I−T;II−P;III−S;IV−Q (I) Let the sides a and b are the roots of x2−5x+3=0, then a+b=5 and ab=3 Alos, ∠C=
π
3
∴
1
2
=
a2+b2−c2
2ab
⇒ab=(a+b)2−2ab−c2⇒c=4 Now, rR=
abc
2(a+b+c)
=
3×4
2(5+4)
=
2
3
(II) We have a=
r1(r2+r3)
√r1r2+r2r3+r3r1
=
8×36
24
=12 (III) Give 1≤a≤10 1 Also, for roots of opposite sign 2a2−7a+3<0 or
1
2
<a<3⇒1 or 2 ∴ Desired probability =
2
10
=
1
5
(IV) (α,β) lies on the director circle of the ellipse i. e. on x2+y2=9 So, we can assume α=3cosθ,β=3sinθ Thus F=12cosθ+9sinθ=3(4cosθ+3sinθ) ⇒−15≤F≤15