Tag Archives: Complex Numbers

Math

Prove that, ${\left( {\frac{{1 + \sin \theta + i\cos \theta }}{{1 + \sin \theta – i\cos \theta }}} \right)^n} =$

$\cos n\left( {\frac{\pi }{2} – \theta } \right) + i\sin n\left( {\frac{\pi }{2} – \theta } \right)$

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LHS = ${\left[ {\frac{{(1 + \sin \theta + i\cos \theta )(\sin \theta + i\cos \theta )}}{{\{ 1 + (\sin \theta – i\cos \theta )\} (\sin \theta + i\cos \theta )}}} \right]^n}$

$ = {\left[ {\frac{{(1 + \sin \theta + i\cos \theta )(\sin \theta + i\cos \theta )}}{{(\sin \theta + i\cos \theta ) + \{ {{\sin }^2}\theta – {{(i\cos \theta )}^2}\} }}} \right]^n}$

$ = {\left[ {\frac{{(1 + \sin \theta + i\cos \theta )(\sin \theta + i\cos \theta )}}{{(\sin \theta + i\cos \theta ) + ({{\sin }^2}\theta + {{\cos }^2}\theta )}}} \right]^n}$

$ = {\left[ {\frac{{(1 + \sin \theta + i\cos \theta )(\sin \theta + i\cos \theta )}}{{(1 + \sin \theta + i\cos \theta )}}} \right]^n}$

$ = {(\sin \theta + i\cos \theta )^n}$

$ = {\left\{ {\cos \left( {\frac{\pi }{2} – \theta } \right) + i\sin \left( {\frac{\pi }{2} – \theta } \right)} \right\}^n}$

$ = \cos n\left( {\frac{\pi }{2} – \theta } \right) + i\sin n\left( {\frac{\pi }{2} – \theta } \right)$

= RHS

Mathematics

$\frac{{1 – i\sin \theta }}{{1 + i\cos \theta }} = z$

${\mathop{\rm Im}\nolimits} (z) = 0$

$\theta = ?$

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$z = \frac{{1 – i\sin \theta }}{{1 + i\cos \theta }} \times \frac{{1 – i\cos \theta }}{{1 – i\cos \theta }}$

$ = \frac{{1 – i(\cos \theta + \sin \theta ) + {i^2}\sin \theta \cos \theta }}{{1 – {i^2}{{\cos }^2}\theta }}$

$ = \frac{{1 – \sin \theta \cos \theta – i(\cos \theta + \sin \theta )}}{{1 + {{\cos }^2}\theta }}$

$\because {\mathop{\rm Im}\nolimits} (z) = 0,\cos \theta + \sin \theta = 0$

$\Rightarrow \tan \theta = -1 $

$\Rightarrow \theta = n\pi – \frac {\pi}{4}$,$n\in I$

IIT JEE

$\omega = z\bar z – 2z + 2$

$\left| {\frac{{z + i}}{{z – 3i}}} \right| = 1$

$Re(\omega ) $=min.

$n_{min.} \in N =?$

$\omega ^n \in R$

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Let z and $\omega $ be two complex numbers such that $\omega  = z\bar z – 2z + 2$, $\left| {\frac{{z + i}}{{z – 3i}}} \right| = 1$ and $Re(\omega )$ has minimum value. Then, the minimum value of $n \in N $ for which $\omega ^n $ is real, is equal to _ _ _ _ . Continue reading $\omega = z\bar z – 2z + 2$

$\left| {\frac{{z + i}}{{z – 3i}}} \right| = 1$

$Re(\omega ) $=min.

$n_{min.} \in N =?$

$\omega ^n \in R$ →

IIT JEE

Let $\theta_1,\theta_1,……..,\theta_{10}$ be …

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Let $\theta_1,\theta_1,……..,\theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+……..+\theta_{10}=2\pi$. Define the complex numbers $z_1=e^{i\theta_1},z_k=z_{k-1}e^{i\theta_k}$ for k=2, 3, …….., 10 where $i=\sqrt {-1}$. Consider the statements P and Q given below:

P: $|z_2-z_1|+|z_3-z_2|+………..+|z_{10}-z_9|+|z_1-z_{10}|\leq 2\pi $

Q: $|z_2^2-z_1^2|+|z_3^2-z_2^2|+………..+|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2|\leq 4\pi $

Then,

(A) P is TRUE and Q is FALSE
(B) Q is TRUE and P is FALSE
(C) both P and Q are TRUE
(D) both P and Q are FALSE Continue reading Let $\theta_1,\theta_1,……..,\theta_{10}$ be … →