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# West Bengal Institute of Technology (WBIT) 2010-1st Sem B.Tech Electronics and Communications Engineering Electronics & Comm ( - ) Mathematics - Question Paper

Thursday, 18 July 2013 01:10Web

Name: .....................................................................

RoU.No. :..................................................................

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

CS /B.Tech(NEW) / SEM- l/M-101/2010-11 2010-11 MATHEMATICS-!

Full Marks : 70

Time Allotted : 3 Hours

The figures in the margin indicate fixll 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

If a, ft_are the roots_,of the equation x2-3x + 2 = 0

0 a P

poo

is

then

1 - a a

b)

a)

c) - 6    d)    3.

ii) If y = eax+b,    then ( y5 )Q =

a) aeb    b)    a 5e b

c) ''abeax    d)    ndne of these.

ill) If Rolle's theorem is applied tof(x) = x ( x2 - 1 ) in [ 0, 1 ] , then C =

 a) 1 b) d) c) - V3 Iv) If u + v = x , uv = y , then b) c) u + v d) n/2 sin 7 0 d0 is v) The value of - Jt/2 6.4.2 a) b) 7.5.3.1 c) 0 d) vi) The sequence { ( - 1 ) n [ is a) convergent b) c) divergent d) 0 J_ Vs " ( U, V ) _ {X, y) uv u v ' i 3 i=L L7 2.(6.4.2 ) 7.5.3.1 oscitiatoiy none of these.

vii) If ct = 3* - 2j + fc , = 2* - fc , then ( x ) . "Be is equal to

a) 'i + j + fc

d) 0.

cos 0 sin 0

viii) The matrix

is

- sin 0 cos0

b) skew-symmetric

a) symmetric

c) singular

d) orthogonal.

ix) The value of t for which

= [x + 3y)'i + [y-2z)j + (x + tz ) % is solenoidal is

a) 2

b)

c) 0

d) 1.

x) The distance between two parallel planes x + 2y - z = 4 and 2x + 4y - 2z = 3 is

a)

c)

b)

24

d) none of these.

24

xi)    In the M.V. theorem /( h ) = /( o ) + hfr{ oh } ;

0 < 0 < 1 if/( x) =    and h = 3, then value of 0 is

1

a) 1    b) 3

c)    d) none of these.

<2 ,

xii)    The series I is convergent if

a) p> 1    b) p > 1 c) p < 1    d) p < 1.

GROUP - B ( Short Answer Type Questions )

Answer any three of the following. 3x5= 15

2.    If y = ( x2 - 1 ) n, then show that

( *2 - 1 ) y n + 2 + 2xy n + j - n ( n + 1 ) y n = 0. Hence find y n ( 0 ).

3.    Using M.V.T. prove that

' / x> tan_1x> ; *v~2 - 0 < x < n/2.

X T X    '

1 + a 1    l 1 1 + b 1

1 1    1 + c

1 1 1

1 + d -J

1

1

= abcd[l+i+I+I + I|

Test the nature of the series

1 12

1.2

3.5

. , .    , 3.5.7 J T.......

If ,    are three vectors, then show that

[    tx-g -?x-a] M (o b c]2

If u > tanshow that + H |jj -I sin 2u.

GROUP -C ( Long Answer Type Questions )

Answer any three of the following.

3x15 = 45

8. a)

Determine the conditions under which the system of equations x + y + z = I, x + 2y-z = b, 5x+7y+az = b 2, admits of

i)    only one solution

ii)    no solution

iii) many solutions.

b) Find the eigenvalues ancj the corrsponding eigen-

vecgors of the matrix A =

1 2 1 V 0 1 J

c) Find whether the following series is convergent

22 2 Vi 1 2 " 1

( 4 _ 4 V 34 3

23 2

3) If/ ( x) - x2, g ( x) = x3 on [ 1, 2 J , is Cauchy's mean Value theorem applicable ? If so, find

9.

d0 , show that

cos n8 cos 0

b) If/n =

(n-1) [In + In_2) = 2 sin (n-1)0.

Hence evaluate J ( 4 cos 20 - 3 ) d0.

c) If r = | | , where ~r = xi + yj + ztc , prove that (rn) = nrn_2r.

10. a) Find d( u. v) . where u. = x2 - 2y 2, v-2 x2-y2

a (r, e)

and x = r cos 0 , y = sin 0.

b)    Verify Green's theorem for F = (xy + y2)'i+x2j where the curve C in bounded by y = x and y = x2.

a X %

c)    Evaluate: J J J x3 y2 zdzdy dx.

0 0 0

11. a) Find the maxima and minima of the function x 3 + y 3 3x + 12y + 20. Also find the saddle point.

b) State Cayley- Hamilton theorem and verify the same for

1 2

the matrix A =

-2 - 1

c) Show that Curl Vf =0,

where[ x, y, z) = x2y + 2xy + z2.

xy(x2-y 2)

12. a) Given the function =

x2 + y2 . (x, y)*[ 0,0)

= 0

Find from definition/ ( 0, 0 ) and/ ( 0, 0 ) Is / = f ?

JxyJyx

b) Integrate by Charging the order of integration

a 2a-x

xy dy dx.

xz/a

c) If F (p, v, t) = 0 , show that

 f dp ' du s I dt J X v constant l dP J

dt

t constant X t du I p constant    '

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