SATHEE: Complex Numbers 5 Question 5

Complex Numbers 5 Question 5

5. The minimum value of $| a + b ω + c ω^{2} |$ , where $a, b$ and $c$ are all not equal integers and $ω (\neq 1)$ is a cube root of unity, is

(a) $\sqrt{3}$

(b) $\frac{1}{2}$

(d) 0

$(2005, 1 M)$

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Answer:

Correct Answer: 5. (c)

Solution:

Let $z = | a + b ω + c ω^{2} |$

$\Rightarrow z^{2} = {| a + b ω + c ω^{2} |}^{2} = (a^{2} + b^{2} + c^{2} - a b - b c - c a)$

or $z^{2} = \frac{1}{2} (a - b)^{2} + (b - c)^{2} + (c - a)^{2}$

Since, $a, b, c$ are all integers but not all simultaneously equal.

$\Rightarrow$ If $a = b$ then $a \neq c$ and $b \neq c$

Because difference of integers $=$ integer

$\Rightarrow (b - c)^{2} \geq 1$ {as minimum difference of two consecutive integers is $Extra close brace or missing open brace$ also $(c - a)^{2} \geq 1$

and we have taken $a = b \Rightarrow (a - b)^{2} = 0$

From Eq. (i), $z^{2} \geq \frac{1}{2} (0 + 1 + 1)$

$\Rightarrow z^{2} \geq 1$

Hence, minimum value of $| z |$ is 1 .

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Complex Numbers 5 Question 5

5. The minimum value of $| a + b ω + c ω^{2} |$ , where $a, b$ and $c$ are all not equal integers and $ω (\neq 1)$ is a cube root of unity, is

Answer:

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Complex Numbers 5 Question 5

5. The minimum value of |a+bω+cω2|, where a,b and c are all not equal integers and ω(≠1) is a cube root of unity, is

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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5. The minimum value of $| a + b ω + c ω^{2} |$ , where $a, b$ and $c$ are all not equal integers and $ω (\neq 1)$ is a cube root of unity, is