<p>Name:______</p><p>Teacher:______</p><p>MOUNT ALBERT GRAMMAR SCHOOL</p><p>Mathematics with Calculus</p><p>Scholarship</p><p>TERM 3 2014</p><p>QUESTION BOOKLET</p><p>100 marks Time Allowed: Three hours</p><p>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.</p><p>Achievement Criteria</p><p>Overall Performance:</p><p>GOOD LUCK!</p><p>DS/14 Page 1 of 7 DS/14 Page 2 of 7 QUESTION ONE (20 marks)</p><p>(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</p><p> axes. If the rectangle has a perimeter p(q), prove that the maximum value of p(q) = 4 a2 +b2 .</p><p>(Include a test to show that p(q) is maximum and not minimum).</p><p>(b) V is the volume of a cylindrical steel rod radius r and length l (i.e. V = pr 2l ) at a temperature of</p><p> dr q°C . As it is heated the rod expands with its radius increasing at a rate given by = kr and its dq</p><p> dl length increasing at a rate given by = kl where k is a constant (called the coefficient of linear dq</p><p> dV expansion of steel). Prove that the volume of the rod increases at a rate given by = 3kV . dq</p><p> 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.</p><p>(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.</p><p>(iii) Find f ¢¢(x) and hence show that one of the stationary points is a maximum point and the other is a minimum point.</p><p>(iv) In this question the maximum value of f (x) < its minimum value. Show how this can be </p><p> by sketching a possible graph of y = f (x) showing its main features.</p><p>DS/14 Page 3 of 7 QUESTION TWO (20 marks)</p><p> 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).</p><p>(ii) Find the area bounded by the curve x + y = a and the axes.</p><p> d 2 y a (iii) Show that = and hence comment on the concavity of the curve (giving reasons dx2 2x x for your answer).</p><p>(b) In the diagram the shaded area is rotated around the x-axis to make a belt (used for driving a pulley).</p><p>The shaded area is enclosed by the parabola y = x2</p><p> 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</p><p> y = f (x) around the x-axis between x = b and</p><p> b x = a (where b > a) is given by p y2dx. òa -a a</p><p> 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><p> p (ii) By adding the integrals in (i) show that each one is equal to . 12</p><p>DS/14 Page 4 of 7 QUESTION THREE (20 marks)</p><p>(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</p><p> sin2q (ii) = 2 sinq</p><p> sin2q (iii) lim q®0 sinq</p><p>(b) Note: Do not use a calculator in this question.</p><p>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> 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 ( )</p><p>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 ).</p><p>(c) Note: Do not use a calculator in this question. </p><p> p 2 p 2 + 2 (i) Use the result cos = to help you prove that cos = . 4 2 8 2</p><p> p 2 + 2+ 2 (ii) Use the result in (i) to help you prove that cos = . 16 2</p><p> p (iii) Use the result in (i) and (ii) to write down the exact value of cos . 32</p><p>DS/14 Page 5 of 7 QUESTION FOUR (20 marks)</p><p>(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 )</p><p>(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</p><p>(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 ¥</p><p> and argument of S¥ . a Note: a + ar + ar2 + ar 3 +..... = provided r <1 1- r</p><p>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</p><p>(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).</p><p>(ii) Hence find (in terms of a and b) the solutions of the equation f (z) = 0.</p><p>DS/14 Page 6 of 7 QUESTION FIVE (20 marks)</p><p> x2 y2 (a) The ellipse in this question refers to the standard ellipse with the equation + =1. a2 b2</p><p>(i) By differentiating the equation of the ellipse find the gradient of the tangent to the ellipse </p><p> at (x1, y1) and hence show that the equation of the tangent to the ellipse at (x1, y1) is </p><p> xx yy 1 + 1 =1. a2 b2</p><p>P is the point (acosq, bsinq) on the ellipse and Q is the point </p><p>æ -a3 sinq b3 cosq ö ç , ÷. è a4 sin2 q + b4 cos2 q a4 sin2 q +b4 cos2 q ø</p><p>(ii) Show that Q is also on the ellipse.</p><p>(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: </p><p> bycosq - axsinq = a4 sin2 q +b4 cos2 q . Hence (or otherwise) show that these tangents are perpendicular to each other.</p><p>(iv) By squaring the equations in (iii) and then adding them show that these tangents intersect </p><p> at a point on the circle x2 + y2 = a2 + b2 for all values of q. </p><p>(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.</p><p>(ii) Prove that y = (x - a)(x -b) is the equation of only one half of a rectangular hyperbola. </p><p>State the equations of its asymptotes. (Note: There is no need to repeat any work already done in (i)).</p><p>DS/14 Page 7 of 7</p>