Level 3 Calculus, 2005

Transcription

1 For Supervisor s Level 3 Calculus, Sketch graphs and find equations of conic sections Credits: Three 9.30 am Wednesda 1 November 005 Check that the National Student Number (NSN) on our admission slip is the same as the number at the top of this page. Make sure that ou have a cop of the Formulae and Tables booklet L3-CALCF. You should answer ALL the questions in this booklet. Show ALL working for ALL questions. If ou need more space for an answer, use the page(s) provided at the back of this booklet and clearl number the question. Check that this booklet has pages 15 in the correct order and that none of these pages is blank. YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION. For Achievement Sketch graphs of conic sections. Achievement Criteria Achievement with Merit Solve problems involving conic sections. Achievement with Ecellence Solve more diffi cult conic section problems. Find equations of conic sections from given information. Overall Level of Performance (all criteria within a column are met) New Zealand Qualifi cations Authorit, 005 All rights reserved. No part of this publication ma be reproduced b an means without the prior permission of the New Zealand Qualifi cations Au thor i t.

2 You are advised to spend 0 minutes answering the questions in this booklet. Show ALL working. QUESTION ONE ] - Sketch the graph of 9 3 g + = 1. If ou need to redraw this graph, use page 1 or 13. Label an intercepts and an asmptotes

3 3 QUESTION TWO Sketch the graph of = 0. Label an intercepts and an asmptotes. If ou need to redraw this graph, use page 1 or

4 QUESTION THREE Sketch the graph of the curve defined b = sec t,, = 3 tan t. Label an intercepts and an asmptotes. If ou need to redraw this graph, use page 1 or

5 5 QUESTION FOUR (a) Find the equation of the conic section shown:

6 (b) Find the equation of the conic section described below. A hperbola: centre at (0,0), distance between the vertices is the equation of one of its asmptotes is =

7 7 (c) Find the equation of the conic section shown: 8 8

8 8 QUESTION FIVE A wok has a vertical cross section which is parabolic. It also has a horizontal cross section which is circular. 0 cm 1 cm The wok is 0 cm wide and has a depth of 1 cm. The wok is filled with water to a depth of 8 cm. What is the surface area in cm of the water in the wok?

9 9 QUESTION SIX The Fibonacci Chocolate Compan makes Easter eggs with an elliptical cross section. These eggs are ver difficult to stack, so the compan cuts the bottom and the top off the egg leaving a smmetrical shape as shown. The cut-down egg has a width of cm and a height of 1 cm. The cut surface is 18 cm wide. cm 1 cm 18 cm Calculate the original height in cm of an uncut egg.

10 QUESTION SEVEN Find the equation of the tangent to the curve ] g = 1 at the point (7,3).

11 11 QUESTION EIGHT The sketch shows part of a parabola and two tangents. m Points A and B are where the tangents touch the parabola. The parabola has a width of m at the top of the sketch. Point C is the verte of the parabola. A C B 5. m The height of this parabolic section is 5. m. The distance between points C and D is m. m Calculate the distance in metres between points A and B. D

12 1 If ou have made a mistake and need to redraw a graph, use the appropriate cop printed here and clearl number the question

13 13 If ou have made a mistake and need to redraw a graph, use the appropriate cop printed here and clearl number the question

14 1 Etra paper for continuation of answers if required. Clearl number the question. Question number

15 15 Etra paper for continuation of answers if required. Clearl number the question. Question number

16