Instructions for Section B

Transcription

1 MATHMETH EXAM 2 SECTION B Answer all questions in the spaces provided. Instructions for Section B In all questions where a numerical answer is required, an eact value must be given unless otherwise specified. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Question 1 (11 marks) Let f :[0, 8π] R, f( ) = 2 cos 2 + π. a. Find the period and range of f. b. State the rule for the derivative function f. 1 mark c. Find the equation of the tangent to the graph of f at = π. 1 mark SECTION B Question 1 continued TURN OVER

2 2016 MATHMETH EXAM 2 12 d. Find the equations of the tangents to the graph of f :[0, 8π] R, f( ) = 2 cos 2 + π that have a gradient of 1. e. The rule of f can be obtained from the rule of f under a transformation T, such that T : R R, T y = 0 a y + π b Find the value of a and the value of b. 3 marks f. Find the values of, 0 8π, such that f () = 2 f () + π. SECTION B continued

3 2013 MATHMETH (CAS) EXAM 2 12 SECTION 2 Answer all questions in the spaces provided. Instructions for Section 2 In all questions where a numerical answer is required, an eact value must be given unless otherwise specified. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Question 1 (1) Trigg the gardener is working in a temperature-controlled greenhouse. During a particular 24-hour time interval, the temperature (T C) is given by T(t) = cos πt, 0 t 24, where t is the 8 time in hours from the beginning of the 24-hour time interval. a. State the maimum temperature in the greenhouse and the values of t when this occurs. b. State the period of the function T. 1 mark c. Find the smallest value of t for which T = 26. SECTION 2 Question 1 continued

4 MATHMETH (CAS) EXAM 2 d. For how many hours during the 24-hour time interval is T 26? SECTION 2 Question 1 continued TURN OVER

5 2013 MATHMETH (CAS) EXAM 2 14 Trigg is designing a garden that is to be built on flat ground. In his initial plans, he draws the graph of y = sin() for 0 2π and decides that the garden beds will have the shape of the shaded regions shown in the diagram below. He includes a garden path, which is shown as line segment PC. The line through points P 2 π, 3 y 3 2 and C(c, 0) is a tangent to the graph of y = sin() at point P. 1 P 2π 3, 3 2 O X C(c, 0) 2π 1 e. i. Find dy d when = 2 π. 3 1 mark ii. Show that the value of c is π. 1 mark 3 SECTION 2 Question 1 continued

6 MATHMETH (CAS) EXAM 2 In further planning for the garden, Trigg uses a transformation of the plane defined as a dilation of factor k from the -ais and a dilation of factor m from the y-ais, where k and m are positive real numbers. f. Let X, P and C be the image, under this transformation, of the points X, P and C respectively. i. Find the values of k and m if X P = 10 and X C = 30. ii. Find the coordinates of the point P. 1 mark SECTION 2 continued TURN OVER

7 2010 MATHMETH(CAS) EXAM 2 16 Question 3 An ancient civilisation buried its kings and queens in tombs in the shape of a square-based pyramid, WABCD. The kings and queens were each buried in a pyramid with WA = WB = WC = WD = 10 m. Each of the isosceles triangle faces is congruent to each of the other triangular faces. The base angle of each of these triangles is, where < <. 4 2 Pyramid WABCD and a face of the pyramid, WAB, are shown here. W W A Z D Y B A Z B C Z is the midpoint of AB. a. i. Find AB in terms of. ii. Find WZ in terms of = b. Show that the total surface area (including the base), S m 2, of the pyramid, WABCD, is given by S = 400(cos 2 () + cos () sin ()). SECTION 2 Question 3 continued

8 MATHMETH(CAS) EXAM 2 c. Find WY, the height of the pyramid WABCD, in terms of. d. The volume of any pyramid is given by the formula Volume = 1 3 area of base vertical height. Show that the volume, T m 3, of the pyramid WABCD is cos 4 6 2cos. 1 mark Queen Hepzabah s pyramid was designed so that it had the maimum possible volume. e. Find dt and hence find the eact volume of Queen Hepzabah s pyramid and the corresponding value of. d 4 marks SECTION 2 Question 3 continued TURN OVER

9 2010 MATHMETH(CAS) EXAM 2 18 Queen Hepzabah s daughter, Queen Jepzibah, was also buried in a pyramid. It also had WA = WB = WC = WD = 10 m. The volume of Jepzibah s pyramid is eactly one half of the volume of Queen Hepzabah s pyramid. The volume of Queen Jepzibah s pyramid is also given by the formula for T obtained in part d. f. Find the possible values of, for Jepzibah s pyramid, correct to two decimal places. Total 13 marks SECTION 2 continued

10 2008 MATHMETH(CAS) EXAM 2 22 Question 4 The graph of f : ( π, π) (π, 3π) R, f () = tan is shown below. 2 y O 2 3 a. i. Find f ' π. 2 ii. Find the equation of the normal to the graph of y = f () at the point where = π 2. iii. Sketch the graph of this normal on the aes above. Give the eact ais intercepts = 6 marks SECTION 2 Question 4 continued

11 MATHMETH(CAS) EXAM 2 π b. Find the eact values of ( π, π) ( π, 3 π) such that f '() = f '. 2 Let g() = f ( a). c. Find the eact value of a ( 1, 1) such that g(1) = 1. Let h :( π, π) ( π, 3 π) R, h() = sin tan d. i. Find h' (). ii. Solve the equation h' () = 0 for ( π, π) ( π, 3 π). (Give eact values.) = 3 marks SECTION 2 Question 4 continued TURN OVER

12 2008 MATHMETH(CAS) EXAM 2 24 e. Sketch the graph of y = h() on the aes below. Give the eact coordinates of any stationary points. Label each asymptote with its equation. Give the eact value of the y-intercept. y O Total 15 marks END OF QUESTION AND ANSWER BOOK