117476
If \(n\) is an integer other than a multiple of 3 , then the value of \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}}\) is
1 1
2 -1
3 0
4 3
Explanation:
C If \(\mathrm{n}\) is not a multiple of 3 Let, \(\mathrm{n}=3 \mathrm{~m}+1\) or \(\mathrm{n}=3 \mathrm{~m}+2\), where \(\mathrm{m} \in \mathrm{z}\) If \(n=3 m+1\) then \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}} =1+\omega^{3 \mathrm{~m}+1}+\omega^{6 \mathrm{~m}+2}\) \(=1+\omega+\omega^2=0 end{aligned}\) If \(n=3 m+2\) then \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}} =1+\omega^{3 \mathrm{~m}+2}+\omega^{6 \mathrm{~m}+4}\) \(=1+\omega^2+\omega=0 end{aligned}\) Hence, option (c) is correct.
SRM JEEE-2011
Complex Numbers and Quadratic Equation
117477
The principal value of \(\mathrm{i}^{\mathrm{i}}\) is equal to
1 e
2 \(\mathrm{e}^{-\pi / 2}\)
3 \(\mathrm{e}^{-3 \pi / 2}\)
4 None of these
Explanation:
B Let, \(\quad \mathrm{z}=\mathrm{i}^{\mathrm{i}}\) Taking \(\log\) on booth side \(\log z=i \log i\) \(\log z=i \log e^{i \pi / 2}\) \(\log z=\mathrm{i}^2 \frac{\pi}{2}=-\frac{\pi}{2}\) \(z=e^{-\pi / 2}\) Hence, option (b) is correct.
SRM JEEE-2015
Complex Numbers and Quadratic Equation
117478
The conjugate of a complex number is \(\frac{1}{1-\mathrm{i}}\) Then the complex number is
1 \(-\frac{1}{\mathrm{i}-1}\)
2 \(\frac{1}{i+1}\)
3 \(-\frac{1}{i+1}\)
4 \(\frac{1}{\mathrm{i}-1}\)
Explanation:
B \( \text { Let, } \mathrm{z} \text { is a complex number }\) \(\mathrm{z}=\frac{1}{1-\mathrm{i}} \times \frac{1+\mathrm{i}}{1+\mathrm{i}}\) \(=\frac{1+\mathrm{i}}{(1)^2-(\mathrm{i})^2}=\frac{1+\mathrm{i}}{2}\) \(=\frac{1}{2}+\frac{\mathrm{i}}{2}\) \(\therefore \quad \overline{\mathrm{z}}=\frac{1}{2}-\frac{\mathrm{i}}{2}=\frac{1-\mathrm{i}}{2}=\frac{1-\mathrm{i}}{1^2-\mathrm{i}^2}=\frac{1}{1+\mathrm{i}}\)Ans: b Exp: (B) : Let, \(\mathrm{z}\) is a complex number
SRM JEEE-2016
Complex Numbers and Quadratic Equation
117479
The value of \(4+5\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{365}\) is
1 \(1-\mathrm{i} \sqrt{3}\)
2 \(-1+\mathrm{i} \sqrt{3}\)
3 \(\mathrm{i} \sqrt{3}\)
4 \(-\mathrm{i} \sqrt{3}\)
Explanation:
C The value of \(4+5\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{365}\) We know that imaginary cube root of unity \(\omega=-\frac{1}{2}+i \frac{\sqrt{3}}{2}\) \(\text { Then, } =4+5(\omega)^{334}+3(\omega)^{365}\) \(=4+5\left(\omega^3\right)^{111} \omega+3\left(\omega^3\right)^{121} \cdot \omega^2\) \(=4+5 \omega+3 \omega^2 \quad\left\{\because \omega^3=1\right\}\) \(=1+3+3 \omega+2 \omega+3 \omega^2\) \(=1+3+3 \omega+3 \omega^2+2 \omega\) \(=1+3\left(1+\omega+\omega^2\right)+2 \omega\) \(=1+2 \omega \quad\left\{\because 1+\omega+\omega^2=0\right)\) \(=1+2\left(\frac{-1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}\right) \quad\left\{\because \omega=\frac{-1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}\right\}\) \(=1-1+\mathrm{i} \sqrt{3}=\mathrm{i} \sqrt{3}\)
117476
If \(n\) is an integer other than a multiple of 3 , then the value of \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}}\) is
1 1
2 -1
3 0
4 3
Explanation:
C If \(\mathrm{n}\) is not a multiple of 3 Let, \(\mathrm{n}=3 \mathrm{~m}+1\) or \(\mathrm{n}=3 \mathrm{~m}+2\), where \(\mathrm{m} \in \mathrm{z}\) If \(n=3 m+1\) then \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}} =1+\omega^{3 \mathrm{~m}+1}+\omega^{6 \mathrm{~m}+2}\) \(=1+\omega+\omega^2=0 end{aligned}\) If \(n=3 m+2\) then \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}} =1+\omega^{3 \mathrm{~m}+2}+\omega^{6 \mathrm{~m}+4}\) \(=1+\omega^2+\omega=0 end{aligned}\) Hence, option (c) is correct.
SRM JEEE-2011
Complex Numbers and Quadratic Equation
117477
The principal value of \(\mathrm{i}^{\mathrm{i}}\) is equal to
1 e
2 \(\mathrm{e}^{-\pi / 2}\)
3 \(\mathrm{e}^{-3 \pi / 2}\)
4 None of these
Explanation:
B Let, \(\quad \mathrm{z}=\mathrm{i}^{\mathrm{i}}\) Taking \(\log\) on booth side \(\log z=i \log i\) \(\log z=i \log e^{i \pi / 2}\) \(\log z=\mathrm{i}^2 \frac{\pi}{2}=-\frac{\pi}{2}\) \(z=e^{-\pi / 2}\) Hence, option (b) is correct.
SRM JEEE-2015
Complex Numbers and Quadratic Equation
117478
The conjugate of a complex number is \(\frac{1}{1-\mathrm{i}}\) Then the complex number is
1 \(-\frac{1}{\mathrm{i}-1}\)
2 \(\frac{1}{i+1}\)
3 \(-\frac{1}{i+1}\)
4 \(\frac{1}{\mathrm{i}-1}\)
Explanation:
B \( \text { Let, } \mathrm{z} \text { is a complex number }\) \(\mathrm{z}=\frac{1}{1-\mathrm{i}} \times \frac{1+\mathrm{i}}{1+\mathrm{i}}\) \(=\frac{1+\mathrm{i}}{(1)^2-(\mathrm{i})^2}=\frac{1+\mathrm{i}}{2}\) \(=\frac{1}{2}+\frac{\mathrm{i}}{2}\) \(\therefore \quad \overline{\mathrm{z}}=\frac{1}{2}-\frac{\mathrm{i}}{2}=\frac{1-\mathrm{i}}{2}=\frac{1-\mathrm{i}}{1^2-\mathrm{i}^2}=\frac{1}{1+\mathrm{i}}\)Ans: b Exp: (B) : Let, \(\mathrm{z}\) is a complex number
SRM JEEE-2016
Complex Numbers and Quadratic Equation
117479
The value of \(4+5\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{365}\) is
1 \(1-\mathrm{i} \sqrt{3}\)
2 \(-1+\mathrm{i} \sqrt{3}\)
3 \(\mathrm{i} \sqrt{3}\)
4 \(-\mathrm{i} \sqrt{3}\)
Explanation:
C The value of \(4+5\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{365}\) We know that imaginary cube root of unity \(\omega=-\frac{1}{2}+i \frac{\sqrt{3}}{2}\) \(\text { Then, } =4+5(\omega)^{334}+3(\omega)^{365}\) \(=4+5\left(\omega^3\right)^{111} \omega+3\left(\omega^3\right)^{121} \cdot \omega^2\) \(=4+5 \omega+3 \omega^2 \quad\left\{\because \omega^3=1\right\}\) \(=1+3+3 \omega+2 \omega+3 \omega^2\) \(=1+3+3 \omega+3 \omega^2+2 \omega\) \(=1+3\left(1+\omega+\omega^2\right)+2 \omega\) \(=1+2 \omega \quad\left\{\because 1+\omega+\omega^2=0\right)\) \(=1+2\left(\frac{-1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}\right) \quad\left\{\because \omega=\frac{-1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}\right\}\) \(=1-1+\mathrm{i} \sqrt{3}=\mathrm{i} \sqrt{3}\)
117476
If \(n\) is an integer other than a multiple of 3 , then the value of \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}}\) is
1 1
2 -1
3 0
4 3
Explanation:
C If \(\mathrm{n}\) is not a multiple of 3 Let, \(\mathrm{n}=3 \mathrm{~m}+1\) or \(\mathrm{n}=3 \mathrm{~m}+2\), where \(\mathrm{m} \in \mathrm{z}\) If \(n=3 m+1\) then \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}} =1+\omega^{3 \mathrm{~m}+1}+\omega^{6 \mathrm{~m}+2}\) \(=1+\omega+\omega^2=0 end{aligned}\) If \(n=3 m+2\) then \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}} =1+\omega^{3 \mathrm{~m}+2}+\omega^{6 \mathrm{~m}+4}\) \(=1+\omega^2+\omega=0 end{aligned}\) Hence, option (c) is correct.
SRM JEEE-2011
Complex Numbers and Quadratic Equation
117477
The principal value of \(\mathrm{i}^{\mathrm{i}}\) is equal to
1 e
2 \(\mathrm{e}^{-\pi / 2}\)
3 \(\mathrm{e}^{-3 \pi / 2}\)
4 None of these
Explanation:
B Let, \(\quad \mathrm{z}=\mathrm{i}^{\mathrm{i}}\) Taking \(\log\) on booth side \(\log z=i \log i\) \(\log z=i \log e^{i \pi / 2}\) \(\log z=\mathrm{i}^2 \frac{\pi}{2}=-\frac{\pi}{2}\) \(z=e^{-\pi / 2}\) Hence, option (b) is correct.
SRM JEEE-2015
Complex Numbers and Quadratic Equation
117478
The conjugate of a complex number is \(\frac{1}{1-\mathrm{i}}\) Then the complex number is
1 \(-\frac{1}{\mathrm{i}-1}\)
2 \(\frac{1}{i+1}\)
3 \(-\frac{1}{i+1}\)
4 \(\frac{1}{\mathrm{i}-1}\)
Explanation:
B \( \text { Let, } \mathrm{z} \text { is a complex number }\) \(\mathrm{z}=\frac{1}{1-\mathrm{i}} \times \frac{1+\mathrm{i}}{1+\mathrm{i}}\) \(=\frac{1+\mathrm{i}}{(1)^2-(\mathrm{i})^2}=\frac{1+\mathrm{i}}{2}\) \(=\frac{1}{2}+\frac{\mathrm{i}}{2}\) \(\therefore \quad \overline{\mathrm{z}}=\frac{1}{2}-\frac{\mathrm{i}}{2}=\frac{1-\mathrm{i}}{2}=\frac{1-\mathrm{i}}{1^2-\mathrm{i}^2}=\frac{1}{1+\mathrm{i}}\)Ans: b Exp: (B) : Let, \(\mathrm{z}\) is a complex number
SRM JEEE-2016
Complex Numbers and Quadratic Equation
117479
The value of \(4+5\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{365}\) is
1 \(1-\mathrm{i} \sqrt{3}\)
2 \(-1+\mathrm{i} \sqrt{3}\)
3 \(\mathrm{i} \sqrt{3}\)
4 \(-\mathrm{i} \sqrt{3}\)
Explanation:
C The value of \(4+5\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{365}\) We know that imaginary cube root of unity \(\omega=-\frac{1}{2}+i \frac{\sqrt{3}}{2}\) \(\text { Then, } =4+5(\omega)^{334}+3(\omega)^{365}\) \(=4+5\left(\omega^3\right)^{111} \omega+3\left(\omega^3\right)^{121} \cdot \omega^2\) \(=4+5 \omega+3 \omega^2 \quad\left\{\because \omega^3=1\right\}\) \(=1+3+3 \omega+2 \omega+3 \omega^2\) \(=1+3+3 \omega+3 \omega^2+2 \omega\) \(=1+3\left(1+\omega+\omega^2\right)+2 \omega\) \(=1+2 \omega \quad\left\{\because 1+\omega+\omega^2=0\right)\) \(=1+2\left(\frac{-1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}\right) \quad\left\{\because \omega=\frac{-1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}\right\}\) \(=1-1+\mathrm{i} \sqrt{3}=\mathrm{i} \sqrt{3}\)
117476
If \(n\) is an integer other than a multiple of 3 , then the value of \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}}\) is
1 1
2 -1
3 0
4 3
Explanation:
C If \(\mathrm{n}\) is not a multiple of 3 Let, \(\mathrm{n}=3 \mathrm{~m}+1\) or \(\mathrm{n}=3 \mathrm{~m}+2\), where \(\mathrm{m} \in \mathrm{z}\) If \(n=3 m+1\) then \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}} =1+\omega^{3 \mathrm{~m}+1}+\omega^{6 \mathrm{~m}+2}\) \(=1+\omega+\omega^2=0 end{aligned}\) If \(n=3 m+2\) then \(1+\omega^{\mathrm{n}}+\omega^{2 \mathrm{n}} =1+\omega^{3 \mathrm{~m}+2}+\omega^{6 \mathrm{~m}+4}\) \(=1+\omega^2+\omega=0 end{aligned}\) Hence, option (c) is correct.
SRM JEEE-2011
Complex Numbers and Quadratic Equation
117477
The principal value of \(\mathrm{i}^{\mathrm{i}}\) is equal to
1 e
2 \(\mathrm{e}^{-\pi / 2}\)
3 \(\mathrm{e}^{-3 \pi / 2}\)
4 None of these
Explanation:
B Let, \(\quad \mathrm{z}=\mathrm{i}^{\mathrm{i}}\) Taking \(\log\) on booth side \(\log z=i \log i\) \(\log z=i \log e^{i \pi / 2}\) \(\log z=\mathrm{i}^2 \frac{\pi}{2}=-\frac{\pi}{2}\) \(z=e^{-\pi / 2}\) Hence, option (b) is correct.
SRM JEEE-2015
Complex Numbers and Quadratic Equation
117478
The conjugate of a complex number is \(\frac{1}{1-\mathrm{i}}\) Then the complex number is
1 \(-\frac{1}{\mathrm{i}-1}\)
2 \(\frac{1}{i+1}\)
3 \(-\frac{1}{i+1}\)
4 \(\frac{1}{\mathrm{i}-1}\)
Explanation:
B \( \text { Let, } \mathrm{z} \text { is a complex number }\) \(\mathrm{z}=\frac{1}{1-\mathrm{i}} \times \frac{1+\mathrm{i}}{1+\mathrm{i}}\) \(=\frac{1+\mathrm{i}}{(1)^2-(\mathrm{i})^2}=\frac{1+\mathrm{i}}{2}\) \(=\frac{1}{2}+\frac{\mathrm{i}}{2}\) \(\therefore \quad \overline{\mathrm{z}}=\frac{1}{2}-\frac{\mathrm{i}}{2}=\frac{1-\mathrm{i}}{2}=\frac{1-\mathrm{i}}{1^2-\mathrm{i}^2}=\frac{1}{1+\mathrm{i}}\)Ans: b Exp: (B) : Let, \(\mathrm{z}\) is a complex number
SRM JEEE-2016
Complex Numbers and Quadratic Equation
117479
The value of \(4+5\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{365}\) is
1 \(1-\mathrm{i} \sqrt{3}\)
2 \(-1+\mathrm{i} \sqrt{3}\)
3 \(\mathrm{i} \sqrt{3}\)
4 \(-\mathrm{i} \sqrt{3}\)
Explanation:
C The value of \(4+5\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{365}\) We know that imaginary cube root of unity \(\omega=-\frac{1}{2}+i \frac{\sqrt{3}}{2}\) \(\text { Then, } =4+5(\omega)^{334}+3(\omega)^{365}\) \(=4+5\left(\omega^3\right)^{111} \omega+3\left(\omega^3\right)^{121} \cdot \omega^2\) \(=4+5 \omega+3 \omega^2 \quad\left\{\because \omega^3=1\right\}\) \(=1+3+3 \omega+2 \omega+3 \omega^2\) \(=1+3+3 \omega+3 \omega^2+2 \omega\) \(=1+3\left(1+\omega+\omega^2\right)+2 \omega\) \(=1+2 \omega \quad\left\{\because 1+\omega+\omega^2=0\right)\) \(=1+2\left(\frac{-1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}\right) \quad\left\{\because \omega=\frac{-1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}\right\}\) \(=1-1+\mathrm{i} \sqrt{3}=\mathrm{i} \sqrt{3}\)
