Let `P_n(ix) =1+2x+3x^2+............+(n+1)x^n` be a polynomial such that n is even.The number of real roots of `P_n(x)=0` is

Let P_n(x) =1+2x+3x^2+............+(n+1)x^n be a polynomial such that n is even.The number of real roots of P_n(x)=0 is

Let P(x) = a_(0) + a_(1)x^(2) + a_(2)x^(4) + ..........+ a_(n) x^(2n) be a polynomial in a real variable x with 0 lt a_(0) lt a_(1) lt a_(2) lt ……………. lt a_(n) The function P(x) has :

In the given figure graph of : y =p (x) = x ^(n)+a_(1) x ^(n-1) +a_(2) x ^(n-2)+ …..+ a _(n) is given. The minimum number of real roots of equation (p' (x))^(2) +p(x)p''(x) =0 are:

In the given figure graph of : y =p (x) = x ^(n)+a_(1) x ^(n-1) +a_(2) x ^(n-2)+ …..+ a _(n) is given. The minimum number of real roots of equation (p' (x))^(2) +p(x)p''(x) =0 are:

Let P(x) be a polynomial such that p(x)-p'(x)= x^(n) , where n is a positive integer. Then P(0) equals-

In the given figure graph of : y =p (x) = x ^(n)+a_(1) x ^(n-1) +a_(2) x ^(n-2)+ ….. + a _(n) is given. The product of all imaginary roots of p(x) =0 is:

In the given figure graph of : y =p (x) = x ^(n)+a_(1) x ^(n-1) +a_(2) x ^(n-2)+ ….. + a _(n) is given. The product of all imaginary roots of p(x) =0 is:

Let n be a positive integer with f(n) = 1! + 2! + 3!+.........+n! and p(x),Q(x) be polynomial in x such that f(n+2)=P(n)f(n+1)+Q(n)f(n) for all n >=1 , Then p(2)=

Let n be a positive integer with f(n) = 1! + 2! + 3!+.........+n! and p(x),Q(x) be polynomial in x such that f(n+2)=P(n)f(n+1)+Q(n)f(n) for all n >= 1, Then (a) P(x) = x+3 (b) Q(x) = -x-2 (c) P(x) = -x-2 (d) Q(x) = x+3