If the distance between the plane, 23x – 10y – 2z + 48 = 0 and the plane

containing the lines $${{x + 1} \over 2} = {{y - 3} \over 4} = {{z + 1} \over 3}$$

and $${{x + 3} \over 2} = {{y + 2} \over 6} = {{z - 1} \over \lambda }\left( {\lambda \in R} \right)$$

is equal to $${k \over {\sqrt {633} }}$$, then k is equal to ______.

If C_r $$ \equiv $$ ²⁵C_r and
C₀ + 5.C₁ + 9.C₂ + .... + (101).C₂₅ = 2²⁵.k, then k is equal to _____.

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 ______.

A wire of length L and mass per unit length 6.0 × 10^–3 kgm^–1 is put under tension of 540 N. Two consecutive frequencies that it resonates at are : 420 Hz and 490 Hz. Then L in meters is :