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# West Bengal Institute of Technology (WBIT) 2009 B.C.A Computer Application BM101 Mathematics ( ) - Question Paper

Wednesday, 17 July 2013 10:55Web

Name:

Roll No. :.............................................................

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

CS/BCA/SEM-l/BM-101/2009-10

2009 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 :

10 x 1 = 10

i) The value of x,l2 ~2 ls

a) 1    b) 4

c) 0    d) 2.

2

ii)    The value of J dx    is equal to

1

a) 1    b) 2

c) 3    d) 0.

iii)    If x = - 1 Is a root of the equation x2 - x - k = 0, then the value of k is

a) 1    b) 0

c) <2    d) 2.

i

CS/BCA/SEM-1/BM-101/2009-10

tv) If a, fJ, y be the roots of the equation

x3 - 3x2 + 6x - 2 = 0, then a + (S + y is

a) 2    b) 1

c) 3    d) none of these.

v)    If A { 1, 2, 3, 4 } and B = { 2, 4, 6 }, then A A B is

a) { 1, 2 }    b) { 1, 2, 3, 6 }

c) { 1, 3, 6 }    d) { 6 } .

vi)    What is the order of the matrix B, if [ 3 4 2 ]

B = (2 10 3 6 9]?

a) 1 x 5    b) 1 x 3

c) 3 x 5    d) 5 x 3.

vii)    The degree of the polynomial

/(x) = ( x2 + x-2 ) / ( *-1 ) is

a) 0    b) 1

c) 2    d) 3.

d

viii)    If y = log x2, the value of is

i 2    w 2

3)    w - P*

c) I    d) 2x.

2 5 L 0

0

t

3

1 3 1 J

ix) The value of t for which the matrix

is

singular, is . 3

a) " o

b)

d) - 2.

i 3

c) 2

l0 ( 1 + x)1/x is equal to

Mm

x)

a) 1

C) oo    d) 0.

xi)    If a, p, y be the roots of the equation

x3 - 3x2 + 6x - 2 = 0, then Zap is

a) 3    b) 6

c) 2    d) none of these.

xii)    If A { 1, 2, 3 } and B = { 2, 3, 6 }, then A (J B is

a) { 1, 2, 3 }    b) { 2, 3 }

c) { 1, 2, 3, 6 }    d) none of these.

b)

GROUP -B ( Short Answer Type Questions )

3 x 5 = 15

n/2

!;

sin x

dx.

2. Evaluate the Integral

3.    If u = log r and r2 = x2 + y2 + z2 , prove that

T d2u d 2u d 2u 1

r2L dx2 + dy2 + dz2 J * L

4.    In a survey of 320 persons, number of persons taking tea is 210, tairing milk is 100 and coffee is 70. Number of persons who take tea and milk is 50, milk and coffee is 30, tea and coffee is 50. The number of persons taking all three together is 20. Find the number of people who take neither tea nor coffee nor milk.

- 3 4 1

as a sum of a symmetric and a

Express

2 3 0

1 4 5 _

skew-symmetric matrix.

6. If a, P, y he the roots of the equation x3 + 2x2 + 3x + 4 = 0, then find the equation whose roots are

1_

Y '

3 x 15 = 45 sin 6

is

7. a)

GROUP -C (Long Answer Type Questions)

Answer any three of the following

cos 6

Verify whether the matrix

orthogonal.

b) Solve the following system of linear equations by using Cramer's Rule :

2x + 5y + 3z = 9

3x + y + 2z = 3

x + 2y - z = 6

1 2 3 L 2 3 1 J

and B =

c) If A =

cos 0 - sin 0 . sin 0 sin 0 . cos 20

d) Show that

sin 0 ( sin 0 + cos 0 )

11

2    - 2 , find AB.

3    3 . cos 0 - sin 0

. sin 0 sin 0 - sin 0 ( sin 0 + cos 0 ) 0

2 + 5 + 4 + 4

8. a) Evaluate any two :

tan 2x - x

lim

i)

U)

x -+ o 3x- sin x

lim *logVl +X x -* o sjn 2 x

m Um 1 - cos (x- a )

X-* a ( X- a)z */2

b) Evaluate I x2 sin x dx.

o

with respect to x'

c) Dififerentiate

(l+x1)

9.    a) If A ** { a, b, c, d, e}, B = { c, a, e, g} and

C = { b, e, f, g }, then show that

(a ub) nc = (A no u(b nc).

, then find A 2 and show that

i -1 i

b)    If A = 2-10

_ 1 o 0 J A2 = A~ 1.

c)    Find the maxima and minima of x3 - 6x2 + 9x- 8.

5 + 5 + 5

10.    a) Determine whether the function

fix,y) = 2,; 2 if (x> y) * ( 0, 0 )

x t y

= 0    if (x. y) = ( 0, 0 )

is continuous at the origin.

b)    Apply Descartes rule of signs to find the nature of roots of the equation

x4 + 2x2 + x-12 = 0

c)    State Cauchy's mean value theorem.    5 + 5 + 5

11.    a) Find the value of a and b for which the system of

equations

x + 2y + z = 1 2x + y + 3z = b

x + ay + 3z = b + 1

. ' i has (i) unique solution, (ii) many solutions.

11205

b) Solve the following system of equations by matrix inversion method

x + y + z = 6 x - 2y + z = 0 2x - y + z = 3

r 2 -4 6

c) Find out the rank of the matrix

2 3-1 L 3 1 2 _

5 + 5 + 5

12. a) If u = cos 1 { ( x + y ) / + Vy~ } , then show that

du

du

+ dtj +2 cotu = 0

b)    If PSQ be a focal chord of a conic with focus S and semi latus rectum L, then prove that

1/SP+ 2/SQ = 2/L.

c)    Find the point on the conic 6/r =1+4 cos 0 whose vertical angle is n/3.    8 + 4 + 3

11205    7

+ 5 + 5