MHT CET · Maths · Determinants
If \(w=\frac{-1-i \sqrt{3}}{2}\) where \(i=\sqrt{-1}\), then the value of \(\left|\begin{array}{ccc}1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w\end{array}\right|\) is
- A -1
- B 0
- C 1
- D 3
Answer & Solution
Correct Answer
(B) 0
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \omega=\frac{-1-\mathrm{i} \sqrt{3}}{2}, \omega^2=\frac{1-3+2 \mathrm{i} \sqrt{3}}{4}=\frac{-1+\mathrm{i} \sqrt{3}}{2}, \\ & \omega^3=1, \omega^6=1 \\ & \left|\begin{array}{ccc}1 & \omega & \omega^2 \\ \omega & \omega^2 & 1 \\ \omega^2 & 1 & \omega\end{array}\right| \\ & =1\left(\omega^3-1\right)-\omega\left(\omega^2-\omega^2\right)+\omega^2\left(\omega-\omega^4\right) \\ & =1\left(\omega^3-1\right)+0+\omega^2\left(\omega-\omega^4\right) .\end{aligned}\)
\(\begin{aligned} & =\omega^3-1+\omega^3-\omega^6 \\ & =2 \omega^3-\omega^6-1 \\ & =2(1)-1-1 \\ & =0\end{aligned}\)
\(\begin{aligned} & =\omega^3-1+\omega^3-\omega^6 \\ & =2 \omega^3-\omega^6-1 \\ & =2(1)-1-1 \\ & =0\end{aligned}\)
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 \(x=\frac{1-t^2}{1+t^2}\) and \(y=\frac{2 a t}{1+t^2}\), then \(\frac{d y}{d x}=\)MHT CET 2021 Medium
- If \(1-\cos \theta=\sin \theta \cdot \sin \frac{\theta}{2}\), then the value of \(\theta\) isMHT CET 2025 Medium
- \(\int \frac{x+1}{x\left(1+x \mathrm{e}^x\right)^2} \mathrm{~d} x=\)
(where \(C\) is a constant of integration.)MHT CET 2022 Medium - If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are three vectors, \(|\overline{\mathrm{a}}|=2,|\overline{\mathrm{b}}|=4,|\overline{\mathrm{c}}|=1\), \(|\overline{\mathrm{b}} \times \overline{\mathrm{c}}|=\sqrt{15}\) and \(\overline{\mathrm{b}}=2 \overline{\mathrm{c}}+\lambda \overline{\mathrm{a}}\), then the value of \(\lambda\), isMHT CET 2023 Medium
- The maximum value of z = 6x +8y subject to is…..MHT CET 2019 Hard
- If \(f(x)=3[x]+\{x+1\}\), where \([x]\) is greatest integer function of \(x\) and \(\{x\}\) is fractional part function of \(x\), then \(f(-1.32)=\)MHT CET 2021 Easy
More PYQs from MHT CET
- For the reaction, \(A+3 B \rightarrow 2 C\) rate of consumption of \(A\) is \(1.4 \mathrm{~mol~} \mathrm{dm}^{-3}\) minute \({ }^{-1}\). Calculate rate of consumption of B .MHT CET 2025 Easy
- Calculate the number of unit cells in 10.8 g metal \(\left(\varrho a^3=7.2 \times 10^{-22} \mathrm{~g}\right)\)MHT CET 2024 Easy
- Carbonated water is an example of a solution ofMHT CET 2024 Easy
- Identify the ore of magnesium from following?MHT CET 2020 Easy
- Birds living in the same habitat generally differ in their niches because they_________.MHT CET 2023 Easy
- Match the Column I with Column II and select the correct option.
Select the correct option:Column I (Part of the alimentary canal) Column II (shape) i. Duodenum a. Worm - like ii. Stomach b. U - shaped iii. Tongue c. J - shaped iv. Appendix d. Triangular MHT CET 2024 Medium