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# West Bengal Institute of Technology (WBIT) 2008 B.C.A Computer Application BM101 Mathematics ( ) - Question Paper

Wednesday, 17 July 2013 11:00Web

CS/BCA/SEM- 1/BM-101/08/(09)    3

ENGINEERING & MANAGEMENT EXAMINATIONS, DECEMBER

MATHEMATICS -1

SEMESTER - 1

' I Full Marks : 70

Time : 3 Hours ]

GROUP - A ( Multiple Choice Type Questions)

10 x 1 - 10

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

i)    If A = { 1, 2, 3, 4, 8 }, B = { 2. 4, 6, 7 }, then A A B is

a)    {2, 4}

b)    { 1, 2, 3, 4, 6, 7. 8 }

e) <(

d) {1, 3, 6, 7, 8}.

ii)    Hm ( 1 + x)1/x is equal to

x-0

b)

d)

a) 1

c) 0 d

1

x

log ( 1/x) x log e.

b)

d)

is

b)

d)

x4

2x.

til) dx ( l0g a X ) 1S eqUal t0 a)

c) ( 1 /x) log a e

H 2 u

iv) If y = log x2, the value of \

\

v) The matrix A =

 1 1 <2 V2 1 1 V2 V2

is an

r

a) orthogonal matrix    b)

c) identity matrix    d)

vi) Derivative of x4 with respect to x2    is

a) 4x3    b)

c) 2x    d)

v

idempotent matrix none of these.

2x2

4.

vii)    If the rocts of the equation ax2+bx+c = 0(a*0) are reed and unequal, then its discriminant D satisfies

a)    D > 0 and D = a perfect square

b)    D = 0

c)    D > 0 and D* a perfect square

d)    D < 0.

viii)    If A = { 1, 2, 3}, B = {2, 3, 6}, then A U Bis

a) { 1. 2, 3}    b) { 2, 3 }

c) { 1, 2, 3, 6}    d) none of these.

ix)    If a, p, y be the roots of x3 - 3x2 + 6x - 2 = 0, then X aP is

a) 3    b) 6

c) 2    d) none of these.

x)    If / ( x) = 3 + 2x ; when x > 0    /

= - 3 - 2x ; when x < 0,

then lim f ( x) is

* -* o

a)

c)

3

0

b)

d)

none of these.

xi) If /(x) =

; when 0

= l ; when x = 0, then

a)    /( x) is continuous at x = 0

b)    fix) is continuous, but not differentiable at x = 0

c)    ' / ( x) is discontinuous at x = 0

d)    none of these.

2

xii) The value of J | x |dx is

- l

b) 5

d) none of these.

a) 3 c) 5/2

GROUP -B ( Short Answer Type Questions )

Answer any three of the following.    3x5= 15

If a, p, y be the roots of the equation, x3 + px2 + qx + r = 0, then find the value of la3.

2.

x+ y

- l

3. If u = tan

Ji+yTy

Prove that the set of even integers (including zero ) forms an additive group.

, , , 3u du l-o , then show that x + y 3y = 4 sm

jt/2

r

Vcos x

dx.

5. Evaluate

V sin x + Vcos x

 ' 9 1 f l 5 ' 6. If P = ,4 3 , and 9 = <7 12 ,

null matrix.

, find the matrix R so that 5P + 3 Q + 2 R is a

GROUP -C (Long Answer Type Questions )

Answer any three of the following questions.

State Rolles Theorem.

a)

b)

c)

a)

b)

Differentiate n times the following equation :

( 1 + x2 ) y2 + ( 2x-1 ) y j = 0.

If y = sin ( m sin _ 1 x ) , show that

( 1-x2) yn + 2-(2n+ l)xy+1 + (m2-n2)yn =0. 4 + 5 + 6

* If pth, qth and rth terms of an A.P. are P, Q and R respectively, show that p(Q-R) + q(R-P) + r(P-9) = 0.

Show that the centroid of a triangle with vertices ( x t . y t ) . ( *2 , y 2 ) and

(*3- y3) is

y i + y 2 + y

Xj + X2 + X3

Find the equation of a straight line through the point of intersection of lines 2x - 3y + 4 = 0 and 3x + 4y - 5 = 0 and that is perpendicular to the line 6x - 7y + 8 = 0.    5 + 5 + 5

c)

a)

X    K

Show that cos x>l--2~ if 0 < x < g

y 2 ,, 2

. x2 + 2*0

x + y

b) If f[x, y)

0 ,x2 + y2 = 0.

then show that f ( 0, 0 ) = fyx ( 0, 0 ) .

x2

4 + 6 + 5

dx.

Evaluate

(x2 + a2)(x2 + b2)

Reduce the following equation to its canonical form and determine the nature of the conic represented by it :    .

6x2 - 5xy - 6y 2 + 14x + 5y + 4 = 0.

Find the equation of the ellipse one of whose foci is ( - 1, 1 ), eccentricity is 0-5 and the corresponding directrix is y = x + 3.    9 + 6

b)

11.    a) Solve the equation by Cardans method, 2x3 + 3x2 + 3x + 1 =0.

b)    Let G = {aeR/-i<a<l}. Define a binary operation on G by a b =    V a, b e G. Show that ( G, 0 ) is a group.

c)    Find the nature of the roots x4 + qx2 + rx - s = 0 by Descartes rule of signs (where q, r, s, being positive ).    15

12.    a) If by a trnasformatkm of one rectangular axis to another with same origin the

expression ax+ by changes to a1 x' + bAy1,

2 2

prove that a2 + ba*a,+b')

oo

r dx

J ( x+ 1 ) ( x +

b) Show that I (-.T nr~2l = lo2

0

1 + 2 t + n

c) Use the method of integration to evaluate Lt ;-JTTT-- k > 0.

n -o    n

15

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