Download VTU BE 2020 Jan CSE Question Paper 15 Scheme 4th Sem 15MATDIP41 Additional Mathematics II

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15MATDIP41
Fourth Semester ME, Degree Examination, Dee.2019/Jan.2020
Additional Mathematics - II

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Time: 3 hrs. Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a. Find the rank of the matrix by
by applying elementary row transformations.

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I 4
b. Find the inverse of the matrix
3
using Caylery-Hamilton theorem.
c. Solve the following system of equations by Gauss elimination method.

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+ y + 4z = 12, 4x+ 11 - z = 33, 8x 3y + 2z 20
OR
-1 -3 -1
2 3 -1
2 a. Find the rank of the matrix by reducing it to echelon form. (06 Marks)

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0 1 1
1 1
- 1

_ '1

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1.
7 0
b.
Find the eigen values of A = -2 6
-

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2 (05 Marks)
0 -2 5
c. Solve by Gauss elimination method: x + y + z = 9, x 2y + 3z = 8, 2x -4- y - z = 3
(05 marks)
NIoduk-2

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3
a.
b.
Solve
d'y d' y

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dy
+6
,
+11
? + 6y = 0 (05 Marks)

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(05 Marks)
dx dx
-
dx
Solve y" 4y' +13y = cos 2x

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c. Solve by the method of undetermined coefficients y" + 3y' + 2y =12x
2
(06 Marks)
OR
4

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a.
' d
Solve d y
+
y

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-- 5-- + oy =
dx
2
dx
(05 Marks)

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b.
Solve y" + 4y' -12y = e
2
' -3 sin 2x (05 Marks)
c Solve by the method of variation of parameter

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d y
+ y = tan x (06 Marks)
dx
Module-3
5 a. Find the Laplace transform of

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i) e
-2
sin h 4t ii) e
-2
'(2cos5t -sin 5t)

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(06 Marks)
b. Find the Laplace transform of fit) = t
2
0 < t < 2 and f(t + 2) = f(t) for t > 2. (05 Marks)
7

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I 2 3 2
A= 2 3 5 1
"c3
V
1 3 4 5

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(06 Marks)
(05 Marks)
(05 Marks)
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15MATDIP41
Fourth Semester ME, Degree Examination, Dee.2019/Jan.2020
Additional Mathematics - II

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Time: 3 hrs. Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE full question from each module.
Module-1
1 a. Find the rank of the matrix by
by applying elementary row transformations.

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

I 4
b. Find the inverse of the matrix
3
using Caylery-Hamilton theorem.
c. Solve the following system of equations by Gauss elimination method.

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+ y + 4z = 12, 4x+ 11 - z = 33, 8x 3y + 2z 20
OR
-1 -3 -1
2 3 -1
2 a. Find the rank of the matrix by reducing it to echelon form. (06 Marks)

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0 1 1
1 1
- 1

_ '1

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1.
7 0
b.
Find the eigen values of A = -2 6
-

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2 (05 Marks)
0 -2 5
c. Solve by Gauss elimination method: x + y + z = 9, x 2y + 3z = 8, 2x -4- y - z = 3
(05 marks)
NIoduk-2

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3
a.
b.
Solve
d'y d' y

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dy
+6
,
+11
? + 6y = 0 (05 Marks)

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(05 Marks)
dx dx
-
dx
Solve y" 4y' +13y = cos 2x

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c. Solve by the method of undetermined coefficients y" + 3y' + 2y =12x
2
(06 Marks)
OR
4

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a.
' d
Solve d y
+
y

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-- 5-- + oy =
dx
2
dx
(05 Marks)

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b.
Solve y" + 4y' -12y = e
2
' -3 sin 2x (05 Marks)
c Solve by the method of variation of parameter

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d y
+ y = tan x (06 Marks)
dx
Module-3
5 a. Find the Laplace transform of

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i) e
-2
sin h 4t ii) e
-2
'(2cos5t -sin 5t)

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(06 Marks)
b. Find the Laplace transform of fit) = t
2
0 < t < 2 and f(t + 2) = f(t) for t > 2. (05 Marks)
7

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I 2 3 2
A= 2 3 5 1
"c3
V
1 3 4 5

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(06 Marks)
(05 Marks)
(05 Marks)
1 of 2
7 a. Find the inverse Laplace transform or i)

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Module-4
2s ?I
s
2
+ 4s + 29

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15MATDI P41
c. Express f(t) = ft
0 < t <4
interms of unit step function and hence find L[t(t)]. (05 Marks)
L5 t > 4

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OR
cos at ? cos bt
(06 Marks) 6
a.
Find the Laplace transform of i) t cosat ii)

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b. Given f(t) =
t
E 0 < t< a/2
4
where f(t + ) = f(t). Show that L[f(t)] = tan h ?

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as
?E a/2 < t < a S 4 )
(05 Marks)
c. Express f(t) =
0 < t < I

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1 < t < 2
t > 2
interms of unit step function and hence find L[f(t)].

(05 Marks)

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+ 2 4s ?1
ii) s + , (06 Marks)
s
-
+ 36 s

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-
+25
s
-
+1

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b. Find the inverse Laplace transform of log
s
2
+ 4
(05 Marks)

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c.
Solve by using Laplace transforms y" + 4y' + 4y = given that y(0) = 0, y'(0) = 0.
(05 Marks)
OR
8

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a. Find the inverse Laplace transformof
1
(06 Marks)
(s + )(s + 2)(5 + )
(5 + a

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b. Find the inverse Laplace transthrm of cot--I
b
(05 Marks)
C.
Using Laplace transforms solve the differential equation y"' + 2y" ? ? 2y = 0 given

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y(0) = y'(0) = 0 and y"(0) = 6. (05 Marks)
Module-5
9 a. State and prove Baye's theorem. (06 Marks)
b. The machines A, B and C produce respectively 60%, 30%, 10% of the total number of items _
of a factory. The percentage of defective output of these machines are respectively 2%, 3%

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and 4%. An item is selected at random and is found defective. Find the probability that the
item was produced by machine "C". (05 Marks)
e.
The probability that a team wins a match is 3/5. If this team play 3 matches in a tournament,
what is the probability that i) win all the matches ii) lose all the matches. (05 Marks)

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OR
10 a. If A and B are any two events of S. which are not mutually exclusive then
P(Au B) = P(A) + P(B) ? P(AnB). (06 Marks)
b.
If A and B are events with `P(AnB) = 7/8, P(AnB) = 1/4, P(g?)= 5/8. Find P(A), P(B) and

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P(A nii). (05 Marks)
C.
The probability that 4: person A solves the problem is 1/3, that of B is 1/2 and that of C is
3/5. If the problem is simultaneously assigned to all of them what is the probability that the
problem is solved? . _.

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-
(05 Marks)
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