ADDITONAL MATHEMATICS

2002 – 2011
CLASSIFIED CALCLUS
Compiled & Edited
By

Dr. Eltayeb Abdul Rhman

www.drtayeb.tk

First Edition
2011
 6

6 The curves y = x2 and 3y = –2x2 + 20x – 20 meet at the point A. For

Examiner’s
y Use
y = x2

3y = –2x2 + 20x – 20

O x

(i) Show that the x-coordinate of A is 2. [1]

(ii) Show that the gradients of the two curves are equal at A. [3]

(iii) Find the equation of the tangent to the curves at A. [1]

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9 A body moves in a straight line such that, t s after passing through a fixed point O, its For
displacement from O is s m. The velocity v ms–1 of the body is such that v = 5cos4t. Examiner’s
Use

(i) Write down the velocity of the body as it passes through O. [1]

(ii) Find the value of t when the acceleration of the body is first equal to 10 ms–2. [4]

(iii) Find the value of s when t = 5. [4]

兰 (7x + 8) dx.
1
3
(b) (i) Find [2]

兰
8 1
(ii) Hence evaluate (7x + 8) 3 dx. [2]
0

dy
10 (a) A curve is such that dx = ae1–x – 3x2, where a is a constant. At the point (1, 4), the gradient
of the curve is 2.

(i) Find the value of a. [1]

(ii) Find the equation of the curve. [5]

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12 Answer only one of the following two alternatives. For

Examiner’s
EITHER Use

The equation of a curve is y = (x – 1)(x2 – 6x + 2).

(i) Find the x-coordinates of the stationary points on the curve and determine the nature of each
of these stationary points. [6]

(ii) Given that z = y2 and that z is increasing at the constant rate of 10 units per second, find the
rate of change of y when x = 2. [2]

(iii) Hence find the rate of change of x when x = 2. [2]

OR

The diagram shows a cuboid with a rectangular base of sides x cm and 2x cm. The height of the
cuboid is y cm and its volume is 72 cm3.

y cm

x cm
2x cm

(i) Show that the surface area A cm2 of the cuboid is given by
216
A = 4x2 + x .
[3]

(ii) Given that x can vary, find the dimensions of the cuboid when A is a minimum. [4]

(iii) Given that x increases from 2 to 2 + p, where p is small, find, in terms of p, the corresponding
approximate change in A, stating whether this change is an increase or a decrease. [3]

1 Find the value of k for which the x-axis is a tangent to the curve

y = x2 + (2k + 10)x + k 2 + 5. [3]

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12 Answer only one of the following two alternatives. For

Examiner’s
EITHER Use

The tangent to the curve y = 3x3 + 2x2 – 5x + 1 at the point where x = –1 meets the y-axis at the
point A.

(i) Find the coordinates of the point A. [3]

The curve meets the y-axis at the point B. The normal to the curve at B meets the x-axis at the
point C. The tangent to the curve at the point where x = –1 and the normal to the curve at B meet
at the point D.

(ii) Find the area of the triangle ACD. [7]

OR

y y = x (x – 3)2

P R

O Q x

The diagram shows the curve y = x (x – 3)2 . The curve has a maximum at the point P and
touches the x-axis at the point Q. The tangent at P and the normal at Q meet at the point R. Find
the area of the shaded region PQR. [10]

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For
Examiner’s
dy Use
1 (i) Given that y = sin 3x, find . [1]
dx

(ii) Hence find the approximate increase in y as x increases from π to π + p, where p is small.
9 9
[2]

dy k(x + 5) , where k is a constant to be found. [3]

5 (i) Given that y = x 2x + 15, show that = –––––––
dx 2x + 15

兰 兰
5
(ii) Hence find x+5 x+5
––––––– dx and evaluate ––––––– dx. [3]
2x + 15 –3 2x + 15

O Q x

y = x sin x

The diagram shows part of the curve y = x sin x and the normal to the curve at the point

冢 冣
P π , π . The curve passes through the point Q(π, 0).
2 2
(i) Show that the normal to the curve at P passes through the point Q. [4]

(ii) Given that d (x cos x) = cos x – x sin x, find

dx 兰x sin xdx. [3]

(iii) Find the area of the shaded region. [5]

© UCLES 2011 0606/21/M/J/11

 8

7 A particle moves in a straight line so that, t s after passing through a fixed point O, its velocity, For
60 .
v ms–1, is given by v = ––––––– Examiner’s
(3t + 4)2 Use

(i) Find the velocity of the particle as it passes through O. [1]

(ii) Find the acceleration of the particle when t = 2. [3]

(iii) Find an expression for the displacement of the particle from O, t s after it has passed
through O. [4]

© UCLES 2011 0606/22/M/J/11

 14

11 Answer only one of the following two alternatives. For

Examiner’s
EITHER Use

(a) Using an equilateral triangle of side 2 units, find the exact value of sin 60° and of cos 60°.
[3]
(b)
P S
60° 60°

x cm x cm

Q y cm R

PQRS is a trapezium in which PQ = RS = x cm and QR = y cm.

Angle QPS = angle RSP = 60° and QR is parallel to PS.

(i) Given that the perimeter of the trapezium is 60 cm, express y in terms of x. [2]

(ii) Given that the area of the trapezium is A cm2 , show that
(30x – x2)
A= 3 . [3]
2
(iii) Given that x can vary, find the value of x for which A has a stationary value and
determine the nature of this stationary value. [4]

OR
r cm

h cm For a sphere of radius r:

4 3
Volume = πr
3
Surface area = 4π r2

The diagram shows a solid object in the form of a cylinder of height h cm and radius r cm on top
of a hemisphere of radius r cm. Given that the volume of the object is 2880 π cm3,

(i) express h in terms of r, [2]

(ii) show that the external surface area, A cm2, of the object is given by
A = 5 π r2 + 5760 π
3 r . [3]
Given that r can vary,

(iii) find the value of r for which A has a stationary value, [4]

(iv) find this stationary value of A, leaving your answer in terms of π, [2]

(v) determine the nature of this stationary value. [1]

© UCLES 2011 0606/22/M/J/11

 7

5 A particle moves in a straight line such that its displacement, x m, from a fixed point O at time For
t s, is given by x = 3 + sin 2t, where t ⭓ 0. Examiner’s
Use

(i) Find the velocity of the particle when t = 0. [2]

(ii) Find the value of t when the particle is first at rest. [2]

(iii) Find the distance travelled by the particle before it first comes to rest. [2]

3π
(iv) Find the acceleration of the particle when t = . [2]
4

dy k(x + 5) , where k is a constant to be found. [3]

5 (i) Given that y = x 2x + 15, show that = –––––––
dx 2x + 15

兰 兰
5
(ii) Hence find x+5 x+5
––––––– dx and evaluate ––––––– dx. [3]
2x + 15 –3 2x + 15

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冕 冕 (e + 1) dx.
2
For
8 (a) Find (ex + 1)2 dx and hence evaluate x 2 [6] Examiner’s
0 Use

2
4 A curve has equation y = (3x2 + 15) 3 . Find the equation of the normal to the curve at the point
where x = 2. [6]

dy 1
(b) A curve is such that dx = (4x +1)– 2 . Given that the curve passes through the point with
coordinates (2, 4.5), find the equation of the curve. [5]

© UCLES 2011 0606/11/O/N/11

 12

10 Answer only one of the following two alternatives. For

Examiner’s
EITHER Use

A B
E F

r cm
D G

θ
θ θ r cm

C
The figure shows a sector ABC of a circle centre C, radius 2r cm, where angle ACB is 3θ radians.
The points D, E, F and G lie on an arc of a circle centre C, radius r cm. The points D and G are
the midpoints of CA and CB respectively. Angles DCE and FCG are each θ radians. The area of
the shaded region is 5 cm2.

(i) By first expressing θ in terms of r, show that the perimeter, P cm, of the shaded region is
given by P = 4r + 8r . [6]
(ii) Given that r can vary, show that the stationary value of P can be written in the form
k 2 , where k is a constant to be found. [4]
(iii) Determine the nature of this stationary value and find the value of θ for which it occurs. [2]

OR
A

10 cm E

r cm
θ
θ D
O C

The figure shows a sector OAB of a circle, centre O, radius 10 cm. Angle AOB = 2θ radians
π
where 0 < θ < 2 . A circle centre C, radius r cm, touches the arc AB at the point D. The lines OA
and OB are tangents to the circle at the points E and F respectively.

(i) Write down, in terms of r, the length of OC. [1]

10 sin θ
(ii) Hence show that r = 1 + sin θ . [2]
dr 10
(iii) Given that θ can vary, find dθ when r = 3 . [6]
π
(iv) Given that r is increasing at 2 cms–1, find the rate at which θ is increasing when θ = 6 . [3]

© UCLES 2011 0606/11/O/N/11

 9

7 (i) Show that

冢4 – x冣
2
can be written in the form px
– 12 1
+ q + rx 2 , where p, q and r are
For
Examiner’s
Use
x
integers to be found. [3]

dy 冢4 – x 冣
2
(ii) A curve is such that = for x > 0. Given that the curve passes through the
dx x
point (9, 30), find the equation of the curve. [5]

dy
9 (i) Given that y = x sin 4x, find . [3]
dx

冕 x cos 4x dx and evaluate 冕

π
8
(ii) Hence find x cos 4x dx. [6]
0

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11 Answer only one of the following two alternatives. For

Examiner’s
EITHER Use

A curve has equation y = e–x (Acos 2x + Bsin 2x). At the point (0, 4) on the curve, the gradient of
the tangent is 6.

(i) Find the value of A. [1]

(ii) Show that B = 5. [5]

π
(iii) Find the value of x, where 0 < x < 2 radians, for which y has a stationary value. [5]

OR
1n(x2 – 1)
A curve has equation y = , for x > 1.
x2 – 1

dy k x(1 – 1n(x2 –1))

(i) Show that = , where k is a constant to be found. [4]
dx (x2 – 1)2
(ii) Hence find the approximate change in y when x increases from 5 to 5 + p, where p is
small. [2]

(iii) Find, in terms of e, the coordinates of the stationary point on the curve. [5]

x x
6 A curve has equation y = 6 cos 2 + 4 sin 2 , for 0 ⬍ x ⬍ 2π radians.

(i) Find the x-coordinate of the stationary point on the curve. [5]

(ii) Determine the nature of this stationary point. [2]

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 11

10 (a) Differentiate tan (3x + 2) with respect to x. [2] For

Examiner’s
Use

2
(b) Differentiate (√⎯x + 1) 3 with respect to x. [3]

ln (x 3 – 1)
(c) Differentiate with respect to x. [3]
2x + 3

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11 A particle moves in a straight line so that, t s after leaving a fixed point O, its velocity v ms–1 is For
given by v = 3e2t + 4t. Examiner’s
Use

(i) Find the initial velocity of the particle. [1]

(ii) Find the initial acceleration of the particle. [3]

(iii) Find the distance travelled by the particle in the third second. [4]

2 Given that the straight line y = 3x + c is a tangent to the curve y = x2 + 9x + k, express k in

terms of c. [4]

11 It is given that y # (x ! 1)(2x 0 3)3
2.

dy
(i) Show that can be written in the form kx √2x − 3 and state the value of k. [4]
dx
Hence

(ii) find, in terms of p, an approximate value of y when x # 6 ! p, where p is small, [3]

6
(iii) evaluate  x √2x − 3 dx. [3]
2

0606/2/M/J/04

0606/13/O/N/11
 5

1 dy kx For
4 (i) Given that y = x2 + 3 , show that dx = (x2 + 3)2 , where k is a constant to be found. [2] Examiner’s
Use

冕 冕
3
6x 6x
(ii) Hence find (x2 + 3)2 dx and evaluate (x2 + 3)2 dx. [3]
1

11
y
(2, 3.5)

(5, 1.4)
A B

O 2 p 5 x

The diagram shows part of a curve, passing through the points (2, 3.5) and (5, 1.4). The gradient of
a
the curve at any point (x, y) is −


3 , where a is a positive constant.
x

(i) Show that a # 20 and obtain the equation of the curve. [5]

The diagram also shows lines perpendicular to the x-axis at x # 2, x # p and x # 5. Given that the
areas of the regions A and B are equal,

(ii) find the value of p. [5]

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12 Answer only one of the following two alternatives. For

Examiner’s
EITHER Use
y

B
P (1,1n 2)
y =1n (x+1) – 1n x

O C A x

The diagram shows part of the curve y = 1n (x +1) – 1n x. The tangent to the curve at the point
P (1, 1n 2) meets the x-axis at A and the y-axis at B. The normal to the curve at P meets the
x-axis at C and the y-axis at D.

(i) Find, in terms of 1n 2, the coordinates of A, B, C and D. [8]

Area of triangle BPD 1
(ii) Given that = , express k in terms of 1n 2. [3]
Area of triangle APC k

OR

A curve has equation y = xex. The curve has a stationary point at P.

(i) Find, in terms of e, the coordinates of P and determine the nature of this stationary point. [5]

The normal to the curve at the point Q (1, e) meets the x-axis at R and the y-axis at S.

(ii) Find, in terms of e, the area of triangle ORS, where O is the origin. [6]

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8 A particle travels in a straight line so that, t s after passing through a fixed point O, its velocity, For

()
v ms–1, is given by v = 12cos t .
3
Examiner’s
Use

(i) Find the value of t when the velocity of the particle first equals 2 ms–1. [2]

(ii) Find the acceleration of the particle when t = 3. [3]

(iii) Find the distance of the particle from O when it first comes to instantaneous rest. [4]

8 A curve has the equation y # (ax ! 3) ln x, where x p 0 and a is a positive constant. The
normal to the curve at the point where the curve crosses the x-axis is parallel to the line 5y ! x # 2.
Find the value of a. [7]

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 12

10 For
y Examiner’s
Use

y = x3 – 9x2 + 24x + 2
Q

P
O x

The diagram shows part of the curve y = x3 – 9x2 + 24x + 2 cutting the y-axis at the point P. The
curve has a minimum point at Q.

(i) Find the coordinates of the point Q. [4]

(ii) Find the area of the region enclosed by the curve and the line PQ. [6]

2 A curve has gradient e4x + e–x at the point (x, y). Given that the curve passes through the point (0, 3),
find the equation of the curve. [4]

x−

1 Given that y = , find
x
2 + 5

d
y
(i) an expression for ,
d
x

(ii) the x-coordinates of the stationary points.

[4]

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 14

12 Answer only one of the following two alternatives. For

Examiner’s
EITHER Use

y=ln x
P (e, 1)

O (1, 0) Q x

The diagram shows part of the curve y = ln x cutting the x-axis at the point (1, 0). The normal to
the curve at the point P(e, 1) cuts the x-axis at the point Q.

1
冢
(i) Show that Q is the point e + e , 0 . 冣 [4]
d
(ii) Show that dx (x ln x) = 1 + ln x. [1]

冕
(iii) Hence find ln xdx and the area of the shaded region. [5]

OR

A y = e x cos x
(0,1)

( , 0)
2
O B x

The diagram shows part of the curve y = e x cos x, cutting the x-axis at the point 冢2π, 0冣 . The
normal to the curve at the point A(0, 1) cuts the x-axis at the point B.

(i) Find the coordinates of B. [4]

d
(ii) Show that dx [e x (cos x + sin x)] = 2e x cos x. [2]

冕
(iii) Hence find e x cos xdx and the area of the shaded region. [4]

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 4

8
S 1m R

Y
1m
qx m

P xm Q
X

The diagram shows a square PQRS of side 1 m. The points X and Y lie on PQ and QR respectively such
that PX = x m and QY = qx m, where q is a constant such that q > 1.

(i) Given that the area of triangle SXY is A m2, show that
A = 12 (1 – x + qx 2). [3]

(ii) Given that x can vary, show that QY = YR when A is a minimum and express the minimum value
of A in terms of q. [4]

9 Given that y = (x – 5) 2x + 5 ,
dy kx
(i) show that can be written in the form and state the value of k, [4]
dx 2x + 5
(ii) find the approximate change in y as x decreases from 10 to 10 – p, where p is small, [2]
(iii) find the rate of change of x when x = 10, if y is changing at the rate of 3 units per second at this
instant. [2]

6 (i) Show that

d
dx (
1
cos x
– sin x )
can be written in the form
1 –
k
sin x
and state the value of k. [4]

π–

∫
(ii) Hence evaluate 4 2 dx. [3]
0 1 – sin x

2x + 4
10 A curve has the equation y

=

.
x−2

d
y k
(i) Find the value of k for which

=

. [2]
d
x (x − 2)
2

(ii) Find the equation of the normal to the curve at the point where the curve crosses the x-axis. [4]

A point (x, y) moves along the curve in such a way that the x-coordinate of the point is increasing at
a constant rate of 0.05 units per second.

(iii) Find the corresponding rate of change of the y-coordinate at the instant that y # 6. [3]

0606/1/M/J/02
 5

11 A curve has the equation y # xe
2x.

(i) Find the x-coordinate of the turning point of the curve. [4]

d
2
y
(ii) Find the value of k for which
2


=

ke
2x(1 + x). [3]
d
x

(iii) Determine whether the turning point is a maximum or a minimum. [2]

12

y = x 2– 6x +10

O x

The diagram shows part of the curve y = x 2 – 6x + 10 passing through the points P and Q. The curve
has a minimum point at P and the gradient of the line PQ is –2. Calculate the area of the shaded region.
[11]

OR

A particle travels in a straight line, starting from rest at point A, passing through point B and coming to
rest again at point C. The particle takes 5 s to travel from A to B with constant acceleration. The motion
of the particle from B to C is such that its speed, v ms–1, t seconds after leaving A, is given by
1
v= (20 – t)3 for 5  t  T.
225

(i) Find the speed of the particle at B and the value of T.

(ii) Find the acceleration of the particle when t = 14.
(iii) Sketch the velocity-time curve for 0  t  T.
(iv) Calculate the distance AC.
[11]

0606/2/M/J/02
 5

12 Answer only one of the following two alternatives.

EITHER

y
B
y # 2 sin x ! 4 cos x
A

O x

The diagram shows part of the curve y # 2 sin x ! 4 cos x, intersecting the y-axis at A and with
its maximum point at B. A line is drawn from A parallel to the x-axis and a line is drawn from B
parallel to the y-axis. Find the area of the shaded region. [11]

OR

y
y #  1! 4x

P(2, 3)

O x

The diagram shows part of the curve y

=

√1 + 4x, intersecting the y-axis at A. The tangent to the
curve at the point P(2, 3) intersects the y-axis at B. Find the area of the shaded region ABP. [11]

8
3 Express


1  √x 
  3√x + 2  dx in the form a + b√2, where a and b are integers. [6]

0606/1/M/J/03
 5

12 Answer only one of the following two alternatives.

EITHER

A particle moves in a straight line so that, t s after leaving a fixed point O, its velocity, v ms01, is
given by v =  10(1 − e −2 t ).
1

(i) Find the acceleration of the particle when v # 8. [4]

(ii) Calculate, to the nearest metre, the displacement of the particle from O when t # 6. [4]

(iii) State the value which v approaches as t becomes very large. [1]

(iv) Sketch the velocity-time graph for the motion of the particle. [2]

OR

d sin θ
(i) By considering sec θ as (cos θ)01 show that  (sec θ) = . [2]
dθ cos 2 θ
(ii) The diagram shows a straight road joining two points, P and Q, 10 km apart. A man is at point
A, where AP is perpendicular to PQ and AP is 2 km. The man wishes to reach Q as quickly as
possible and travels across country in a straight line to meet the road at point X, where angle
PAX # θ radians.

A
θ
2 km
X
P Q
10 km

The man travels across country along AX at 3 km h01 but on reaching the road he travels at
5 km h01 along XQ. Given that he takes T hours to travel from A to Q, show that

T =   2 sec θ + 2 − 2 tan θ  . [4]

3 5

(iii) Given that θ can vary, show that T has a stationary value when PX # 1.5 km. [5]

11 A particle travels in a straight line so that, t seconds after passing a fixed point A on the line, its
acceleration, a ms–2, is given by a = –2 – 2t. It comes to rest at a point B when t = 4.

(i) Find the velocity of the particle at A. [4]

(ii) Find the distance AB. [3]

(iii) Sketch the velocity-time graph for the motion from A to B. [1]

0606/2/M/J/04
 5

12 Answer only one of the following two alternatives.

EITHER

B
O x

The diagram, which is not drawn to scale, shows part of the graph of y = 8 – e2x, crossing the y-axis
at A. The tangent to the curve at A crosses the x-axis at B. Find the area of the shaded region bounded
by the curve, the tangent and the x-axis. [10]

OR

A piece of wire, of length 2 m, is divided into two pieces. One piece is bent to form a square of side x m
and the other is bent to form a circle of radius r m.

(i) Express r in terms of x and show that the total area, A m2, of the two shapes is given by

(π + 4)x2 – 4x + 1
A = ––––––––––––––– .
π
[4]

Given that x can vary, find

(ii) the stationary value of A, [4]

(iii) the nature of this stationary value. [2]

8
1 A curve has the equation
= y .–––––
2x – 1

dy
(i) Find an expression for ––– . [3]
dx

(ii) Given that y is increasing at a rate of 0.2 units per second when x = – 0.5, find the corresponding
rate of change of x. [2]

© 0606/01/M/J/05
 4

1 A curve has the equation y = (x – 1)(2x – 3)8. Find the gradient of the curve at the point where x = 2.
[4]

5 (i) Differentiate xln x – x with respect to x. [2]

(ii)
y

y = ln x

O 3 x

The diagram shows part of the graph of y = ln x. Use your result from part (i) to evaluate the
area of the shaded region bounded by the curve, the line x = 3 and the x –axis. [4]

e2x
6 A curve has the equation y = –––– , for 0 < x < π.
sin x

dy
(i) Find ––– and show that the x-coordinate of the stationary point satisfies 2 sin x – cos x = 0. [4]
dx

(ii) Find the x-coordinate of the stationary point. [2]

3 The line y = 3x + k is a tangent to the curve x2 + xy + 16 = 0.

(i) Find the possible values of k. [3]

(ii) For each of these values of k, find the coordinates of the point of contact of the tangent with the
curve. [2]

dy
9 A curve is such that
dx 冢 2 冣 2 冢
= 2 cos 2x – π . The curve passes through the point π , 3 . 冣
(i) Find the equation of the curve. [4]

(ii) Find the equation of the normal to the curve at the point where x = 3π . [4]
4

0606/02/M/J/05
 4

7 A particle moves in a straight line, so that, t s after leaving a fixed point O, its velocity, v m s–1, is given
by
v = pt2 + qt + 4,

where p and q are constants. When t = 1 the acceleration of the particle is 8 m s–2. When t = 2 the
displacement of the particle from O is 22 m. Find the value of p and of q. [7]

1 + sin x dy 1
8 (i) Given that y = ––––––– , show that –– = ––––––– . [5]
cos x dx 1 – sin x

(ii)
y
2
y = ––––––
1 – sinx

O 3π 5π x
––– –––
4 4

2
The diagram shows part of the curve y = ––––––– . Using the result given in part (i), find the
1 – sin x
3π 5π
area of the shaded region bounded by the curve, the x-axis and the lines x = ––– and x = ––– . [3]
4 4

4 (a) Differentiate etan x with respect to x. [2]

e
1–
2 1–2x dx.
(b) Evaluate [4]
0

0606/01/M/J/06
 5

9 A cuboid has a total surface area of 120 cm2. Its base measures x cm by 2x cm and its height is h cm.

(i) Obtain an expression for h in terms of x. [2]

Given that the volume of the cuboid is V cm3,

4x3 .
(ii) show that V = 40x – ––– [1]
3
Given that x can vary,
4x
(iii) show that V has a stationary value when h = –– . [4]
3

2 Find the equation of the normal to the curve y = 2x + 4 at the point where x = 4. [5]
x–2

9
5 A curve has the equation y = x + .
x
dy d2y
(i) Find expressions for and . [4]
dx dx2
(ii) Show that the curve has a stationary value when x = 9. [1]

(iii) Find the nature of this stationary value. [2]

11 A particle, moving in a straight line, passes through a fixed point O with velocity 14 ms–1. The
acceleration, a ms–2, of the particle, t seconds after passing through O, is given by a = 2t – 9. The
particle subsequently comes to instantaneous rest, firstly at A and later at B. Find

(i) the acceleration of the particle at A and at B, [4]

(ii) the greatest speed of the particle as it travels from A to B, [2]

(iii) the distance AB. [4]

0606/02/M/J/06
 6

11 Answer only one of the following two alternatives.

EITHER
y

y = 3 sin x + 4 cos x

O π x
2
π
The graph shows part of the curve y = 3sin x + 4 cos x for 0  x  radians.
2
(i) Find the coordinates of the maximum point of the curve. [5]

(ii) Find the area of the shaded region. [5]

OR
y

C 12
y=
(3x + 2)2

O B x

12
The diagram, which is not drawn to scale, shows part of the curve y = , intersecting the
(3x + 2)2
y-axis at A. The tangent to the curve at A meets the x-axis at B. The point C lies on the curve and BC is
parallel to the y-axis.

(i) Find the x-coordinate of B. [4]

(ii) Find the area of the shaded region. [6]

6 (i) Differentiate x2 ln x with respect to x. [2]

冕 4x ln x dx = e + 1.
e
(ii) Use your result to show that 2 [4]
1

0606/01/M/J/07
 6

12 Answer only one of the following two alternatives.

EITHER

A curve has equation y = (x2 – 3)e–x.

(i) Find the coordinates of the points of intersection of the curve with the x-axis. [2]

(ii) Find the coordinates of the stationary points of the curve. [5]

(iii) Determine the nature of these stationary points. [3]

OR

A particle moves in a straight line such that its displacement, s m, from a fixed point O at a time t s, is
given by

s = ln(t + 1) for 0
t
3,

s = 12 ln (t – 2) – ln(t + 1) + ln 16 for t > 3.

Find

(i) the initial velocity of the particle, [2]

(ii) the velocity of the particle when t = 4, [2]

(iii) the acceleration of the particle when t = 4, [2]

(iv) the value of t when the particle is instantaneously at rest, [2]

(v) the distance travelled by the particle in the 4th second. [2]

1 Differentiate with respect to x

(i) 1 + x3 , [2]

(ii) x2 cos 2x. [3]

9 (a) Variables x and y are related by the equation y = 5x + 2 – 4e–x.

dy
(i) Find –– . [2]
dx
(ii) Hence find the approximate change in y when x increases from 0 to p, where p is small. [2]

(b) A square of area A cm2 has a side of length x cm. Given that the area is increasing at a constant rate
of 0.5 cm2 s–1, find the rate of increase of x when A = 9. [4]

0606/01/M/J/08
 4

7
y

P (x, y)

4 2
y=
x2

O x

4 2
The diagram shows part of the curve y = . The point P (x, y) lies on this curve.
x2
(i) Write down an expression, in terms of x, for (OP)2. [1]
dS
(ii) Denoting (OP)2 by S, find an expression for . [2]
dx
(iii) Find the value of x for which S has a stationary value and the corresponding value of OP. [3]

10 (a) Find

(i) 兰 (2x12– 1) 4 dx, [2]

(ii) 兰x(x –1) dx.

2 [3]

dy 3( x + 1)
(b) (i) Given that y = 2( x − 5 ) x + 4 , show that = . [3]
dx x+4

(ii) Hence find 兰 (xx++14) dx. [2]

The diagram shows the curve y = 4x – x2, which crosses the x-axis at the origin O and the point A.
The tangent to the curve at the point (1, 3) crosses the x-axis at the point B.

(i) Find the coordinates of A and of B. [5]

(ii) Find the area of the shaded region. [5]

0606/02/M/J/08
 5

11 Answer only one of the following two alternatives.

EITHER
ln x , w here x > 0.
A curve has equation y = –––
x2
(i) Find the exact coordinates of the stationary point of the curve. [6]
d2y a ln x + b , where a and b are integers.
(ii) Show that –––2 can be written in the form –––––––– [3]
dx x4
(iii) Hence, or otherwise, determine the nature of the stationary point of the curve. [2]

OR
dy π π π
A curve is such that –– = 6 cos 2x + –  for – – ⭐ x ⭐ 5π
–– . The curve passes through the point  – , 5.
dx 2 4 4 4
Find

(i) the equation of the curve, [4]

(ii) the x-coordinates of the stationary points of the curve, [3]

(iii) the equation of the normal to the curve at the point on the curve where x = 3π
–– . [4]
4

0606/11/M/J/10
 4

5
y
y = 6 – 3–
x
A

B
O 3 x

The diagram shows part of the curve y = 6 – 3–x which passes through the point A where x = 3. The
normal to the curve at the point A meets the x-axis at the point B. Find the coordinates of the point B.
[5]

5 Given that a curve has equation y = x2 + 64 x , find the coordinates of the point on the curve where
d2y
–––2 = 0. [7]
dx

0606/13/M/J/10
 6

11 A particle moves in a straight line such that its displacement, x m, from a fixed point O on the line at
time t seconds is given by x = 12{1n (2t + 3)}. Find

(i) the value of t when the displacement of the particle from O is 48 m, [3]

(ii) the velocity of the particle when t = 1, [3]

(iii) the acceleration of the particle when t =1. [3]

12 Answer only one of the following two alternatives.

EITHER

y π, 7
B ––
4

A C
y=5

O x

dy
The diagram shows part of a curve for which –– = 8 cos 2x. The curve passes through the
dx
π
point B 冸 – , 7冹. The line y = 5 meets the curve at the points A and C.
4
(i) Show that the curve has equation y = 3 + 4 sin 2x. [3]

(ii) Find the x-coordinate of the point A and of the point C. [4]

(iii) Find the area of the shaded region. [5]

OR
dy
A curve is such that –– = 6e3x – 12. The curve passes through the point (0, 1).
dx
(i) Find the equation of the curve. [4]

(ii) Find the coordinates of the stationary point of the curve. [3]

(iii) Determine the nature of the stationary point. [2]

(iv) Find the coordinates of the point where the tangent to the curve at the point (0, 1) meets the
x-axis. [3]

0606/13/M/J/10
 6

5 Given that a curve has equation y = x2 + 64 x , find the coordinates of the point on the curve where
d2y
–––2 = 0. [7]
dx

3 The volume V cm3 of a spherical ball of radius r cm is given by V = 4– πr3. Given that the radius is
3
1– cm s–1, find the rate at which the volume
increasing at a constant rate of π is increasing when V = 288π.
[4]

P(1, 8)

y = x3 – 6x2 + 8x + 5

O x

The diagram shows part of the curve y = x3 – 6x2 + 8x + 5. The tangent to the curve at the point P(1, 8)
cuts the curve at the point Q.

(i) Show that the x-coordinate of Q is 4. [6]

(ii) Find the area of the shaded region. [6]

0606/21/M/J/10
 4

x+2 dy k(x + 4)
9 (i) Given that y = ––––––––– , show that –– = –––––––––3 , where k is a constant to be found. [5]
(4x + 12)½ dx (4x + 12) /2
13
(ii) Hence evaluate ∫1
x+4
–––––––––3 dx.
(4x + 12) /2
[3]

9 A particle starts from rest and moves in a straight line so that, t seconds after leaving a fixed point O, its
velocity, v ms–1, is given by

v = 4 sin 2t.

(i) Find the distance travelled by the particle before it first comes to instantaneous rest. [5]

(ii) Find the acceleration of the particle when t = 3. [3]

R Q(x, y)

y = 12 – 2x

O P x

The diagram shows part of the line y = 12 – 2x. The point Q (x, y) lies on this line and the points P and
R lie on the coordinate axes such that OPQR is a rectangle.

(i) Write down an expression, in terms of x, for the area A of the rectangle OPQR. [2]

(ii) Given that x can vary, find the value of x for which A has a stationary value. [3]

(iii) Find this stationary value of A and determine its nature. [2]

0606/22/M/J/10
 9

9 A body moves in a straight line such that, t s after passing through a fixed point O, its For
displacement from O is s m. The velocity v ms–1 of the body is such that v = 5cos4t. Examiner’s
Use

(i) Write down the velocity of the body as it passes through O. [1]

(ii) Find the value of t when the acceleration of the body is first equal to 10 ms–2. [4]

(iii) Find the value of s when t = 5. [4]

dy
10 (a) A curve is such that dx = ae1–x – 3x2, where a is a constant. At the point (1, 4), the gradient
of the curve is 2.

(i) Find the value of a. [1]

(ii) Find the equation of the curve. [5]

兰 (7x + 8) dx.
1
3
(b) (i) Find [2]

兰
8 1
(ii) Hence evaluate (7x + 8) 3 dx. [2]
0

0606/11/M/J/11
 14

12 Answer only one of the following two alternatives. For

Examiner’s
EITHER Use

The equation of a curve is y = (x – 1)(x2 – 6x + 2).

(i) Find the x-coordinates of the stationary points on the curve and determine the nature of each
of these stationary points. [6]

(ii) Given that z = y2 and that z is increasing at the constant rate of 10 units per second, find the
rate of change of y when x = 2. [2]

(iii) Hence find the rate of change of x when x = 2. [2]

OR

The diagram shows a cuboid with a rectangular base of sides x cm and 2x cm. The height of the
cuboid is y cm and its volume is 72 cm3.

y cm

x cm
2x cm

(i) Show that the surface area A cm2 of the cuboid is given by
216
A = 4x2 + x .
[3]

(ii) Given that x can vary, find the dimensions of the cuboid when A is a minimum. [4]

(iii) Given that x increases from 2 to 2 + p, where p is small, find, in terms of p, the corresponding
approximate change in A, stating whether this change is an increase or a decrease. [3]

0606/11/M/J/11
 14

12 Answer only one of the following two alternatives. For

Examiner’s
EITHER Use

The tangent to the curve y = 3x3 + 2x2 – 5x + 1 at the point where x = –1 meets the y-axis at the
point A.

(i) Find the coordinates of the point A. [3]

The curve meets the y-axis at the point B. The normal to the curve at B meets the x-axis at the
point C. The tangent to the curve at the point where x = –1 and the normal to the curve at B meet
at the point D.

(ii) Find the area of the triangle ACD. [7]

OR

y y = x (x – 3)2

P R

O Q x

The diagram shows the curve y = x (x – 3)2 . The curve has a maximum at the point P and
touches the x-axis at the point Q. The tangent at P and the normal at Q meet at the point R. Find
the area of the shaded region PQR. [10]

0606/12/M/J/11
 12

For
Examiner’s
dy Use
1 (i) Given that y = sin 3x, find . [1]
dx

(ii) Hence find the approximate increase in y as x increases from π to π + p, where p is small.
9 9
[2]

dy k(x + 5) , where k is a constant to be found. [3]

5 (i) Given that y = x 2x + 15, show that = –––––––
dx 2x + 15

兰 兰
5
(ii) Hence find x+5 x+5
––––––– dx and evaluate ––––––– dx. [3]
2x + 15 –3 2x + 15

O Q x

y = x sin x

The diagram shows part of the curve y = x sin x and the normal to the curve at the point

冢 冣
P π , π . The curve passes through the point Q(π, 0).
2 2
(i) Show that the normal to the curve at P passes through the point Q. [4]

(ii) Given that d (x cos x) = cos x – x sin x, find

dx 兰x sin xdx. [3]

(iii) Find the area of the shaded region. [5]

0606/21/M/J/11
 8

7 A particle moves in a straight line so that, t s after passing through a fixed point O, its velocity, For
60 .
v ms–1, is given by v = ––––––– Examiner’s
(3t + 4)2 Use

(i) Find the velocity of the particle as it passes through O. [1]

(ii) Find the acceleration of the particle when t = 2. [3]

(iii) Find an expression for the displacement of the particle from O, t s after it has passed
through O. [4]

7 (i) Differentiate x sin x with respect to x. [2]

π–
(ii) Hence evaluate ∫0
2
x cos x dx. [4]

0606/22/M/J/11
 14

11 Answer only one of the following two alternatives. For

Examiner’s
EITHER Use

(a) Using an equilateral triangle of side 2 units, find the exact value of sin 60° and of cos 60°.
[3]
(b)
P S
60° 60°

x cm x cm

Q y cm R

PQRS is a trapezium in which PQ = RS = x cm and QR = y cm.

Angle QPS = angle RSP = 60° and QR is parallel to PS.

(i) Given that the perimeter of the trapezium is 60 cm, express y in terms of x. [2]

(ii) Given that the area of the trapezium is A cm2 , show that
(30x – x2)
A= 3 . [3]
2
(iii) Given that x can vary, find the value of x for which A has a stationary value and
determine the nature of this stationary value. [4]

OR
r cm

h cm For a sphere of radius r:

4 3
Volume = πr
3
Surface area = 4π r2

The diagram shows a solid object in the form of a cylinder of height h cm and radius r cm on top
of a hemisphere of radius r cm. Given that the volume of the object is 2880 π cm3,

(i) express h in terms of r, [2]

(ii) show that the external surface area, A cm2, of the object is given by
A = 5 π r2 + 5760 π
3 r . [3]
Given that r can vary,

(iii) find the value of r for which A has a stationary value, [4]

(iv) find this stationary value of A, leaving your answer in terms of π, [2]

(v) determine the nature of this stationary value. [1]

0606/22/M/J/11
 4

10 A curve has the equation y # x
3 ln x, where x p 0.

d
y
(i) Find an expression for . [2]
d
x
Hence

(ii) calculate the value of ln x at the stationary point of the curve, [2]

(iii) find the approximate increase in y as x increases from e to e ! p, where p is small, [2]

(iv) find

x
2 ln x

d
x. [3]

11 A car moves on a straight road. As the driver passes a point A on the road with a speed of 20 ms–1, he
notices an accident ahead at a point B. He immediately applies the brakes and the car moves with an
acceleration of a ms–2, where a = 3t – 6 and t s is the time after passing A. When t = 4, the car passes
2
the accident at B. The car then moves with a constant acceleration of 2 ms–2 until the original speed of
20 ms–1 is regained at a point C. Find

(i) the speed of the car at B, [4]

(ii) the distance AB, [3]
(iii) the time taken for the car to travel from B to C. [2]

Sketch the velocity-time graph for the journey from A to C. [2]

0606/2/O/N/02
 5

12 Answer only one of the following two alternatives.

EITHER

lm
rm

The diagram shows a greenhouse standing on a horizontal rectangular base. The vertical semicircular
ends and the curved roof are made from polythene sheeting. The radius of each semicircle is r m and
the length of the greenhouse is l m. Given that 120 m2 of polythene sheeting is used for the greenhouse,
express l in terms of r and show that the volume, V m3, of the greenhouse is given by
πr 3
V = 60r – . [4]
2

Given that r can vary, find, to 2 decimal places, the value of r for which V has a stationary value. [3]

Find this value of V and determine whether it is a maximum or a minimum. [3]

OR
y
y = x2 ln x

O Q x

The diagram shows part of the curve y = x2 ln x, crossing the x-axis at Q and having a minimum point
at P.
dy
(i) Find the value of at Q. [4]
dx
1
(ii) Show that the x-coordinate of P is . [3]
e
d2 y
(iii) Find the value of at P. [3]
dx2

8 Given that y

=
ln x , find

2x + 3
d
y
(i) , [3]
d
x

(ii) the approximate change in y as x increases from 1 to 1 ! p, where p is small, [2]

(iii) the rate of change of x at the instant when x # 1, given that y is changing at the rate of
0.12 units per second at this instant. [2]

0606/2/O/N/02
 6

12 Answer only one of the following two alternatives.

EITHER

r cm

x cm

5r cm 5r cm
4 4

A piece of wire, 125 cm long, is bent to form the shape shown in the diagram. This shape encloses
a plane region, of area A cm
2, consisting of a semi-circle of radius r cm, a rectangle of length x cm
and an isosceles triangle having two equal sides of length 5r cm.
4

2
2
(i) Express x in terms of r and hence show that A

=

125r − πr − 7r . [6]
2 4
Given that r can vary,

(ii) calculate, to 1 decimal place, the value of r for which A has a maximum value. [4]

OR

30 cm

h cm
r cm

12 cm

The diagram shows the cross-section of a hollow cone of height 30 cm and base radius 12 cm and a
solid cylinder of radius r cm and height h cm. Both stand on a horizontal surface with the cylinder
inside the cone. The upper circular edge of the cylinder is in contact with the cone.

(i) Express h in terms of r and hence show that the volume, V cm
3, of the cylinder is given by
V

=
π(30r
2 −
25
r
3). [4]

Given that r can vary,

(ii) find the volume of the largest cylinder which can stand inside the cone and show that, in this
case, the cylinder occupies
94

of the volume of the cone. [6]

[The volume, V, of a cone of height H and radius R is given by V

=

31


π R
2
H.]

0606/1/O/N/03
 5

3 The diagram shows part of the curve y = 3sin 2x + 4cos x.

O π x
2

Find the area of the shaded region, bounded by the curve and the coordinate axes. [5]

A curve has the equation y = e
2
x + 3e
−
2
x.

1 1

(i) Show that the exact value of the y-coordinate of the stationary point of the curve is 2√3. [4]

(ii) Determine whether the stationary point is a maximum or a minimum. [2]

(iii) Calculate the area enclosed by the curve, the x-axis and the lines x # 0 and x # 1. [4]

9 A curve has the equation y = 2x – 4 .

x+3

(i) Obtain an expression for dy and hence explain why the curve has no turning points. [3]
dx
The curve intersects the x-axis at the point P. The tangent to the curve at P meets the y-axis at the
point Q.

(ii) Find the area of the triangle POQ, where O is the origin. [5]

© 0606/1/O/N/04
 5

12

A particle, travelling in a straight line, passes a fixed point O on the line with a speed of 0.5 ms–1. The
acceleration, a ms–2, of the particle, t s after passing O, is given by a = 1.4 – 0.6t.

(i) Show that the particle comes instantaneously to rest when t = 5. [4]

(ii) Find the total distance travelled by the particle between t = 0 and t = 10. [6]

6 A particle starts from rest at a fixed point O and moves in a straight line towards a point A. The
velocity, v ms–1, of the particle, t seconds after leaving O, is given by v = 6 – 6e–3t . Given that the particle
reaches A when t = ln 2, find

(i) the acceleration of the particle at A, [3]

(ii) the distance OA. [4]

8 A curve has the equation y = (x + 2) x − 1 .

dy kx
(i) Show that ––– = , where k is a constant, and state the value of k. [4]
dx x −1

5 x
(ii) Hence evaluate dx. [4]
2 x −1

dy
3 (i) Given that y = 1 + ln (2x – 3), obtain an expression for ––– . [2]
dx
(ii) Hence find, in terms of p, the approximate value of y when x = 2 + p, where p is small. [3]

d2y
10 A curve is such that –––– = 6x – 2. The gradient of the curve at the point (2, –9) is 3.
dx2
(i) Express y in terms of x. [5]
16
(ii) Show that the gradient of the curve is never less than – –– . [3]
3

0606/02/O/N/04
 5

12 Answer only one of the following two alternatives.

EITHER
y

S (0, 15)
y = x2 – 10x + 24

Q (4, 0)
R O T x
(3.75, 0)

The diagram, which is not drawn to scale, shows part of the curve y = x2 – 10x + 24 cutting the x-axis at
Q(4, 0). The tangent to the curve at the point P on the curve meets the coordinate axes at S(0, 15) and at
T (3.75, 0).

(i) Find the coordinates of P. [4]

The normal to the curve at P meets the x-axis at R.

(ii) Find the coordinates of R. [2]

(iii) Calculate the area of the shaded region bounded by the x-axis, the line PR and the curve PQ. [5]

OR
π
A curve has the equation y = 2cos x – cos 2x, where 0 < x
– .
2
dy d2y
(i) Obtain expressions for ––– and –––– . [4]
dx dx2
(ii) Given that sin 2x may be expressed as 2sin x cos x, find the x-coordinate of the stationary point of the
curve and determine the nature of this stationary point. [4]

∫ // y dx.
π
2
(iii) Evaluate [3]
π
3

7 The function f is defined for the domain –3 ⭐ x ⭐ 3 by

2
( )
f(x) = 9 x – 1 – 11.
3

(i) Find the range of f. [3]

(ii) State the coordinates and nature of the turning point of

(a) the curve y = f(x),

(b) the curve y =
f(x)
.

[4]

0606/02/O/N/05
 6

12

y y = 4 – e–2 x

A O C x

The diagram shows part of the curve y = 4 – e–2x which crosses the axes at A and at B.

(i) Find the coordinates of A and of B. [2]

The normal to the curve at B meets the x-axis at C.

(ii) Find the coordinates of C. [4]

(iii) Show that the area of the shaded region is approximately 10.3 square units. [5]

3 Evaluate
π
– π
∫ sin(2x + –6 )dx.
0
6 [4]

0606/01/O/N/06
 4

6 A curve has equation y = x3 + ax + b, where a and b are constants. The gradient of the curve at the
point (2, 7) is 3. Find

(i) the value of a and of b, [5]

(ii) the coordinates of the other point on the curve where the gradient is 3. [2]

1
7 (a) Find the value of m for which the line y = mx – 3 is a tangent to the curve y = x + and find
x
the x-coordinate of the point at which this tangent touches the curve. [5]

(b) Find the value of c and of d for which {x : – 5 < x < 3} is the solution set of x2 + cx < d. [2]

6 A curve is such that dy = 6 , and (6, 20) is a point on the curve.

dx 4x + 1
(i) Find the equation of the curve. [4]

A line with gradient – 12 is a normal to the curve.

(ii) Find the coordinates of the points at which this normal meets the coordinate axes. [4]

x
−
11 The equation of a curve is y = xe 2 .
x
dy 1 −
(i) Show that = ( 2 − x )e 2 . [3]
dx 2
d2y
(ii) Find an expression for 2 . [2]
dx
The curve has a stationary point at M.

(iii) Find the coordinates of M. [2]

(iv) Determine the nature of the stationary point at M. [2]

0606/02/O/N/06
 5

11 Answer only one of the following two alternatives.

EITHER

A curve has the equation y = xe2x.

2
(i) Obtain expressions for dy and d y2 . [5]
dx dx
(ii) Show that the y-coordinate of the stationary point of the curve is – 1 . [3]
2e
(iii) Determine the nature of this stationary point. [2]

OR

(i) Show that ( )

ln x
d –––
dx x2
1 – 2 lnx
= –––––––
x3
. [3]

(ii) Show that the y-coordinate of the stationary point of the curve y = ––– 1 .
ln x is –– [3]
x2 2e
(iii) Use the result from part (i) to find ∫( )
ln
–––
x3
x
dx. [4]

1 The two variables x and y are related by the equation yx2 = 800.

(i) Obtain an expression for dy in terms of x. [2]

dx
(ii) Hence find the approximate change in y as x increases from 10 to 10 + p, where p is small. [2]

9 A particle travels in a straight line so that, t s after passing through a fixed point O, its speed, v ms–1, is
()
given by v = 8cos –2 .
t

(i) Find the acceleration of the particle when t = 1. [3]

The particle first comes to instantaneous rest at the point P.

(ii) Find the distance OP. [4]

0606/01/O/N/07
 5

10
y

X
O x

The diagram shows part of the curve y = 4 x – x. The origin O lies on the curve and the curve
intersects the positive x-axis at X. The maximum point of the curve is at M. Find

(i) the coordinates of X and of M, [5]

(ii) the area of the shaded region. [4]

d2y dy
8 A curve is such that 2
= 4e–2x. Given that = 3 when x = 0 and that the curve passes through the
dx dx
point (2, e–4), find the equation of the curve. [6]

0606/02/O/N/07
 5

12 Answer only one of the following two alternatives.

EITHER

x2
A curve has equation y = .
x+1
(i) Find the coordinates of the stationary points of the curve. [5]

The normal to the curve at the point where x = 1 meets the x-axis at M. The tangent to the curve at the
point where x = –2 meets the y-axis at N.

(ii) Find the area of the triangle MNO, where O is the origin. [6]

OR

A curve has equation y = ex – 2 – 2x + 6.

(i) Find the coordinates of the stationary point of the curve and determine the nature of the stationary
point. [6]

The area of the region enclosed by the curve, the positive x-axis, the positive y-axis and the line x = 3
is k + e – e–2.

(ii) Find the value of k. [5]

4 (i) Differentiate x ln x with respect to x. [2]

(ii) Hence find 兰ln x dx. [3]

© UCLES 2008 0606/01/O/N/08

www.xtremepapers.net
 5

10
y

y = x 3 − 8x 2 + 16x

O x

The diagram shows part of the curve y = x3 – 8x2 + 16x.

(i) Show that the curve has a minimum point at (4, 0) and find the coordinates of the maximum
point. [4]

(ii) Find the area of the shaded region enclosed by the x-axis and the curve. [4]

A particle moves in a straight line so that t seconds after passing a fixed point O its acceleration,
a ms–2, is given by a = 4t – 12. Given that its speed at O is 16 ms–1, find

the values of t at which the particle is stationary, [5]

the distance the particle travels in the fifth second. [5]

4 The line y = 5x – 3 is a tangent to the curve y = kx2 – 3x + 5 at the point A. Find

(i) the value of k, [3]

(ii) the coordinates of A. [2]

7 (i) Find
dx ( e3x .
d– xe3x – –––
3 ) [3]

(ii) Hence find ∫xe3xdx. [3]

0606/02/O/N/08
 4

2x
8 A curve has equation y = .
x2 + 9

(i) Find the x-coordinate of each of the stationary points of the curve. [4]

(ii) Given that x is increasing at the rate of 2 units per second, find the rate of increase of y when
x = 1. [3]

5 Two variables, x and y, are related by the equation

32
y = 6x2 + .
x3
dy
(i) Obtain an expression for . [2]
dx
(ii) Use your expression to f ind the approximate change in the value of y when x increases from
2 to 2.04. [3]
 6

12 Answer only one of the following two alternatives.

EITHER

3 ()
(i) State the amplitude of 1 + sin –x . [1]

()
(ii) State, in radians, the period of 1 + sin –x .
3
[1]

y = 1.5
A B

()
y = 1 + sin –x
3

O x

The diagram shows the curve y = 1 + sin –x

3 () meeting the line y = 1.5 at points A and B. Find

(iii) the x-coordinate of A and of B, [3]

(iv) the area of the shaded region. [6]

OR

A particle moves in a straight line such that t s after passing through a fixed point O, its velocity,
v m s–1, is given by v = k cos 4t, where k is a positive constant. Find

(i) the value of t when the particle is first instantaneously at rest, [1]

(ii) an expression for the acceleration of the particle t s after passing through O. [2]

Given that the acceleration of the particle is 12 m s–2 when t = 3π ,

8
(iii) find the value of k. [2]

Using your value for k,

(iv) sketch the velocity-time curve for the particle for 0 艋 t 艋 π, [2]

(v) find the displacement of the particle from O when t = π . [4]

 5
8 y

S P

y = 27 – x2

R Q
O t units x

The diagram shows part of the curve y = 27 – x2. The points P and S lie on this curve. The points Q
and R lie on the x-axis and PQRS is a rectangle. The length of OQ is t units.

(i) Find the length of PQ in terms of t and hence show that the area, A square units, of PQRS is given
by
A = 54t – 2t 3. [2]

(ii) Given that t can vary, find the value of t for which A has a stationary value. [3]

(iii) Find this stationary value of A and determine its nature. [3]

5
π π
( )
A curve has the equation y = 2x sin x + . The curve passes through the point P , a .
3 2
(i) Find, in terms of π, the value of a. [1]

(ii) Using your value of a, find the equation of the normal to the curve at P. [5]
 6

12 Answer only one of the following two alternatives.

EITHER
1
dy x
The point P(0, 5) lies on the curve for which = e 2 . The point Q, with x-coordinate 2, also lies
dx
on the curve.

(i) Find, in terms of e, the y-coordinate of Q. [5]

The tangents to the curve at the points P and Q intersect at the point R.

(ii) Find, in terms of e, the x-coordinate of R. [5]

OR y

¹x
y = e² + 5

D C

O B x
1
x
The diagram shows part of the curve y = e 2 + 5 crossing the y-axis at A. The normal to the curve
at A meets the x-axis at B.

(i) Find the coordinates of B. [4]

The line through B, parallel to the y-axis, meets the curve at C. The line through C, parallel to the
x-axis, meets the y-axis at D.

(ii) Find the area of the shaded region. [6]

 4

11 (i) Find 冕 11+ x dx. [2]

2x dy A Bx
(ii) Given that y = , show that = + , where A and B are to be found. [4]
1+ x dx 1+ x 冢 1+ x 冣
3

冕冢 冕冢
3
x x
(iii) Hence find dx and evaluate dx. [4]
1+ x 冣 1+ x 冣
3 3
0

x cm A cm2
4x cm

x cm

The figure shows a rectangular metal block of length 4x cm, with a cross-section which is a square of
side x cm and area A cm2. The block is heated and the area of the cross-section increases at a constant
rate of 0.003 cm2s–1. Find

(i) dA in terms of x, [1]

dx
(ii) the rate of increase of x when x = 5, [3]

(iii) the rate of increase of the volume of the block when x = 5. [4]

0606/11/O/N/10
 6

12 Answer only one of the following two alternatives.

EITHER
dy
A curve is such that = 4x2 – 9. The curve passes through the point (3, 1).
dx
(i) Find the equation of the curve. [4]

The curve has stationary points at A and B.

(ii) Find the coordinates of A and of B. [3]

(iii) Find the equation of the perpendicular bisector of the line AB. [4]

OR
dy
A curve has the equation y = Ae2x + Be–x where x  0. At the point where x = 0, y = 50 and = – 20.
dx
(i) Show that A = 10 and find the value of B. [5]

(ii) Using the values of A and B found in part (i), find the coordinates of the stationary point on the
curve. [4]

(iii) Determine the nature of the stationary point, giving a reason for your answer. [2]

5
π π
( )
A curve has the equation y = 2x sin x + . The curve passes through the point P , a .
3 2
(i) Find, in terms of π, the value of a. [1]

(ii) Using your value of a, find the equation of the normal to the curve at P. [5]

0606/11/O/N/10
 4

11 (i) Find 冕 11+ x dx. [2]

2x dy A Bx
(ii) Given that y = , show that = + , where A and B are to be found. [4]
dx 1+ x 冢 1+ x 冣
3
1+ x

冕冢 冕冢
3
x x
(iii) Hence find dx and evaluate dx. [4]
1+ x 冣 1+ x 冣
3 3
0

x cm A cm2
4x cm

x cm

The figure shows a rectangular metal block of length 4x cm, with a cross-section which is a square of
side x cm and area A cm2. The block is heated and the area of the cross-section increases at a constant
rate of 0.003 cm2s–1. Find

(i) dA in terms of x, [1]

dx
(ii) the rate of increase of x when x = 5, [3]

(iii) the rate of increase of the volume of the block when x = 5. [4]

0606/12/O/N/10
 6

12 Answer only one of the following two alternatives.

EITHER
dy
A curve is such that = 4x2 – 9. The curve passes through the point (3, 1).
dx
(i) Find the equation of the curve. [4]

The curve has stationary points at A and B.

(ii) Find the coordinates of A and of B. [3]

(iii) Find the equation of the perpendicular bisector of the line AB. [4]

OR
dy
A curve has the equation y = Ae2x + Be–x where x  0. At the point where x = 0, y = 50 and = – 20.
dx
(i) Show that A = 10 and find the value of B. [5]

(ii) Using the values of A and B found in part (i), find the coordinates of the stationary point on the
curve. [4]

(iii) Determine the nature of the stationary point, giving a reason for your answer. [2]

0606/12/O/N/10
 4

( )
1 2
9 (a) Find x 3 – 3 dx. [3]
dy
(b) (i) Given that y = x x 2 + 6 , find . [3]
dx
(ii) Hence find  x2 + 3
x2 + 6
dx. [2]

10 A particle travels in a straight line so that, t s after passing through a fixed point O, its displacement s m
from O is given by s = ln(t2 + 1).

(i) Find the value of t when s = 5. [2]

(ii) Find the distance travelled by the particle during the third second. [2]

(iii) Show that, when t = 2, the velocity of the particle is 0.8 ms–1. [2]

(iv) Find the acceleration of the particle when t = 2. [3]

© UCLES 2010 0606/13/O/N/10

www.XtremePapers.net
 5

12 Answer only one of the following two alternatives.

EITHER

A curve has the equation y = A sin 2x + B cos 3x. The curve passes through the point with coordinates
and has a gradient of – 4 when x = .
π ( )
π
12
,3
3
(i) Show that A = 4 and find the value of B. [6]
π
(ii) Given that, for 0  x  3 , the curve lies above the x-axis, find the area of the region enclosed by the
π
curve, the y-axis and the line x = . [5]
3
OR

C
A

B
O x
y = 4x2 – 2x3

The diagram shows the curve y = 4x2 – 2x3. The point A lies on the curve and the x-coordinate of A is 1.
The curve crosses the x-axis at the point B. The normal to the curve at the point A crosses the y-axis at the
point C.

(i) Show that the coordinates of C are (0, 2.5). [5]

(ii) Find the area of the shaded region. [6]

0606/13/O/N/10
A particle moves in a straight line so that, at time t s after passing a fixed point O, its velocity is v ms–1,
where

v = 6t + 4 cos 2t.

Find

the velocity of the particle at the instant it passes O, [1]

the acceleration of the particle when t = 5, [4]

the greatest value of the acceleration, [1]

the distance travelled in the fifth second. [4]

x cm
7
x cm

45 cm

x cm

60 cm

A rectangular sheet of metal measures 60 cm by 45 cm. A scoop is made by cutting out squares, of side
x cm, from two corners of the sheet and folding the remainder as shown.

(i) Show that the volume, V cm3, of the scoop is given by

V = 2700x – 165x 2 + 2x 3. [2]

(ii) Given that x can vary, find the value of x for which V has a stationary value. [4]

A particle moves in a straight line so that, at time t s after passing a fixed point O, its velocity is v ms–1,
where

v = 6t + 4 cos 2t.

Find

(i) the velocity of the particle at the instant it passes O, [1]

(ii) the acceleration of the particle when t = 5, [4]

(iii) the greatest value of the acceleration, [1]

(iv) the distance travelled in the fifth second. [4]

0606/21/O/N/10
 3

1 The two variables x and y are such that y = 10 .

(x + 4)3
dy
(i) Find an expression for . [2]
dx
(ii) Hence find the approximate change in y as x increases from 6 to 6 + p, where p is small. [2]

dy
2 Find the equation of the curve which passes through the point (4, 22) and for which = 3x(x – 2).
dx
[4]

10 The equation of a curve is y = x 2ex. The tangent to the curve at the point P(1, e) meets the y-axis at the
point A. The normal to the curve at P meets the x-axis at the point B. Find the area of the triangle OAB,
where O is the origin. [9]

0606/23/O/N/10
ADDITONAL MATHEMATICS
2002 – 2011
CLASSIFIED CALCLUS
Compiled & Edited
By

Dr. Eltayeb Abdul Rhman

www.drtayeb.tk

First Edition
2011