Let zr1 - r - 4 be complex numbers such that -zr-r-1 and -30 z1-20 z2-15 z3-12 z4-k-z1z2z3-z2z3z4-z3z4z1-z4z1z2- Then value of k equals -

The correct option is D |z_4z_1z_2| |z_1| = √(2), |z_2| = √(3), |z_3| = √(4), |z_4| = √(5) | 30z_1 + 20 z_2 + 15 z_3 + 12 z _4 | = k | z_1z_2z ...

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Properties of Modulus

Let zr1 ≤ r...

Question

Let $z_{r} (1 \leq r \leq 4)$ be complex numbers such that $| z_{r} | = \sqrt{r + 1} a n d | 30 z_{1} + 20 z_{2} + 15 z_{3} + 12 z_{4} | = k | z_{1} z_{2} z_{3} + z_{2} z_{3} z_{4} + z_{3} z_{4} z_{1} + z_{4} z_{1} z_{2} |$ . Then value of $k$ equals ?

| z_{1} z_{2} z_{3} |

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| z_{2} z_{3} z_{4} |

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| z_{3} z_{4} z_{1} |

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| z_{4} z_{1} z_{2} |

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Solution

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z_{r} (i \leq r \leq 4)

| Z_{r} | = \sqrt{r + 1}

| 30 z_{1} + 20 z_{2} + 15 z_{3} + 12 z_{4} | = K | z_{1} z_{2} z_{3} + z_{2} z_{3} z_{4} + z_{3} z_{4} z_{1} + z_{4} z_{1} z_{2} |

z_{K}

| z_{k} | = \sqrt{K + 1} & | 30 z_{1} + 20 z_{2} + 15 z_{3} + 12 z_{4} | = K | \sum z_{1} z_{2} z_{3} |

Z_{r} = c o s (\frac{π}{2^{r}}) + i s i n (\frac{π}{2^{r}})

r = 1, 2, 3, . . . .

Z_{1} Z_{2} Z_{3} . . . . . . . \infty =

Z_{r} = c o s \frac{π}{3^{r}} + i s i n \frac{π}{3^{r}}, r = 1, 2, 3, . . . . . .

z_{1} z_{2} z_{3} . . . . . . . . \infty = i

| z_{i} | = i, i = 1, 2, 3, 4

| 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,

∣ ∣ ∣ \frac{1}{{¯ ¯ ¯ z}_{1}} + \frac{4}{{¯ ¯ ¯ z}_{2}} + \frac{9}{{¯ ¯ ¯ z}_{3}} + \frac{16}{{¯ ¯ ¯ z}_{4}} ∣ ∣ ∣ .

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