Partial Derivatives And Euler's Equation MCQ Level - 2

# Partial Derivatives And Euler's Equation MCQ Level - 2

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## 10 Questions MCQ Test Topic wise Tests for IIT JAM Physics | Partial Derivatives And Euler's Equation MCQ Level - 2

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Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 1

### If then the value of is

Detailed Solution for Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 1

u = sin-1(x + y/(√x + √y))

Put x = xt and y = yt

u = sin-1[(xt-yt)/(√xt + √yt)]

sinu = t1/2[(x+y)/(√x + √y)]

= t1/2 f(x,y)

This function sin u is homogeneous with degree 1/2.

By Euler's theorem

xux + yuy = G(u) = nf(u)/f,(u) = 1/2 tan u

xux + yuy = 1/2 tan u

x2uxx + 2xyuyy + y2uyy = G(u)[G'(u) - 1]

= 1/2 tan u [(sec2u - 2)/2]

= 1/4 tan u [(tan2u - 1)/1]

= 1/4 * sin u/cos u [(sin2u - cos2u)/cos2u]

x2uxx + 2xyuyy + y2uyy = -(sinu cos2u)/4ucos3u

Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 2

### then the value of is equal to

Detailed Solution for Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 2

We have  v is homogeneous function of degree n then The correct answer is: n(n - 1)

Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 3

### The accompanying figure shows the graph of an unspecified function of f(x, y) and partial derivatives fx(x, y) and fy(x, y). Determine which is which and explain. Detailed Solution for Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 3

The correct answer is: II - f(xy),  I - fx(x, y), III - fy(x, y)

Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 4 Detailed Solution for Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 4

We are given  f(u) is homogeneous function degree 1, then    [By dividing with cos3 u] The correct answer is: Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 5

If u = f(t) and v = f(t) and  t = φ (x, y)  then

Detailed Solution for Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 5  The correct answer is: Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 6

If w = f(u, v) where u = x + y and v = x – y then Detailed Solution for Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 6

We have w = f(u, v)  The correct answer is: Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 7

If f is differentiable and z = u + f(u2v2), then

Detailed Solution for Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 7

Let w = u2v2
then z = f(w) + u So, The correct answer is: Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 8

If then the value of is

Detailed Solution for Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 8 f(u) is homogeneous function of degree 2 Now, let g(u) = 2tan u The correct answer is: 2tan u(2sec2u –1)

Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 9

If f(x, y, z) = 0 then the value of equal to

Detailed Solution for Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 9

(1) Differentiate with respect to y, I get:

0+F2+F3∂z/∂y=0

So

F3 ∂z/∂y = −F2

(2) Differentiate with respect to x, I get:

F1 + F2 ∂y/∂x + 0 = 0

So F2 ∂y/∂x = −F1

(3) Differentiate with respect to z, I get:

F1 ∂x/∂z +0 + F3 = 0

4) After some manipulations with the Fi, I get to the conclusion that

∂z/∂y∗∂y/∂x∗∂x/∂z=−1, so when evaluated with x, z, y respectively, conclusion is still true

Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 10

Use the information the figure to find the first order partial derivatives of f at the point (1, 2) Detailed Solution for Partial Derivatives And Euler's Equation MCQ Level - 2 - Question 10

The correct answer is: ## Topic wise Tests for IIT JAM Physics

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