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\(\because \) zr is complex root of equation z4 + z3 + 2 = 0 where r = 1, 2, 3, 4.

\(\therefore \sum z_r = -\frac ba = -\frac{1}1 = -1\)

\(\sum z_i z_j = \frac Ca = \frac 01 = 0\)

\(\sum z_i z_jz_k = \frac{-d}a = \frac 01 = 0\)

\(\prod z_r = \frac ea = \frac 21 = 2\)

\(\prod\limits _{r = 1}^{4} (2z_r + 1) = 16 \prod\limits_{r=1}^4 Z_Rr + 8 \sum z_i z_jz_k + 4\sum z_iz_j + 2\sum z_r + 1\)

\(= 16 \times 2 +8\times 0+ 4\times 0+ 2\times-1 + 1\)

\(= 32 - 2+ 1\)

\(= 31\)

Option (B) is correct.

\(\sum\limits_{i =1}^4 \sum\limits_{j =1}^4 z_i z_j = 0\)

\(\therefore Im\left(\sum\limits_{i =1}^4 \sum\limits_{j=1}^4 z_iz_j\right) = 0\)

Option (D) is also correct.

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