12. Let x, y and z be independent variables and let the functions fi, f2 and f3 of x, y and z be defined by
fi(x,y,z)=y2+z2 f2(x,y,z)=x2+z2 f(x,y,z)=x2+y2.
What is the determinant of the Jacobian matrix of this system of functions?
(A) -16
(B) -16xyz
(C) 0
(D) 16
(E) 16xyz
13. Consider the line L in R3 which is the intersection of two planes with equations
x-y-z=2
and
x+y+z=1.
Which point P = (x,y,z) on L is closest to the origin?
(C) 2'4'4
(D)( 2 '4'4
2'44
2 4'4
4
14. Consider the differential equation
4y=0
where y is a function of the variable t and y" denotes the second derivative of y.
Letting c and c2 denote real constants, what is the general solution y(t) of the differential equation?
(A) ce2t+c2e-2t
(Bce+c2e-t
(C) c1 cos(t)+c2 sin(t)
(D) c1cos(2t)+c2sin(2t)
(E) ce2tcos(t)+c2e-2t sin(t)