Consider the following reactions :

$$NaCl{\rm{ }} + {\rm{ }}{K_2}C{r_2}{O_7} + \mathop {{H_2}S{O_4}}\limits_{(conc.)} $$$$ \to $$ (A) + side products

(A) + NaOH $$ \to $$ (B) + Side products

$$\left( B \right){\rm{ }} + \mathop {{H_2}S{O_4}}\limits_{(dilute)} + {\rm{ }}{H_2}{O_2}$$ $$ \to $$ (C) + Side products

The sum of the total number of atoms in one molecule each of (A) and (B) and (C) is

Let $${a_1}$$ , $${a_2}$$ , $${a_3}$$ ,....... be a G.P. such that
$${a_1}$$ < 0, $${a_1}$$ + $${a_2}$$ = 4 and $${a_3}$$ + $${a_4}$$ = 16.
If $$\sum\limits_{i = 1}^9 {{a_i}} = 4\lambda $$, then $$\lambda $$ is equal to:

The coefficient of x⁷ in the expression
(1 + x)¹⁰ + x(1 + x)⁹ + x²(1 + x)⁸ + ......+ x¹⁰ is:

The number of ordered pairs (r, k) for which
6.³⁵C_r = (k² - 3). ³⁶C_{r + 1}, where k is an integer, is :