If C₀, C₁, C₂, _ _ _ _ _, C_n are the coefficients in the expansion of (1 + x)ⁿ, then what is the value of C₁ + C₂ + C₃ + _ _ _ _ _ + C_n?

If a + b + c = 4 and ab + bc + ca = 0, then what is the value of the following determinant?

$\left| {\begin{array}{*{20}{c}} {{a}}&{{b}}&{{c}}\\ {{b}}&{{c}}&{{a}}\\ {{c}}&{{a}}&{{b}} \end{array}} \right|$

The number of integer values of k, for which the equation 2sinx = 2k + 1 has a solution, is

If a₁, a₂, a₃, _ _ _ _ _, a₉ are in GP, then what is the value of the following determinant?

$\left| {\begin{array}{*{20}{c}} {{ln\:a_1}}&{{ln\:a_2}}&{{ln\:a_3}}\\ {{ln\:a_4}}&{{ln\:a_5}}&{{ln\:a_6}}\\ {{ln\:a_7}}&{{ln\:a_8}}&{{ln\:a_9}} \end{array}} \right|$