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# West Bengal Institute of Technology (WBIT) 2010-2nd Sem B.C.A Computer Application - BM201 Mathematics ( ) - Question Paper

Wednesday, 17 July 2013 10:20Web

Name:

..................................................

Invigilator's Signature:.................................

CS/BCA/SEM-2/BM-201/2010

2010 MATHEMATICS

. . *

Time Allotted : 3 Hours    Full Marks: 70

The figures in the margin indicate full marks.

Candidates are required to give their answers in their own words

as far as practicable.

GROUP -A (Multiple Choice Type Questions )

1. Choose the correct alternatives for any ten of the following:    10x1 = 10

1) The basis of a vector space contains

a)    linearly independent set of vectors

b)    linearly dependent set of vectors

c)    scalars only

d)    none of these.

ii) Thie solution of =r=0 is

dx

' '    v    x        (

a) y-e*    b) y = 0

c) y = sin x    d) y = logex.

111) UJ (3, 1 ) = jc( 1,2 ) + y ( 0, 3 ) then the values of x and y are respectively

a) (3, - 5)    : b) (3, 1 )

c) (3,-5/3)    d) (3, -5/2 ).

. iv) lim ( 3n + 1 ) / ( 2n - 3 ) is

n -*oo

a) o    b) I

. 2 - - . 2 ...

c) 1    d) -i

3'

v)    The value of (l/D2)3) is

a) x5    b) J_

20

c) 20    d) J-x5

20

vi)    2)l/np is divergent if

a) P A !    b) p > 1

c)    p< 1    d) p = 1.

vii)    If P = { 2, 4, 6, 7, 8, 9 }, 0 = { 1, 2, 6, 9 }, then P - 0 is

a)    {4, 7, 8}

b)    {4, 6, 8, 9}    '

O    U}

d)    { 2, 4, 6, 7, 8, 9}.

a) -e2x    b) xe2*

c) -xe2x    d) -xe3x.    ,

Ix) Integrating factor of + y x is

dx

a) e~x    b) ex

c) x2    d) none of these.

x)    The differential equation + ay* - x is

a)    linear of degree 2

b)    non-linear of order one and degree 4

c)    non-linear of order one and degree 2

d)    none of these.

xi)    If vectors ( a, 0, 1 ), ( 0, 1, 0 ), ( 1, a, 1 ) of a vector space 1R3 over JR be linearly dependent, then the value of a is

a)    2 ..

b)    3    y

c)    1

d)    none of these.

CS/BCA/ SEM-2 / BM-201 /2010

xii) Auxiliary equation of the differential equation dr

d 2y .

- + 4y-sinx is

a)    y = cos 2x + sin 2x

' ' ' . ' . '

b)    ymct cos 2x + c2 sin 2x

c)    y-Cj cos x +sin 2x

-        V    '            *

d)    noneofthese.

xiii} The general solution.of log = x-y is

d*

a) ey-x-c    b) e*+ey-c

c) ex*y = c    d) e*~y - c.

xiv) If S and T be two subspaces of a vector space V, then which of the following is also a subspace of V ?

a) sur    b) S-T

c) t-s    d) snr.

GROUP -B (Short Answer Type Questions )

Answer any three of the folloying. 3x5=15

2.    Show that the sequence |2 + (-.1)" 1/nj is convergent.

3.    Solve : 5*--2 + y-x2+e3jt

dx2 dx

CS/BCA/SEM*2/BM-2Q1 /2010

4.    Find the value of x for which the vectors ( 1, 2, 1 ). ( x, 3, 1 ) and ( 2, x, 0) become linearly independent.

5.    Find the value of the limit ton (4n3 + 6n -7)/(n3 - 2n2 +l).

- ' - . , f

6.    Find a basis and the dimension of S n T, where S and T are

subspaces of J?3 defined by

S = {( x, y, z ) G J?3 : 2x + y + 3z = 0}

and T = {{x,y,z)ER3:x + 2y + z = 0}

GROUP-C (Long Answer Type Questions )

Answer any three of the following. 3 x 15 = 45

7. a) Show that

is

/n" + l Vn2 + 2 Vn2 + n

'    + -T"r.~ -I ' + ... +

.2

convergent and converges to 1.

b) Show that the sequence V2,V2 +t/2,-2+-y/2 +>/,... converges to 2.    8 + 7

8. Solve the following equations :    3x5

f .

a)    [C?-2D+i)ymx&inx

b)     12 logx dx2 x ' dx P

c) 3& + 2-JL--4 dx x + 1 y

9.    a) Prove that a subset S of a vector space V over I? is a

subspace if and only if ax + py e S for all a, p 4 R and x, y e S.

b)    Prove that the vectors { ( 1. 2, 2 ), ( 2, 1, 2 ), ( 2, 2, 1 )} are linearly independent in R3.

c)    Find the basis and the dimension of the subspace W of R3 where    -

W =:.{ ( x, y, z ) e JR3: x + y + z *0}    5 + 5 + 5

10.    a) Solve ( px - y ) ( py + x ) = a2p, by using the

substitution x2-u, y2 u; where p-.

. d*

b) Obtain the general solution and singular solution of the equation y-px+-y/o2p2+b2 .    7 + 8

CS / BGA/SEM-2 / BM-201 /2010

c) Find the matrix of the linear transformation T oil. V3 (R ) defined as

T ( a, b, c ) = (.2b + c, a 4b, 3a ) with respect to the ordered basis B where    '

B = {( 1. 1. 1 ), ( 1,1,0), (1,0,0)}.    3 + 6 + 6

12. a) Prove that the sequence {an} is monotonically increasing and bounded when    ,

an - (3n +1)/(n + 2)

b)    State D Alemberts Ratio Test.

c)    If a, 0, y form a basis of a vector space V, then provd that a + y, 2a + 3{J 4y and a + 2p + 3y also form a basis of the vector space V.        8 + 2 + 5

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