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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 + ꢅ = ꢆ ; ꢁꢀ + ꢃꢂ + ꢇꢄ + ꢁꢅ = ꢆ ; ꢈꢀ + ꢉꢂ + ꢁꢄ = ꢆ

ꢑ ꢃ

ꢀ

[6 ꢊꢋꢌꢍꢎ]

ꢈ

(

b) Find the eigen values and eigen vectors of the matrix ꢏ = ꢐꢆ

ꢁ

ꢉꢓ.

ꢆ

ꢆ ꢒ

Q.2

(

a) Find the ꢔ^ꢕ^ꢖ order derivative of ꢂ = ꢗꢘꢙ^ꢃꢀ .

[6 ꢊꢋꢌꢍꢎ]

b) Using Taylor’s theorem, express the polynomial ꢚ(ꢀ) = ꢁꢀ^ꢈ + ꢛꢀ^ꢁ + ꢀ − ꢉ

in the powers of (ꢀ − ꢑ).

(

Q.3 Solve any TWO:

ꢜ

ꢄ

ꢜꢄ

ꢜꢂ

ꢀ^ꢁꢢꢂ

ꢁ

if ꢄ = ꢝꢞꢟ^ꢠ^ꢑ

ꢡ _ꢀ_ꢢ_ꢂ ꢣ .

(

a) Evaluate

[6 ꢊꢋꢌꢍꢎ]

ꢀ

b) If ꢄ is a homogeneous function of degree ꢔ in ꢀ ꢋꢤꢥ ꢂ , prove that

ꢜ

^ꢁꢄ

_ꢁ+ ꢁꢀꢂ

ꢜ^ꢁꢄ

ꢀ^ꢁ_ꢜ

+ ꢂ^ꢁ_ꢁ = ꢔ(ꢔ − ꢑ)ꢄ.

[6 ꢊꢋꢌꢍꢎ]

ꢀ

ꢜꢀꢜꢂ

ꢜꢂ

ꢜꢦ ꢜꢦ ꢜꢦ

c) If ꢦ = ꢚ(ꢂ − ꢄ , ꢄ − ꢀ , ꢀ − ꢂ) , show that_ꢜ_ꢀ +_ꢜ_ꢂ +_ꢜ_ꢄ = ꢆ.

(

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Q.4

a) Expand ꢚ(ꢀ, ꢂ) = ꢧ^ꢀ^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 ꢀ^ꢨ ꢂ^ꢔꢄ^ꢩ , when ꢀ + ꢂ + ꢄ = ꢗ .

[4 ꢊꢋꢌꢍꢎ]

Q.5

a) Evaluate the integral ꢪ = ∫_ꢑ ∫_ꢑ

b) Change the order of integration and evaluate ∫^ꢁ ∫

ꢭ

ꢬ ꢫꢂ ꢫꢀ

ꢀꢂ

(

[6 ꢊꢋꢌꢍꢎ]

ꢮ

ꢁ

ꢀ

ꢗ

ꢘꢙꢂ

ꢂ

(

ꢫꢀ ꢫꢂ.

[6 ꢊꢋꢌꢍꢎ]

ꢆ

c) Evaluate the integral ꢪ = ∫ ∫ ∫_ꢆ^ꢀ^ꢢ^ꢂ ꢧ^ꢀ^ꢢ^ꢂ^ꢢ^ꢄ ꢫꢄ ꢫꢀ ꢫꢂ .

ꢭ

ꢀ

ꢆ

Q.6

(a) State D’ Alembert’s ratio test, and hence check the convergence of the series:

ꢔ^ꢁ

ꢑ

∞

∑

_ꢯ_ꢑꢡ_ꢁ_ꢔ +_ꢔ_ꢁꢣ.

[6 ꢊꢋꢌꢍꢎ]

ꢔ

(

b) State Cauchy’s root test, and hence check the convergence of the series:

[

(ꢁꢔꢢꢑ) ꢀ ]^ꢔ

∑

(ꢀ > ꢆ)

_ꢔꢔ^ꢰ^ꢑ

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