West Bengal Institute of Technology (WBIT) 2008-1st Sem B.Tech Electronics and Communications Engineering Electronics & Comm ( - ) Mathematics - Question Paper
/
CS/B.Tech/SEM-l/M-101/08/(09) 3
ENGINEERING & MANAGEMENT EXAMINATIONS, DECEMBER - 2008
MATHEMATICS
SEMESTER - 1
Time : 3 Hours )
GROUP - A ( Multiple Choice Type Questions)
1. Choose the correct alternatives for any ten of the following :
1) The value of Um * is
x-o 6 tx
[ Full Marks : 70
10 x 1 = 10
1_
2
b)
a) 0
d) none of these.
c) 1
ii) The sequence
is
b) oscillatory d) monotonic decreasing.
a) monotonic increasing c) divergent r
The distance between the two parallel planes x + 2y - z = 4 and 2x + 4y - 3z = 3 is
ill)
M A_ b) 24
a)
V24
11
y[24
d) none of these.
c)
rm
+ 5x + 3
c) 15.sin | + 5x + 3
iv) n 01 derivative of sin ( 5x + 3 ) is
a) 5 n.cos ( 5x + 3 ) b) 5 n.sin
\
d) - sin ( 5x + 3 ).
/
CS/B.Tech/SEM-1 /M-101/08/(09) 4
v) If u ( x, y ) = tan ~ 1 j , then the value of x + y is
a) 0 b) 2 u[x, y)
c) u (x, y) d) none of these.
vi) Iff (x) is continuous in [ a, a + h ] , derivable in (a, a + h) then /( a+ h) -/( a) = hf[ a + 9h), where
a) 6 is any real b) 0 < 0 < 1
c) 0 > 1 d) 0 is an integer.
of J J (x +
vii) The value of J J (x+ y ) dxdy =
i o
a) 2 b) 3
c) 1 d) 0.
viii) The series is convergent if
a) p 1 b) p > 1
c) p < 1 d) p 1.
ix) Value of J x dy where C is the arc cut off from the parabola y 2 = x from the c
of jx
point ( 0, 0 ) to ( 1, - 1 ) is
1 1 - 3 b) 3
c) 0 . d) none of these.
x) sin 2 x dx =
o
/ 7 8 15 b) 15
, 8xt 4
c) 15 d> IS -
xi) If u + v = x, uv = y , then =
a) u-v b) uv
c) u + v d) u/v.
xil) If/( x) = 1 , x * is continuous at x = | then/ ( ) =
a) ~ b) 1
c) - 1 d) 0.
xiii) The value of the constant p, so that the vector function
/ = ( x + 3y ) \ + ( y - 2z ) J + (x + p z) fc is solenoidal, is
a) - 1 b) 2
c) -2 d) 1.
xiv) If c? = 31 - 2 + fc, 1? = 2 - fc , then ( 0? x ) . 01 is equal to
a) 0 b) 1
c) | d) - 1.
Um xv
xv) The limit * -*0 - v5 does not exist.
-*o x + y 2
0 True b) False.
GROUP -B ( Short Answer Type Questions )
Answer any three of the following. 3x5= 15
' n/2
2. Prove that if, / n = J xn sin x dx, then Jfl + n(n-l)Jn_2 = n(rt/2)ri"1 .
3. Test the convergence of the series
T1, (Vn4+l-Vn4-l).
n = 1
4. If/( x) = sin "1 x, 0 < a < b < 1 , use mean value theorem to prove
b - a . _ i , . _ i . b - a , ...... _ < sin 1 b - sin 1 a < . . .
V(1 -a2) V(1 -b2) .
5. Show that lQX = ( - 1 ) n x (log x - 1 - 1/2 - 1/3 - ...... - 1/n )
6. Find the values of a and b such that
0 ( 1 + a cos 0 ) - b sin 0 ,
Um -~3 - = 1.
e -o 0
7. Find the equation of the sphere having the circle x2 + y2 + z2 - 10y + 4z - 8 = 0, x+y + z = 6asa great circle.
GROUP - C ( Long Answer Type Questions )
Answer any three of the following. 3 x 15 = 45
8. a) Using mean value theorem prove that
X 71
<tan-1x<x, 0 < x < n 5
. Q N UU1 A N A | v V A N a
CS/B.Tech/SEM-1 /M-101/08/(09) 7
I
b) If z is a function of x and y and x=r cos 0, y = r sin 0 then prove that
, 3 2z d 2z _ 9 2z 1 dz 192z 3x2 dy2 ~ dr2 + r dr + r2 302 '
If/f h) =/(0) + hf'(O) + 2j f"( Qh ), 0 < 0 < 1, fix) = 1 / ( 1 + x ) and h = 7, find 0. 5
c)
2 o
9. a)
Show that for the function fix, y) = <
x w2
fxy(0,0)=fyx(0,0). 5
b)
State comparison test for convergence of an infinite series. Test the convergence of any one of the following series :
c)
|
D |
6 |
8 |
10 | |
|
1.3.5 T |
4.5.7 |
+ 5.7.9 | ||
|
ID |
1 + |
2 p 2! |
3 p + 3! |
4 P + 4! + |
Find the extreme values, if any, of the following function : /( x, y ) = x3 + y3 - 3axy.
n/2
x/2
J cos 8 x dx.
cos n x dx. Hence evaluate
Obtain the reduction formula for
of If'y
b)
c)
Compute the value of JJ y dxdy where R is the region in the first quadrant
R
X 2 u 2
bounded by the ellipse = 1. 5
n/2 .
Obtain the reduction formula for J sin m x cos n x dx, where m, n are positive
o
integers ( m > 1, n > 1 ) . Hence evaluate
n/2
n/4
I
o
V*g+ iil *!
11. a) If u = sin " 1 *\ J X i + i then verify whether the following indentlfy is ture :
i 2
+ + VdP =
92u _ a2u d2u tanu f 13 tan2u) -
3 + 2xyte + y dP = ~MT [l2+ 12 ) 5
b) Find the angle between the surfaces x3 + y 3 + z3 - 3xyz = 5 and
x2 y + y2 z + z 2 x- 5xyz = 8 at the point ( 1, 0, 1 ). 5
c) Evaluate [ 7* "7* j where ~? = a cos u'l + a sin u + bu fc. 5
12. a) A variable plane passes through a fixed point ( a, b, cj and meets the coordinate
axes at A, B, C. Show that the locus of the point of intersection of the plane
a b c . K
through A, B. C and parallel to the coordinate planes Is - + - + - = 1. &
. x - 1 u - 2 z-3___i
b) Show that the straight lines 2~ = 3 = 4 311(1
x ~ - - - i~ are coplanar.
4 4 5
222
c) Find the length of the perimeter of the asteroid. *3 + x3 = a3 .
Determine also the length of the cycloid x = a ( 6 + sin 6 ), y = a I 1 - cos 6 ).
00
13. a) Discuss the convergence of the series C2 * Is !t absolutely
n - 1
n * 1
convergent ?
b) Find the directional derivative of J ( x, y. z 1 = x2 yz + 4xz 2 at the point ( 1. 2. - 1 ) in the direction of the vector 2*1 - J - 2%. 5
c) Find the moment of inertia of a thin uniform lamina in the form of an ellipse
2 2 -/ e
+ iL = 1 about its major and minor axes respectively. 5
a2 b*
END
11701 ( 13/12 )
+ X* I
|
Attachment: |
| Earning: Approval pending. |
