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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