of f ( x) .
(a) increasing,
(b) decreasing.
(7 marks)
(4 marks)
(V106A023)
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(V106A018)
Find the stationary point of the graph of y = xex. State whether it is a maximum point, a minimum
point or neither.
(5 marks)
It is given that f ( x)  ( x  1)2 e x .
(a) Find the range of values of x such that f ( x) is
(i) increasing,
(ii) decreasing.
(b) Hence, find the local extrema of f ( x) .
(7 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106A024)
Applications of Differentiation
(V106A028)
x 1
.
x2  3
(a) Find the range of x such that
dy
 0,
(i)
dx
dy
 0.
(ii)
dx
(b) Hence, find the local extrema of y.
Consider the function y 
Determine the range of x for which the curve y 
x 1
is
ex
(a) concave upward,
(b) concave downward.
(5 marks)
(V106A029)
(7 marks)
Show that the curve y  1  x 2 is concave upward for any values of x.
(V106A025)
(4 marks)
Consider the function y   x( x 2  6 x  15) .
(V106A030)
(a) Find the range of x such that
(i)
dy
 0,
dx
(ii)
dy
 0.
dx
Show that the curve y = –x4 – 3(x + 1)2 + 4 is concave downward for any values of x.
(4 marks)
(V106A031)
3
Use the second derivative test to find the maximum / minimum point(s) of the curve y  x 2 ( x  3) .
(b) Hence, find the local extrema of y.
(7 marks)
§6.3
Second Derivative Test and Point of Inflexion
(6 marks)
(V106A032)
Use the second derivative test to find the maximum / minimum point(s) of the curve , where
0  x  2 .
(V106A026)
Determine the range of x for which the curve y = 2x3 – 3x2 is
(6 marks)
(a) concave upward,
(b) concave downward.
© Pearson Education Asia Limited 2019
(5 marks)
(V106A033)
Use the second derivative test to find the maximum / minimum point(s) of the curve
y  cos2 x  sin 2 x , where 0  x   .
(V106A027)
Determine the range of x for which the curve y  2 x3  9 x 2  3 is
(6 marks)
(a) concave upward,
(b) concave downward.
(V106A034)
(5 marks)
Use the second derivative test to find the maximum / minimum point(s) of the curve y  4 x 2  1 .
(5 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106A035)
Use the second derivative test to find the maximum / minimum point(s) of the curve
§6.4
y  e2 x  2e x  3 .
(V106A041)
(5 marks)
Applications of Differentiation
Curve Sketching
Let f ( x)  4 x 2 . Determine whether the graph of y  f ( x) has reflectional symmetry about the
y-axis or has rotational symmetry about the origin or neither.
(V106A036)
Show that the curve y = xe–x does not have minimum points.
(2 marks)
(5 marks)
(V106A042)
Let f ( x)  2 x 4  3x 2  1 . Determine whether the graph of y  f ( x) has reflectional symmetry
(V106A037)
Find the point(s) of inflexion of the curve y = x(x2 – 3x + 3).
about the y-axis or has rotational symmetry about the origin or neither.
(5 marks)
(V106A038)
(2 marks)
(V106A043)
Let f ( x)  6 x5  5 x3  x . Determine whether the graph of y  f ( x) has reflectional symmetry
2
 1
Find the point(s) of inflexion of the curve y  1   .
 x
about the y-axis or has rotational symmetry about the origin or neither.
4
(5 marks)
(V106A039)
Find the point(s) of inflexion of the curve y 
(2 marks)
(V106A044)
1
.
2
x 9
x5
. Determine whether the graph of y  f ( x) has reflectional symmetry about
x2  2 x  5
the y-axis or has rotational symmetry about the origin or neither.
(2 marks)
Let f ( x) 
(5 marks)
(V106A040)
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3x 2  2
Find the point(s) of inflexion of the curve y  2
.
x 1
(V106A045)
Find, if any, the equations of the vertical and horizontal asymptotes of the curve y 
(5 marks)
x3
.
x2
(4 marks)
(V106A046)
Find, if any, the equations of the vertical and horizontal asymptotes of the curve y 
x( x  1)
.
x 1
(4 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106A047)
Find, if any, the equations of the vertical and horizontal asymptotes of the curve y 
Applications of Differentiation
(V106A052)
2 x3  x 2  7 x  1
.
2 x3  10
(4 marks)
(V106A048)
Using the information given in the following table, sketch the curve y  f(x), where f(x) is a
polynomial.
x < –2 x –2
–2 < x < 3
x=3 3<x<7
x=7
x>7
f ( x)
1
–3
4

f ( x)
+
0
–
0
+
0
–
(2 marks)
2 x3  x 2  2
Find the equation of the oblique asymptote of the curve y 
.
x2  1
(3 marks)
(V106A049)
(V106A053)
Using the information given in the following table, sketch the curve y = f(x), where f(x) is a
polynomial.
.
x 3 x 3
x3  2 x  2
Find the equations of all the asymptotes of the curve y 
.
x2 1
3  x  2 3 1 x  2 3 1 2 3 1  x  2 3 x  2 3
.
(5 marks)
f ( x)
f ( x )
+
2
0
+
2 3  x  2 3 1
x  2 3 1
–
–2
0
f ( x)
f ( x )
5
(V106A050)
Using the information given in the following table, sketch the curve y = f(x), where f(x) is a
polynomial.
x<0
x=0
0<x<2
x=2
2<x<3 x=3
x>3
f ( x)
–12
16
15
f ( x )
+
+
+
0
–
0
+
0
+
0
–
–
2 3 1  x  3 3 x  3 3 x  3 3
+
0
+
+
(2 marks)
§6.5
(2 marks)
Global Extrema and Optimization Problems
(V106A054)
Find the global extrema of f (x) = x – ln x for
© Pearson Education Asia Limited 2019
(V106A051)
Using the information given in the following table, sketch the curve y = f(x), where f(x) is a
polynomial.
x < –1
x = –1
–1 < x < 0
x=0
0<x<3
x=3
x>3
f ( x)
10
5
–22
–
f ( x )
+
0
–
–
0
+
1
 x  2.
2
(5 marks)
(V106A055)
Find the global extrema of f(x) = exx–e for x  e .
(5 marks)
(2 marks)
(V106A056)
If P = x3 – 3xy + 3 and x + y = 3, find the maximum value of P for 5  x  0 .
(5 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106A057)
The figure shows a cone of base radius x and height 3 – x. Find the value of x such
that the volume of the cone attains its maximum.
(6 marks)
Applications of Differentiation
(V106A061)
The figure shows a trapezium ABDE composed of a triangle ABC and a
rectangle ACDE. AB = k, BC = CD and ABD   , where k is a positive
constant. Find  such that the perimeter P ( ) of the trapezium attains
its maximum. (Give your answer correct to3 significant figures.)
(5 marks)
(V106A058)
The figure shows a cuboid of length ex – 2, width 5 – ex and height ex. Find the
value of x such that the volume of the cuboid attains its maximum. (Give your
(V106A062)
In the figure, A and B are points on the horizontal ground and the vertical wall
answer correct to 3 significant figures.)
respectively. A, B and O are on the same vertical plane. An ant at A is crawling
along the straight path AO with an uniform speed of 2 cm/s .
Meanwhile, another ant at B is climbing up vertically with an uniform speed of
(6 marks)
1 cm/s. It is given that OA  20 cm and OB  10 cm . Let x cm be the distance
between the two ants after t seconds, where 0  t  10 .
(V106A059)
The figure shows a rectangle with perimeter 24 cm. E is a point on CD and
AB  x cm . As x varies, find the maximum area of △AEB.
(a) Express x in terms of t.
(b) Find the shortest distance between the two ants.
(6 marks)
(6 marks)
6
(V106A060)
The volume and the total surface area of a sphere are V cm3 and A cm2 respectively. If the radius of
the sphere is r cm (0  r  3) , find the value of r such that the difference between the values of V and
A is the greatest.
(V106A063)
The figure shows a cylinder of radius x cm and height (8 – 2x) cm. Find the value
of x such that the volume of the cylinder attains its maximum.
(6 marks)
(5 marks)
© Pearson Education Asia Limited 2019
(V106A064)
The figure shows a semi-circle of centre O with diameter 20 cm. ABCD is a
rectangle inscribed in the semi-circle and BOC   , where 0   

2
.
(a) Express the area of ABCD in terms of  .
(b) Find the maximum area of ABCD.
(6 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
§6.6
Question Bank
Rates of Change
Chapter 6
Applications of Differentiation
(V106A068)
AB and CD are two buildings with height 80 m and they are 50 m apart. Mary and John take lifts
(V106A065)
In an experiment, the height H (in m) of a plant is recorded after it
from the ground to the top of the buildings from A and C respectively. The lifts of which Mary and
1
1
has grown for t years. A scientist finds that H  t  e1t  e  .
2e
2
of elevation of John from Mary when John is 60 m above the ground.
Find the rate of change of the height at t = 1. (Give your answer
correct to 3 significant figures.)
(2 marks)
John taken are at constant speeds of 2 m/s and 3 m/s respectively. Find the rate of change of the angle
(3 marks)
(V106A069)
Pat and Peter are schoolmates. They go home from school at the same time. Peter walks due east at a
constant speed of 0.8 m/s while Pat walks due south at a constant speed of 0.6 m/s.
(V106A066)
In the figure, an inverted right conical paper cup of base radius 3 cm and height
8 cm is standing vertically. Water is poured into the cup at a constant rate of 10
cm3/s. When the depth of water is 2 cm, find the rate of change of the depth of
water.
(4 marks)
7
(V106A067)
A new planet has been formed and it orbits around a star. The average surface temperature S C of the
planet is given by S = –201 + 2e0.02t – 4e0.01t, where t is the time in year.
How many years does the planet orbit around the star such that the rate of change of the average
surface temperature is 4 C /year? (Give your answer correct to 2 decimal places.)
(3 marks)
(a) Find the distance between Pat and Peter after 1 minute.
(b) Find the rate of change of distance between Pat and Peter after 1 minute.
(5 marks)
(V106A070)
The rate of inflation of a spherical balloon is 0.1 m3/s. When the radius of the balloon is 1 m, find the
rates of change of the radius and the surface area of the balloon.
(6 marks)
(V106A071)
In the condition of very low temperature, a metal cube contracts such that the length of each edge of
the cube decreases at a rate of 0.01 cm/h and the shape of the cube remains unchanged. When the
length of each edge becomes 10 cm, find the rate of change of the volume of the cube.
(3 marks)
(V106A072)
© Pearson Education Asia Limited 2019
It is given that cos  3tan  . Find
d

d
1

  rad/s .
when   ,   and
dt
4
dt
6
2
(3 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
Applications of Differentiation
Level 2 Questions
(V106A073)
A point moves along a straight line and its displacement s at time t is given by s  2t  t  t . Find
the velocity and the acceleration of the point at t  2 .
(4 marks)
3
2
§6.1
Tangents to Curves
(V106B001)
(V106A074)
An object moves along a straight line. Its displacement s from a reference point after t seconds is
Let C be the curve y 
given by s  t 3  2t 2  t  3 . Find the range of t such that the velocity is negative.
(a) Find
(3 marks)
(V106A075)
A particle moves along a straight line and its initial displacement is 2 cm from a fixed point O. Then,
t seconds later, the displacement s cm of the particle is given by s  2kt 3  kt 2  k , where k is a
positive constant. Find the range of t such that the acceleration is negative.
(4 marks)
8
(V106A076)
A particle moves along a straight line. Its displacement s cm from a fixed point O after t seconds is
given by s  t  2sin t , where 0  t  2 . Find the range of t such that the acceleration is negative.
(3 marks)
5
1
, where x  .
2x 1
2
dy
.
dx
(2 marks)
(b) Find the equations of the tangents to C which are parallel to the straight line 10 x  y  2  0 .
(2 marks)
(V106B002)
If the tangents to the curve cos2 x – 2y + 1 = 0 (0  x  2 ) are parallel to the straight line x + 2y = 2,
find the equations of the tangents.
(4 marks)
(V106B003)
If the equation of the tangent to the curve y  x3  3x  1 at the point (a, b) is y  9 x  15 , find the
point (a, b).
(4 marks)
(V106A077)
A particle moves along a straight rail. Its displacement s m after t s is given by s  t 4  4t 2  8 .
Find the time at which the particle has zero acceleration.
(3 marks)
(V106B004)
If the equation of the tangent to the curve y  x ln x at the point (a, b) is y  x  1 , find the
point (a, b).
(3 marks)
© Pearson Education Asia Limited 2019
(V106B005)
If the tangent to the curve C : y  2 x3  4 x  9 at A(–1, 11) intersects the curve C at B, find the
coordinates of B.
(5 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106B006)
(V106B011)
Find the equations of the tangents drawn from the external point (8, 0) to the curve C : x 2  y 2  16 .
It is given a curve y  ( x  3)2 e2 x .
(6 marks)
Applications of Differentiation
(a) Find all the stationary points of the curve.
(2 marks)
(V106B007)
(b) Determine whether each of them is a maximum point or a minimum point.
Find the equations of the tangents drawn from the external point (2, 1) to the curve
(3 marks)
C : y  3x 3  2 x  3 .
(V106B012)
(6 marks)
(V106B008)
It is given a curve y  e2 x  10e x  12 x .
(a) Find all the stationary points of the curve.
Find the equations of the tangents drawn from the external point (6, 0) to the curve C : 4 x  y 2  16 .
(3 marks)
(b) Determine whether each of them is a maximum point or a minimum point.
(6 marks)
§6.2
(3 marks)
9
(V106B013)
Let f (x) = x4 – 4x4ln x, where x > 0.
(a) Find f ( x ) .
Local Extrema and First Derivative Test
(V106B009)
(2 marks)
It is given a curve y  x3  5x 2  3x  9 .
(b) Hence, using the first derivative test, find the local extrema of f(x).
(2 marks)
(a) Find all the stationary points of the curve.
(3 marks)
(V106B014)
(b) Determine whether each of the points is a maximum point or a minimum point.
(3 marks)
(V106B010)
It is given a curve y 
© Pearson Education Asia Limited 2019
(a) Find
Using the first derivative test, find the local extrema of y  e x sin x , where 0  x  2 .
x 1
x2  1
.
dy
.
dx
(2 marks)
(5 marks)
(b) Find the stationary point of the curve and determine whether it is a maximum point or a
minimum point.
(4 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106B015)
It is given a curve y = cos2 x + sin x, where 0  x  2 .
Applications of Differentiation
(V106B019)
It is given that f ( x)  ax  bx ln x , where a and b are positive constants and x > 1.
The following table shows some values of f (x) and the signs of f ( x ) with respect to x.
(a) Find all the stationary points of the curve.
(3 marks)
(b) Determine whether each of them is a maximum point or a minimum point.
1<x<e
x=e
f (x)
(3 marks)
e < x < e2
e
f ( x )
+
0
x = e2
x > e2
0



(a) Find the values of a and b.
(V106B016)
(3 marks)
Let f ( x)  ax  bx  2 x  1 . If f(x) attains its local extrema at x = –2 and x = –1, find the values of
3
2
a and b.
(4 marks)
(b) Find the turning point of the graph of y = f (x) and state whether it is a maximum point or a
minimum point.
(2 marks)
(V106B020)
(V106B017)
5
 3
If P   ,   is a stationary point of the curve C : y  ax2  bx  1 , find the values of a and b.
4
 2
(5 marks)
It is given a curve y 
(a) Find
x2  a
for x  1 , where a is a real constant.
x 1
dy
in terms of a.
dx
10
(2 marks)
(V106B018)
(b) The curve has a stationary point at x  3 .
(i) Find the value of a.
(ii) Find the local extrema of the curve.
x 2  kx  1
It is given a curve y 
, where k is a real constant.
x2  2
dy
(a) Find
in terms of k.
dx
(6 marks)
(2 marks)
(b) The curve has a stationary point at x = –1.
(i) Find the value of k.
(ii) Find the maximum point and minimum point of the curve.
© Pearson Education Asia Limited 2019
(6 marks)
§6.3
Second Derivative Test and Point of Inflexion
(V106B021)
It is given that the equation of a curve is y = x2ln x, where x > 0.
(a) Find
dy
d2y
and 2 .
dx
dx
(4 marks)
(b) Find the stationary point of the curve and state whether it is a maximum point or a minimum
point.
(4 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106B022)
It is given that the equation of a curve is y 
(a) Find
Applications of Differentiation
(V106B025)
It is given that the equation of a curve is y = sin 2x – cos 2x, where 0  x   .
x 1
.
e2 x
dy
d2y
(a) Find
and
.
dx
dx 2
dy
d2y
and 2 .
dx
dx
(2 marks)
(4 marks)
(b) Find the stationary point of the curve and state whether it is a maximum point or a minimum
point.
(4 marks)
(b) Find the stationary points of the curve and determine whether each of them is a maximum point
or a minimum point.
(4 marks)
(V106B026)
(V106B023)
It is given that the equation of a curve is y = sin2 x + 3sin x, where 0  x  2 .
1
It is given that the equation of a curve is y  x 2 x  1 , where x   .
2
(a) Find
(a) Find
dy
d2y
and
.
dx
dx 2
(2 marks)
(4 marks)
11
(b) Find the stationary point of the curve and determine whether it is a maximum point or a
minimum point.
(4 marks)
(b) Find the stationary points of the curve and determine whether each of them is a maximum point
or a minimum point.
(4 marks)
(V106B027)
It is given that the equation of a curve is y 
(V106B024)
It is given that the equation of a curve is y  ( x  1) x  1 , where x  1 .
(a) Find
dy
d2y
and
.
dx
dx 2
(a) Find
dy
d2y
and
.
dx
dx 2
x  ln x
, where x > 0.
x
dy
d2y
and 2 .
dx
dx
(4 marks)
(b) Find the maximum / minimum point(s) of the curve.
(4 marks)
© Pearson Education Asia Limited 2019
(b) Find the stationary point of the curve and determine whether it is a maximum point or a
minimum point.
(4 marks)
(3 marks)
(c) Find the point of inflexion of the curve.
(2 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106B028)
It is given that the equation of a curve is y  x  2sin x , where 0  x  2 .
(V106B031)
Consider a curve y  kx  e4x , where k is a constant. It is given that the curve has a turning point at
2
(a) Find
Applications of Differentiation
dy
d y
and
.
dx
dx 2
(2 marks)
(b) Find the maximum / minimum point(s) of the curve.
x
1
ln 2 .
4
(a) Find the value of k.
(3 marks)
(c) Find the point of inflexion of the curve.
(2 marks)
(2 marks)
(b) Find
d2y
. Hence, determine whether the turning point is a maximum point or a minimum point.
dx 2
(3 marks)
(V106B029)
(V106B032)
1
It is given that the equation of a curve is y  (3x  1) 5 .
(a) Find
Consider the curve y 
dy
d2y
and
.
dx
dx 2
(2 marks)
(a) Find
xk
, where k is a real constant.
x  x 1
2
dy
in terms of k.
dx
(2 marks)
(b) Find, if any, the maximum / minimum point(s) of the curve.
(2 marks)
12
(c) Find the point of inflexion of the curve.
(2 marks)
(V106B030)
(b) It is given that the curve has a stationary point at x  1 .
(i) Find the value of k.
d2y
.
dx 2
(iii) Determine whether y attains its local maximum at x = 1.
(ii) Hence, find
x2  9
It is given that the equation of a curve is y  2
.
x 9
(6 marks)
dy
d2y
(a) Find
and
.
dx
dx 2
(V106B033)
Consider a curve y  ( x  k ) ln x , where k is a constant and x > 0. It is given that the curve has a
(2 marks)
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(b) Find the maximum / minimum point(s) of the curve.
turning point at x  e 2 .
(a) Find the value of k.
(3 marks)
(c) Find the point(s) of inflexion of the curve.
(2 marks)
(2 marks)
(b) Find
d2y
. Hence, determine whether the turning point is a maximum point or a minimum point.
dx 2
(3 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
Applications of Differentiation
(V106B034)
(V106B037)
The curve y  ( x  a)( x  1)2 has a stationary point at x  1 .
It is given that f ( x)  2 x 4  6 x 2 .
(a) Find the value of a.
(a) By considering f ( x) and f ( x) , write down the axis of symmetry for the curve y  f ( x) .
(2 marks)
(b) Find
d2y
.
dx 2
(1 mark)
(2 marks)
(b) (i) Find the x- and y-intercepts of the curve.
(ii) Find the turning point(s) of the curve.
(iii) Find the point(s) of inflexion of the curve.
(9 marks)
(c) Find the maximum / minimum point(s) of the curve.
(3 marks)
(c) Hence, sketch the curve y  f ( x) .
(2 marks)
(d) Find the point of inflexion of the curve.
(2 marks)
(V106B038)
§6.4
Curve Sketching
x2  7 x  6
and its graph y  f ( x) .
x
(a) Find the domain of f ( x) .
Consider the function f ( x) 
13
(V106B035)
A polynomial f (x) has the following properties:
x<0 x=0 0<x<1 x=1 1<x<2 x=2 2<x<3 x=3 x>3
f ( x)
1
–3
–1
1
f ( x )
–
–
–
0
+
+
+
0
–

f ( x)
+
+
+
+
+
0
–
–
–
(a) Write down all the maximum and minimum points, and the point(s) of inflexion of the graph of
y  f ( x) .
(3 marks)
(1 mark)
(b) Find the x- and y-intercepts of the graph.
(2 marks)
(c) Find the range of values of x for which the graph is
(i) increasing,
(ii) decreasing.
Hence, find the turning point(s) of the graph.
(b) Sketch the graph of y  f ( x) .
(4 marks)
(2 marks)
(d) Find the asymptote(s) of the graph.
(4 marks)
© Pearson Education Asia Limited 2019
(V106B036)
A polynomial f (x) has the following properties:
x<1
1<x<4
x=4 4<x<7
x=7
x>7
x 1
f ( x)
9
–45
–99
f ( x )
+
0
–
–
–
0
+
f ( x)
–
–
–
0
+
+
+
(a) Write down all the maximum and minimum points, and the point(s) of inflexion of the graph of
y  f ( x) .
(3 marks)
(b) Sketch the graph of y  f ( x) .
(2 marks)
(e) Using the results of (a) to (d), sketch the graph.
(2 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106B039)
Consider f ( x) 
Applications of Differentiation
(V106B041)
x2  x  1
.
x2  2 x  2
(a) Find the x- and y-intercepts.
x
.
2
x 9
Consider the curve y 
(a) Find
(i) the domain of f(x),
(ii) the x- and y-intercepts of the curve y = f(x).
(2 marks)
(3 marks)
(b) (i) Find the maximum and / or minimum point(s).
(ii) Find the points of inflexion.
(9 marks)
(b) Show that the curve does not have maximum and minimum points.
(2 marks)
(c) Find the asymptote(s) of the curve.
(2 marks)
(c) Find the asymptotes of the curve.
(3 marks)
(d) Sketch the curve.
(2 marks)
(d) Find the range of values of x for which the curve is
(i) concave upward,
(4 marks)
(V106B042)
It is given that the curve y = x3 + px2 + qx + 1, where p and q are constants, and A( 1, t ) and
B ( h, k ) are the turning points.
(2 marks)
(a) Show that 2p – q = 3.
(ii) concave downward.
(e) Sketch the curve.
(2 marks)
14
(V106B040)
Consider the curve y = x5 – 5x4 + 5x3.
(b) Express p and q in terms of h.
(a) Find the x- and y-intercepts.
(c) If h = 3, find A and B.
(2 marks)
(2 marks)
(b) Find the maximum and / or minimum point(s).
(2 marks)
(d) Sketch the curve.
(5 marks)
(2 marks)
(c) Find the points of inflexion.
(4 marks)
§6.5
(2 marks)
(V106B043)
A shop can sell W kg of sugar per day. W kg is related to the selling price of the sugar $x per kg. It is
Global Extrema and Optimization Problems
(d) Sketch the curve.
© Pearson Education Asia Limited 2019
given that W  e
35 x
6
for x  5 and the cost of the sugar is $5 per kg.
(a) Express the daily profit $P in selling the sugar in terms of x.
(1 mark)
(b) Find the maximum daily profit in selling the sugar. (Give your answer in exact value.)
(4 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106B044)
In the figure, ABCD is a rectangle with AB = 32 cm and BC = 4 cm.
DA and DC are produced to E and F such that B lies on EF.Find the
minimum length of EF.
(6 marks)
Applications of Differentiation
(V106B047)
In each day, a company can produce x tons of copper and y tons of nickel, where 0  x  6 and
21  10 x 2  x3
. The company can earn $1000 and $5000 after selling 1 ton of copper and 1 ton of
5( x 2  2)
nickel respectively. If all copper and nickel produced in a day are sold-out, the daily profit ($P) can
qx  r 

be found by P  1000  p  2
 , where p, q and r are real constants.
x 2

y
(a) Find the values of p, q and r.
(1 mark)
(V106B045)
3
x
 625 . The
50
ship, with a constant speed of x km/h, now takes a journey of 500 km and the total travelling cost is
$E.
(a) Express E in terms of x.
(1 mark)
If a ship travels at a constant speed of x km/h, the travelling cost per hour ($C) is C 
2
dE
d E
and
. Hence, find the speed of the ship such that the total travelling cost of the
dx
dx 2
journey is minimum.
(b) Find
(b) If all copper and nickel produced in this day can be sold-out, how many tons of copper and
nickel should the company produce in order to get the maximum daily profit?
(5 marks)
(V106B048)
After putting a certain chemical into a cup of liquid, the surface temperature S(t) C of the cup can be
modelled by S (t )  26  (t  1)2 e  t , where t (in min) is the time after the chemical has been put into
the liquid and λ is a positive constant. It is given that S(5) = S(17).
15
(a) Find the exact value of λ.
(6 marks)
(c) If the speed of the ship lies between 10 km/h and 20 km/h inclusive, find the minimum total
travelling cost of the journey.
(2 marks)
(V106B046)
x2 y 2

 1 , where x  0 and y  0 . A vertical line and a
4 9
horizontal line are drawn through point P. If the vertical line and the horizontal line cut the x-axis and
the y-axis at M and N respectively, find the maximum area of △MNP.
Let P be a moving point on the curve
© Pearson Education Asia Limited 2019
(6 marks)
(2 marks)
(b) Will the surface temperature of the cup drop to 6C? Explain your answer.
(5 marks)
(V106B049)
The figure shows a right solid cone with base radius r cm, height h cm and slant
height cm . It is known that the curved surface area of the cone is 20π cm2.
(a) Express
in terms of r.
(1 mark)
3
(b) Let V cm be the volume of the cone.
(i)
1
Prove that V   400r 2  r 6 .
3
(ii) As r varies, could the volume of the cone be 60 cm3? Explain your
answer.
(8 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106B050)
In the figure, a container without a lid consists of two parts. The lower part is a
hollow hemisphere of radius r cm. The upper part is a right cylindrical pipe of
base radius r cm and height h cm. It is known that the area of the outer surface
of the container is 32  cm2.
(a) Prove that h 
Applications of Differentiation
(V106B052)
A sewage treatment plant accidentally discharges untreated sewage into a lake and then causes a
temporary decrease in the amount of algae in the lake. After the accident has happened for t days, the

10
50 

amount of algae A(t) (in units) is given by A(t )  200 1 
.
2
 t  5 (t  5) 
16
r.
r
(2 marks)
dA(t )
d 2 A(t )
(a) Find
and
.
dt
dt 2
3
(4 marks)
(b) Let V cm be the capacity of the container.
1 

Prove that V  16r  r 3   .
3 

(ii) Amy claims that the capacity of the container can be greater than 135 cm3. Do you agree?
Explain your answer.
(i)
(b) How many days does it take for the amount of algae to attain the minimum after the accident?
(2 marks)
(c) What will be the amount of algae in the lake after a very long time?
(2 marks)
(8 marks)
(V106B053)
The figure shows a triangle ABC with AB = 2x, BC = x + 1, AC = 2 and BAC  
(V106B051)
In an experiment, a toxin is added into a certain bacterial colony. A researcher finds that the
16
population (in millions) of the bacterial colony is given by P (t ) 
e
0.1t
10
, where t (in min) is the
 e0.02 t


0    .
2

(a) Express cos in terms of x.
time after the toxin is added.
(2 marks)
(b) Find the maximum value of  as x varies.
(a) Find the initial population of the bacterial colony.
(2 marks)
(b) Find the maximum population of the bacterial colony. (Give your answer correct to the nearest
thousand.)
(5 marks)
(5 marks)
(V106B054)
In the figure, ABO is a straight line and POA  90 . P is
moving towards O and the angle between PA and PB is .
Suppose PO  y m .
© Pearson Education Asia Limited 2019
(a) Express tan  in terms of y.
(2 marks)
(b) Find
d
. Hence, find the maximum value of . (Give your
dy
answer correct to 3 significant figures.)
(5 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
§6.6
Question Bank
Chapter 6
(V106B058)
Rates of Change
Consider the curve C : y  4e x , where x  0 . It is given that P (m, n) is a point lying on C. Let A be
(V106B055)
A chemical is added into a solution such that the temperature (T C) of the solution can be modelled
by T 
Applications of Differentiation
800
 25 , where t (in min) is the time after the chemical being added into the
t  18t  205
the area of rectangle PMON, where M (m, 0) , N (0, n) and O is the origin.
(a) Express A in terms of m.
2
solution.
(a) Find the initial temperature of the solution.
(1 mark)
(b) Find the rate of change of the temperature of the solution when the chemical has been just added
for 15 min.
(2 marks)
(1 mark)
(b) If P moves along C,
(i) find the maximum value of A;
(ii) OM increases at a constant rate of 0.5 unit per second, find the rate of change of A when
m = 0.5.
(7 marks)
(V106B059)
(V106B056)
In an experiment, a scientist measures some variables T, x and y (in units) against time (in seconds).
The displacement (s cm) of a toy car after t seconds can be modelled by s  10t  0.5t 2 where
 y  (2 x  1) ln x
It is known that the relationship between the variables is given by the system 
. When
2
2
T  y  x
(a) If the velocity of the toy car at time T doubles the velocity at time 2T, find T.
0  t  20 .
(3 marks)
(b) Does the acceleration of the toy car at time T also double the acceleration at time 2T? Explain
17
x = 1, x increases at a rate of 2 units/s.
your answer.
(a) Find the rate of change of y when x  1 .
(2 marks)
(2 marks)
(b) Find the rate of change of T when x  1.
(2 marks)
(V106B057)
The figure shows a triangle ABC with AB = x cm, BC = 4 cm, AC = 5 cm and
ACB   .
© Pearson Education Asia Limited 2019
(a) Express x in terms of  .
(2 marks)
1
(b) If  increases at a rate of
rad/s, find the rate of change of x when
7


3
.
(2 marks)
(V106B060)
In the figure, a hemispherical bowl of radius 13 cm contains
some water of depth h cm . The volume of water, V cm3, in the
1
bowl is given by V   h 2 (39  h) . Let A cm2 be the area of
3
the water surface.
(a) Express A in terms of h.
(2 marks)
(b) Now, more water is poured into the bowl such that
dV
1

A , where t is measured in seconds.
dt 20
Find the rate of change of the water level.
(3 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106B061)
After t years, the population P (in thousands) of a certain city is given by
144
P  20 
.
(t  2)(t  8)
(V106B064)
(a) Find the rate of change of the population at t = 4.
(a) Express A in terms of x.
Applications of Differentiation
The figure shows a triangle PQR with PQ  (x – 1) cm, PR  x2 cm
and QPR 

3
. Let A cm2 be the area of △PQR.
(2 marks)
(b) According to a certain research, the relationship between the population P (in thousands) and
the total volume (in m3) of sewage disposed per year is given by S  500 P2  4P  180 .
(2 marks)
1
cm/s.
3
Find the rate of change of the area of △PQR when x  2 .
(b) It is given that x increases at a rate of
Find the rate of change of the total volume of sewage disposed per year at t = 4.
(3 marks)
(3 marks)
(V106B065)
(V106B062)
After t hours, the relationship between the area A m2 covered by algae in a certain lake and the
2
3e2 t
being inflated for t min, the radius is given by r 
.
1  e2t
(a) Find the rate of change of the volume at t = ln 2. (Give your answer correct to 2 decimal places.)
(3 marks)
(b) If the balloon is being inflated for a very long period of time, find the volume of the balloon.
(3 marks)

 
 1
nearby air temperature  C can be modelled by A  20 ln   1   (  20)e 20 , where
  4  2
4    40 .
18
(a) Express
4
The radius of a balloon is r cm and its volume V cm3 can be found by V   r 3 . When the balloon is
3
dA
d
in terms of  and
.
dt
dt
(2 marks)
1
(b) At a certain moment, the air temperature is 20C and drops at a rate of C per hour.
2
Find the rate of change of the area covered by the algae at that moment.
(Give your answer correct to 4 decimal places.)
(V106B066)
In Figure 1, a rectangular tank with dimension 20 cm 15 cm  8 cm is placed on a horizontal table.
The tank is full of water initially. The tank is then lifted up slowly. Let θ be the angle inclined of the
tank with the table as shown in Figure 2. The rate of change of θ is 0.1 rad/s .
(2 marks)
© Pearson Education Asia Limited 2019
(V106B063)
The figure shows an isosceles triangle ABC with AC = BC = x cm, AB = 8 cm and
Figure 1
ACB   .
Figure 2
(a) Find the time T (in s) at which half of the water is remaining in the tank.
3
1
It is given that x increases at a rate of
cm/s while  decreases at a rate of
8
2
rad/s.
When the triangle becomes equilateral, find the rate of change of its area.
(3 marks)
(4 marks)
(b) In the period 0  t  T ,
(i) express the volume V (in cm3) of water remaining in the tank in terms of θ,
(ii) find the rate of change of the volume of water remaining in the tank after 1 second .
(4 marks)
(Give your answers correct to 4 decimal places if necessary.)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106B067)
The figure shows an inverted vessel in the shape of a right circular cone. The base radius and the
height of the vessel are 6 cm and 15 cm respectively. Let V cm3 and h cm be the volume and the
depth of the water in the vessel respectively.
(a) Express V in terms of h.
(2 marks)
(b) Water has been leaking out of the vessel through the apex for t min. The depth of the water is
given by h 
15
t
4
.
2e  1
Find the rate of change of volume of the water in the vessel at t = 4.
(Give your answer correct to 2 decimal places.)
(3 marks)
Applications of Differentiation
(V106B069)
A rod MN with length 200 cm is leaned against a cupboard
fixed on the wall with cross-section of rectangle ABCD,
where AB = 20 cm and BC = 120 cm. The rod MN lies in the
same plane of the cross-section of rectangle ABCD. The rod
touches the edge of the top of cupboard and the floor at the
same time. Suddenly, the rod slides down so that N moves
toward E. Let x cm and y cm be the horizontal distances from
M and N respectively from the wall.
(a) When CN = 90 cm, the rate of change of θ is –0.5 rad/s. Find the rate of change of x at this
moment.
(3 marks)
(V106B068)
A container in the form of a right square pyramid with height 16 cm is held
vertically. The length of one side of square base is 24 cm. Water is now poured
into the container.
19
(a) Let A cm2 be the wet surface area of the container (excluding the base) and h
cm be the depth of water in the container. Prove that A 
15
(32  h)h .
4
(4 marks)
(b) The depth of water in the container increases at a constant rate of 2 cm/s. Find the rate of change
of the wet surface area of the container (excluding the base) when the volume of water in the
container is 2322 cm3.
(3 marks)
(b) Peter claims that N is moving toward E at a speed faster than the horizontal speed M is leaving
the wall. Do you agree? Explain your answer.
(4 marks)
© Pearson Education Asia Limited 2019
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
Structural Questions
(V106C003)
Consider the curve y 
(V106C001)
x2
.
x2  5
(a) Find the x- and y-intercepts of the curve y = f(x).
Let f ( x) 
x
5 x 2  48 x
, where k is a real constant. The curve has a vertical asymptote
2x  k
5
.
2
(a) Find the value of k.
(2 marks)
(b) Find
(i) the range of values of x for which f(x) is increasing,
(ii) the maximum and minimum points of the curve y = f(x).
(1 mark)
(b) (i) Find the x- and y-intercepts of the curve.
(ii) Find the turning points of the curve.
(6 marks)
(5 marks)
(c) Find the asymptote(s) of the curve y = f(x).
(1 mark)
(c) Find the range of values of x for which the curve is
(i) concave upward,
(ii) concave downward.
(d) Sketch the curve y = f(x).
(3 marks)
(2 marks)
x  x3
. Using the results obtained in (a) – (d), sketch the curve y  g ( x) on the
x2  5
same graph in (d).
(3 marks)
(e) Let g ( x) 
Applications of Differentiation
(d) Find the oblique asymptote of the curve.
2
(1 mark)
(e) Using the results of (a) – (d), sketch the curve.
(2 marks)
20
(V106C004)
(V106C002)
Consider the curve y  x  a 
x  3x  10
.
1 x
(a) Find the domain of g(x).
It is given that g ( x) 
2
4
, where a and b are real constants. It is given that the curve has a
xb
vertical asymptote x = 2 and an oblique asymptote y = x + 3.
(a) Find the values of a and b.
(1 mark)
(b) (i) Find the x- and y-intercepts of the curve y = g(x).
(ii) Find the turning points of the curve.
(2 marks)
(b) Find the turning point(s) of the curve.
(4 marks)
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(4 marks)
(c) Find the range of values of x for which the curve is
(i) concave upward,
(c) Find the range of values of x for which the curve is
(i) concave upward,
(ii) concave downward.
(ii) concave downward.
(3 marks)
(4 marks)
(d) Find the asymptotes of the curve.
(d) Using the above results, sketch the curve.
(2 marks)
(2 marks)
(e) Using the results of (a) – (d), sketch the curve y = g(x).
(2 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106C005)
Consider the function f ( x) 
x3
, where k is a real constant. Let C1 be the curve y  f ( x) , where
xk
its vertical asymptote is x = 3.
Applications of Differentiation
(V106C007)
In a certain country, the total number N(t) (in ten thousands) of imported cars can be estimated by
N (t )  3  ae k t  15e2k t , where a and k are positive constants and t (in years) is the time since the
import of foreign cars is permitted. The following figure shows the graph of N(t) against t.
(a) Find the value of k.
(1 mark)
(b) Find the horizontal asymptote of the curve C1.
(1 mark)
(c) Find f ( x ) . Hence, find the interval(s) of x for which f ( x) is decreasing.
(2 marks)
(d) Find the range of values of x for which the curve C1 is concave upward.
(2 marks)
The government of the country initially does not permit the import of foreign cars. The car market is
dominated by imported cars when the total number of imported cars is 4.76 ten thousand or above.
(2 marks)
From the figure, the car market would be dominated by imported cars for a period of 22.86 years.
(e) Sketch the curve C1.
(f) Let C2 be the curve y = g(x), where g ( x)  f ( x  2)  2 .
(a) (i) Find the value of a.
(ii) Find the value of k correct to 1 significant figure.
(i) Find the equation of the curve C2.
(ii) Sketch the curve C2.
(6 marks)
(4 marks)
21
For (b), take the round-off value of k obtained in (a).
(b) (i) Find N (t ) and N (t ) .
(V106C006)
Hence, find the value of t such that the total number of imported cars attains its maximum.
Mary is remoulding a clay. She keeps the shape of the clay as a solid cylinder and volume (V cm3) of
the clay unchanged, and its radius is decreasing at a constant rate of 0.5 cm per minute.
After time t minutes, the base radius and the height of the clay are r cm and h cm respectively.
(Give your answer correct to 3 significant figures.)
(a) Show that
dh h
 .
dt r
(4 marks)
(b) Let S cm2 be the total surface area of the cylinder.
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(i)
Show that
dS
  ( h  2r ) .
dt
(ii) At the beginning, the height of the clay is 2.2 times of the base radius. Mary claims that the
total surface area of the clay increases during the whole period of remoulding. Do you
agree? Explain your answer.
(7 marks)
(ii) If the total number of imported cars reaches 5.5 ten thousand, the government will not
permit further import of foreign cars. Explain whether the government will take such
action.
(7 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106C008)
In a greenhouse, the weight of tomatoes W(x) (in tons) produced per month can be modelled by
W ( x)  ( x  Ax  B)e
2

x
10
, where x (in C) is the uniform temperature of the greenhouse and
15  x  35 . At 25 C, the weight of tomatoes produced per month is 8.210 tons. At 30C, the weight
of tomatoes produced per month is 3.734 tons.
(a) Find the values of A and B. (Give your answers correct to the nearest integer.)
(2 marks)
For (b) – (c), take the round-off values of A and B obtained in (a).
(b) Find
(i)
dW ( x )
,
dx
(ii) the maximum weight of tomatoes that can be produced per month. (Give your answer
correct to 4 significant figures.)
(5 marks)
(c) Owing to the cost of maintaining different temperatures of the greenhouse, the net profit $P(x)
of selling all tomatoes produced per month is given by P( x) 
1000W ( x)
.
x
22
The owner of the greenhouse claims the following:
‘The maximum net profit per month can be obtained when the weight of tomatoes per month
attains its maximum.’
Determine whether the owner is correct. Explain your answer.
Applications of Differentiation
(V106C009)
Initially, the pH value of the water in a lake is 5.5. Owing to the leakage of chemical waste from a
factory, the lake has been contaminated for t days and the pH value P(t) of the water can be modelled
by P(t )  a  t e k t , where a and k are positive constants.
(a) Find the value of a.
(1 mark)
(b) It is known that P(30)  P(15)  1.85334 . Find the value of k correct to 1 decimal place.
(2 marks)
For (c), take the round-off value of k obtained in (b).
dP(t )
d 2 P (t )
(c) (i) Find
and
.
dt
dt 2
(ii) After how many days will the pH value attain its maximum?
(iii) Jack claims that the rate of change of the pH value attains its minimum on the 15th day
after the leakage. Do you agree? Explain your answer.
(9 marks)
(V106C010)
A tunnel company has reduced the tunnel fee since 1st January. The number of vehicles N(t) (in
thousands) passing through the tunnel each day can be modelled by N (t ) 
20
, where t is the
1  Ae  k t
number of days since fee reduction, and A, k are positive constants .
(3 marks)
(a) Find
dN (t )
in terms of A and k. Hence, prove that N(t) is increasing for t  0 .
dt
(3 marks)
© Pearson Education Asia Limited 2019
(b) There are 4000 vehicles passing through the tunnel on 1st January and 13 000 vehicles on 11th
January. Find the values of A and k. (Give your answers correct to 1 decimal place if necessary.)
(2 marks)
For (c), take the round-off values of A and k obtained in (b).
(c) Find
(i) the number of vehicles passing through the tunnel each day after many years,
(ii) the day on which the rate of increase of the number of vehicles passing through the tunnel
each day attains its maximum.
(7 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
Applications of Differentiation
(V106C011)
A factory discharges chemical waste into a river. The concentration M (mg/L) of the chemical waste
(V106C013)
In the figure, ABCD is a square with side of length a. P is a moving point on AD
in the river is given by M  kxe0.02 x , where k is a positive constant and x is the number of days after
such that PCD   , where 0   
the waste is discharged into the river.
2

(a) Show that S  a 1  tan  
cos 

2
2
dM
d M
and
in terms of k.
dx
dx 2
(ii) Find the value of x such that the concentration of the chemical waste in the river attains its
maximum. Hence, if k = 8, find the maximum concentration of the chemical waste in the
(a) (i)
Find
river.
(8 marks)
(b) If M > 15, the factory will be fined for discharging the chemical waste into the river. Find the
range of k such that the factory will not be fined.
(3 marks)
(V106C012)
P is a moving point on the circle x2 + y2 = 1 as shown in the figure. The
23


tangent to the circle at P cuts the x-axis at the point Q. Let   0    
2

be the angle subtended by OP and the positive x-axis.

4
. Let AP = x, PC = y and S = x + 2y.

.

(3 marks)
(b) Find  such that S attains its minimum.
(4 marks)
(c) Suppose a = 100 and  decreases at a rate of 0.03 rad/s. Find the rate of change of x when S
attains its minimum.
(2 marks)
(V106C014)
In a remote control plane competition, the remote control plane has to slide on the ground and then
fly to hit the target T. In Figure 1, the horizontal distance between starting point P and the target is
12 m and the target is 5 m above the ground. The plane can slide on the ground for x m at a constant
speed of 0.5 m/s to point Q and then fly to target T at a constant speed of 0.3 m/s along a straight path
to hit the target T.
(a) Find the coordinates of P in terms of  .
(1 mark)
(b) Find the equation of tangent to the circle at P.
(3 marks)
(c) Suppose  increases at a rate of
1

rad/s and Q is moving along the positive x-axis. When   ,
4
6
find the moving speed of Q.
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(3 marks)
Figure 1
Figure 2
(a) Let t s be the time that the plane travels from P to T.
(i) Express t in terms of x.
(ii) When t is minimum, show that QT = 6.25 m.
(6 marks)
(b) In Figure 2, the plane is flying from Q to T with QT equals to the value mentioned in (a)(ii). Let
MPQ   and MQR   , where M is the position of the plane.
(i)
Show that MQ 
165 tan 
.
4(4  3tan  )
(ii) Find the rate of change of  when   0.2 rad . (Give your answer correct to 4 decimal
places.)
(7 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106C015)
Consider the curve C : y  e x , where x  0 . Let P be a moving point lying on C. The tangent to C
at P cuts the y-axis at the point Q while the horizontal line passing through P cuts the y-axis at the
point R.
 2 m  
(a) Denote the x-coordinate of P be m. Prove that the y-coordinate of Q is 
 e
 2 
m
.
(3 marks)
(b) Find the greatest area of △PQR.
(4 marks)
(c) Let O be the origin. It is given that OP increases at a rate exceeding
e2 (16  e4 )
16  e4
Applications of Differentiation
(V106C017)
In the figure, two thin identical rods OP and PR, each of length  cm , are
placed on a coordinate plane. OP and PR are hinged at P(x, y) (y > 0) such
that O and R lie on the positive x-axis. O is fixed while R moves along the
positive x-axis.
1
cm/s . Let OPR   .
P is falling vertically at the rate of
3
2
(a) When  
, find
3
(i) the rate of change of  ,
(ii) the rate of change of the area of △OPR,
in terms of  .
units per
(6 marks)
(b) Now, a circle centred at Q is inscribed in △OPR, where the radius
minute. Someone claims that the area of △PQR increases at a rate higher than 1 square unit per
is r cm, as shown in the figure.
minute when the x-coordinate of P is 4. Do you agree? Explain your answer.
When  
(4 marks)
2
, find the rate of change of the radius of the circle.
3
(4 marks)
24
(V106C016)
△ABC is a triangle with AC  3 and ABC 

. Let BAC   , AB  x and BC  y . It is
3



rad/s .
given that  varies from
to
at a constant rate of
12
6
2
(a) Express x and y in terms of  .
(a) (i)
(3 marks)
Show that the area of △ABC is
0  
d2
(sin 2  2sin  ) , where
2

.
2
(ii) Find the maximum area of △ABC in terms of d.
2
(b) Let S m be the area of △ABC.
Show that
(V106C018)
△ABC is an isosceles triangle with AB = AC. H is the circumcentre of
△ABC. Let BAC   and HA = d.
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dS
 2

 3 sin 
 2  .
d
 3

(7 marks)
Hence, find the value of  such that S attains its maximum.
(6 marks)
(c) Find the range of  such that the rate of change of the area of △ABC is positive.
(b) Suppose D and E are two points outside △ABC such that
AD  BD  AE  CE . If the circumcentres of △ABD and △ACE are mid-points of AB and
AC respectively. When the area of △ABC attains its maximum, will the area of pentagon
(3 marks)
ADBCE also attain its maximum? Explain your answer.
(5 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106C019)
The figure shows a hexagon PQRSTU, where PQ = QR = RS = ST = TU = UP = 10 cm and QR // UT.
Let PS = x cm (0 < x < 30) and M cm2 be the area of the hexagon.
(d) When y attains its maximum, P slides towards T at a rate of
Applications of Differentiation
1
m/s . At that moment, find the
16
rate of change of the shortest distance between Q and the horizontal rail AB.
(3 marks)
(V106C021)
A restaurant reopens after renovation. The number of customers N(t) can be modelled by
N (t ) 
(a) Show that M 
600
, where t is the number of days since the reopening, k and r are constants. According
1  ke  r t
to some records, 85 customers and 255 customers visited the restaurant respectively on the 1st day
( x  10) 300  20 x  x 2
.
2
and the 4th day since the reopening.
(3 marks)
(b) Find the value of x such that M attains its maximum. Hence, find the maximum value of M.
(4 marks)
(c) If 5  x  10 , find the maximum value of M.
(d) When PS = 10 cm, P is moving away from S at a rate of
(a) Find the values of k and r. (Give your answers correct to 1 decimal place.)
(2 marks)
For (b) – (d), take the round-off values of k and r obtained in (a).
(b) Show that N(t) is increasing for t  0 .
(2 marks)
(3 marks)
1
cm/s. At that moment, find the rate of
2
(c) Find the day on which the rate of increase of the number of customers attains its maximum.
(5 marks)
25
(d) After many days, N(t) will approach the number Na. Find Na.
change of the area of the hexagon.
(2 marks)
(2 marks)
(V106C020)
The figure shows a straight rod PQ of length 8 m resting on a
vertical wall ST of height 1 m. The point P at the bottom of
the rod slides along the horizontal rail AB such that
(V106C022)
In the figure, rods AB and AC are hinged at A, and B is fixed. The
lengths of rods AB and AC are 30 cm and 40 cm respectively, and
ABC   . A, B and C lie on the same vertical plane. As C moves,
the height of △ABC decreases at a constant rate of 1 cm/s . Suppose
PRQ = 90. Let PT = x m ( 0  x  63 ) and RT = y m.
the distance between B and C is x cm.
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(a) (i)
(a) Show that x 2  60 x cos   700  0 .
Express PS in terms of x.
(ii) Hence, or otherwise, prove that y 
8x
x2  1
(2 marks)
x.
(b) Express
(3 marks)
dy
d2y
(b) Find
and 2 .
dx
dx
d
in terms of .
dt
(2 marks)
(c) When x = 50, find
(4 marks)
(c) Find the maximum value of y.
(3 marks)
(i) the rate of change of  ,
(ii) the rate of change of x.
(6 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
Question Bank
Chapter 6
(V106C023)
In the figure, a straight line L touches a circle of centre O at the point
P. AOB is a diameter of the circle and OA = 4 cm. D is a point on L
such that AD is perpendicular to L. Let AD = x cm (0 < x < 8) and
S cm 2 be the area of △ADP.
Applications of Differentiation
(c) In reality, instead of the shortest route obtained in (b), the man would choose the route ADB as
shown in the following figure.
(a) (i) Express DP in terms of x.
(ii) Express S in terms of x.
(3 marks)
Give a reason why he would choose the route ADB.
(b) Find the maximum value of S.
(5 marks)
(c) When AD = 2 cm, its length decreases at a rate of
1
cm/s . At that moment, find the rate of
3
change of the area of △ADP.
(1 mark)
(V106C025)
In the figure, the shaded area represents a river and the river bank is a straight line. C and D are two
(2 marks)
points on the bank. A man carrying an empty bucket walks from A to P which lies on CD. He uses
1 unit of energy per metre to carry the empty bucket. When he reaches P, he fills the bucket with
(V106C024)
water and continues to carry the bucket to B. He uses
6
units of energy per metre to carry the bucket
5
26
with water. It is known that AC = 45 m, BD  10 5 m and CD = 80 m. Let x m be the distance
between C and P (0  x  80) and E units be the total energy used by the man.
In the figure, the shaded region represents a river and the river bank is a straight line. C and D are
two points on the bank. A man carrying an empty bucket walks from A to P which lies on CD. When
he reaches P, he fills the bucket with water and continues to carry the bucket to B. It is known that
AC = 60 m, BD = 40 m and CD = 200 m. Let x m be the distance between C and P (0  x  200) and
T m be the total distance travelled by the man.
© Pearson Education Asia Limited 2019
(a) (i)
Express T in terms of x.
(ii) Prove that
dT

dx
x
x  3600
2
(a) (i)

x  200
x  400 x  41 600
2
.
Express E in terms of x. Hence, prove that
(ii) It is given that
(4 marks)
d 2E

dx 2
k1
( x  2025)
the values of k1 and k2.
2
3
2
dE
dx
 0.
x  60
k2

( x  160 x  6900)
2
3
2
, where k1, k2 are constants. Find
(6 marks)
(b) Hence, find CP such that the total distance travelled by the man is minimum.
(4 marks)
HKDSE Mathematics in Action (Extended Part) Module 2 Volume 1
(b) Given that
Question Bank
Chapter 6
dE
 0 only when x = 60 for 0  x  80 , find the minimum value of E.
dx
(2 marks)
(c) The man pours out all water from the bucket at B. Then, he carries the empty bucket and travels
from B to Q which also lies on CD. After filling the bucket with water at Q, he finally transports
the bucket to A. The energy per metre, used by him to carry the bucket with or without water, is
the same as before. If he wants to use the smallest amount of energy, will the position of Q be
the same as that of P? Explain your answer.
(3 marks)
Applications of Differentiation
27
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