West Bengal Institute of Technology (WBIT) 2009-1st Sem B.Tech Electronics and Communications Engineering Electronics & Comm ( - ) Mathematics - Question Paper
Name:...........................................................................................................................................
RoU.No.:.............................................................................../
. , -y
Invigilator's Signature:..............................................
CS/B.Tech/SEM- 1/M-101/2009-10 2009 MATHEMATICS
Time Allotted : 3 Hours Pull 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)
j 1. Choose the correct alternatives for any ten of the following :
' 10x1 = 10
V1+V2 + V3 +.....+ Vn
n Vn
is
1) The value of lim n -<
a) 0 b) 1
ii) Which of the following functions obeys Rolle's theorem in [ 0, it J ?
a) x b) sin x
c) cosx d) tanx.
ill) If C is the circle x2 + y 2 = 4, then J x2 dx is
. , < c '
a) 0 b) |
c) 3 d) 1.
iv) The equation x2 + y2 = a2, z = 0 represents
a) circle b) cylinder
c) sphere d) right circular cylinder.
v) If 0 be an angle between the vectors
~a = 6* + 2j + 3fc and "# = 2 - 9j + 6fc, then
\
/ 12 A v77/
0 0 = cos 1 b) 0 = sin 1
' 12
c) 0 = tan" 1 y jj J d) none of these.
vi) If Cauchy's mean value theorem is applicable to the function/( x) x and g ( x) = x2, then the value of C is
a) 3/2 ' b) 0
c) 1/2 d) - 3/2.
d 2 J
vii) If y 2 = 4ax ( a is a real constant), then " dy2 is
, 2a .. 2a
v b) p
2a 2a
3 d) -
y3 y
11701
CS/B.Tech/SEM-1 /M-101/2009-10
viii) The law of mean is given by
a) -/'(c)
b> -/'(<=)
c) mfctial mJlle) dl /(M:/(a) ,/|c,
3 I r 0 )
ix) If x = r cos 0 and y - r sin 0 , then the value of j x1 y ) is
b) r d) - r.
a)
c)
0
I
r
x) The series
is
n
a = 1
|
a) convergent c) oscillatory |
b) divergent d) none of these. |
xsin , xO
xi) The function / ( x) =
_ <
is
, x = 0
0
a) continuous and differentiable at x = 0
b) continuous but not differentiable at x = 0
c) neither continuous nor differentiable at x = 0
d) none of these.
CS/B.Tech/SEM-1 /M-101 /2009-10
jdi) If/( x, y ) = tan ( x / y ), then x | + y is
b) cot ( x / y )
0 tan ( x / y )
d) none of these.
c) 0
xiii) The moment of inertia of a thin uniform rod of mass M and length 2a about an axis perpendcular to the rod at its centre is
Ma2
2
Ma2 4
Ma-
b)
d)
a)
3
c) Ma2
X ~ 1 U 2+1
xiv) The point of intersection of the line = 3 = _ } with the plane x + 2y - z = 5 is
b) ( 0. 1. 3 ) d) none of these.
a) ( 1. 1. 1 )
f C. A \
(5 j ZA 1
3 .'3
c)
it/2
/
xv) The reduction formula of I n =
cos n x dx is
0
* , n- 1 r
i r n - 1 T C) I n~ n I n - 2
n
n n - 1 1 d) none of these.
GROUP -B ( Short Answer Type Questions)
Answer any three of the following. 3x5=15
2. If y = ( x2 - 1 ) n, then show that
(*2- i)vn*2*2*yn+i-.nin+ 1) yn - -
3. is~i, are three vectors, then show the
[ ~a x I>, if x ~t, x ] = [ ~dt, it, ] 2 , where symbols have their usual meanings.
4. Test the convergence of the series
. 92 2"J 42 I1 d2 f?
5. A. B, C and D are points ( a. 3, - 1 ), ( 3, 5. - 3 ), ( 1, 2, 3,) and ( 3, 5, 7 ) respectively. If AB is perpendicular to CD, then find the value of a.
dti du 1 _n , then prove that x-+y~+TcotM ~ 0 ox dy 2
-i x+v 6. If = cos J-
7. Verify Rolle's theorem for the function /(x) = |x|,- 1 <x< 1 .
CS/B.Tech/SEM-1 /M-101 /2009-10
GROUP -C ( Long Answer Type Questions )
Answer any three of the following.
3 x 15 = 45
8. a) Examine continuity and differentiability of
'l '
/ ( x ) at x = 0, when / ( x ) = x sin /(0) = 0.
; ( x * 0 ) and
2xy
b) Show that /(x, _y) =
0 for (x,j>) = (0,0)
is not continuous at ( 0. 0 ) c) Find the extrema of the function
/ ( x, y ) = x3 + 3xy 2 - 3y 2 - 3x2 + 4.
5 + 5 + 5
n/2
9. a) Obtain a reduction formula for
sin nxdxand evaluate
ji/2
sin 5 x dx.
b) If z = f ( x, y ) where x = eu, cos v, y = e u sin v then show that
dz dz
dz
v--hx = e
du dv dy
c) Prove that the function f{x)= | x - 11, 0 < x < 2, is continuous at x = 1, but not differentiable there. Is it continuous and derivable atx=0? 5 + 5 + 5
CS/B.Tech/SEM-1 /M-101/2009-10 State Leibnitz's theorem for Alternating Series and test convergence of the series 1 - + 2 ~ 2 + ...........
Define absolute and conditional convergence of Series. Also show that the series CS is absolutely
b)
n = 1
convergent. 6 + 9
11. a)
A particle moves on the curve x = 212, y = t2 - 4t, z = 3t - 5, where t is the time. Find the components of velocity and acceleration at time t = 1 in the direction 'i - 3j + 2fc.
b)
Find the angles between the lines whose direction cosines are given by the equations I + m + n = 0 and
c)
I2 + m2 - n2 = 0.
Find the shortest distance between the lines
1 "-2" 1 and 7 ~ - 6 = ~T~
/ .
5 + 5 + 5
12. a) Find the n-th derivative of y = ( ax + b ) m , number.
b) Test the convergence of the series
m is any
c) Find :
- 3 xyz ) . 5 + 5 + 5
div and curl where = grad (x3 + y3 + z3
13. a) Find the whole length of the loop of the curve 9y2=(x-2)(x-5)2.
X
2 x -
b) Evaluate J Jsinx+y) dx dy.
0 0
6 + 6 + 3
c) State Greens Theorem.
11701 8
|
Attachment: |
| Earning: Approval pending. |
