Cape pure math 2007 unit 2 paper 2

Cape pure math 2007 unit 2 paper 2

1. (XK) ‘rest cove 02234020
MAYAUNE 2007

CARIBBEAN EXAMINATIONS COUNCIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS.

UNIT2 ~ PAPER 02

FORM TP 2007250

2 hours

30 MAY 2007 (p.m)

This examination paper consists of THREE sections: Module 1, Module 2 and Module 3.

Each section consists of 2 questions
“The maximum mark foreach section is 40.
“The maximum mark for this examination is 120.
‘This examination consists of 5 pages.

2. Answer ALL questions from the THREE sections

3. Unless otherwise stated in the question, any numerical answer tha is not
‘exact MUST be writen correct 0 thre significant figures.

Examination Materials

Mathematical formulae and tables
Electronic calculator
Graph paper

Copyright © 2006 Caribbean Examinations Council®
All sighs reserved
(02234020/CAPE 2007

2. Section A (Module 1)

Answer BOTH questions.

(a) Solve, for x> 0, he equation 3 logy = 2 08,8 5: LS marks

© 60 Copyandeompleethetable below or values 2and e* using calculator, where
neccesary. Approximate all value to 2 decimal places.

x Juolo | os | 10 | us | 20 | 2s | 30
x 1.00] 141 | 2.00 400 8.00
eo | 2m 037 | 022 005

LS marks}

Gi) Om the same pair of axes and using a scale of 4 em for 1 unit onthe x-axis,

on for 1 unit on the yaxis, draw the graphs of the two curves y = 2% and

yoetfor-1Sx<3,xe R US marks]

Gi Use your graphs to find

a) the value of satisfying 20 [ 2 marks}
D) the range of values of x for which 2e <0. 12 marks}
‘Total 20 marks
2. (@ Showihat for m2 2, tan "x= tant race tan ni [3 marks}
de à
© Find E when y= tan" 13 marks}
© Laso [tant xds.n22
(6) By using the result in (a) above, show that fy + fy (7 marks}
(ii) Hence evaluate a [7 marks)
Total 20 marks
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(02234020/CAPE 2007

3. . @
©
4 ©
o

Section B (Module 2)

Answer BOTH questions.

‘The sequence (u,) is given by u = 1 and.

ine Dune he

Prove by Mathematical Induction that, nt Yn € N 19 marks}

Given thatthe sum of the first terms ofa series, $, is 932%,

find the mah term of S LS mart]
D show that Sis a geometric progression 1 2marks}
find the fit tem and common ratio of S [ 2marksl

Gv) deduce the sum to infinity ofS 1 2marks}
‘Total 20 marks

“The function fis given y x +1. Show tha
(fi) =O as a root ein the interval, 1) LA marks
i) ifs is a fest approximation to off) = Din (0,1) the Newton-Raphson

Sf
Qing xa

LS marks}

method gives a second approximation x in (0, Ds

John’s father gave him a loan of $10 800 to buy a car. The loan was o be repaid by
12 unequal monthly instalments, starting with an initial payment of SP in the first
month. There is no interest charged on the loan, but the instalments increase by $60
per month

Show that P= $70. LS marks}
Gi) Find in terms of n, << 12, an expression forthe remaining debt onthe loan
after John has paid the m-th instalment. L 6 marks]
Total 20 marks

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(02234020/CAPE 2007

4. Section € (Module 3)

Answer BOTH questions.

(2) A bag contains 5 white marbles and 5 black marbles. Six marbles are chosen at
random.

(Determine the number of ways of selecting the six marbles if there are no

restrictions. [2 marks}

(6) Find the probability that the marbles chosen contain more black marbles than
white marbles [A marks}

®) The table below summarises the programme preference of 100 television viewers.

‘Televison | Number of | Number of | un
Preference | Males | Females
Matlock 20 10 30
News 1 is 2
Friends is 20 5
Tol 2 8 100.
Determine the probability that a person selected at random
@ isafemale [2 marks}
(i) isa male or likes watching the News. [4 marks}
(ii) isa female that likes watching Friends [2 marks}
Gv) does not like watching Matlock [2 marks}

(©) The table below lists the probability distribution of the number of accidents per week
on a particular highway.

Number of Accidents
Per Week op! a
Probabiity [025| o [oxo] p [ozo|o15

6) Caleulate the value of. [2 marks]

Gi) Determine the probability that there are more than 3 accidents in a week

[2 marks}

‘Total 20 marks

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(02234020/CAPE 2007

5. (2) A system of equations is given by

x+y+2=10
Br 2y432=38
Rey ma

where cis a real number.

Lt mark}
[A mark
13 mar
U1 mar}

[4 marks}

(Write the system in matrix form.
Gi) Weite down the augmented mati.
Qi) Reduce the augmented matrix to chelon form.
6%) Deduce the value of for which the system is consistent
(©) Find ALL solutions corresponding o this value o
oat
©) Given a=|-1 01
x aa
find
GA where Fis the 3 x 3 Identity matrix and kis a cal number 3 marks]

the values of £ for which | &1-A |

END OF TEST

LT marks}

Total 20 marks

(02234020/CAPE 2007