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Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7

Question 1.
In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 1
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 2
It is not a homogeneous function
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 3
∴ It is a homogeneous function with degree 3.
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 4
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 5
∴ It is homogeneous function of degree 0.
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 15
∴ It is not a homogeneous function

Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7

Question 2.
Prove that f(x, y) = x3 – 2x2 y + 3xy2 + y3 is homogeneous; what is the degree? Verify Euler’s Theorem for f.
Solution:
f (x, y) = x3 – 2x2y + 3xy2 + y3
f(tx, ty) = t3x3 – 2(t2x2)(ty) + 3(tx)(t2y2) + t3y3
= t3 [x3 – 2x2y + 3xy2 + y3]
f(tx, ty) = t3 f(x, y)
‘f’ is a homogeneous function of degree 3. By Euler’s theorem, we have
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 16
∴ Euler’s Theorem verified

Question 3.
Prove that g(x, y) = x log (\(\frac{y}{x}\)) is homogeneous; what is the degree? Verify Euler’s Theorem for g.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 17
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 18
∴ Euler’s Theorem verified

Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7

Question 4.
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 19
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 20

Question 5.
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 21
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 22
∴ ‘f’ is a homogeneous function of degree 1. By Euler’s theorem, we have
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 23

Question 6.
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 24
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 25

Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7

Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 Additional Problems

Question 1.
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 255
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 29
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 26

Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7

Question 2.
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 27
Solution:
R.H.S. is not a homogeneous and hence
Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.7 28

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