This set of Engineering Mathematics Question Bank focuses on “Euler’s Theorem – 2”.

1. In euler theorem x ^{∂z}⁄_{∂x} + y ^{∂z}⁄_{∂y} = nz, here `n` indicates

a) order of z

b) degree of z

c) neither order nor degree

d) constant of z

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Explanation: Statement of euler theorem is “if z is an homogeneous function of x and y of order `n` then x

^{∂z}⁄

_{∂x}+ y

^{∂z}⁄

_{∂y}= nz ”.

2. If z = x^{n} f(^{y}⁄_{x}) then

a) y ^{∂z}⁄_{∂x} + x ^{∂z}⁄_{∂y} = nz

b) 1/y ^{∂z}⁄_{∂x} + 1/x ^{∂z}⁄_{∂y} = nz

c) x ^{∂z}⁄_{∂x} + y ^{∂z}⁄_{∂y} = nz

d) 1/x ^{∂z}⁄_{∂x} + 1/y ^{∂z}⁄_{∂y} = nz

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Explanation: Since the given function is homogeneous of order n , hence by euler’s theorem

x

^{∂z}⁄

_{∂x}+ y

^{∂z}⁄

_{∂y}= nz.

3. Necessary condition of euler’s theorem is

a) z should be homogeneous and of order n

b) z should not be homogeneous but of order n

c) z should be implicit

d) z should be the function of x and y only

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Explanation:

Answer `a` is correct as statement of euler’s theorem is “if z is an homogeneous function of x and y of order `n` then x

^{∂z}⁄

_{∂x}+ y

^{∂z}⁄

_{∂y}= nz”

Answer `b` is incorrect as z should be homogeneous.

Answer `c` is incorrect as z should not be implicit.

Answer `d` is incorrect as z should be the homogeneous function of x and y not non-homogeneous functions.

4. If

a) 0

b) zln(z)

c) z^{2} ln(z)

d) z

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5. If

a) 2 tan(z)

b) 2 cot(z)

c) tan(z)

d) cot(z)

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6. Value of is ,

a) -2.5 u

b) -1.5 u

c) 0

d) -0.5 u

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7. If f1(x,y) and f2(x,y) are homogeneous and of order `n` then the function f3(x,y) = f1(x,y) + f2(x,y) satisfies euler’s theorem.

a) True

b) False

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9. If z = Sin^{-1} (^{x}⁄_{y}) + Tan^{-1} (^{y}⁄_{x}) then x ^{∂z}⁄_{∂x} + y ^{∂z}⁄_{∂y} is

a) 0

b) y

c) 1 + ^{x}⁄_{y} Sin^{-1} (^{x}⁄_{y})

d) 1 + ^{y}⁄_{x} Tan^{-1} (^{y}⁄_{x})

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Explanation: Given z = Sin

^{-1}(

^{x}⁄

_{y}) + Tan

^{-1}(

^{y}⁄

_{x})

Let, u = Sin

^{-1}(

^{x}⁄

_{y}) and v = Tan

^{-1}(

^{y}⁄

_{x}) hence z = u + v

Now, let u’ = Sin(u) =

^{x}⁄

_{y}= f

^{x}⁄

_{y}) hence u’ satisfies euler’s theorem,

10. If f(x,y)is a function satisfying euler’ s theorem then

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11. If is

a) Sin(4u) – Cos(2u)

b) Sin(4u) – Sin(2u)

c) Cos(4u) – Sin(2u)

d) Cos(4u) – Cos(2u)

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12. If

a) u ln(u)

b) u ln(u)^{2}

c) u [1+ln(u)].

d) 0

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**Sanfoundry Global Education & Learning Series – Engineering Mathematics.**

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