Mount Albert Grammar School

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 7 DS/14 Page 2 of 7 QUESTION ONE (20 marks)

(a) The diagram shows that P(acosq, bsinq ) is any y Q x2 y2 point on the first quadrant of the ellipse + =1. a2 b2 x Q, R and S are also points on the ellipse such that R S PQRS is a rectangle with its sides parallel to the

axes. If the rectangle has a perimeter p(q), prove that the maximum value of p(q) = 4 a2 +b2 .

(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 = pr 2l ) at a temperature of

dr q°C . As it is heated the rod expands with its radius increasing at a rate given by = kr and its dq

dl length increasing at a rate given by = kl where k is a constant (called the coefficient of linear dq

dV expansion of steel). Prove that the volume of the rod increases at a rate given by = 3kV . dq

ax + b c (c) In this question f (x) = + where a, b and c are positive non-zero constants. c ax + b (i) 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 3 of 7 QUESTION TWO (20 marks)

dy - y (a) (i) By solving the differential equation = show that x + y = a given that y = 0 dx x at x = a (where a > 0).

(ii) Find the area bounded by the curve x + y = a and the axes.

d 2 y a (iii) Show that = and hence comment on the concavity of the curve (giving reasons dx2 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 = x2

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

b x = a (where b > a) is given by p y2dx. òa -a a

p p p 3 1 3 1 (c) (i) Use the substitution x = - u to show that ò p n dx = ò p n dx. 2 6 cot x +1 6 tan x +1

p (ii) By adding the integrals in (i) show that each one is equal to . 12

DS/14 Page 4 of 7 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. sin2q (i) = 0 sinq

sin2q (ii) = 2 sinq

sin2q (iii) lim q®0 sinq

(b) Note: Do not use a calculator in this question.

æ p ö æ p ö (i) Prove that tanç + A÷´ tanç - A÷ =1 è 4 ø è 4 ø æ p ö æ p ö 2 and that tanç + A÷+ tanç - A÷ = è 4 ø è 4 ø cos2A

p p æ p p ö 2 (ii) If A = and so 2A = = ç - ÷ show that = 2 2 3 -1 . 24 12 è 4 6 ø cos2A ( )

7p 5p (iii) Use the results from (i) and (ii) to show that tan and tan are the solutions of 24 24 7p 5p x2 - 2 2 3 -1 x +1= 0. Hence find the exact values of tan and tan . ( ) 24 24 (Note: a2 + b - 2a b = a - b ).

(c) Note: Do not use a calculator in this question.

p 2 p 2 + 2 (i) Use the result cos = to help you prove that cos = . 4 2 8 2

p 2 + 2+ 2 (ii) Use the result in (i) to help you prove that cos = . 16 2

p (iii) Use the result in (i) and (ii) to write down the exact value of cos . 32

DS/14 Page 5 of 7 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 + z2 + z3 +... 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¥ . p (iii) Use the Cartesian form of cis to help you find S in polar form. Hence find the modulus 3 ¥

and argument of S¥ . a Note: a + ar + ar2 + ar 3 +..... = provided r <1 1- r

2 (c) If f (z) = z4 - 2(a+ b) z3 + 2(a +b)2 z2 - 2(a + b)(a2 +b2 ) z + (a2 +b2 ) 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.

DS/14 Page 6 of 7 QUESTION FIVE (20 marks)

x2 y2 (a) The ellipse in this question refers to the standard ellipse with the equation + =1. a2 b2

(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

xx yy 1 + 1 =1. a2 b2

P is the point (acosq, bsinq) on the ellipse and Q is the point

æ -a3 sinq b3 cosq ö ç , ÷. è a4 sin2 q + b4 cos2 q a4 sin2 q +b4 cos2 q ø

(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 - axsinq = a4 sin2 q +b4 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 x2 + y2 = a2 + b2 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 7 of 7