The smallest positive integer n for which ( 1 + i )^2n = ( 1 - i )^2n is: a.1 b.2 c.3 d.4

Answers 2

Answer:

answer is b from this

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Answer:

[tex]\qquad\qquad\qquad\boxed{ \bf{ \:(b) \: \: \: 2 \: \: }} \\ \\ [/tex]

Explanation:

Given that,

[tex]\qquad\sf \: {(1 + i)}^{2n} = {(1 - i)}^{2n} \\ \\ [/tex]

can be further rewritten as

[tex]\qquad\sf \: \dfrac{{(1 + i)}^{2n}}{{(1 - i)}^{2n}} = 1 \\ \\ [/tex]

[tex]\qquad\sf \: {\bigg(\dfrac{1 + i}{1 - i} \bigg) }^{2n} = 1 \\ \\ [/tex]

On rationalizing the denominator, we get

[tex]\qquad\sf \: {\bigg(\dfrac{1 + i}{1 - i} \times \dfrac{1 + i}{1 + i} \bigg) }^{2n} = 1 \\ \\ [/tex]

[tex]\qquad\sf \: {\bigg(\dfrac{(1 + i)^{2} }{ {1}^{2} - {i}^{2} } \bigg) }^{2n} = 1 \\ \\ [/tex]

[tex]\qquad\boxed{ \sf{ \: \because \: (x + y)(x - y) = {x}^{2} - {y}^{2} \: }} \\ \\ [/tex]

[tex]\qquad\sf \: {\bigg(\dfrac{ {1}^{2} + {i}^{2} + 2i}{ 1 - ( - 1) } \bigg) }^{2n} = 1 \: \: \: \: \boxed{ \sf{ \: \because \: {i}^{2} = - 1 \: }}\\ \\ [/tex]

[tex]\qquad\boxed{ \sf{ \: \because \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy \: }} \\ \\ [/tex]

[tex]\qquad\sf \: {\bigg(\dfrac{ 1 - 1+ 2i}{2 } \bigg) }^{2n} = 1 \: \: \: \: \boxed{ \sf{ \: \because \: {i}^{2} = - 1 \: }}\\ \\ [/tex]

[tex]\qquad\sf \: {\bigg(\dfrac{ 2i}{2 } \bigg) }^{2n} = 1 \: \\ \\ [/tex]

[tex]\qquad\sf \: {(i)}^{2n} = 1 \: \\ \\ [/tex]

[tex]\qquad\sf \: {(i)}^{2n} = {(i)}^{4} \: \\ \\ [/tex]

[tex]\qquad\sf \: 2n = 4 \: \\ \\ [/tex]

[tex]\qquad\sf\implies \sf \: n = 2 \: \\ \\ [/tex]

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How many vocal cords does a person have? a.3 b.1 c.2 d.5

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There are 2 vocal cords, which produce sound in humans

Explanation:

c is correct option

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Velocity of sound waves in air is 330 m/s . For a particle sound wave in air, a path difference of 40 cm is equivalent to phase difference of 1.6pi . The frequency of this wave is a.165 Hz b.660 Hz c.150 Hz d.330 Hz

Answer:

b.660 Hz

Explanation:

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If A is a square matrix such that A^2 = A , then det(A) equals a.- 1 or 1 b.- 2 or 2 c.- 3 or 3 d.None of these
Option d) None of these ✔️

answer img

Answer:

[tex]\qquad\qquad\boxed{ \bf{ \: \: \:(d) \: \: \: None\:of\:these \: \: \: }} \\ \\ [/tex]

Explanation:

Given that,

[tex]\qquad\sf \: {A}^{2} = A \\ \\ [/tex]

Taking determinant on both sides, we get

[tex]\qquad\sf \: | {A}^{2} | = |A| \\ \\ [/tex]

can be rewritten as

[tex]\qquad\sf \: |A.A | = |A| \\ \\ [/tex]

[tex]\qquad\sf \: |A| |A| = |A| \\ \\ [/tex]

[tex]\qquad\sf \: |A| |A| - |A| = 0 \\ \\ [/tex]

[tex]\qquad\sf \: |A| (|A| -1)= 0 \\ \\ [/tex]

[tex]\qquad\sf \: |A| = 0 \: \: \: or \: \: \: |A| -1= 0 \\ \\ [/tex]

[tex]\qquad\sf\implies \sf \: |A| = 0 \: \: \: or \: \: \: |A| = 1\\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information

If A and B are square matrices of order n, then

[tex]\sf \: |AB| = |A| \: |B| \\ \\ [/tex]

[tex]\sf \: |adjA| = { |A| }^{n - 1} \\ \\ [/tex]

[tex]\sf \: |A \: adjA| = { |A| }^{n} \\ \\ [/tex]

[tex]\sf \: | {A}^{ - 1} | = { |A| }^{ - 1} \\ \\ [/tex]

[tex]\sf \: |A| = |A'| \\ \\ [/tex]

[tex]\sf \: |adj(adj \: A)| = { |A| }^{ {(n - 1)}^{2} } \\ \\ [/tex]

Transistor is a: | Physics Questions a.current - controlled current device b.current - controlled voltage device c.voltage - controlled current device d.voltage - controlled voltage device

Explanation:

This electric field determines the flow of the charge carriers in a transistor. So, we can conclude that a transistor is a current-controlled voltage device. Hence, option (B) is the correct answer.

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