This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Euler’s Theorem – 1”.

1. f(x, y) = x^{3} + xy^{2} + 901 satisfies the Eulers theorem

a) True

b) False

View Answer

Explanation: The function is not homogenous and hence does not satisfy the condition posed by eulers theorem.

2. find the value of f_{y} at (x,y) = (0,1)

a) 101

b) -96

c) 210

d) 0

View Answer

Explanation: Using Euler theorem

xf

_{x}+ yf

_{y}= n f(x, y)

Substituting x = 0; n=-96 and y = 1 we have

f

_{y}= -96. f(0, 1) = -96.(1⁄1)

= – 96.

3. A non-polynomial function can never agree with eulers theorem

a) True

b) false

View Answer

Explanation: Counter example is the function

.

4. Find the value of f_{x} at (1,0)

a) 23

b) 16

c) 17(sin(2) + cos(1⁄2) )

d) 90

View Answer

Explanation : Using Eulers theorem we have

xf

_{x}+ yf

_{y}= nf(x, y)

Substituting (x,y)=(1,0) we have

f_{x} = 17f(1, 0)

17 (sin(2) + cos(1⁄2)).

5. For a homogenous function if critical points exist the value at critical points is

a) 1

b) equal to its degree

c) 0

d) -1

View Answer

Explanation: Using Euler theorem we have

xf

_{x}+ yf

_{y}= nf(x, y)

At critical points f

_{x}= f

_{y}= 0

f(a, b) = 0(a, b) → criticalpoints.

6. For homogenous function with no saddle points we must have the minimum value as

a) 90

b) 1

c) equal to degree

d) 0

View Answer

Explanation: Substituting f

_{x}= f

_{y}= 0 At critical points in euler theorem we have

nf(a, b) = 0 ⇒ f(a, b) = 0(a, b) → criticalpoints.

7. For homogenous function the linear combination of rates of independent change along x and y axes is

a) Integral multiple of function value

b) no relation to function value

c) real multiple of function value

d) depends if the function is a polynomial

View Answer

Explanation: Eulers theorem is nothing but the linear combination asked here, The degree of the homogenous function can be a real number. Hence, the value is integral multiple of real number.

8. A foil is to be put as shield over a cake (circular) in a shape such that the heat is even along any diameter of the cake.

Given that the heat on cake is proportional to the height of foil over cake, the shape of the foil is given by

a) f(x, y) = sin(y⁄x)x^{2} + xy

b) f(x, y) = x^{2} + y^{3}

c) f(x, y) = x^{2}y^{2} + x^{3}y^{3}

d) not possible by any analytical function

View Answer

Explanation:Given that the heat is same along lines we need to choose a homogenous function.

Checking options we get that only option satisfies condition for homogenity.

9. f(x, y) = sin(y⁄x)x^{3} + x^{2}y find the value of f_{x} + f_{y} at (x,y)=(4,4)

a) 0

b) 78

c) 4^{2} . 3(sin(1) + 1)

d) -12

View Answer

Explanation: Using Euler theorem we have

xf

_{x}+ yf

_{y}= nf(x, y)

Substituting (x,y)=(4,4) we have

4f_{x} + 4f_{y} = 3f(4, 4) = 3⁄4(4^{3} . sin(1) + 4^{3})

= 4^{2} . 3(sin(1) + 1).

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

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