If `lim_(x rarr 0) (x+ln(sqrt(x^2 +1)-x))/x^3 = p/q` where p and q are coprimes then find the value of `|p^2-q^2|.`

If 2+ i sqrt3 is a root of the equation x^(2)+px+q=0 , where p and q are real then find the value of p and q.

If lim_(x rarr 0) (1 + px)^(q/x) = e^(4), where p,q in N , then

If the value of lim_(x->0)( ((3/ x)+1)/((3/x)-1))^(1/x) can be expressed in the form of e^(p/q), where p and q are relative prime then find the value of (p+q) .

lim_(x rarr0)(p^(x)-q^(x))/(tan x)

If x^(2)-2x-1 is a factor of px^(3)+qx^(2)+1 , (where p , q are integers) then find the value of p +q.

consider f(x)==cos^-1x+xos((sqrt3 x+sqrt(1-x^2))/2) , if sum_(r=1)^100 f(1-(1/10)^r)=p/q pi where p and q relatively prome number then find the value of p-16q

consider f(x)==cos^-1x+xos((sqrt3 x+sqrt(1-x^2))/2) , if sum_(r=1)^100 f(1-(1/10)^r)=p/q pi where p and q relatively prome number then find the value of p-16q

If lim_(x rarr 0) (1+px)^(q//x) = e^(4) , where p, q, in N, then :

If the area bounded by y = ||x|^(2) - 4|x| + 3|| and the x-axis from x = 1 to x = 3 is p/q (where, p & q are coprime) then the value of p + q is

If the area bounded by y = ||x|^(2) - 4|x| + 3|| and the x-axis from x = 1 to x = 3 is p/q (where, p & q are coprime) then the value of p + q is