The number of integral values of m for which the equation

(1 + m² )x² – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is :

If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0, 10), then the distance of R from the origin is :

Let the number 2,b,c be in an A.P. and
A = $$\left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4 & {{b^2}} & {{c^2}} \cr } } \right]$$. If det(A) $$ \in $$ [2, 16], then c lies in the interval :

Let ƒ : R $$ \to $$ R be a differentiable function satisfying ƒ'(3) + ƒ'(2) = 0.
Then $$\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + f(3 + x) - f(3)} \over {1 + f(2 - x) - f(2)}}} \right)^{{1 \over x}}}$$ is equal to