[Shirsendu Roy]

 

(1-p)^{2}=p^{2}/16

or, 1-p=p/4

or, 1=p/4+p

or, 1=5p/4

or, p=4/5

or, 5+4=9

[Shirsendu Roy]

a is true, b is true

a is true by differential equations

b is true by differential equations and integration

[Shirsendu Roy]

Let d=gcd{(a-1),(a^{p}-1/(a-1))}

then d=1 where both are prime to each other

or, d|{-(a-1)+(a-1)(a^{p}-1/(a-1))}

or,d|{a^{p}-1-(a-1)}

or, d|pq where q is some integer by fermat's theorem

or, d=p.

[Shirsendu Roy]

x^5+x-10=0

There is only one sign change then there is only one real root

f(1)<0,f(2)>0 then the real root lies between 1 and 2

here Let rational number gives zero's of nth degree polynomial f(x) =a0+a1x+a2x^2+....+anx^n of integer coefficients

If p/q is rational zero of f(x), being p and q relatively prime integers, qnot equals 0then none of these integers is a zero of f(x).

then root is irrational

[Shirsendu Roy]

Let points be p1,p2,....,p8

placing point p8 at the centre of the circle and placing other seven points on the periphery of the circle, at equal distances.

Now we have one point at the centre seven points at the periphery of the circle. The heptagon which will be formed by joining the seven points on the circle will have all its side equal in length. The length of the side of the heptagon is less than 2pi/7 cm which is less than 1 cm.

Now if we replace the position of any point then too we get a pair of points whose distance from each other is less than 1 cm.

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