PARTIAL DIFFERENTIATION IMPORTANT MCQ's

PARTIAL DIFFERENTIATION IMPORTANT MCQ's

1. f(x, y) = x2 + xyz + z Find fx at (1,1,1)

a) 0
b) 1
c) 3
d) -1

2. f(x, y) = sin(xy) + x2 ln(y) Find fyx at (0, π⁄2)
a) 33
b) 0
c) 3
d) 1

3. f(x, y) = x2 + y3 ; X = t2 + t3; y = t3 + t9 Find df⁄dt at t=1.
a) 0
b) 1
c)-1
d) 164

4. f(x, y) = sin(x) + cos(y) + xy2; x = cos(t); y = sin(t) Find df⁄dt at t = π⁄2
a) 2
b)-2
c) 1
d) 0

5. f(x, y, z, t) = xy + zt + x2 yzt; x = k3 ; y = k2; z = k; t = √k
Find df⁄dt at k = 1
a) 34
b) 16
c) 32
d) 61

6. The existence of first order partial derivatives implies continuity.
a) True
b) False

7. The gradient of a function is parallel to the velocity vector of the level curve.
a) True
b) False

8. f(x, y) = sin(y + yx2) / 1 + x2 Value of fxy at (0,1) is
a) 0
b) 1
c) 67
d) 90

9. f(x, y) = sin(xy + x3y) / x + x3 Find fxy at (0,1).
a) 2
b) 5
c) 1
d) undefined

10. Differentiation of function f(x,y,z) = Sin(x)Sin(y)Sin(z)-Cos(x) Cos(y) Cos(z) w.r.t ‘y’ is?
a) f’(x,y,z) = Cos(x)Cos(y)Sin(z) + Sin(x)Sin(y)Cos(z)
b) f’(x,y,z) = Sin(x)Cos(y)Sin(z) + Cos(x)Sin(y)Cos(z)
c) f’(x,y,z) = Cos(x)Cos(y)Cos(z) + Sin(x)Sin(y)Sin(z)
d) f’(x,y,z) = Sin(x)Sin(y)Sin(z) + Cos(x)Cos(y)Cos(z)

11. 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

12. 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

13. 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

14. If f(x,y) = x+y⁄y , x ∂z⁄∂x + y ∂z⁄∂y = ?
a) 0
b) 1
c) 2
d) 3

15. Does function f(x,y) = Sin−1[(x√+y√)x+y√] can be solved by euler’ s theorem.
a) True
b) False

16. 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

17. If u = xx + yy + zz , find du⁄dx + du⁄dy + du⁄dz at x = y = z = 1.
a) 1
b) 0
c) 2u
d) u

18. If u=x2tan−1(yx)−y2tan−1(xy) then ∂2u∂x∂y is?
a) x2+y2x2−y2
b) x2−y2x2+y2
c) x2x2+y2
d) y2x2+y2

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+2xy∂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. Find the approximate value of [0.982 + 2.012 + 1.942](1⁄2).
a) 1.96
b) 2.96
c) 0.04
d) -0.04

21. The happiness(H) of a person depends upon the money he earned(m) and the time spend by him with his family(h) and is given by equation H=f(m,h)=400mh2 whereas the money earned by him is also depends upon the time spend by him with his family and is given by m(h)=√(1-h2). Find the time spend by him with his family so that the happiness of a person is maximum.
a) √(1⁄3)
b) √(2⁄3)
c) √(4⁄3)
d) 0