The equation `[1 x y][(1,3,1),(0,2,-1),(0,0,1)] [(1),(x),(y)]=[0]` has
i)for y=0 (p) rational roots
ii) for y=-1 (q) irrational roots
(r) integral roots

Options

A. `{:("(i)",(ii)),((p),"(r)"):}`
B. `{:("(i)",(ii)),((q),"(p)"):}`
C. `{:("(i)",(ii)),((p),"(q)"):}`
D. `{:("(i)",(ii)),((r),"(p)"):}`

Answer

C

Explanation

`[(1,x,y)][(1,3,1),(0,2,-1),(0,0,1)]=[(1),(x),(y)]=10`
or `[(1,3+2x,1-x+y)] [(1),(x),(y)]=[0]`
or `[1+3x+2x^(2)+y-xy+y^(2)]=[0]`
or `2x^(2)+y^(2)+y+3x-xy+1=0`
if `y=0, 2x^(2)+3x+1=0`
or `(2x+1) (x+1)=0`
or `x=-1//2, -1` (rational roots)
If `y=-1, 2x^(2)+4x+1=0`
or `x=(-4 pm sqrt(12))/4 =(-2 pm sqrt(3))/2` (irrational roots)