IIT JEE , AIEEE Forum - Mathematics , Algebra - (1) If a, b, c are natural no. than find out the probability of a2+b2+c2 is divisible by 7

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(1) If a, b, c are natural no. than find out the probability of a2+b2+c2 is divisible by 7.

(2) Find out the minimum non negative value of a, b, c if the polynomials x⁴+ax³+bx²+cx+d has real root.

(3) Find out the maximum value of (x-a)(x-b)(x-c)(ax+by+cz).If it is given that all a,b,c,(x-a),(x-b),(x-c) are positive.

(4) If f(x) = (a/x) + (b/x+b)-(c/x+c)-(d/x+d), and f(x) is divisible by x2, than find out the value of (1/a) + (1/b)-(1/c)-(1/d).

(5) Find all +ve integers a, b such that each of the equations x²-ax+b=0 and x²-bx+a=0 has distinct +ve integrals root.