Reduce \((\frac{1}{1-4i}-\frac{2}{1+i})(\frac{3-4i}{5+i})\) to the standard form .
Reduce \((\frac{1}{1-4i}-\frac{2}{1+i})(\frac{3-4i}{5+i})\) to the standard form .
Solution and Explanation
\((\frac{1}{1-4i}-\frac{2}{1+i})(\frac{3-4i}{5+i})\)\(=[\frac{(1+i)-2(1-4i)}{(1-4i)(1+i)}][\frac{3-4i}{5+i}]\)
\(=[\frac{1+i-2+8i}{1+i-4i-4i^2}][\frac{3-4i}{5-3i}]=[\frac{-1+9i}{5-3i}][\frac{3-4i}{5+i}]\)
\([\frac{-3+4i+27i-36^2}{25+5i-15i-3i^2}]=\frac{33+31i-4i}{28-10i}=\frac{33+31i}{14-5i}\)
\(=\frac{(33+3li)}{2(14-5i)}×\frac{(14+5i)}{(14-5i)} \) \([on\,multiplaying\,numerator\,and\,denominator\,by\,(14+5i)]\)
\(\frac{462+165i+434i+155i^2}{2[(14)^2(5i)^2]}=\frac{307+599i}{2(196-25i^2)}\)
\(=\frac{307+599i}{2(221)}=\frac{307+599i}{442}=\frac{307}{442}+\frac{599i}{442}\)
This is the required standard form.
Top Questions on Complex Numbers and Quadratic Equations
- The minimum value of \(| z+ \frac{3+4i}{2}|, |z| \leq 1\) is,
- JEE Main - 2024
- Mathematics
- Complex Numbers and Quadratic Equations
Let α,β be the roots of the equation, ax2+bx+c=0.a,b,c are real and sn=αn+βn and \(\begin{vmatrix}3 &1+s_1 &1+s_2\\1+s_1&1+s_2 &1+s_3\\1+s_2&1+s_3 &1+s_4\end{vmatrix}=\frac{k(a+b+c)^2}{a^4}\) then k=
- WBJEE - 2023
- Mathematics
- Complex Numbers and Quadratic Equations
- For $z \in C$ if the minimum value of $(| z -3 \sqrt{2}|+| z - p \sqrt{2} i |)$ is $5 \sqrt{2}$, then a value of $p$ is ______
- JEE Main - 2023
- Mathematics
- Complex Numbers and Quadratic Equations
- The remainder when $(11)^{1011}+(1011)^{11}$ is divided by $9$ is
- JEE Main - 2023
- Mathematics
- Complex Numbers and Quadratic Equations
- If the shortest between the lines $\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}$ and $\frac{x-\lambda}{3}=\frac{y-2 \sqrt{6}}{4}=\frac{z+2 \sqrt{6}}{5}$ is $6$ , then the square of sum of all possible values of $\lambda$ is
- JEE Main - 2023
- Mathematics
- Complex Numbers and Quadratic Equations
Questions Asked in CBSE Class XI exam
- Distinguish between the following pairs of sentences.
(i) He was visibly moved.
(ii) He was visually impaired.- CBSE Class XI
- The adventure
- How, according to you, can peace and liberty be maintained in a state?
- CBSE Class XI
- The tale of melon City
Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?
- CBSE Class XI
- Pressure
- Describe the change in hybridisation (if any) of the Al atom in the following reaction.
\(AlCl_3+Cl^- \rightarrow AlCl_4^-\)- CBSE Class XI
- Hybridisation
- Why did the narrator of the story want to forget the address?
- CBSE Class XI
- The address
Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.