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UNIVERSITY OF MORATUWA Name

Department of Mathematics

MA(1012) Mathematics Group .........................................

Mid Semester Examination

Level 1 Semester 1 2009 Registration No:......................................................

Time allowed: 1 hour

Answer all questions in the given spaces

Q(1). If , and are the roots of the equation 03 cbxax then the determinant of

is .

Q(2). Find the value of for which the system of linear equations has unique solution:

2)1(

)1(

1)1(

zyx

zyx

zyx

then =..

Q(3)Find the rank of matrix

835

753

132

A , then rank of A ...

Q(4) Let

123

012

001

A and 321 ,, XXX be three columns of matrix X such that

1

3

2

,

9

3

2

,

0

0

1

321 AXXAAX then matrix X is

Q(5) Find the value of the following determinant

25250222324

1

.....

.......

.

21

.....

22

.....

23

.....22.....01223.......101

24......210

x

..

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Q(6) a) Number of primes between 40 and 100 =

b) Number of twin primes between 40and 100 =

Q(7) Let )x(f = 12 x

Given ,0 find the value so that 3x Min 10},2{ xf

Q(8) Find the solution to .652 xx

Q(9) Let 21

21 xxf

xxg sin

174 35 xxxxh

Find the formula for xhgf

Q(10) fxxx 111 4

Where f is a polynomial in . Find that f .

Q(11). p: x is a Mathematician

q: x is clever.

r: x is rich.

Write )()]([ prrqqp in words.

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Q(12). Define a sequence of sets ,.......,, 210 XXX by 0X and for 0n

}.{1 nnn XXX . Write down 321 ,, XXX .

1X =

2X =

3X =

Q(13). Let )},(),,(),,(),,(),,{( cbbbcabaaaR be a relation on the set }.,,,{ dcba What is

the minimum number of elements which need to be added to R in order that r is

becomes

(i) reflexive (ii) symmetric (iii) transitive

Q(14). Let family of set defined as

}4,|),{( myxandRyxyxAm for .Rm

Find (i) AU Rm .

Q(15). Give examples to disprove the followings

(i) )()( BABA (ii) )()()( CABACBA