20 MOST IMPORTANT EULER'S THEOREM MCQ'S

20 MOST IMPORTANT EULER'S THEOREM MCQ'S

1. f(x, y) = x3 + xy2 + 901 satisfies the Euler’s theorem.
a) True
b) False

2. f(x, y)=x3+y3x99+y98x+y99 find the value of fy at (x,y) = (0,1).
a) 101
b) -96
c) 210
d) 0

3. A non-polynomial function can never agree with euler’s theorem.
a) True
b) false

4. f(x,y)=x9.y8sin(x2+y2xy)+cos(x3x2y+yx2)x11.y6 Find the value of fx at (1,0).
a) 23
b) 16
c) 17(sin(2) + cos(1⁄2))
d) 90

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

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

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

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

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

10. 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

11. If z = xn 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

12. 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

13. If z=ex2+y2x+y then, x∂z∂x+y∂z∂y is?
a) 0
b) zln(z)
c) z2 ln⁡(z)
d) z

14. If z=sin−1x3+y3+z3x+y+z then, x∂z∂x+y∂z∂y.
a) 2 tan(z)
b) 2 cot(z)
c) tan(z)
d) cot(z)

15. Value of x∂u∂x+y∂u∂y if u=Sin−1(yx)(x√+y√)x3+y3 is?
a) -2.5 u
b) -1.5 u
c) 0
d) -0.5 u

16. 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

17. If z=ln(x2+y2x+y)−ex2+y2x+y then find x∂z∂x+y∂z∂y.
a) x∂z∂x+y∂z∂y=x2+y2x+yex2+y2x+y
b) x∂z∂x+y∂z∂y=1−x2+y2x+yex2+y2x+y
c) x∂z∂x+y∂z∂y=1+x2+y2x+yex2+y2x+y
d) x∂z∂x+y∂z∂y=−x2+y2x+yex2+y2x+y

18. 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)

19. If f(x,y)is a function satisfying euler’ s theorem then?
a) x2∂2f∂x2+2xy∂2f∂x∂y+y2∂2f∂y2=n(n−1)f
b) 1x2∂2f∂x2+2/xy∂2f∂x∂y+1y2∂2f∂y2=n(n−1)f
c) x2∂2f∂x2+2xy∂2f∂x∂y+y2∂2f∂y2=nf
d) y2∂2f∂x2+2xy∂2f∂x∂y+x2∂2f∂y2=n(n−1)f

20. If u=Tan−1(x3+y3x+y) then, x2∂2u∂x2+y2∂2u∂y+2xy∂2u∂x∂y is?
a) Sin(4u) – Cos(2u)
b) Sin(4u) – Sin(2u)
c) Cos(4u) – Sin(2u)
d) Cos(4u) – Cos(2u)