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DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE  
RAIGAD-402 103.  
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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1) All questions are compulsory.  
2) Use of non-programmable calculator is allowed.  
3) Figures to right indicate full marks.  
4) Illustrate your answer with neat sketches, diagram etc. whatever necessary.  
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.  
(
c)  
d)  
Find the Eigen values & Eigen vectors of the following matrix  
4
1
6
3
6
2 ]  
[
−
1 −4 −3  
(
Find ꢃꢄꢅ by using Cayley-Hamilton Theorem  
2
Where ꢃ = [−1  
1
−1  
2
−1  
1
−1]  
2
Q.2 Attempt any three from the following:  
4 X 3 = 12  
(a)  
If ꢆ = log(ꢀꢇ + ꢁꢇ + ꢂꢇ − 3ꢀꢁꢂ) then prove that  
ꢎ
ꢍ ꢆ = −  
ꢉ
ꢉ
ꢉ
ꢏ
ꢈꢉ  
ꢊ +  
+
ꢑ .  
ꢉꢋ  
ꢉꢌ  
(ꢊꢐꢋꢐꢌ)  
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(b)  
If ꢒ(ꢆ) is a homogeneous function of degree ꢓ in ꢀ & ꢁ, then show that  
ꢉꢑ  
ꢔ
ꢑ + 2ꢀꢁ  
ꢉꢑꢔ  
ꢉꢑꢔ  
ꢘ(ꢔ)  
ꢀꢎ ꢉ  
+ ꢁꢎ ꢑ = ꢕ(ꢆ) ꢖ ꢕ′(ꢆ) − 1 ꢗ where ꢕ(ꢆ) = ꢓ  
.
ꢊ
ꢉꢊꢉꢋ  
ꢉꢋ  
ꢘ`(ꢔ)  
ꢊꢑ  
ꢊ
ꢐꢋꢑ  
ꢐꢋ  
(
c)  
d)  
Verify Euler’s theorem for ꢆ =  
.
(
If ꢂ = ꢒ(ꢆ, ꢙ) where ꢆ = ꢚꢀ + ꢛꢁ ꢜꢓꢝ ꢙ = ꢚꢁ − ꢛꢀ, then show that  
ꢉꢑ  
ꢌ
ꢉꢑꢌ  
ꢉꢋꢑ  
ꢉꢑꢌ  
ꢉ ꢌ  
ꢑ
= (ꢚꢎ + ꢛꢎ) ꢈꢉꢔꢑ + ꢉꢞꢑꢍ .  
+
ꢊꢑ  
ꢉ
Q.3 Attempt any three from the following:  
4 X 3 = 12  
(a)  
Show that ꢟ. ꢟ′ = 1 If ꢀ = ꢠꢔꢡꢢꢣꢙ & ꢁ = ꢠꢔꢣꢤꢓꢙ .  
ꢥ
Expand ꢒ(ꢀ, ꢁ) = sin(ꢀꢁ) in the powers of (ꢀ − 1) & (ꢁ − ꢎ ) by using Taylor’s  
(b)  
Theorem.  
(
c)  
d)  
Discus the maxima & minima of ꢀꢁ(ꢜ − ꢀ − ꢁ) .  
ꢦꢧ  
ꢨꢧ  
ꢩꢧ  
ꢌꢑ  
(
If ꢆ = ꢊꢑ + ꢋꢑ +  
where ꢀ + ꢁ + ꢂ = 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  
ꢪ
(a)  
Evaluate  
∫ ꢀꢇ√4ꢀ − ꢀꢎ ꢝꢀ .  
ꢫ
ꢁꢎ(4 − ꢀ) = ꢀ(ꢀ − 2)ꢎ .  
(b)  
(c)  
(d)  
Trace the curve  
Trace the curve  
Trace the curve  
ꢬ = ꢜ 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  
ꢅ
ꢅ
∫ ꢀꢎꢠꢊꢋ ꢝꢀ ꢝꢁ.  
∫
ꢫ
ꢋ
(
Change to polar co-ordinates & evaluate it  
√ꢎꢦꢊꢄꢊꢑ  
(ꢀꢎ+ꢁꢎ) ꢝꢀ ꢝꢁ  
∫
ꢎ
ꢦ
∫
ꢫ
ꢫ
ꢅ ꢌ (ꢊꢐꢌ)  
∫ꢄꢅ ∫ꢫ ∫(ꢊꢄꢌ) (ꢀ + ꢁ + 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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** END ***  
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