MCQ 511 Mark
If $\omega (\ne 1)$is a cube root of unity and ${{(1+\omega )}^{7}}=A+B\omega $, then $A$ and $B$ are respectively, the numbers [IIT 1995]
View full question & answer→MCQ 521 Mark
One of the cube roots of unity is [MP PET 1994, 2003]
- ✓
$\frac{-1+i\sqrt{3}}{2}$
- B
$\frac{1+i\sqrt{3}}{2}$
- C
$\frac{1-i\sqrt{3}}{2}$
- D
$\frac{\sqrt{3}-i}{2}$
AnswerCorrect option: A. $\frac{-1+i\sqrt{3}}{2}$
View full question & answer→MCQ 531 Mark
If $\omega $ is a complex cube root of unity, then for positive integral value of$n$, the product of $\omega .{{\omega }^{2}}.{{\omega }^{3}}........{{\omega }^{n}}$, will be [Roorkee 1991]
- A
$\frac{1-i\sqrt{3}}{2}$
- B
$-\frac{1-i\sqrt{3}}{2}$
- C
- ✓
View full question & answer→MCQ 541 Mark
The roots of the equation ${{x}^{4}}-1=0$, are [MP PET 1986]
AnswerCorrect option: B. $1,\,-1,i,-i$
View full question & answer→MCQ 551 Mark
If $z=\frac{\sqrt{3}+i}{2}$, then the value of ${{z}^{69}}$ is [RPET 2002]
View full question & answer→MCQ 561 Mark
If $\alpha ,\beta ,\gamma $ are the cube roots of $p(p<0)$, then for any $x,y$ and $z,\,\,\frac{x\alpha +y\beta +z\gamma }{x\beta +y\gamma +z\alpha }=$ [IIT 1989]
- ✓
$\frac{1}{2}(-1+i\sqrt{3})$
- B
$\frac{1}{2}(1+i\sqrt{3})$
- C
$\frac{1}{2}(1-i\sqrt{3})$
- D
AnswerCorrect option: A. $\frac{1}{2}(-1+i\sqrt{3})$
View full question & answer→MCQ 571 Mark
${{\left( -\frac{1}{2}+\frac{\sqrt{3}}{2}i \right)}^{1000}}=$
- A
$\frac{1}{2}+\frac{\sqrt{3}}{2}i$
- B
$\frac{1}{2}-\frac{\sqrt{3}}{2}i$
- ✓
$-\frac{1}{2}+\frac{\sqrt{3}}{2}i$
- D
AnswerCorrect option: C. $-\frac{1}{2}+\frac{\sqrt{3}}{2}i$
View full question & answer→MCQ 581 Mark
The cube roots of unity when represented on the Argand plane form the vertices of an [IIT 1988; Pb. CET 2004]
MCQ 591 Mark
The value of $\frac{a+b\omega +c{{\omega }^{2}}}{b+c\omega +a{{\omega }^{2}}}+\frac{a+b\omega +c{{\omega }^{2}}}{c+a\omega +b{{\omega }^{2}}}$ will be [BIT Ranchi 1989; Orissa JEE 2003]
View full question & answer→MCQ 601 Mark
If $x=a+b,y=a\omega +b{{\omega }^{2}},z=a{{\omega }^{2}}+b\omega $, then the value of ${{x}^{3}}+{{y}^{3}}+{{z}^{3}}$ is equal to [Roorkee 1977; IIT 1970]
- A
${{a}^{3}}+{{b}^{3}}$
- ✓
$3({{a}^{3}}+{{b}^{3}})$
- C
$3({{a}^{2}}+{{b}^{2}})$
- D
AnswerCorrect option: B. $3({{a}^{3}}+{{b}^{3}})$
View full question & answer→MCQ 611 Mark
If $x=a+b,y=a\alpha +b\beta $ and $z=a\beta +b\alpha ,$ where $\alpha $and $\beta $ are complex cube roots of unity, then $xyz$ = [IIT 1978; Roorkee 1989; RPET 1997]
- A
${{a}^{2}}+{{b}^{2}}$
- ✓
${{a}^{3}}+{{b}^{3}}$
- C
${{a}^{3}}{{b}^{3}}$
- D
${{a}^{3}}-{{b}^{3}}$
AnswerCorrect option: B. ${{a}^{3}}+{{b}^{3}}$
View full question & answer→MCQ 621 Mark
If $\omega $ is a cube root of unity, then a root of the equation $\left| \begin{matrix} x+1 & \omega & {{\omega }^{2}} \\ \omega & x+{{\omega }^{2}} & 1 \\ {{\omega }^{2}} & 1 & x+\omega \\ \end{matrix} \right|=0$ is [MNR 1990; MP PET 1999]
- A
$x=1$
- B
$x=\omega $
- C
$x={{\omega }^{2}}$
- ✓
$x=0$
View full question & answer→MCQ 631 Mark
The product of all the roots of ${{\left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right)}^{3/4}}$ is [MNR 1984; EAMCET 1985]
- A
$-1$
- ✓
- C
$\frac{3}{2}$
- D
$-\frac{1}{2}$
View full question & answer→MCQ 641 Mark
If $\omega $ is a complex cube root of unity, then $(1+\omega )(1+{{\omega }^{2}})$ $(1+{{\omega }^{4}})(1+{{\omega }^{8}})...$to $2n$ factors = [AMU 2000]
View full question & answer→MCQ 651 Mark
If $\omega $ is a complex cube root of unity, then $(x-y)(x\omega -y)$ $(x{{\omega }^{2}}-y)=$
- A
${{x}^{2}}+{{y}^{2}}$
- B
${{x}^{2}}-{{y}^{2}}$
- ✓
${{x}^{3}}-{{y}^{3}}$
- D
${{x}^{3}}+{{y}^{3}}$
AnswerCorrect option: C. ${{x}^{3}}-{{y}^{3}}$
View full question & answer→MCQ 661 Mark
If $x=a,y=b\omega ,z=c{{\omega }^{2}}$, where $\omega $ is a complex cube root of unity, then $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=$ [AMU 1983]
View full question & answer→MCQ 671 Mark
If $\omega $ is a cube root of unity, then the value of ${{(1-\omega +{{\omega }^{2}})}^{5}}+{{(1+\omega -{{\omega }^{2}})}^{5}}=$ [IIT 1965; MP PET 1997; RPET 1997]
View full question & answer→MCQ 681 Mark
If w is a complex cube root of unity, then $(1-\omega )(1-{{\omega }^{2}})$ $(1-{{\omega }^{4}})(1-{{\omega }^{8}})=$
View full question & answer→MCQ 691 Mark
If $\alpha $and $\beta $ are imaginary cube roots of unity, then ${{\alpha }^{4}}+{{\beta }^{4}}$ + $\frac{1}{\alpha \beta }=$ [IIT 1977]
View full question & answer→MCQ 701 Mark
If $\omega $ is a cube root of unity, then ${{(1+\omega )}^{3}}-{{(1+{{\omega }^{2}})}^{3}}=$
- ✓
- B
$\omega $
- C
${{\omega }^{2}}$
- D
View full question & answer→MCQ 711 Mark
Square of either of the two imaginary cube roots of unity will be
- A
- ✓
Other imaginary cube root of unity
- C
Sum of two imaginary roots of unity
- D
AnswerCorrect option: B. Other imaginary cube root of unity
View full question & answer→MCQ 721 Mark
If $n$ is a positive integer not a multiple of 3, then $1+{{\omega }^{n}}+{{\omega }^{2n}}$ = [MP PET 2004]
View full question & answer→MCQ 731 Mark
${{(27)}^{1/3}}=$
- A
- B
$3,\,\,3i,\,3{{i}^{2}}$
- ✓
$3,\,3\omega ,\,3{{\omega }^{2}}$
- D
AnswerCorrect option: C. $3,\,3\omega ,\,3{{\omega }^{2}}$
View full question & answer→MCQ 741 Mark
If $\omega $ is a cube root of unity, then $(1+\omega -{{\omega }^{2}})$ $(1-\omega +{{\omega }^{2}})$ = [MNR 1990; MP PET 1993, 2002]
View full question & answer→MCQ 751 Mark
The two numbers such that each one is square of the other, are [MP PET 1987]
- A
$\omega ,\,{{\omega }^{3}}$
- B
$-i,\,\,i$
- C
$-1,\,1$
- ✓
$\omega ,\,\,{{\omega }^{2}}$
AnswerCorrect option: D. $\omega ,\,\,{{\omega }^{2}}$
View full question & answer→MCQ 761 Mark
If $i{{z}^{4}}+1=0$, then $z$ can take the value [UPSEAT 2004]
AnswerCorrect option: B. $\cos \frac{\pi }{8}+i\,\sin \frac{\pi }{8}$
View full question & answer→MCQ 771 Mark
If $\frac{1}{x}+x=2\cos \theta ,$ then ${{x}^{n}}+\frac{1}{{{x}^{n}}}$ is equal to [UPSEAT 2001]
- ✓
$2\cos n\theta $
- B
$2\sin n\theta $
- C
$\cos n\,\theta $
- D
$\sin \,n\theta $
AnswerCorrect option: A. $2\cos n\theta $
View full question & answer→MCQ 781 Mark
If n is a positive integer, then ${{(1+i)}^{n}}+{{(1-i)}^{n}}$ is equal to [Orissa JEE 2003]
- A
${{(\sqrt{2})}^{n-2}}\cos \left( \frac{n\pi }{4} \right)$
- B
${{(\sqrt{2})}^{n-2}}\sin \left( \frac{n\pi }{4} \right)$
- ✓
${{(\sqrt{2})}^{n+2}}\cos \left( \frac{n\pi }{4} \right)$
- D
${{(\sqrt{2})}^{n+2}}\sin \left( \frac{n\pi }{4} \right)$
AnswerCorrect option: C. ${{(\sqrt{2})}^{n+2}}\cos \left( \frac{n\pi }{4} \right)$
View full question & answer→MCQ 791 Mark
${{\left( \frac{1+\sin \theta +i\,\cos \theta }{1+\sin \theta -i\,\cos \theta } \right)}^{n}}$= [Kerala (Engg.) 2002]
- ✓
$\cos \left( \frac{n\pi }{2}-n\theta \right)+i\,\sin \left( \frac{n\pi }{2}-n\theta \right)$
- B
$\cos \left( \frac{n\pi }{2}+n\theta \right)+i\,\sin \left( \frac{n\pi }{2}+n\theta \right)$
- C
$\sin \left( \frac{n\pi }{2}-n\theta \right)+i\,\cos \left( \frac{n\pi }{2}-n\theta \right)$
- D
$\cos \,n\left( \frac{\pi }{2}+2\theta \right)+i\,\sin \,n\left( \frac{\pi }{2}+2\theta \right)$
AnswerCorrect option: A. $\cos \left( \frac{n\pi }{2}-n\theta \right)+i\,\sin \left( \frac{n\pi }{2}-n\theta \right)$
View full question & answer→MCQ 801 Mark
Given $z={{(1+i\sqrt{3})}^{100}},$ then $\frac{\operatorname{Re}(z)}{\operatorname{Im}(z)}$ equals [AMU 2002]
- A
- B
- ✓
$\frac{1}{\sqrt{3}}$
- D
$\sqrt{3}$
AnswerCorrect option: C. $\frac{1}{\sqrt{3}}$
View full question & answer→MCQ 811 Mark
The value of i1/3 is [UPSEAT 2002]
- ✓
$\frac{\sqrt{3}\,+i}{2}$
- B
$\frac{\sqrt{3}\,-i}{2}$
- C
$\frac{1+i\sqrt{3}}{2}$
- D
$\frac{1-i\sqrt{3}}{2}$
AnswerCorrect option: A. $\frac{\sqrt{3}\,+i}{2}$
View full question & answer→MCQ 821 Mark
$\frac{{{(\cos \alpha +i\,\sin \alpha )}^{4}}}{{{(\sin \beta +i\,\cos \beta )}^{5}}}=$ [RPET 2002]
- A
$\cos (4\alpha +5\beta )+i\,\sin (4\alpha +5\beta )$
- B
$\cos (4\alpha +5\beta )-i\,\sin (4\alpha +5\beta )$
- ✓
$\sin (4\alpha +5\beta )-i\cos (4\alpha +5\beta )$
- D
AnswerCorrect option: C. $\sin (4\alpha +5\beta )-i\cos (4\alpha +5\beta )$
View full question & answer→MCQ 831 Mark
If ${{x}_{n}}=\cos \,\left( \frac{\pi }{{{4}^{n}}} \right)+i\,\sin \,\left( \frac{\pi }{{{4}^{n}}} \right)\,,$ then ${{x}_{1}}.\,{{x}_{2}}.\,{{x}_{3}}....\infty =$ [EAMCET 2002]
- ✓
$\frac{1+i\sqrt{3}}{2}$
- B
$\frac{-1+i\sqrt{3}}{2}$
- C
$\frac{1-i\sqrt{3}}{2}$
- D
$\frac{-1-i\sqrt{3}}{2}$
AnswerCorrect option: A. $\frac{1+i\sqrt{3}}{2}$
View full question & answer→MCQ 841 Mark
${{\left[ \frac{1+\cos (\pi /8)+i\,\sin (\pi /8)}{1+\cos (\pi /8)-i\,\sin (\pi /8)} \right]}^{8}}$ is equal to [RPET 2001]
View full question & answer→MCQ 851 Mark
The value of $\frac{(\cos \alpha +i\,\sin \alpha )\,(\cos \beta +i\,\sin \beta )}{(\cos \gamma +i\,\sin \gamma )\,(\cos \,\delta +i\,\sin \delta )}$ is [RPET 2001]
- A
$\cos (\alpha +\beta -\gamma -\delta )-i\,\sin (\alpha +\beta -\gamma -\delta )$
- ✓
$\cos (\alpha +\beta -\gamma -\delta )+i\,\sin (\alpha +\beta -\gamma -\delta )$
- C
$\sin (\alpha +\beta -\gamma -\delta )-i\,\cos (\alpha +\beta -\gamma -\delta )$
- D
$\sin (\alpha +\beta -\gamma -\delta )+i\,\cos (\alpha +\beta -\gamma -\delta )$
AnswerCorrect option: B. $\cos (\alpha +\beta -\gamma -\delta )+i\,\sin (\alpha +\beta -\gamma -\delta )$
View full question & answer→MCQ 861 Mark
${{(\sin \theta +i\,\cos \theta )}^{n}}\,$is equal to [RPET 2001]
- A
$\cos n\theta +i\,\sin n\theta $
- B
$\sin n\theta +i\,\cos n\theta $
- ✓
$\cos n\left( \frac{\pi }{2}-\theta \right)+i\,\sin n\left( \frac{\pi }{2}-\theta \right)$
- D
AnswerCorrect option: C. $\cos n\left( \frac{\pi }{2}-\theta \right)+i\,\sin n\left( \frac{\pi }{2}-\theta \right)$
View full question & answer→MCQ 871 Mark
We express $\frac{{{(\cos 2\theta -i\sin 2\theta )}^{4}}{{(\cos 4\theta +i\sin 4\theta )}^{-5}}}{{{(\cos 3\theta +i\sin 3\theta )}^{-2}}{{(\cos 3\theta -i\sin 3\theta )}^{-9}}}$ in the form of $x+iy$, we get [Karnataka CET 2001]
- ✓
$\cos 49\theta -i\,\sin 49\theta $
- B
$\cos 23\theta -i\,\sin 23\theta $
- C
$\cos 49\theta +i\,\sin 49\theta $
- D
$\cos 21\theta +i\,\sin 21\theta $
AnswerCorrect option: A. $\cos 49\theta -i\,\sin 49\theta $
View full question & answer→MCQ 881 Mark
The value of ${{\left[ \frac{1-\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}}{1-\cos \frac{\pi }{10}-i\sin \frac{\pi }{10}} \right]}^{10}}=$ [Karnataka CET 2001]
View full question & answer→MCQ 891 Mark
${{(-\sqrt{3}+i)}^{53}}$ where ${{i}^{2}}=-1$ is equal to [AMU 2000]
AnswerCorrect option: C. ${{2}^{53}}\,\left( \frac{\sqrt{3}}{2}+\frac{1}{2}i \right)$
View full question & answer→MCQ 901 Mark
If $\cos \alpha +\cos \beta +\cos \gamma =0=$$\sin \alpha +\sin \beta +\sin \gamma $ then $\cos 2\alpha +\cos 2\beta +\cos 2\gamma $ equals [RPET 2000]
View full question & answer→MCQ 911 Mark
If $\sin \alpha +\sin \beta +\sin \gamma =0=$$\cos \alpha +\cos \beta +\cos \gamma ,$ then the value of ${{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma $ is [RPET 1999]
View full question & answer→MCQ 921 Mark
${{\left( \frac{\cos \theta +i\sin \theta }{\sin \theta +i\cos \theta } \right)}^{4}}$equals [RPET 1996]
- A
$\sin 8\theta -i\cos 8\theta $
- B
$\cos 8\theta -i\sin 8\theta $
- C
$\sin 8\theta +i\cos 8\theta $
- ✓
$\cos 8\theta +i\sin 8\theta $
AnswerCorrect option: D. $\cos 8\theta +i\sin 8\theta $
View full question & answer→MCQ 931 Mark
The value of expression $\left( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2} \right)$ $\,\left( \cos \frac{\pi }{{{2}^{2}}}+i\sin \frac{\pi }{{{2}^{2}}} \right)$........to $\infty $ is [Kurukshetra CEE 1998]
View full question & answer→MCQ 941 Mark
If ${{\left( \frac{1+\cos \theta +i\sin \theta }{i+\sin \theta +i\cos \theta } \right)}^{4}}=\cos n\theta +i\sin n\theta $, then $n$ is equal to [EAMCET 1986]
View full question & answer→MCQ 951 Mark
${{\left( \frac{1+\cos \varphi +i\sin \varphi }{1+\cos \varphi -i\sin \varphi } \right)}^{n}}=$
- A
$\cos n\varphi -i\sin n\varphi $
- ✓
$\cos n\varphi +i\sin n\varphi $
- C
$\sin n\varphi +i\cos n\varphi $
- D
$\sin n\varphi -i\cos n\varphi $
AnswerCorrect option: B. $\cos n\varphi +i\sin n\varphi $
View full question & answer→MCQ 961 Mark
If $(\cos \theta +i\sin \theta )(\cos 2\theta +i\sin 2\theta )........$ $(\cos n\theta +i\sin n\theta )=1$, then the value of $\theta $ is[Karnataka CET 1992; Kurukshetra CEE 2002]
- A
$4m\pi $
- B
$\frac{2m\pi }{n(n+1)}$
- ✓
$\frac{4m\pi }{n(n+1)}$
- D
$\frac{m\pi }{n(n+1)}$
AnswerCorrect option: C. $\frac{4m\pi }{n(n+1)}$
View full question & answer→MCQ 971 Mark
If $a=\sqrt{2i}$ then which of the following is correct [Roorkee 1989]
- ✓
$a=1+i$
- B
$a=1-i$
- C
$a=-(\sqrt{2})i$
- D
AnswerCorrect option: A. $a=1+i$
View full question & answer→MCQ 981 Mark
The following in the form of $A+iB$ ${{(\cos 2\theta +i\sin 2\theta )}^{-5}}$ ${{(\cos 3\theta -i\sin 3\theta )}^{6}}$${{(\sin \theta -i\cos \theta )}^{3}}$ in the form of $A+iB$ is [MNR 1991]
- A
$(\cos 25\theta +i\sin 25\theta )$
- B
$i(\cos 25\theta +i\sin 25\theta )$
- C
$i\,(\cos 25\theta -i\sin 25\theta )$
- ✓
$(\cos 25\theta -i\sin 25\theta )$
AnswerCorrect option: D. $(\cos 25\theta -i\sin 25\theta )$
View full question & answer→MCQ 991 Mark
The value of $\frac{4(\cos {{75}^{o}}+i\sin {{75}^{o}})}{0.4(\cos {{30}^{o}}+i\sin {{30}^{o}})}$ is
- A
$\frac{\sqrt{2}}{10}(1+i)$
- B
$\frac{\sqrt{2}}{10}(1-i)$
- C
$\frac{10}{\sqrt{2}}(1-i)$
- ✓
$\frac{10}{\sqrt{2}}(1+i)$
AnswerCorrect option: D. $\frac{10}{\sqrt{2}}(1+i)$
View full question & answer→MCQ 1001 Mark
The roots of ${{(2-2i)}^{1/3}}$ are
- ✓
$\sqrt{2}\left( \cos \frac{\pi }{12}-i\sin \frac{\pi }{12} \right),\sqrt{2}\left( -\sin \frac{\pi }{12}+i\cos \frac{\pi }{12} \right),-1-i$
- B
$\sqrt{2}\left( \cos \frac{\pi }{12}+i\sin \frac{\pi }{12} \right),\sqrt{2}\left( -\sin \frac{\pi }{12}-i\cos \frac{\pi }{12} \right)\,,\,1+i$
- C
$1+\sqrt{2}i,-1-i,-2-2i$
- D
AnswerCorrect option: A. $\sqrt{2}\left( \cos \frac{\pi }{12}-i\sin \frac{\pi }{12} \right),\sqrt{2}\left( -\sin \frac{\pi }{12}+i\cos \frac{\pi }{12} \right),-1-i$
View full question & answer→