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# Partial Derivatives And Euler's Equation NAT

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

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*Answer can only contain numeric values
QUESTION: 1

### then the value of x2fxx + 2xyfxy + y2fyy =

Solution:    *Answer can only contain numeric values
QUESTION: 2

### If u(x, y) = eαx+βy satisfy the condition uxx - 7uxy + 12uyy = 0. α2 - 7αβ + β2 = ____

Solution:

We are given with u(x,y)=eαx+ βy

ux=αeαx+ βy , uy= βeαx+ βy

uxx2eαx+ βy , uyy2eαx+ βy

uxy= αβeαx+ βy

put these values in the given relation, we get

uxx-7uxy+12uyy=0

eαx+ βy2-7αβ+12β2]=0

eαx+ β ≠0=> α2-7αβ+12β2=0

*Answer can only contain numeric values
QUESTION: 3

### Let f = yx, what is at x = 2, y = 1

Solution:

f = yx *Answer can only contain numeric values
QUESTION: 4

If  z = 2(ax + by)2 – (x2 + y2)  and  a2 + b2 = 1 then find the value of Solution:

Explanation : dz/dx = 4a(ax+by) - 2x

d2z/dx2 = 4a2 - 2

dz/dy = 4b(ax+by) - 2y

d2z/dy2 = 4b2 - 2

d2z/dx+ d2z/dx2 = 4a2 - 2 + 4b2 - 2

=> 4(a2+b2) - 4

As we know that a2 + b2 = 1

=> 4(1) - 4

=> 0

*Answer can only contain numeric values
QUESTION: 5 Solution: v is homogeneous function of degree 3 *Answer can only contain numeric values
QUESTION: 6

If u = log(tan x + tan y + tan z), then (sin 2x) ux + (sin 2y)uy + sin 2z)uz is equal to

Solution:     *Answer can only contain numeric values
QUESTION: 7

If  f(x, y) = x3y – xy3  then Solution: 1 – 12 = –11 *Answer can only contain numeric values
QUESTION: 8

If w = x2cos xy, then Find the value of a + b.

Solution:

We have  = 1 - 0 = 1 a + b = 1

*Answer can only contain numeric values
QUESTION: 9

If v = log(x2 + y2) then vxx + vyy equal to

Solution: *Answer can only contain numeric values
QUESTION: 10

If  z = eax+by f(ax - by) then the value of The value of n is

Solution: = 2abz

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