0 1 a_ Find the rank of the matrix:
2 3 5 4 -
cr, A = 0 2 3 4 by elementary row transformations. (08 Marks)
CS
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4 8 13 12t'.1)
4 .3 kr)
tj)
OC
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EC u
C rr
tu
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2b.
c.
a.
Solve by Gauss elimination method
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2x + v + 4z ---- 124x+ lly?z= 33
8x ? 3y + 2z = 20
Find all the eigen values for the matrix A
OR
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Reduce the matrix8
?6
2
? 6
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7-4
2
-- 4
3
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(06 Marks)(06 Marks)
1 2 3 2
2 3 5 1 into its echelon form and hence find its rank. (06 Marks)
1 3 4 5
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b. Applying Gauss elimination method, solve the system of equations2x + 5y + 7z = 52
2x+y?z= 0
x + y + z = 9
7
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?20
c. Find all the eigen values for the matrix A =
2 6 ? 2
0 ? 2 5
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Module-2(06 Marks)
(08 Marks)
a.
b.
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c.a.
b.
Solve
Solve
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SolveSolve
Solve
d
4
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y 2d y d2
y
0
=
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(06 Marks)(06 Marks)
(08 Marks)
(06 Marks)
(06 Marks)
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dx4
dx
3
dx
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2d2 y 6dy
+ 9y 5e
-2
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' =dx
2
dx
2
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d y+by the method of variation y = sec x of parameters.
dx
-
OR
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d'y0 y=
dx
y" + 3y' + 2y = I 2x
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2USN
3
4
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17MATDIP41
Fourth Semester B.E. Degree Examination, Dec.2019/Jan.2020
Additional Mathematics - II
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Time: 3 hrs. Max. Marks: 100Note: Answer any FIVE full questions, choosing ONE full question from each module.
ti Module-1
0 1 a_ Find the rank of the matrix:
2 3 5 4 -
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cr, A = 0 2 3 4 by elementary row transformations. (08 Marks)CS
4 8 13 12
t'.1)
4 .3 kr)
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tj)OC
E
C u
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C rrtu
2
b.
c.
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a.Solve by Gauss elimination method
2x + v + 4z ---- 12
4x+ lly?z= 33
8x ? 3y + 2z = 20
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Find all the eigen values for the matrix AOR
Reduce the matrix
8
?6
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2? 6
7
-4
2
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-- 43
(06 Marks)
(06 Marks)
1 2 3 2
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2 3 5 1 into its echelon form and hence find its rank. (06 Marks)1 3 4 5
b. Applying Gauss elimination method, solve the system of equations
2x + 5y + 7z = 52
2x+y?z= 0
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x + y + z = 97
?2
0
c. Find all the eigen values for the matrix A =
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2 6 ? 20 ? 2 5
Module-2
(06 Marks)
(08 Marks)
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a.b.
c.
a.
b.
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SolveSolve
Solve
Solve
Solve
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d4
y 2d y d
2
y
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0=
(06 Marks)
(06 Marks)
(08 Marks)
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(06 Marks)(06 Marks)
dx
4
dx
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3dx
2
d2 y 6dy
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+ 9y 5e-2
' =
dx
2
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dx2
d y+
by the method of variation y = sec x of parameters.
dx
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-OR
d'y
0 y=
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dxy" + 3y' + 2y = I 2x
2
USN
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34
l oft
17MATDIP41
c. Solve by the method of undetermined coefficients :
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y" ? 4y' + 4y = eX
(08 Marks)
Module-3
5 a. Find the Laplace transforms of sin5t cos2t (06 Marks)
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b. Find the Laplace transforms of (3t + 4)3
(06 Marks)
sin 2t 0 < t <
c. Express f(t)
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0 tinin terms of unit step function and hence find L[f(t)]. (08 Marks)
OR
. 1
6 a. Find the Laplace transforms of
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t(06 Marks)
b. Find the Laplace transform of 2' + t sin t (06 Marks)
c. If f(t) = t
2
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0 < t < 2 and I(t + 2) = fft) , for t > 2, find L[Rt)j. (08 Marks)Module-4
7 a_ Find the Laplace Inverse of
(08 Marks)
(06 Marks)
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(06 Marks)(06 Marks)
(06 Marks)
(08 Marks)
(s +1)(s ?1)(s + 2)
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b. Find the inverse Laplace transform of ,3s + 7
s
-
? 2s ?3
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c. Solve y" + 2y` ? 3y = sin t, y(0) = 0, y1
(0)
OR
8 a. Find the inverse Laplace transform of
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+ a\
~s+b 1
b. Find the inverse Laplace transform of
4s ?1
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?s' + 25
c. Find the inverse Laplace of y" 5y` + 6y = e' with y(0) = yr(0) = 0.
log
Module-5
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9 a. State and prove Addition theorem on probability_ (05 Marks)
b. A student A can solve 75% of the problems given in the book and a student B can solve
70%. What is the probability that A or B can solve a problem chosen at random. (06 Marks)
c. Three machines A, B, C produce 50%, 30% and 20% of the items in a factory. The
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percentage of defective outputs of these machines are 3, 4 and 5 respectively. If an item isselected at random, what is the probability that it is defective? If a selected item is defective,
vvrhat is the probability that it is from machine A? (09 Marks)
OR
10 a. Find the probability that the birth days of 5 persons chosen at random will fall in 12 different
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calendar months. (05 Marks)b. A box A contains 2 white balls and 4 black balls. Another box B contains 5 white balls and
7 black balls. A ball is transferred from box A to box B. Then a ball is drawn from box B.
Find the probability that it is white. (06 Marks)
c. State and prove Baye's theorem.
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(09 Marks)* * * * *
2
of 2
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