Download App

Complex Numbers(p-1) — Maths (commerce) STD 11 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 11 Commerce / ArtsMaths (commerce)Complex Numbers(p-1)4 Marks
Question
Find the square root of the following complex numbers:
2(1 – √3i)
✓

Answer

Let $\sqrt{2(1-\sqrt{3} i)}=\mathrm{a}+\mathrm{bi}$, where $\mathrm{a}, \mathrm{b} \in \mathrm{R}$
Squaring on both sides, we get
$
\begin{aligned}
& 2(1-\sqrt{ } 3 i)=(a+b i)^2 \\
& 2(1-\sqrt{ } 3 i)=a^2+b^2 i^2+2 a b i \\
& 2-2 \sqrt{ } 3 i=\left(a^2-b^2\right)+2 a b i \quad \ldots . .\left[\because i^2=-1\right]
\end{aligned}
$
Equating real and imaginary parts, we get
$
\begin{array}{ll}
& a^2-b^2=2 \text { and } 2 a b=-2 \sqrt{3} \\
\therefore & a^2-b^2=2 \text { and } b=-\frac{\sqrt{3}}{a} \\
\therefore & a^2-\left(-\frac{\sqrt{3}}{a}\right)^2=2 \\
& \text { } \\
\therefore \quad & a^2-\frac{3}{a^2}=2 \\
\therefore & a^4-3=2 a^2 \\
\therefore & a^4-2 a^2-3=0 \\
\therefore & \left(a^2-3\right)\left(a^2+1\right)=0 \\
\therefore & a^2=3 \text { or } a^2=-1 \\
& B a \in R
\end{array}
$
$
\begin{array}{ll}
\therefore & a^2-\frac{3}{a^2}=2 \\
\therefore & a^4-3=2 a^2 \\
\therefore & a^4-2 a^2-3=0 \\
\therefore & \left(a^2-3\right)\left(a^2+1\right)=0 \\
\therefore & a^2=3 \text { or } a^2=-1
\end{array}
$
But $\mathrm{a} \in \mathrm{R}$
$
\begin{array}{ll}
\therefore & \mathrm{a}^2 \neq-1 \\
\therefore & \mathrm{a}^2=3 \\
\therefore & \mathrm{a}= \pm \sqrt{3}
\end{array}
$
When $\mathrm{a}=\sqrt{3}, \mathrm{~b}=\frac{-\sqrt{3}}{\sqrt{3}}=-1$
When $\mathrm{a}=-\sqrt{3}, \mathrm{~b}=\frac{-\sqrt{3}}{-\sqrt{3}}=1$
$
\therefore \quad \sqrt{2(1-\sqrt{3} \mathrm{i})}= \pm(\sqrt{3}-\mathrm{i})
$

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 "Solve the Following Question.(4 Marks)" from Complex Numbers(p-1)→Complex Numbers(p-1) — all question types→Maths (commerce) STD 11 Commerce / Arts question papers→Maharashtra Board STD 11 Commerce / Arts question papers→Maharashtra Board question paper generator→

Similar questions

The vertices of a triangle are $A(1,4), B(2,3)$, and $C(1,6)$. Find equations of : altitudes of $\triangle A B C$
→
\begin{aligned}
& f(x)=\frac{\log x-\log 3}{x-3} \text { for } x \neq 3 \\
& =3 \text { for } x=3, \text { at } x=3
\end{aligned}
→
Find the square root of the following complex numbers:
-8 – 6i
→
Two dice are thrown together. What is the probability that sum of the numbers on two dice is 5 or the number on the second die is greater than or equal to the number on the first die?
→
How many numbers between 100 and 1000 have the digit 7 exactly once?
→
Calculate the coefficient of variation for the data given below. [Given √3.3 = 1.8166]

C.I.

 5-15 15-25 25-35 35-45 45-55 55-65 65-75
F

6

7

15

25

8

18

21

→
After a meeting, every participant shakeshands with every other participants. If the number of handshakes is 66, find the number of participants in the meeting.
→
Heena Enterprise sold cosmetics worth $₹ 25,000$ to Leena traders, a retailer. Leena Traders sold it further to Meena Beauty Products for ₹ 30,000. Meena Beauty Product sold it further to the customers for ₹ 40,000 . The rate of GST is $18 \%$. Find: GST Payable by each party
→
Construct a bivariate frequency distribution table of the marks obtained by students in Statistics (X) and English (Y).

Marks in statistics (x)

37

20

46

28

35

26

41

48

32

23

20

39

47

33

27

26

Marks in english (y)

30

32

41

33

29

43

30

21

44

38

47

24

32

31

20

21

Construct a bivariate frequency distribution table for the above data by taking class intervals 20 – 30, 30 – 40, …. etc. for both X and Y. Also find the marginal distributions and conditional frequency distribution of Y when X lies between 30 – 40.

→
Evaluate the following:

$\lim _{x \rightarrow 0}\left[\frac{\left(2^x-1\right)^2}{\left(3^x-1\right) \cdot \log (1+x)}\right]$

→