Mixed

Please post a brief soln to the problem

1) The no. of right triangles with integer sides and inradius r=2013

2)Let a,b,c,d be non zero digits. If n is the no. of 4 digit no.s abcd such that ab+cd is even, then last digit of n is :

3)If the eqn of ax²+y²+bz²+ 2yz+zx+3xy=0 represents a pair of perpendicular planes, then a=?

4)Let f(x) be aone-one polynomial function func such that f(x)f(y)=f(x)+f(y)+f(xy), for all x,y belonging to R-{0}, f(1)≠1, f'(1)=3.

Let g(x) =Ï€4(f(x)+3) - ∫₀^xf(x)dx, then

a) g(x)=0 has exactly 1 root for x belongs to (0,1)
b) g(x)=0 has exactly 2 root for x belongs to (0,1)
c)g(x)≠0 for x belongs to R-{0}
d)g(x)=0 for x belongs to R-{0}

5)Letf₁(x) and f₂(x) be continuous and differentianble
If f₁(0)=f₁(2)=f₁(4)
and f₁(1)+f₁(3)=0=f₂(2)=f₂(4)=.
If f₁(x) and f₂'(x) have no common root, then the min no. of common roots of
f₁'(x)f₂'(x)+f₁(x)f₂''(x) in [0,4] is