JACOBIAN MOST IMPORTANT MCQ's

JACOBIAN MOST IMPORTANT MCQ's

1. The jacobian of p,q,r w.r.t x,y,z given p=x+y+z, q=y+z, r=z is ________

a) 0
b) 1
c) 2
d) -1

2. Given \(u=\frac{yz}{x}, v=\frac{zx}{y}, w=\frac{xy}{z}\) then the value of \(\frac{∂(u,v,w)}{∂(x,y,z)}\) is ________
a) 4
b) -4
c) 0
d) 1

3. If u=x+3y2-z3, v=4x2 yz, w=2z2-xy then \(\frac{∂(u,v,w)}{∂(x,y,z)} \) at (1,1,1).
a) -184
b) -90
c) 20
d) 40

4. If x=rcos⁡θ, y=rsin⁡θ then the value of \(\frac{∂(x,y)}{∂(r,θ)}\) is ________
a) 1
b) 0
c) r
d) \(\frac{1}{r}\)

5. The application of Jacobians is significant in the evaluation of double integral of the form ∬f(x,y) dx dy and triple integral of the form ∭f(x,y,z)dx dy dz by transformation from one system of coordinate to the other.
a) True
b) False

6. If u+v=ex cos⁡y and u-v=ex sin y the value of \(J(\frac{u,v}{x,y})\) is ________
a) e2x
b) \(\frac{e^2x}{2} \)
c) \(\frac{-e^2x}{2} \)
d) 0

7. Which among the following is the definition of Jacobian of u and v w.r.t x and y?
a) \(J(\frac{x,y}{u,v})\)
b) \(J(\frac{u,v}{x,y})\)
c) \(\frac{∂(x,y)}{∂(u,v)} \)
d) \(\frac{∂(u,x)}{∂(v,y)} \)