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QUESTION: 1

If z = xy In (x, y) then

Solution:

QUESTION: 2

Suppose is equal to

Solution:

** u** and

on adding

The correct answer is: *z*

QUESTION: 3

If then the value of

Solution:

*f*(*x*, *y*) is homogeneous function of degree –2

So, using Euler's equation

The correct answer is: 0

QUESTION: 4

If u = sin x/l*sin y/m * sin z/n * cospt satisfy the equation then

Solution:

Again using the given condition

QUESTION: 5

If then equal to

Solution:

The correct answer is

QUESTION: 6

then

Solution:

** f_{1}** is homogeneous of degree

On adding

The correct answers are:

QUESTION: 7

If then

Solution:

We have

** u** is homogeneous function of degree

Now differentiate partially w.r.t. * x* again

The correct answer is:

QUESTION: 8

Find a function *w* = *f*(*x*, *y*) whose first partial derivatives are and and whose value at point (ln2, 0) is ln2.

Solution:

We σw/σx=1+e^{x}cos y

Integrate both w.r.t.x

w(x,y) =x+e^{x} cosy+f(y)

σw/σy=0-e^{x}siny+f’(y)

Given =-e^{x}siny+2y

So,on comparing the above two equation

f’(y)=2y

f(y)=y2+c(on integration)

so, w(x,y)=x+e^{x}cos y+y^{2}+c

Now, using (In2,0)is In2,we get

C=-2

Hence, w(x,y)=x+y^{2}+e^{x}cosy-2

**The correct answer is, option C; w=x+y ^{2}+e^{x}cosy-2**

QUESTION: 9

If u = sin^{-1} [(x^{3} + y^{3} + z^{3})/(ax + by + cz)] then xdu/dx + ydu/dy + zdu/dz equal to

Solution:

So, by Euler's theorem

The correct answer is: 2 tan *u*

QUESTION: 10

The contour on xy plane where partial derivative of x^{2} + y^{2} with respect to y is equal to the partial derivative of 6y + 4x w.r.t. x is

Solution:

So,

**2 y = 4**

The correct answer is:

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