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 :

The value of $$\alpha $$ for which
$$4\alpha \int\limits_{ - 1}^2 {{e^{ - \alpha \left| x \right|}}dx} = 5$$, is:

The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y is :