NATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE SEMESTER 2 EXAMINATION 2010-2011 MA1505 Mathematics 1 April/May 2011 — Time allowed : 2 hours Matriculation Number: INSTRUCTIONS TO CANDIDATES 1. Write your matriculation number neatly in the space above. 2. Do not insert loose papers into this booklet. This booklet will be collected at the end of the examination. 3. This examination paper contains a total of EIGHT (8) questions and comprises THIRTY- THREE (33) printed pages. 4. Answer ALL 8 questions. The marks for each question are indicated at the beginning of the question. The maximum score is 80 marks. 5. Write your solution in the space below each question. 6. Calculators may be used. However, you should lay out systematically the various steps in your calculations. 7. This is a CLOSED BOOK examination. One A4-sized helpsheet is allowed. For official use only. Do not write in the boxes below. Question 1 2 3 4 5 6 7 8 Marks
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NATIONAL UNIVERSITY OF SINGAPORE FACULTY … university of singapore faculty of science semester 2 examination 2010-2011 ... 4.answer all 8 questions.
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NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 2 EXAMINATION 2010-2011
MA1505 Mathematics 1
April/May 2011 — Time allowed : 2 hours
Matriculation Number:
INSTRUCTIONS TO CANDIDATES
1. Write your matriculation number neatly in the space above.
2. Do not insert loose papers into this booklet. This booklet will be collected at the end of theexamination.
3. This examination paper contains a total of EIGHT (8) questions and comprises THIRTY-THREE (33) printed pages.
4. Answer ALL 8 questions. The marks for each question are indicated at the beginning of thequestion. The maximum score is 80 marks.
5. Write your solution in the space below each question.
6. Calculators may be used. However, you should lay out systematically the various steps inyour calculations.
7. This is a CLOSED BOOK examination. One A4-sized helpsheet is allowed.
For official use only. Do not write in the boxes below.
Question 1 2 3 4 5 6 7 8
Marks
MA1505
Question 1 (a) [5 marks]
The curve C in the xy-plane is given parametrically by the equations
C : x = e2t + 2 , y = e2t − 8t, where t is any real number.
Find all local extreme points P (x, y), if any, on C, and determine whether
these points are local maximum or local minimum.
2
MA1505
(More space for the solution to Question 1 (a).)
3
MA1505
Question 1 (b) [5 marks]
The region R lies in the first quadrant and is bounded by the y-axis and the
curves y = e2x and y = 2ex. If R is revolved about the line y = 4, find the
volume of the solid generated.
(Give the exact volume in terms of π.)
4
MA1505
(More space for the solution to Question 1 (b).)
5
MA1505
Question 2 (a) [5 marks]
Suppose k is a positive constant such that the power series