West Bengal Institute of Technology (WBIT) 2010-1st Sem B.Tech Electronics and Communications Engineering Electronics
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
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 ( U, V ) _ uv u v ' i 3 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
c) d) none of these.
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
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
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 =
