Let `a ,b in N and a > 1.` Also `p` is a prime number. If `a x^2+b x+c=p` for any intergral values of `x ,` then prove that `ax^2+b x+c!=2p` for any integral value of `xdot`

Let f(x)=Ax^2+Bx+C where A, B, C are three real constants, if f(x) is integer for integral values of x, then prove that each of 2A, (A+B) and C is an integer. Conversely, if each of 2A, (A+B) , C is an integer then f(x) will be integer for integral values of x.

If a ,b ,c are three distinct real numbers in G.P. and a+b+c=x b , then prove that either x 3.

Suppose a,b,c are in A.P and a^2 , b^2 , c^2 are in G.P if a

If a^2, b^2, c^2 are in A.P , then prove that 1/(b+c), 1/(c+a),1/(a+b) are in A.P

If (x + 1) and (x + 2) be any two factors of the polynomial (x^(3)+3x^(2)+2ax+b) , then determine the value of a and b.

Number of positive integral ordered pairs of (x,y), so that 8,x,y are in H.P. is a. 0 b. 1 c. 2. d. 3

If x^2+p x-444p=0 has integral roots where p is prime number, then find the value of p .

If a,b,c are in G. P. and x, y are A.M. of a, b and b, c respectively, then the value of ((a)/(x)+ (c )/(y)) is-

If a^(x) = b^(y) = c^(z) and a, b, c, are in G.P., prove that x, y, z are in H.P.