Question:
If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}(i=\sqrt{-1})$
then $\left(1+i z+z^{5}+i z^{8}\right)^{9}$ is equal to
-1
1
0
$(-1+2 \mathrm{i})^{9}$
Correct Option: 1
Solution:
$z=\frac{\sqrt{3}}{2}+\frac{i}{2}=\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}$
$\Rightarrow \mathrm{z}^{5}=\cos \frac{5 \pi}{6}+\mathrm{i} \sin \frac{5 \pi}{6}=\frac{-\sqrt{3}+\mathrm{i}}{2}$
and $z^{8}=\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}=-\left(\frac{1+i \sqrt{3}}{2}\right)$
$\Rightarrow\left(1+i z+z^{5}+\mathrm{iz}^{8}\right)^{9}=\left(1+\frac{\mathrm{i} \sqrt{3}}{2}-\frac{1}{2}-\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}-\frac{\mathrm{i}}{2}+\frac{\sqrt{3}}{2}\right)^{9}$
$=\left(\frac{1+\mathrm{i} \sqrt{3}}{2}\right)^{9}=\cos 3 \pi+\mathrm{i} \sin 3 \pi=-1$
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- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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