SATHEE: Complex Numbers 2 Question 22

Complex Numbers 2 Question 22

23. Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y (x, y \in R, i = \sqrt{- 1})$ of the equation $s z + t \bar{z} + r = 0$ , where $\bar{z} = x - i y$ . Then, which of the following statement(s) is (are) TRUE?

(2018 Adv.)

(a) If $L$ has exactly one element, then $| s | \neq | t |$

(b) If $| s | = | t |$ , then $L$ has infinitely many elements

(d) If $L$ has more than one element, then $L$ has infinitely many elements

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Answer:

Correct Answer: 23. (c)

Solution:

We have,

$s z + t \bar{z} + r = 0$

On taking conjugate

$\bar{s} \bar{z} + \bar{t} z + \bar{r} = 0$

On solving Eqs. (i) and (ii), we get

$z = \frac{\bar{r} t - r \bar{s}}{| s |^{2} - | t |^{2}}$

(a) For unique solutions of $z$ $| s |^{2} - | t |^{2} \neq 0 \Rightarrow | s | \neq | t |$

It is true

(b) If $| s | = | t |$ , then $\bar{r} t - r \bar{s}$ may or may not be zero.

So, $z$ may have no solutions.

$∴ L$ may be an empty set. It is false.

Hence, it is true.

(d) In this case locus of $z$ is a line, so $L$ has infinite elements. Hence, it is true.

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Complex Numbers 2 Question 22

23. Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y (x, y \in R, i = \sqrt{- 1})$ of the equation $s z + t \bar{z} + r = 0$ , where $\bar{z} = x - i y$ . Then, which of the following statement(s) is (are) TRUE?

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

Complex Numbers 2 Question 22

23. Let s,t,r be non-zero complex numbers and L be the set of solutions z=x+iy(x,y∈R,i=−1) of the equation sz+tz¯+r=0, where z¯=x−iy. Then, which of the following statement(s) is (are) TRUE?

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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23. Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y (x, y \in R, i = \sqrt{- 1})$ of the equation $s z + t \bar{z} + r = 0$ , where $\bar{z} = x - i y$ . Then, which of the following statement(s) is (are) TRUE?