Download Visvesvaraya Technological University (VTU) BE ( Bachelor of Engineering) CSE 2015 Scheme 2020 January Previous Question Paper 4th Sem MATDIP401 Advanced Mathematics II
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MATDIP401
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
i
l and [1
2
, m
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
2
+n
i
n,.
(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)
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
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
-
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.
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
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)
4 a.
Find the angle between the surfaces x
2
+ y
2
+ z
2
= 9 , x
2
+ y
2
? = 3 at (2,-1, 2). (06 Marks)
b.
Find the directional derivatives of the function (I) = x yz + 4xz
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
3
z
2
). (07 Marks)
---)
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)
b.
Show that F = 2xy
2
+ yz)i + (2x
2
y+ xz + 2yz
2
)1+ (2 y
2
z + xy)fc is irrotational and hence
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
Solenoidal. For this value of 'a' show that curl A is also solenoidal.
(07 Marks)
1 of 2
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g
.
0
-
1)
+?? _
g
71
o
csi
0
CO
z
CO
ci
USN
MATDIP401
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
i
l and [1
2
, m
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
2
+n
i
n,.
(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)
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
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
-
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.
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
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)
4 a.
Find the angle between the surfaces x
2
+ y
2
+ z
2
= 9 , x
2
+ y
2
? = 3 at (2,-1, 2). (06 Marks)
b.
Find the directional derivatives of the function (I) = x yz + 4xz
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
3
z
2
). (07 Marks)
---)
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)
b.
Show that F = 2xy
2
+ yz)i + (2x
2
y+ xz + 2yz
2
)1+ (2 y
2
z + xy)fc is irrotational and hence
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
Solenoidal. For this value of 'a' show that curl A is also solenoidal.
(07 Marks)
1 of 2
s + a
+ b
}.
MATDIP401
7 a. Find the inverse Laplace transform of ,
s + 5
s
-
? 6s +13
b. Find 11
-1
{log
c .
Find
8
a.
Using convolution theorem find the Laplace transform of
sks
2 -
I
-
a
-
l
?
b. Solve the differential equation, y"+ 5y' + 6y = 5e
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
2
b. Find the Laplace transform of cos at ? cos bt
t
c. Find the Laplace transform of t
2
sin at .
1
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
(10 Marks)
2 of 2
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This post was last modified on 02 March 2020