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Question

Question: Let \(|{z_i}| = i,i = 1,2,3,4\)and \(|16{z_1}{z_2}{z_3} + 9{z_1}{z_2}{z_4} + 4{z_1}{z_3}{z_4} + {z_2...

Let zi=i,i=1,2,3,4|{z_i}| = i,i = 1,2,3,4and 16z1z2z3+9z1z2z4+4z1z3z4+z2z3z4=48|16{z_1}{z_2}{z_3} + 9{z_1}{z_2}{z_4} + 4{z_1}{z_3}{z_4} + {z_2}{z_3}{z_4}| = 48, then the value of 1z1+4z2+9z3+16z4|\dfrac{1}{{{z_1}}} + \dfrac{4}{{{z_2}}} + \dfrac{9}{{{z_3}}} + \dfrac{{16}}{{{z_4}}}|.
(A). 1
(B). 2
(C). 4
(D). 8

Explanation

Solution

Hint – We know, zi=i,i=1,2,3,4|{z_i}| = i,i = 1,2,3,4, so, z1=1|{z_1}| = 1, z2=2|{z_2}| = 2, z3=3|{z_3}| = 3, z4=4|{z_4}| = 4, Now take out z1z2z3z4|{z_1}||{z_2}||{z_3}||{z_4}| common from the 2nd2_{nd} equation to obtain the required expression.

Complete step-by-step answer:
According to question we have,
zi=i,i=1,2,3,4|{z_i}| = i,i = 1,2,3,4
Therefore, z1=1|{z_1}| = 1, z2=2|{z_2}| = 2, z3=3|{z_3}| = 3, z4=4|{z_4}| = 4.
Also, 16z1z2z3+9z1z2z4+4z1z3z4+z2z3z4=48|16{z_1}{z_2}{z_3} + 9{z_1}{z_2}{z_4} + 4{z_1}{z_3}{z_4} + {z_2}{z_3}{z_4}| = 48
Solving the given equation further by taking z1z2z3z4|{z_1}||{z_2}||{z_3}||{z_4}| common, we get-

16z1z2z3+9z1z2z4+4z1z3z4+z2z3z4=48 =z1z2z3z416z4+9z3+4z2+1z1=48(1)  |16{z_1}{z_2}{z_3} + 9{z_1}{z_2}{z_4} + 4{z_1}{z_3}{z_4} + {z_2}{z_3}{z_4}| = 48 \\\ = |{z_1}||{z_2}||{z_3}||{z_4}||\dfrac{{16}}{{{z_4}}} + \dfrac{9}{{{z_3}}} + \dfrac{4}{{{z_2}}} + \dfrac{1}{{{z_1}}}| = 48 - (1) \\\

Put the value of z1=1|{z_1}| = 1, z2=2|{z_2}| = 2, z3=3|{z_3}| = 3, z4=4|{z_4}| = 4 in equation (1), we get-

=(1)(2)(3)(4)16z4+9z3+4z2+1z1=48 1z1+4z2+9z3+16z4=4824=2  = (1)(2)(3)(4)|\dfrac{{16}}{{{z_4}}} + \dfrac{9}{{{z_3}}} + \dfrac{4}{{{z_2}}} + \dfrac{1}{{{z_1}}}| = 48 \\\ \Rightarrow |\dfrac{1}{{{z_1}}} + \dfrac{4}{{{z_2}}} + \dfrac{9}{{{z_3}}} + \dfrac{{16}}{{{z_4}}}| = \dfrac{{48}}{{24}} = 2 \\\

Hence, the answer is option (B), i.e. 1z1+4z2+9z3+16z4=2|\dfrac{1}{{{z_1}}} + \dfrac{4}{{{z_2}}} + \dfrac{9}{{{z_3}}} + \dfrac{{16}}{{{z_4}}}| = 2.

Note – Whenever such types of questions appear, then always write the information given in the question. Use that information to form equations and solve the question. As, mentioned in the solution, we have found out the values of z1=1|{z_1}| = 1, z2=2|{z_2}| = 2, z3=3|{z_3}| = 3, z4=4|{z_4}| = 4 by using Let zi=i,i=1,2,3,4|{z_i}| = i,i = 1,2,3,4, which is provided in the question and then using these values, we have solved the question.