If the curves, x² – 6x + y² + 8 = 0 and
x² – 8y + y² + 16 – k = 0, (k > 0) touch each other at a point, then the largest value of k is ______.

Let ƒ and g be differentiable functions on R such that fog is the identity function. If for some a, b $$ \in $$ R, g'(a) = 5 and g(a) = b, then ƒ'(b) is equal to :

Let a_n be the n^th term of a G.P. of positive terms.

$$\sum\limits_{n = 1}^{100} {{a_{2n + 1}} = 200} $$ and $$\sum\limits_{n = 1}^{100} {{a_{2n}} = 100} $$,

then $$\sum\limits_{n = 1}^{200} {{a_n}} $$ is equal to :

In a meter bridge experiment S is a standard resistance. R is a resistance wire. It is found that balancing length is $$l$$ = 25 cm. If R is replaced by a wire of half length and half diameter that of R of same material, then the balancing distance $$l'$$ (in cm) will now be________.