If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4x+1=0$,then $(\alpha+\beta)^{-1}+(\beta+\gamma)^{-1}+(\gamma+\alpha)^{-1}$ is equal to

The diameter of the objective of a telescope is $1 \ m$. Its resolving limit for light of wavelength $4538 \ \mathring{A}$ will be:

For which of the following pairs of complexes is the crystal field splitting energy $\Delta_0$ for $II > \Delta_0$ for $I$?

$I$	$II$
$(a). [Cr(H_2O)_6]^{2+}$	$[Cr(H_2O)_6]^{3+}$
$(b). [Fe(H_2O)_6]^{3+}$	$[Fe(CN)_6]^{3-}$
$(c). [Fe(CN)_6]^{3-}$	$[Ru(CN)_6]^{3-}$
$(d). [NiF_6]^{4-}$	$[NiF_6]^{2-}$

$A$ charge $Q$ is divided into two charges $q$ and $Q-q$. The value of $q$ such that the force between them is maximum,is

If $y=3x$ is a tangent to a circle with centre $(1,1)$,then the other tangent drawn through $(0,0)$ to the circle is

Let $a, b, c \in R$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy$ for all $x, y \in R$,then $\sum_{n=1}^{10} f(n)$ is equal to: