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DR. BABASAHEB AMBEDKAR TECHNOLOGICAL UNIVERSITY, LONERE  
End Semester Examination (Supplementary): May 2019  
Branch: B. Tech (Common to all)  
Semester: I  
Subject with code: Engineering Mathematics I (MATH 101)  
Date: 28.05.2019  
Marks: 60  
Duration: 03 Hrs.  
INSTRUCTION: Attempt any FIVE of the following questions. All questions carry equal marks.  
Q.1  
(
a) Solve the equations:  
+ ꢁꢂ + ꢃꢄ + = ꢆ ; ꢁꢀ + ꢃꢂ + ꢇꢄ + ꢁꢅ = ꢆ ; ꢈꢀ + ꢉꢂ + ꢁꢄ = ꢆ  
ꢑ ꢃ  
[6 ꢊꢋꢌꢍꢎ]  
[6 ꢊꢋꢌꢍꢎ]  
(
b) Find the eigen values and eigen vectors of the matrix = ꢐꢆ  
ꢉꢓ.  
ꢆ ꢒ  
Q.2  
(
a) Find the order derivative of = ꢗꢘꢙ .  
[6 ꢊꢋꢌꢍꢎ]  
[6 ꢊꢋꢌꢍꢎ]  
b) Using Taylor’s theorem, express the polynomial ꢚ(ꢀ) = ꢁꢀ + ꢛꢀ + ꢉ  
in the powers of (ꢀ ꢑ).  
(
Q.3 Solve any TWO:  
ꢜꢄ  
ꢜꢂ  
ꢢꢂ  
if = ꢝꢞꢟꢑ  
.  
(
(
a) Evaluate  
,
[6 ꢊꢋꢌꢍꢎ]  
b) Ifis a homogeneous function of degreein ꢋꢤꢥ , prove that  
ꢄ  
+ ꢁꢀꢂ  
ꢄ  
ꢄ  
ꢜ  
+ ꢂ = ꢔ(ꢔ ꢑ)ꢄ.  
[6 ꢊꢋꢌꢍꢎ]  
[6 ꢊꢋꢌꢍꢎ]  
ꢜꢀꢜꢂ  
ꢜꢂ  
ꢜꢦ ꢜꢦ ꢜꢦ  
c) If = ꢚ(ꢂ − ꢄ , ꢀ , ꢂ) , show that + + =.  
(
1
5F8A8CAEE76137CDEFD5C570E101BCD  
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Q.4  
a) Expand ꢚ(ꢀ, ꢂ) = at (ꢑ , ꢑ) by using Taylor’s theorem.  
(
[4 ꢊꢋꢌꢍꢎ]  
(
b) Find the percentage error in the measurement of the area of an ellipse when an error of 1.5 % is made  
in measuring its major and minor axes.  
[4 ꢊꢋꢌꢍꢎ]  
(
c) Find the maximum value of , when + + = .  
[4 ꢊꢋꢌꢍꢎ]  
Q.5  
a) Evaluate the integral =ꢑ  
b) Change the order of integration and evaluate ∫  
ꢬ ꢫꢂ ꢫꢀ  
.
ꢀꢂ  
(
[6 ꢊꢋꢌꢍꢎ]  
ꢘꢙꢂ  
(
(
ꢫꢀ ꢫꢂ.  
[6 ꢊꢋꢌꢍꢎ]  
[6 ꢊꢋꢌꢍꢎ]  
c) Evaluate the integral = ∫ ∫ ∫ ꢫꢄ ꢫꢀ ꢫꢂ .  
Q.6  
(a) State D’ Alembert’s ratio test, and hence check the convergence of the series:  
ꢁ  
+.  
[6 ꢊꢋꢌꢍꢎ]  
[6 ꢊꢋꢌꢍꢎ]  
(
b) State Cauchy’s root test, and hence check the convergence of the series:  
[
(ꢁꢔꢢꢑ) ꢀ ]ꢔ  
(ꢀ > ꢆ)  
ꢑ  
*
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5F8A8CAEE76137CDEFD5C570E101BCD