250+ TOP MCQs on Euler’s Theorem and Answers

Engineering Mathematics Multiple Choice Questions on “Euler’s Theorem – 1”.

1. f(x, y) = x³ + xy² + 901 satisfies the Euler’s theorem.
a) True
b) False
Answer: b
Explanation: The function is not homogenous and hence does not satisfy the condition posed by euler’s theorem.

2. f(x, y)=(frac{x^3+y^3}{x^{99}+y^{98}x+y^{99}}) find the value of f_y at (x,y) = (0,1).
a) 101
b) -96
c) 210
d) 0
Answer: b
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 euler’s theorem.
a) True
b) false
Answer: b
Explanation: Counter example is the function
(f(x, y)=x^9.y^8sin(frac{x^2+y^2}{xy})+cos(frac{x^3}{x^2y+yx^2})x^{11}.y^6).

4. (f(x, y)=x^9.y^8sin(frac{x^2+y^2}{xy})+cos(frac{x^3}{x^2y+yx^2})x^{11}.y^6) Find the value of f_x at (1,0).
a) 23
b) 16
c) 17(sin(2) + cos(1⁄2))
d) 90
Answer: c
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 homogeneous function if critical points exist the value at critical points is?
a) 1
b) equal to its degree
c) 0
d) -1
Answer: c
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) → critical points.

6. For homogeneous function with no saddle points we must have the minimum value as _____________
a) 90
b) 1
c) equal to degree
d) 0
Answer: d
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) → critical points.

7. For homogeneous 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
Answer: c
Explanation: Euler’s theorem is nothing but the linear combination asked here, The degree of the homogeneous 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)x² + xy
b) f(x, y) = x² + y³
c) f(x, y) = x²y² + x³y³
d) not possible by any analytical function
Answer: b
Explanation:Given that the heat is same along lines we need to choose a homogeneous function.
Checking options we get that only option satisfies condition for homogeneity.

9. f(x, y) = sin(y/x)x³ + x²y find the value of f_x + f_y at (x,y)=(4,4).
a) 0
b) 78
c) 4² . 3(sin(1) + 1)
d) -12
Answer: c
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(4³ . sin(1) + 4³)
= 4² . 3(sin(1) + 1).