Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 8 Differentials and Partial Derivatives Ex 8.7 Textbook Questions and Answers, Notes.

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

Question 1.
If each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
(i) f(x, y) = x²y + 6x³ + 7
(ii) h(x, y) = \(\frac { 6x^2y^3-πy^5+9x^4y }{ 2020x^2+2019y^2 }\)
(iii) g(x, y, z) = \(\frac { \sqrt{3x^2+5y^2+z^2} }{ 4x+7y }\)
(iv) U(x, y, z) = xy + sin(\(\frac { y^2-2z^2 }{ xy }\))
Solution:
(i) f(x, y) = x²y + 6x³ + 7
f(λx, λy) = λ³x²y + 6λ³x³ + 7
There is no common λ in this equation.
∴ It is not homogeneous

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

Samacheer Kalvi 12th Maths Guide Chapter 8 Differentials and Partial Derivatives Ex 8.7 1
Thus f is homogeneous with degree 3.
Samacheer Kalvi 12th Maths Guide Chapter 8 Differentials and Partial Derivatives Ex 8.7 2
Thus g is homogeneous with degree 0.
Samacheer Kalvi 12th Maths Guide Chapter 8 Differentials and Partial Derivatives Ex 8.7 3
There is no common λ
∴ It is not homogeneous.

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

Question 2.
Prove that f(x, y) = x³ – 2x²y + 3xy² + y³ is homogeneous. What is the degree? Verify Euler’s Theorem for f.
Solution:
f(x, y) = x³ – 2x²y + 3xy² + y³
f(λx, λy) = λ³x³ -2λ²x²λy + 3λxλ²y² + λ³y³
= λ³ (x³ – 2x²y + 3xy² + y³)
f is a homogeneous function of degree 3
By Euler’s Theorem,
Samacheer Kalvi 12th Maths Guide Chapter 8 Differentials and Partial Derivatives Ex 8.7 4
x \(\frac{\partial f}{\partial x}\) + y \(\frac{\partial f}{\partial y}\) = 3x³ – 4x²y + 3xy² – 2x²y + 6xy² + 3y³
= 3x³ – 6x²y + 9xy² + 3y³
= 3(x³ – 2x²y + 3xy² + y³)
x \(\frac{\partial f}{\partial x}\) + y \(\frac{\partial f}{\partial y}\) = 3 f
We verified the Euler’s Theorem.

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

Question 3.
Prove that g (x, y) = x log(y/x) is homogeneous What is the degree? Verify Eulers Theorem for g.
Solution:
g (x, y) = x log(y/x)
g (tx, ty) = tx log(\(\frac { ty }{ tx }\))
g is a homogeneous function of degree 1.
∴ By Euler’s Theorem,
Samacheer Kalvi 12th Maths Guide Chapter 8 Differentials and Partial Derivatives Ex 8.7 5
Hence verified

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

Question 4.
If u(x, y) = \(\frac { x^2+y^2 }{ \sqrt{x+y} }\), prove that
x \(\frac{\partial u}{\partial x}\) + y\(\frac{\partial u}{\partial y}\) = \(\frac{3}{2}\)u.
Solution:
Samacheer Kalvi 12th Maths Guide Chapter 8 Differentials and Partial Derivatives Ex 8.7 6

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

Question 5.
If v (x, y) = log(\(\frac{x^2+y^2}{x+y}\)) Prove that
x \(\frac{\partial v}{\partial x}\) + y\(\frac{\partial u}{\partial y}\) = 1.
Solution:
Samacheer Kalvi 12th Maths Guide Chapter 8 Differentials and Partial Derivatives Ex 8.7-7
Hence Proved

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

Question 6.
If w(x, y, z) = log (\(\frac{5x^3y^4+7y^2xz^4-75y^3z^4}{x^2+y^2}\)), find x \(\frac{\partial w}{\partial x}\) + y\(\frac{\partial w}{\partial y}\) + z\(\frac{\partial w}{\partial z}\).
Solution:
Samacheer Kalvi 12th Maths Guide Chapter 8 Differentials and Partial Derivatives Ex 8.7 8

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

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