Download VTU BE 2020 Jan CSE Question Paper 15 Scheme 4th Sem MATDIP401 Advanced Mathematics II

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MATDIP401

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Fourth Semester B.E. Degree Examination, Dec.2019/Jan.2020
Advanced Mathematics - II
Time: 3 hrs. Max. Marks:100
Note: Answer any FIVE full questions.

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If [I, ,1n, , n
i
l and [1
2
, m

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2
n 2 ] be the direction cosines of two lines subtending an angle 0
between them then prove that cos 0 = 1
3
l + m, m

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2
+n
i
n,.

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(06 Marks)
Find the angle between two lines whose direction cosines satisfy the relations 1+ m + n = 0
and 21m + 2n1 ? mn =0

(07 Marks)

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Find the co-ordinates of the foot of the perpendicular from A(1, 1,1) to the line joining
B(1, 4, 6) and C(5, 4, 4) .

(07 Marks)
Find the equation of the plane which bisects the line joining (3, 0, 5) and (1, 2, ?1) at right

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angles. (06 Marks)
Show that the points (2, 2, 0), (4, 5, 1), (3, 9, 4) and (0, ?1, ?1) are coplanar. Find the
equation of the plane containing them. (07 Marks)
Find the shortest distance and the equations of the line of shortest distance between the lines:
x

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-
6 y ? 7 z ? 4 x y +9 z ?2
and = = (07 Marks)
3 =-1 ?3 2 4
3 a. Show that the position vectors of the vertices of a triangle a = 4i + 5j + 6k , b = 5 1+ 6J + 4k

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and c = 61+ 41+ 5k form an isosceles triangle.
b. Prove that the points with position vectors 4i +53+ j+ 31+ 93+ 41( and
(06 Marks)
?i+ Sj+4k
are coplanar. (07 Marks)

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c. A particle moves along the curve x
2

2t , y = t ^ ? 4t and z = 3t 5 where t is the time t.
Find the components of velocity and acceleration in the direction of the vector i ? 3 j + 2k at

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t 1. (07 Marks)
4 a.
Find the angle between the surfaces x
2
+ y

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2
+ z
2
= 9 , x
2

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+ y
2
? = 3 at (2,-1, 2). (06 Marks)
b.
Find the directional derivatives of the function (I) = x yz + 4xz

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2
at (1,-2,-1) along 2i ? j? 2k
(07 Marks)
c.
Find div F and curl F at the point (1,-1, 1) where F = NIxy

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3
z
2
). (07 Marks)
---)

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5 a. If r = xi + yj+ zk and r= then prove that, r
(i)
V(ril)= nrn
-2
r V.(rn. r + 3)rn (06 Marks)

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b.
Show that F = 2xy
2
+ yz)i + (2x
2

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y+ xz + 2yz
2
)1+ (2 y
2
z + xy)fc is irrotational and hence

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find a scalar function (i) such that F = Vcb .
(07 Marks)
c. Find the value of the constant 'a' such that A = y(ax
2
+ z) + x(y )3 + 2xy(z ? xy) k is

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Solenoidal. For this value of 'a' show that curl A is also solenoidal.
(07 Marks)
1 of 2
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+? _
g
71
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csi

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0
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CO

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ci
USN

MATDIP401

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Fourth Semester B.E. Degree Examination, Dec.2019/Jan.2020
Advanced Mathematics - II
Time: 3 hrs. Max. Marks:100
Note: Answer any FIVE full questions.
If [I, ,1n, , n

--- Content provided by FirstRanker.com ---

i
l and [1
2
, m
2

--- Content provided by FirstRanker.com ---

n 2 ] be the direction cosines of two lines subtending an angle 0
between them then prove that cos 0 = 1
3
l + m, m
2

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+n
i
n,.

(06 Marks)

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Find the angle between two lines whose direction cosines satisfy the relations 1+ m + n = 0
and 21m + 2n1 ? mn =0

(07 Marks)
Find the co-ordinates of the foot of the perpendicular from A(1, 1,1) to the line joining

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B(1, 4, 6) and C(5, 4, 4) .

(07 Marks)
Find the equation of the plane which bisects the line joining (3, 0, 5) and (1, 2, ?1) at right
angles. (06 Marks)

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Show that the points (2, 2, 0), (4, 5, 1), (3, 9, 4) and (0, ?1, ?1) are coplanar. Find the
equation of the plane containing them. (07 Marks)
Find the shortest distance and the equations of the line of shortest distance between the lines:
x
-

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6 y ? 7 z ? 4 x y +9 z ?2
and = = (07 Marks)
3 =-1 ?3 2 4
3 a. Show that the position vectors of the vertices of a triangle a = 4i + 5j + 6k , b = 5 1+ 6J + 4k
and c = 61+ 41+ 5k form an isosceles triangle.

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b. Prove that the points with position vectors 4i +53+ j+ 31+ 93+ 41( and
(06 Marks)
?i+ Sj+4k
are coplanar. (07 Marks)
c. A particle moves along the curve x

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2

2t , y = t ^ ? 4t and z = 3t 5 where t is the time t.
Find the components of velocity and acceleration in the direction of the vector i ? 3 j + 2k at
t 1. (07 Marks)

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4 a.
Find the angle between the surfaces x
2
+ y
2

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+ z
2
= 9 , x
2
+ y

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2
? = 3 at (2,-1, 2). (06 Marks)
b.
Find the directional derivatives of the function (I) = x yz + 4xz
2

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at (1,-2,-1) along 2i ? j? 2k
(07 Marks)
c.
Find div F and curl F at the point (1,-1, 1) where F = NIxy
3

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z
2
). (07 Marks)
---)
5 a. If r = xi + yj+ zk and r= then prove that, r

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(i)
V(ril)= nrn
-2
r V.(rn. r + 3)rn (06 Marks)
b.

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Show that F = 2xy
2
+ yz)i + (2x
2
y+ xz + 2yz

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2
)1+ (2 y
2
z + xy)fc is irrotational and hence
find a scalar function (i) such that F = Vcb .

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(07 Marks)
c. Find the value of the constant 'a' such that A = y(ax
2
+ z) + x(y )3 + 2xy(z ? xy) k is
Solenoidal. For this value of 'a' show that curl A is also solenoidal.

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(07 Marks)
1 of 2
s + a
+ b
}.

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MATDIP401
7 a. Find the inverse Laplace transform of ,
s + 5

s

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-
? 6s +13
b. Find 11
-1
{log

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c .
Find
8
a.
Using convolution theorem find the Laplace transform of

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sks
2 -
I
-
a

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-
l
?

b. Solve the differential equation, y"+ 5y' + 6y = 5e

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2
' under the condition y(0) = 2, y'(0) =1
using Laplace transform. (10 Marks)
6
a. Find the Laplace transform of, (i) sin 5t cos 2t (ii) (3t + 2

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2
b. Find the Laplace transform of cos at ? cos bt
t
c. Find the Laplace transform of t
2

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sin at .
1
(06 Marks)
(07 Marks)
(07 Marks)

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(06 Marks)
(07 Marks)
(07 Marks)
(10 Marks)
2 of 2

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