Visvesvaraya Technological University (VTU) 2007 B.E MATHS1 - Question Paper

y
- + = (D) logx x c
y y
- =
(iii) The integrating factor of the differential formula cot cos dy y x x
dx
+ = is
(A) cosx (B) - sinx (C) sinx (D) cotx
(iv) The Orthogonal trajectory of xy = c is
(A) x2-y2 = c (B) x2 + y2=c (C) x-y=c2 (D) x2- y=c (1x four = 4)
b Solve three ex tanydx + (1-ex) sec2y dy =0 ( 4)
c Solve two 3
2 3
dy x y
dx x y
+ -
=
+ -
(6)
d obtain the Orthogonal trajectory of the curve y2 = cx3,where c is the parameter . (6)
Q6 a (i)
2
1 1
n
n
- ? ? + ? ?
? ? ? is
(A)Convergent (B) Oscillatory (C) Divergent (D) Conditionally convergent
(ii)If one ( ) lim n
n
n
u l
?8
= then n u ? is convergent for
(A) l <1 (B) l >1 (C) l = one (D) l >-1
(iii) A sequence which is monotonic and bounded is
(A)Absolutely Convergent (B) Oscillatory (C) Divergent (D) Convergent
(iv) The series one 1 one 1
5 nine 13
- + - + - - - - - - - is
(A)Absolutely Convergent (B) Oscillatory
(C) Conditionally convergent (D) convergence (1x four = 4)
b. Determine the nature of the series
3 2
2 5
4 1
n n
? ? +
? ? + ? ?
? (4)
c Show that the series
3/2
1 1
n
n
- ? ? + ? ?
? ?
? is convergent (6)
d. Test the series
2 3
3 five 7
x x x - + - - - - - - - - for
(i)Absolute Convergence (ii ) Conditional convergence. (6)
Q7. a (i) 3 lines are coplanar if
(A) they are concurrent (B) a line is parallel
(C) a line is perpendicular to every of them
(D) they are concurrent and a line is perpendicular to every of them
(ii) The angle ranging from 2 diagonals of a cube is
(A) ? = cos-1(2/3) (B) ? = cos-1(1/3) (C) ? = cos-1(4/3) (D) ? = tan-1(1/3)
(iii) The general formula of the plane is
(A) ax + by =d (B) ax + by- cz = d (C) ax + by + cz + d=0 (D) ax-by = 0
(iv) The angle ranging from the line one three 2
3 two 1
x y z + - +
= =
-
and the plane two three four 0 x y z + + + = is
(A) ( ) one three cos five ? - = (B) ( ) one two s seven in ? - = (C) ( ) one 1 cos two ? - - = (D) ( ) one two sin seven ? - = - (1x four = 4)
b obtain the direction ratios and the direction cosines of the line segment
joining the points P(1,2,-3) and Q(3,0,-4) (4)
c obtain the image of the point (1,2,3) in the line one 3
2 three 1
x y z + -
= =
-
(6)
d . obtain the angle ranging from the lines two 2 15 0 four 29 x y z y z + - + = = + +
and four three 2
4 three 1
x y z + - +
= =
-
(6)
Q.8 a (i) If r xi yj zk
?
= + + , then ?. r
?
is
(A) 0 (B) one (C) two (D) 3
(ii)The value of ? r n is
(A) n(n-1)rn (B) n(n+1)rn-2 (C) n(n+1)rn (D) n(n-1)rn-3
(iii) The directional derivative of xy2 + yz3 at the point (2,-1,1) in the direction of
vector i+2j +2k is
(A) 11/3 (B) 10/3 (C) -11/3 (D) 3/11
(iv) Any motion in which the curl of the velocity vector is zero is stated to be
(A) Rotational (B) Scalar (C) Field (D) Irrotational (1x four = 4)
b obtain the unit tangent vector to the curve r = t2i +2tj-t3k at the points t = ± one (4)
c If f = x 3+y3+z3-3xyz,then obtain ?f and ??f? at the point P(1,-1 ,2) (6)
d If f and g are irrotational vector fields.Show that f x g is a solenoidal vector (6)
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Earning:   Approval pending.