MHT CET · Maths · Complex Number
If \(\mathrm{w}=\frac{-1+\mathrm{i} \sqrt{3}}{2}\), where \(\mathrm{i}=\sqrt{-1}\), then the value of \(\left(3+w+3 w^2\right)^4\) is
- A 16
- B -16
- C 16 w
- D \(16 w^2\)
Answer & Solution
Correct Answer
(C) 16 w
Step-by-step Solution
Detailed explanation
\(\omega\) is a complex cube root of unity
\(\therefore \omega=1 ...(i)\)
\( \therefore 1+\omega+\omega^2=0, \)
\( \therefore \left(3+\omega+3 \omega^2\right)^4 ...(ii)\)
\( =(3+\omega+3(-1-\omega))^4 \)
\( =(3+\omega-3-3 \omega)^4 \)
\( =(-2 \omega)^4 \)
\( =16 \omega^4 \)
\( =16 \omega^3 \times \omega \)
\( =16 \omega\)
...[from (ii)]
...[from (i)]
\(\therefore \omega=1 ...(i)\)
\( \therefore 1+\omega+\omega^2=0, \)
\( \therefore \left(3+\omega+3 \omega^2\right)^4 ...(ii)\)
\( =(3+\omega+3(-1-\omega))^4 \)
\( =(3+\omega-3-3 \omega)^4 \)
\( =(-2 \omega)^4 \)
\( =16 \omega^4 \)
\( =16 \omega^3 \times \omega \)
\( =16 \omega\)
...[from (ii)]
...[from (i)]
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If then the maximum value of is ________MHT CET 2019 Medium
- If both mean and variance of 50 observations \(x_1, x_2, \ldots \ldots, x_{50}\) are equal to 16 and 256 respectively, then mean of \(\left(x_1-5\right)^2,\left(x_2-5\right)^2, \ldots \ldots\left(x_{50}-5\right)^2\) isMHT CET 2023 Medium
- If \(y=\mathrm{A} \cos \mathrm{n} x+\mathrm{B} \sin \mathrm{n} x\), then \(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}=\)MHT CET 2024 Easy
- The value of \(2 \sin ^{-1}\left(\frac{1}{2}\right)+\cot ^{-1}\left(\frac{1}{\sqrt{3}}\right)\) isMHT CET 2022 Easy
- The expression \(((p \wedge q) \vee(p \vee \sim q)) \wedge(\sim p \wedge \sim q)\) is equivalent toMHT CET 2024 Medium
- \(\int_{0}^{1} x(1-x)^{5} d x=\)MHT CET 2020 Medium
More PYQs from MHT CET
- In cyclotron, the time taken by an ion to describe semicircular path in a dee isMHT CET 2020 Medium
- If sum of the slopes of the lines given by is three times their product then …MHT CET 2019 Easy
- An object is immersed in a fluid of refractive index ' \(\mu\) '. In order that the object
becomes invisible when observed from outside, it shouldMHT CET 2020 Easy - What is the \(\mathrm{pH}\) of the solution containing \(1.342 \times 10^{-3} \mathrm{M} \mathrm{H}^{+}\) ions? \((\log 1.342=0.1277)\)MHT CET 2022 Medium
- A uniform metal wire has length ' \(L\) ', mass ' \(M\) ' and cross-sectional area ' \(A\) '. It is under tension ' \(T\) ' and ' \(V\) ' is the speed of transverse wave along the wire. The density of the wireMHT CET 2022 Medium
- To use a transistor as an amplifierMHT CET 2011 Medium