Determine whether the function h ( x , y ) = 6 x 2 y 3 - πy 5 + 9 x 4 y 2020 x 2 + 2019 y 2 is homogeneous or not. If it is so, find the degree.

Given, h ( x , y ) = 6 x 2 y 3 - πy 5 + 9 x 4 y 2020 x 2 + 2019 y 2 h ( tx , ty ) = 6 ( tx ) 2 ( ty ) 3 - π ( ty ) 5 + 9 ( tx ) 4 ( ty ) 2020 ( tx ) 2 + 2019 ( ty ) 2 h ( tx , ty ) = t 5 [ 6 x 2 y 3 - πy 5 + 9 x 4 y ] t 2 [ 2020 x 2 + 2019 y 2 ] Here, h ( tx , ty ) = t 3 h ( x , y ) . Thus, it is a homogeneous function with degree 3 .

Determine whether the function g ( x , y , z ) = 3 x 2 + 5 y 2 + z 2 4 x + 7 y is homogeneous or not. If it is so, find the degree.

Given, g ( x , y , z ) = 3 x 2 + 5 y 2 + z 2 4 x + 7 y g ( tx , ty , tz ) = 3 ( tx ) 2 + 5 ( ty ) 2 + ( tz ) 2 4 ( tx ) + 7 ( ty ) g ( tx , ty , tz ) = t 2 t 3 x 2 + 5 y 2 + z 2 4 x + 7 y Here, g ( tx , ty , tz ) = t 0 g ( x , y , z ) Thus, it is homogeneous function of degree 0 .

Determine whether the function U ( x , y , z ) = xy + sin y 2 - 2 z 2 xy is homogeneous or not. If it is so, find the degree.

Given, U ( x , y , z ) = xy + sin y 2 - 2 z 2 xy U ( tx , ty , tz ) = ( tx ) ( ty ) + sin ( ty ) 2 - 2 ( tz ) 2 ( tx ) ( ty ) U ( tx , ty , tz ) = ( t 2 ) ( xy ) + sin y 2 - 2 z 2 xy Here, U ( x , y , z ) ≠ t p · U ( tx , ty , tz ) . Thus, it is not a homogeneous function.

Prove that g ( x , y ) = xlog y x is homogeneous; what is the degree? Verify Euler’s Theorem for g .

Given, g ( x , y ) = xlog y x g ( tx , ty ) = ( tx ) log ty tx g ( tx , ty ) = t 1 xlog y x g ( tx , ty ) = t 1 [ g ( x , y ) ] ∴ g is homogeneous function of degree 1 . By Euler's theorem, we have x ∂ g ∂ x + y ∂ g ∂ y = g At LHS , we have , x ∂ g ∂ x + y ∂ g ∂ y = x x x y - y x 2 + log y x + y x x y 1 x = x · log y x = g → RHS (Euler’s Theorem verified)

If u ( x , y ) = x 2 + y 2 x + y , prove that x ∂ u ∂ x + y ∂ u ∂ y = 3 2 u .

Given, u ( x , y ) = x 2 + y 2 x + y u ( tx , ty ) = ( tx ) 2 + ( ty ) 2 tx + ty u ( tx , ty ) = t 2 t x 2 + y 2 x + y u ( tx , ty ) = t 3 2 u ( x , y ) u is a homogeneous function of degree 3 2 By Euler's theorem, we have x ∂ u ∂ x + y ∂ u ∂ y = 3 2 u

If v ( x , y ) = log x 2 + y 2 x + y , prove that x ∂ v ∂ x + y ∂ v ∂ y = 1 .

Given, v ( x , y ) = log x 2 + y 2 x + y which is not a homogeneous function. Let f = e v = x 2 + y 2 x + y ⇒ f ( nx , ny ) = n 2 x 2 + n 2 y 2 nx + ny = n x 2 + y 2 x + y So, f is a homogeneous function of degree 1 . By Euler’s theorem, we have x ∂ f ∂ x + y ∂ f ∂ y = nf x ∂ e v ∂ x + y ∂ e v ∂ y = e v ⇒ x e v ∂ v ∂ x + y e v ∂ v ∂ y = e v ∴ x ∂ v ∂ x + y ∂ v ∂ y = 1

E M B I B E

Mathematics Standard 12 Vol II>Chapter 8 - Differentials and Partial Derivatives>Exercise 7

Tamil Nadu Board Solutions for Chapter: Differentials and Partial Derivatives, Exercise 7: Exercise 7

Author:Tamil Nadu Board

Tamil Nadu Board Mathematics Solutions for Exercise - Tamil Nadu Board Solutions for Chapter: Differentials and Partial Derivatives, Exercise 7: Exercise 7

Attempt the practice questions on Chapter 8: Differentials and Partial Derivatives, Exercise 7: Exercise 7 with hints and solutions to strengthen your understanding. Mathematics Standard 12 Vol II solutions are prepared by Experienced Embibe Experts.

Questions from Tamil Nadu Board Solutions for Chapter: Differentials and Partial Derivatives, Exercise 7: Exercise 7 with Hints & Solutions

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Standard 12 Vol II>Chapter 8 - Differentials and Partial Derivatives>Exercise 7>Q 1

Determine whether the function $h (x, y) = \frac{6 x^{2} y^{3} - {πy}^{5} + 9 x^{4} y}{2020 x^{2} + 2019 y^{2}}$ is homogeneous or not. If it is so, find the degree.

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Standard 12 Vol II>Chapter 8 - Differentials and Partial Derivatives>Exercise 7>Q 1

Determine whether the function $g (x, y, z) = \frac{\sqrt{3 x^{2} + 5 y^{2} + z^{2}}}{4 x + 7 y}$ is homogeneous or not. If it is so, find the degree.

MEDIUM

12th Tamil Nadu Board

IMPORTANT

Mathematics Standard 12 Vol II>Chapter 8 - Differentials and Partial Derivatives>Exercise 7>Q 1

Determine whether the function $U (x, y, z) = xy + \sin (\frac{y^{2} - 2 z^{2}}{xy})$ is homogeneous or not. If it is so, find the degree.

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Standard 12 Vol II>Chapter 8 - Differentials and Partial Derivatives>Exercise 7>Q 2

Prove that $f (x, y) = x^{3} - 2 x^{2} y + 3 {xy}^{2} + y^{3}$ is homogeneous; what is the degree? Verify Euler’s Theorem for $f$ .

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Standard 12 Vol II>Chapter 8 - Differentials and Partial Derivatives>Exercise 7>Q 3

Prove that $g (x, y) = xlog (\frac{y}{x})$ is homogeneous; what is the degree? Verify Euler’s Theorem for $g$ .

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Standard 12 Vol II>Chapter 8 - Differentials and Partial Derivatives>Exercise 7>Q 4

If $u (x, y) = \frac{x^{2} + y^{2}}{\sqrt{x + y}}$ , prove that $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = \frac{3}{2} u$ .

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Standard 12 Vol II>Chapter 8 - Differentials and Partial Derivatives>Exercise 7>Q 5

If $v (x, y) = \log (\frac{x^{2} + y^{2}}{x + y})$ , prove that $x \frac{\partial v}{\partial x} + y \frac{\partial v}{\partial y} = 1$ .

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Standard 12 Vol II>Chapter 8 - Differentials and Partial Derivatives>Exercise 7>Q 6

If $w (x, y, z) = \log (\frac{5 x^{3} y^{4} + 7 y^{2} {xz}^{4} - 75 y^{3} z^{4}}{x^{2} + y^{2}})$ , find $x \frac{\partial w}{\partial x} + y \frac{\partial w}{\partial y} + z \frac{\partial w}{\partial z}$ .

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