If ( 1 - i 1 + i )^ 100 = a + ib, then - Vidyadip

MCQ

If ${\left( {\frac{{1 - i}}{{1 + i}}} \right)^{100}} = a + ib$, then

A
$a = 2,b = - 1$
✓
$a = 1,b = 0$
C
$a = 0,b = 1$
D
$a = - 1,b = 2$

✓

Answer

Correct option: B.

$a = 1,b = 0$

b
(b) Given, ${\left( {\frac{{1 - i}}{{1 + i}}} \right)^{100}} = a + ib$; $\left[ {\left( {\frac{{1 - i}}{{1 + i}}} \right) \times \left( {\frac{{1 - i}}{{1 - i}}} \right)} \right] = a + ib$
==> $a + ib = {\left[ {\frac{{{{(1 - i)}^2}}}{2}} \right]^{100}} = {\left[ {\frac{{ - 2i}}{2}} \right]^{100}} = {( - i)^{100}}$
==> $a + ib = {\left[ {{{(i)}^4}} \right]^{25}} = 1 + 0i,$Hence, $a = 1,b = 0$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from 4.1 complex nubers→4.1 complex nubers — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The coefficient of $x-12$ in the expansion of $\left(\frac{x+y}{x^3}\right)^{20}$ is:

The probability that two randomly selected subsets of the set $\{1,2,3,4,5\}$ have exactly two elements in their intersection, is :

In an increasing geometric progression ol positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is $49$. Then the sum of the $4^{\text {th }}, 6^{\text {th }}$ and $8^{\text {th }}$ terms is :-

On the parabola $y=x^2$, the point least distant from the straight line $y=2 x-4$ is:

A person wishes to make up as many different parties as he can out of 20 friends. Each party consists of the same number of friends. How many should be invited at a time:

$\mathop {\lim }\limits_{\theta \to {0^ + }} \,{\left( {\sin \theta } \right)^{\left( {\sin \theta - {{\sin }^2}\theta } \right)}}$ is

Equation of vertical line to the left of y-axis at 5 units from y-axis is:

Number of solution$(s)$ of the equation $ln(1 + sin^2x) = 1 -ln(5 + x^2)$ is -

Let $\mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_{\mathrm{n}}$ be n observations. Let $\mathrm{y}_{\mathrm{i}}=\mathrm{a} \mathrm{x}_{\mathrm{i}}+\mathrm{b} \mathrm{y}_{\mathrm{i}}+\mathrm{b}$ for $\mathrm{i}=1,2,3, \ldots, \mathrm{n}$, where a and b are constants. If the mean of $x_i^{\prime} s$ is 48 and their standard deviation is 12 , the mean of $y_i$ 's 55 and standard deviation of $y_i$ 's is 15 , the values of a and b are:

The number of all $3-$digit numbers $a b c$ (in base $10$ ) for which $(a \times b \times c)+(a \times b)+$ $(b \times c)+(c \times a)+a+b+c=29$ is