7/27/2019 Term I (2011-2012) Unit 2 - Module 2

http://slidepdf.com/reader/full/term-i-2011-2012-unit-2-module-2 1/3

COUVA EAST SECONDARY SCHOOL DECEMBER 2011

UPPER SIX: PURE MATHEMATICS (UNIT 2) : 1Time 1 hours2

MODULE 2 Internal Assessment

GENERAL INSTRUCTIONS

Answer ALL Questions

Unless otherwise stated in the question, any numerical answer that is not exact MUST be written

correct to three significant figures.

The number of marks is given in brackets [ ] at the end of each question or part question.

The total number of marks for this paper is 60.

You are reminded of the need for clear presentation in your answers.

1. It is given that1

f ( ) tan (1 ). x x−

= +

(a) Find f (0) and f (0),′ and show that1

f (0) .2

′′ = − [5]

(b) Hence, find the Maclaurin series for  f ( ) x up to and including the term in 2. x [2]

2. (a) Draw a sketch to illustrate the working of the Newton-Raphson method when dealing with thesolution of an equation f ( ) 0, x = showing the starting position and the first two iterations. [2]

(b) Draw a sketch to illustrate a case where the Newton-Raphson method will fail, showing clearly

the starting position and the relevant iteration(s). [1]

(c) Use the Newton-Raphson method to find the root of the equation

 3 2 x x− =

which lies between 1 and 2. Give your answer correct to 3 decimal places. [4]

3. Prove by induction that for  ,n N ∈

 2 2

2 2 2

1

2 1 .( 1) ( 1)

n

r 

r  n

r r n=

− =+ +∑ [9]

4. (a) Expand13(1 ) x+ in ascending powers of  x, up to and including the term in 2

. x [2]

(b) (i) Hence, or otherwise, expand1

3(8 16 ) x+ in ascending powers of  x, up to and including the

term in 2. x [5]

7/27/2019 Term I (2011-2012) Unit 2 - Module 2

http://slidepdf.com/reader/full/term-i-2011-2012-unit-2-module-2 2/3

(ii) Find the set of values of  x for which the expansion in part (b) (i) is valid. [2]

5. (a) For each of the following series, the sum of the first n terms is given. In each case, state whether 

the series is convergent or not, and find the sum of infinity when it is convergent.

(i)

1

1 1.

3 1

n

r 

r 

un

=

= ++∑ [1]

(ii)

1

( 1)1.

2 2

n

r 

r 

n nu

n=

+= +

+∑ [1]

(iii)

2

2

1

3 1.

4 2

n

r 

r 

nu

n=

+=

+∑ [2]

(b) (i) Show that 2 2 2 2

2 11 1.

( 1) ( 1)

r 

r r r r  

+− ≡

+ +[2]

(ii) Hence find an expression, in terms of n, for  2 2

1

2 1 .( 1)

n

r 

r 

r r =

++∑ [4]

(iii) Find 2 2

2

2 1.

( 1)r 

r 

r r =

∞

+

+∑ [3]

6. (a) An arithmetic progression has first term 8.− The 20th term is three times the 10 th term. Find the

common difference. [3]

(b) Another arithmetic progression has common difference 2. The sum of the first 20 terms is three

times the sum of the first 10 terms. Find the first term. [3]

(c) A geometric progression is such that its 20th term is three times its 10th term. The first term is not

zero, and the common ratio is positive. Find the common ratio, giving your answer to 3

significant figures. [4]

(d) Another geometric progression has non-zero first term and common ratio r , where 0r  > and

1.r  ≠ The sum of the first 20 terms is three times the sum of the first 10 terms. Show that

 2 3 2 0,u u− + =

where10.u r = Hence find the value of r . [5]

2

7/27/2019 Term I (2011-2012) Unit 2 - Module 2

http://slidepdf.com/reader/full/term-i-2011-2012-unit-2-module-2 3/3

END OF TEST

3