Download App

Complex Numbers — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers2 Marks
Question
Express the following complex numbers in the standard form a + ib: $\frac{(2+\text{i})^3}{2+3\text{i}}$
✓

Answer

$\frac{(2+\text{i})^3}{2+3\text{i}}=\frac{2^3+\text{i}^3+3\times2\times\text{i}(2+\text{i})}{2+3\text{i}} \ \big[\therefore \ (\text{a}+\text{b})^3=\text{a}^3+\text{b}^3+3\text{ab}(\text{a}+\text{b})\big]$ $=\frac{(8-\text{i}+6\text{i}(2+\text{i}))}{2+3\text{i}}\times\frac{(2-3\text{i})}{2-3\text{i}}$ (On rationalising the denominator) $=\frac{(8-\text{i}+12\text{i}+6\text{i}^2)(2-3\text{i})}{2^2+3^3}$ $=\frac{(8-6+11\text{i})(2-3\text{i})}{4+9} \ \big(\because \ \text{i}^2=-1\big)$ $=\frac{(2+11\text{i})(2-3\text{i})}{13}$ $=\frac{4-6\text{i}+22\text{i}+33}{13}$ $=\frac{37+16\text{i}}{13}$ $=\frac{37}{13}+\frac{16}{13}\text{i}$ $\therefore\frac{(2+\text{i})^3}{2+3\text{i}}=\frac{37}{13}+\frac{16}{13}\text{i}$

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 "(Each question 2 marks)" from Complex Numbers→Complex Numbers — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the coordinates of the point which divides the line segment joining the points (-2, 3, 5) and (1, -4, 6) in the ratio 2 : 3 internally.
→
In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}+\text{b}^\text{x}+\text{c}^\text{x}-3}{\text{x}}$
→
Solve the following system of inequalities
  1. 8x + 3y $\le$ 100
  2. x $\ge$ 0
  3. y $\ge$ 0
→
Reduce the given equation into the intercept form and find the intercept on the axis. 4x - 3y = 6
→
Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
→
Let A = {x : x $\in$ N}, B = {x : x = 2n, n $\in$ N}, C = {x : x = 2n - 1, n $\in$ N} and D = {x : x is a prime natural number}. Find: $\text{A}\cap\text{D}$
→
Find the conjugate of $\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}$
→
A quadrilateral has vertices (4, 1), (1, 7), (-6, 0) and (-1, -9). Show that the mid-points of the sides of this quadrilateral form a parallelogram.
→
Find the equation of the parabola which is symmetric about the y-axis, and passes through the point (2, -3).
→