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DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE
Semester Examination May - 2019
Branch: First year B. Tech.(ALL)
Semester: I
Subject with Code: Engg. Math-I (BTMA101)
Time: 3 hrs.
Date: 14/05/2019
Max. Marks: 60
Instructions to Students:
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(
(
(
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1) All questions are compulsory.
2) Use of non-programmable calculator is allowed.
3) Figures to right indicate full marks.
5) If some part or parameter is noticed tobe missing you may appropriately
assume it and should mention it clearly.
Q.1 Attempt any three from the following:
4 X 3 = 12
(a)
Reduce the following matrix into normal form & find its rank
1
3
2
4
−1
0
3
−1]
7
[
1 0 −2
(b)
Test the consistency & solve
4ꢀ − 2ꢁ + 6ꢂ = 8 ;
ꢀ + 3ꢂ = −1 ;
15ꢀ − 3ꢁ + 9ꢂ = 21.
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c)
d)
Find the Eigen values & Eigen vectors of the following matrix
4
1
6
3
6
2 ]
[
1 −4 −3
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Find by using Cayley-Hamilton Theorem
2
Where = [−1
1
−1
2
−1
1
−1]
2
Q.2 Attempt any three from the following:
4 X 3 = 12
(a)
If = log(ꢀ + + 3ꢀꢁꢂ) then prove that
ꢍ ꢆ = −
ꢉ
+
+
.
ꢉꢋ
ꢉꢌ
(ꢊꢐꢋꢐꢌ)
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(b)
If ꢒ(ꢆ) is a homogeneous function of degreein &, then show that
ꢑ
+ 2ꢀꢁ
ꢔ
ꢔ
ꢘ(ꢔ)
ꢉ
+ ꢁ = ꢕ(ꢆ) ꢖ ꢕ′(ꢆ) − 1 ꢗ where ꢕ(ꢆ) = ꢓ
.
ꢉꢊꢉꢋ
ꢉꢋ
ꢘ`(ꢔ)
ꢑ
ꢐꢋꢑ
ꢐꢋ
(
c)
d)
Verify Euler’s theorem for =
.
(
If = ꢒ(ꢆ, ꢙ) where = ꢚꢀ + ꢛꢁ ꢜꢓꢝ = ꢚꢁ − ꢛꢀ, then show that
ꢑ
ꢌ
ꢉꢋꢑ
ꢌ
ꢉ ꢌ
= (ꢚ +) ꢈ + ꢉꢞ .
+
ꢑ
Q.3 Attempt any three from the following:
4 X 3 = 12
(a)
Show that ꢟ. ꢟ′ = 1 If =ꢡꢢꢣꢙ & =ꢣꢤꢓꢙ .
Expand ꢒ(ꢀ, ꢁ) = sin(ꢀꢁ) in the powers of (ꢀ 1) & (ꢁ ) by using Taylors
(b)
Theorem.
(
c)
d)
Discus the maxima & minima of ꢀꢁ(ꢜ ꢁ) .
ꢧ
ꢧ
ꢧ
ꢑ
(
If = + +
where + + = 1, then find the stationary values
by using Lagrange’s method of multipliers.
Q.4 Attempt any three from the following:
4 X 3 = 12
(a)
Evaluate
∫ ꢀ√4ꢀ − ꢀ ꢝꢀ .
(4 − ꢀ) = ꢀ(ꢀ − 2) .
(b)
(c)
(d)
Trace the curve
Trace the curve
Trace the curve
ꢬ = cos(2ꢭ) .
ꢀ = ꢜ(ꢮ + ꢣꢤꢓꢮ)
= ꢜ(1 − ꢡꢢꢣꢮ) .
Q.5 Attempt any three from the following:
4 X 3 = 12
(
a)
b)
Change the order of integration & evaluate it
∫ ꢀ ꢝꢀ ꢝꢁ.
(
Change to polar co-ordinates & evaluate it
√ꢎꢦꢊꢄꢊꢑ
(ꢀ+ꢁ) ꢝꢀ ꢝꢁ
ꢅ ꢌ (ꢊꢐꢌ)
() (ꢀ + ꢁ + ꢂ)ꢝꢀ ꢝꢁ ꢝꢂ .
(
c)
d)
Evaluate
(
Find the area outside the circle
inside the cardioide
ꢬ = ꢜ
&
ꢬ = ꢜ(1 + ꢡꢢꢣꢭ).
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