Name:___________________________________
Teacher:_______________________________________
MOUNT ALBERT GRAMMAR SCHOOL
Mathematics with Calculus
Scholarship
TERM 3 2014
QUESTION BOOKLET
100 marks
Time Allowed: Three hours
Instructions:
- Answer all questions.
- Each question is worth 20 marks.
- If you use pencil or twink you have no chance of a remark.
- No dictionaries or translators allowed.
- No borrowing or loaning of equipment.
- All essential working must be shown.
Achievement Criteria
Overall Performance:
GOOD LUCK!
DS/14
Page 1 of 6
QUESTION ONE (20 marks)
(a)
The diagram shows that P ( acosq , bsinq ) is any
point on the first quadrant of the ellipse
x 2 y2
+ = 1.
a2 b2
y
P ( acosq , bsinq )
Q
x
Q, R and S are also points on the ellipse such that
PQRS is a rectangle with its sides parallel to the
S
R
axes. If the rectangle has a perimeter p (q ) , prove
that the maximum value of p (q ) = 4 a 2 + b 2 . (Include a test to show that p (q ) is maximum and
not minimum).
(b)
V is the volume of a cylindrical steel rod radius r and length l (i.e. V = p r 2l ) at a temperature of
q °C . As it is heated the rod expands with its radius increasing at a rate given by
length increasing at a rate given by
dl
= kl where k is a constant (called the coefficient of linear
dq
expansion of steel). Prove that the volume of the rod increases at a rate given by
(c)
In this question f ( x ) =
(i)
dr
= kr and its
dq
dV
= 3kV .
dq
ax + b
c
where a, b and c are positive non-zero constants.
+
c
ax + b
Show that the graph of y = f ( x ) has an intercept on the y-axis but not on the x-axis. Also
state the equation of its asymptote parallel to the y-axis.
(ii)
Find f ¢ ( x ) and hence find the coordinates of its stationary points. Show that the midpoint
of the line joining these stationary points is where the asymptote in (i) cuts the x-axis.
(iii)
Find f ¢¢ ( x ) and hence show that one of the stationary points is a maximum point and the
other is a minimum point.
(iv)
In this question the maximum value of f ( x ) < its minimum value. Show how this can be
by sketching a possible graph of y = f ( x ) showing its main features.
DS/14
Page 2 of 6
QUESTION TWO (20 marks)
(a)
(i)
By solving the differential equation
dy - y
show that
=
dx
x
x + y = a given that y = 0
at x = a (where a > 0 ).
(ii)
Find the area bounded by the curve
(iii)
Show that
x + y = a and the axes.
d2y
a
and hence comment on the concavity of the curve (giving reasons
=
2
dx
2x x
for your answer).
(b)
In the diagram the shaded area is rotated around the
x-axis to make a belt (used for driving a pulley).
The shaded area is enclosed by the parabola y = x 2
shifted a distance k up the y-axis and lies between
k
x = a and x = -a . Find (in terms of a and k) the
volume of the belt.
Note: The volume formed by rotating a curve
y = f ( x ) around the x-axis between x = b and
x = a (where b > a ) is given by
ò
b
a
p y2 dx .
-a
(c)
(i)
(ii)
DS/14
Use the substitution x =
p
2
- u to show that
p
òp
3
6
1
dx =
n
cot x +1
By adding the integrals in (i) show that each one is equal to
Page 3 of 6
a
p
òp
p
12
3
6
.
1
dx .
tan x +1
n
QUESTION THREE (20 marks)
(a)
In (i) and (ii) state the solutions of each equation for 0 £ q £ 2p - if there are no solutions say so.
In (iii) find the limiting value.
(b)
(i)
sin 2q
=0
sinq
(ii)
sin 2q
=2
sinq
(iii)
lim
sin 2q
q ®0 sin q
Note: Do not use a calculator in this question.
(i)
æp
ö
æp
ö
Prove that tan ç + A ÷ ´ tan ç - A ÷ = 1
è4
ø
è4
ø
æp
ö
æp
ö
2
and that tan ç + A ÷ + tan ç - A ÷ =
è4
ø
è4
ø cos 2A
p
(iii)
Use the results from (i) and (ii) to show that tan
(Note:
DS/14
æp p ö
2
= ç - ÷ show that
=2 2
12 è 4 6 ø
cos 2A
p
If A =
24
x2 - 2 2
(c)
and so 2A =
(ii)
(
(
)
3 -1 .
7p
5p
and tan
are the solutions of
24
24
7p
5p
and tan
.
3 -1 x +1 = 0 . Hence find the exact values of tan
24
24
)
a2 + b - 2a b = a - b ).
Note: Do not use a calculator in this question.
p
=
2
p
2+ 2
to help you prove that cos =
.
2
8
2
(i)
Use the result cos
(ii)
Use the result in (i) to help you prove that cos
(iii)
Use the result in (i) and (ii) to write down the exact value of cos
4
Page 4 of 6
p
16
=
2+ 2+ 2
.
2
p
32
.
QUESTION FOUR (20 marks)
(a)
Prove that if z and c are complex numbers such that z = ic then z = -i c (where z and c are the
conjugates of z and c respectively).
(Hint: Suppose that c = a + ib )
(b)
(i)
Describe fully the transformation which the complex number rcisq undergoes in the
Argand Plane when it is multiplied by cisa .
1
1
1
p
If S¥ = 1+ z + z 2 + z 3 +... where z = cis .
2
4
8
3
(ii)
Write down the modulus and argument of the ( n +1) th term in the series for S¥ .
(iii)
Use the Cartesian form of cis
p
3
to help you find S¥ in polar form. Hence find the modulus
and argument of S¥ .
Note: a + ar + ar 2 + ar 3 +..... =
(c)
DS/14
a
provided r < 1
1- r
(
) (
If f ( z) = z 4 - 2 ( a + b) z 3 + 2 ( a + b) z 2 - 2 ( a + b) a 2 + b2 z + a2 + b 2
2
)
2
where a and b are constants
(i)
write f ( z ) as the product of 2 quadratic factors. (Just guessing the factors must be
accompanied by a check to show that your guess is correct).
(ii)
Hence find (in terms of a and b) the solutions of the equation f ( z ) = 0 .
Page 5 of 6
QUESTION FIVE (20 marks)
(a)
x 2 y2
The ellipse in this question refers to the standard ellipse with the equation 2 + 2 = 1.
a b
(i)
By differentiating the equation of the ellipse find the gradient of the tangent to the ellipse
at ( x1, y1 ) and hence show that the equation of the tangent to the ellipse at ( x1, y1 ) is
xx1 yy1
+
= 1.
a2 b2
P is the point ( acosq , bsinq ) on the ellipse and Q is the point
æ
-a 3 sinq
,
ç 4 2
è a sin q + b 4 cos 2 q
ö
÷.
a 4 sin 2 q + b 4 cos 2 q ø
b 3 cosq
(ii)
Show that Q is also on the ellipse.
(iii)
Show that the equation of the tangent at P is : aysinq +bx cosq = ab .
Also show that the equation of the tangent at Q is:
bycosq - ax sinq = a 4 sin 2 q + b 4 cos2 q .
Hence (or otherwise) show that these tangents are perpendicular to each other.
(iv)
By squaring the equations in (iii) and then adding them show that these tangents intersect
at a point on the circle x 2 + y2 = a2 + b 2 for all values of q .
(b)
(i)
Prove that y =
( x - a) (b - x )
is the equation of a semicircle. State the coordinates of its
centre and the length of its radius.
(ii)
Prove that y =
( x - a) ( x - b)
is the equation of only one half of a rectangular hyperbola.
State the equations of its asymptotes.
(Note: There is no need to repeat any work already done in (i)).
DS/14
Page 6 of 6