If x is a positive real number different from 1 such that log_ax, log_bx, log_cx are in AP, then

If z₁, z₂, z₃ are imaginary numbers such that $$|{z_1}|\, = \,|{z_2}|\, = \,|{z_3}|\, = \,\left| {{1 \over {{z_1}}} + {1 \over {{z_2}}} + {1 \over {{z_3}}}} \right|\, = \,1$$, then $$|{z_1} + {z_2} + {z_3}|$$ is

The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together, is

$$1 + {}^n{C_1}\cos \theta + {}^n{C_2}\cos 2\theta + ... + {}^n{C_n}\cos n\theta $$ equals