Download App

4.1 complex nubers — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS4.1 complex nubers1 Mark
MCQ
If ${a^2} + {b^2} = 1,$ then $\frac{{1 + b + ia}}{{1 + b - ia}} = $
  • A
    $1$
  • B
    $2$
  • ✓
    $b + ia$
  • D
    $a + ib$
✓

Answer

Correct option: C.
$b + ia$
c
(c) Given that ${a^2} + {b^2} = 1$, therefore
$\frac{{1 + b + ia}}{{1 + b - ia}} = \frac{{(1 + b + ia)(1 + b + ia)}}{{(1 + b - ia)(1 + b + ia)}}$
$ = \frac{{{{(1 + b)}^2} - {a^2} + 2ia(1 + b)}}{{1 + {b^2} + 2b + {a^2}}}$$ = \frac{{(1 - {a^2}) + 2b + {b^2} + 2ia(1 + b)}}{{2(1 + b)}}$
$ = \frac{{2{b^2} + 2b + 2ia(1 + b)}}{{2\,(1 + b)}} = b + ia$
Trick : Put $a = 0,b = 1$, $\frac{{1 + b + ia}}{{1 + b - ia}} = \frac{{1 + 1 + 0}}{{1 + 1 - 0}} = 1$
But options $(a)$ and $ (c)$  give $1.$
So again put $a = 1,b = 0,\frac{{1 + b + ia}}{{1 + b - ia}} = \frac{{1 + i}}{{1 - i}} = i$.
Which gives $ (c)$ only.
 

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

Out of all the patients in a hospital $89\, \%$ are found to be suffering from heart ailment and $98\, \%$ are suffering from lungs infection. If $\mathrm{K}\, \%$ of them are suffering from both ailments, then $\mathrm{K}$ can not belong to the set :
→
Let the tangents at two points $A$ and $B$ on the circle $x ^{2}+ y ^{2}-4 x +3=0$ meet at origin $O (0,0)$. Then the area of the triangle of $OAB$ is.
→
If the mean deviation about median for the number $3,5,7,2\,k , 12,16,21,24$ arranged in the ascending order, is $6$ then the median is
→
If $\frac{3\pi}{4}<\text{a}<\pi,$ then $\sqrt{2\cot\text{a}+\frac{1}{\sin^\text{a}}}$ is equal to:
→
If $4\text{x} + 3<6\text{x} + 7$, then$\text{ x}\in$
→
If $a \ne 0$ and the line $2bx + 3cy + 4d = 0$ passes through the points of intersection of the parabolas ${y^2} = 4ax$ and ${x^2} = 4ay$, then
→
The roots of the equation $1 - \cos \theta = \sin \theta .\sin \frac{\theta }{2}$ is
→
$\frac{{1 + 7i}}{{{{(2 - i)}^2}}} = $
→
There are five students $S_1, S_2, S_3, S_4$ and $S_5$ in a music class and for them there are five seats $R_1, R_2, R_3, R_4$ and $R_5$ arranged in a row, where initially the seat $R_i$ is allotted to the student $S_i$, $i =1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats.

($1$) The probability that, on the examination day, the student $S_1$ gets the previously allotted seat $R_1$, and $NONE$ of the remaining students gets the seat previously allotted to him/her is

$(A)$ $\frac{3}{40}$ $(B)$ $\frac{1}{8}$ $(C)$ $\frac{7}{40}$ $(D)$ $\frac{1}{5}$

($2$) For $i =1,2,3,4$, let $T _{ i }$ denote the event that the students $S _{ i }$ and $S _{ i +1}$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T _1 \cap T _2 \cap T _3 \cap T _4$ is

$(A)$ $\frac{1}{15}$ $(B)$ $\frac{1}{10}$ $(C)$ $\frac{7}{60}$ $(D)$ $\frac{1}{5}$

Give the answer or quetion ($1$) and ($2$)

→
The value of $\tan\text{x}\sin\Big(\frac{\pi}{2}+\text{x}\Big)\cos\Big(\frac{\pi}{2}-\text{x}\Big)$ is:
→