This set of Engineering Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Euler’s Theorem – 1”.
1. f(x, y) = x3 + xy2 + 901 satisfies the Euler’s theorem.
a) True
b) False
View Answer
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 fy at (x,y) = (0,1).
a) 101
b) -96
c) 210
d) 0
View Answer
Explanation: Using Euler theorem
xfx + yfy = n f(x, y)
Substituting x = 0; n=-96 and y = 1 we have
fy = -96. f(0, 1) = -96.(1⁄1)
= – 96.
3. A non-polynomial function can never agree with euler’s theorem.
a) True
b) false
View Answer
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 fx at (1,0).
a) 23
b) 16
c) 17(sin(2) + cos(1⁄2))
d) 90
View Answer
Explanation: Using Eulers theorem we have
xfx + yfy = nf(x, y)
Substituting (x,y)=(1,0) we have
fx = 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
View Answer
Explanation: Using Euler theorem we have
xfx + yfy = nf(x, y)
At critical points fx = fy = 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
View Answer
Explanation: Substituting fx = fy = 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
View Answer
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
View Answer
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 fx + fy at (x,y)=(4,4).
a) 0
b) 78
c) 42 . 3(sin(1) + 1)
d) -12
View Answer
Explanation: Using Euler theorem we have
xfx + yfy = nf(x, y)
Substituting (x,y)=(4,4) we have
4fx + 4fy = 3f(4, 4) = 3⁄4(43 . sin(1) + 43)
= 42 . 3(sin(1) + 1).
Sanfoundry Global Education & Learning Series – Engineering Mathematics.
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