MCQ 11 Mark
Let $x=\alpha +\beta ,\,y=\alpha \omega +\beta {{\omega }^{2}},\,z=\alpha {{\omega }^{2}}+\beta \omega ,\,\omega $ is an imaginary cube root of unity. Product of xyz is [Orissa JEE 2005]
- A
${{\alpha }^{2}}+{{\beta }^{2}}$
- B
${{\alpha }^{2}}-{{\beta }^{2}}$
- C
${{\alpha }^{3}}+{{\beta }^{3}}$
- ✓
${{\alpha }^{3}}-{{\beta }^{3}}$
AnswerCorrect option: D. ${{\alpha }^{3}}-{{\beta }^{3}}$
View full question & answer→MCQ 21 Mark
If 1, $\omega ,\,{{\omega }^{2}}$ are the cube roots of unity then ${{\omega }^{2}}{{(1+\omega )}^{3}}-(1+{{\omega }^{2}})\omega =$ [Orissa JEE 2005]
View full question & answer→MCQ 31 Mark
If $\omega $ is a cube root of unity but not equal to 1 then minimum value of $|a+b\omega +c{{\omega }^{2}}|$ (where a, b, c are integers but not all equal) is [IIT Screening 2005]
- A
- B
$\frac{\sqrt{3}}{2}$
- ✓
- D
View full question & answer→MCQ 41 Mark
If ${{\tan }^{-1}}(\alpha +i\beta )=x+iy,$ then x = [RPET 2002]
- ✓
$\frac{1}{2}{{\tan }^{-1}}\left( \frac{2\alpha }{1-{{\alpha }^{2}}-{{\beta }^{2}}} \right)$
- B
$\frac{1}{2}{{\tan }^{-1}}\left( \frac{2\alpha }{1+{{\alpha }^{2}}+{{\beta }^{2}}} \right)$
- C
${{\tan }^{-1}}\left( \frac{2\alpha }{1-{{\alpha }^{2}}-{{\beta }^{2}}} \right)$
- D
AnswerCorrect option: A. $\frac{1}{2}{{\tan }^{-1}}\left( \frac{2\alpha }{1-{{\alpha }^{2}}-{{\beta }^{2}}} \right)$
View full question & answer→MCQ 51 Mark
If $\tan (u+iv)=i$, then the value of v is [RPET 2001]
AnswerCorrect option: B. $\infty $
View full question & answer→MCQ 61 Mark
$\cos (x+iy)$is equal to [RPET 2001]
- A
$\sin \,x\,\,\cosh \,y+i\,\cos \,x\,\,\sinh \,y$
- B
$\cos \,x\,\,\cosh \,y+i\,\sin \,x\,\,\sinh \,y$
- ✓
$\cos \,x\,\,\cosh \,y-i\,\sin \,x\,\,\sinh \,y$
- D
AnswerCorrect option: C. $\cos \,x\,\,\cosh \,y-i\,\sin \,x\,\,\sinh \,y$
View full question & answer→MCQ 71 Mark
Which one is correct from the following [RPET 2001]
- ✓
$\sin (ix)=i\,\sinh \,x$
- B
$\cos (ix)=i\,\cosh \,x$
- C
$\sin (ix)=-i\,\sinh \,x$
- D
$\tan (ix)=-i\,\tanh \,x$
AnswerCorrect option: A. $\sin (ix)=i\,\sinh \,x$
View full question & answer→MCQ 81 Mark
The imaginary part of $\cosh (\alpha +i\beta )$is [RPET 2000]
- A
$\cosh \,\alpha \,\,\cos \,\beta $
- ✓
$\sinh \,\alpha \,\,\sin \,\beta $
- C
$\cos \alpha \cosh \beta $
- D
$\cos \alpha \cos \beta $
AnswerCorrect option: B. $\sinh \,\alpha \,\,\sin \,\beta $
View full question & answer→MCQ 91 Mark
$\cosh (\alpha +i\beta )-\cosh (\alpha -i\beta )$ is equal to [RPET 2000]
- A
$2\,\,\sinh \,\alpha \,\,\sinh \,\beta $
- B
$2\,\,\cosh \,\alpha \,\,\cosh \,\beta $
- ✓
$2i\,\,\sinh \,\alpha \,\,\sin \,\beta $
- D
$2\,\,\cosh \,\alpha \,\,\cos \,\beta $
AnswerCorrect option: C. $2i\,\,\sinh \,\alpha \,\,\sin \,\beta $
View full question & answer→MCQ 101 Mark
The value of $\sec h(i\pi )$ is [RPET 1999]
View full question & answer→MCQ 111 Mark
If $\cos (u+iv)=\alpha +i\beta ,$ then ${{\alpha }^{2}}+{{\beta }^{2}}+1$ equals [RPET 1999]
- A
${{\cos }^{2}}u+{{\sinh }^{2}}v$
- B
${{\sin }^{2}}u+{{\cosh }^{2}}v$
- ✓
${{\cos }^{2}}u+{{\cosh }^{2}}v$
- D
${{\sin }^{2}}u+{{\sinh }^{2}}v$
AnswerCorrect option: C. ${{\cos }^{2}}u+{{\cosh }^{2}}v$
View full question & answer→MCQ 121 Mark
$\sinh ix$ is [EAMCET 2002]
- A
$i\sin (ix)$
- ✓
$i\sin x$
- C
$-i\sin x$
- D
$\sin (ix)$
AnswerCorrect option: B. $i\sin x$
View full question & answer→MCQ 131 Mark
The real part of ${{\sin }^{-1}}({{e}^{i\theta }})$ is [RPET 1997]
- ✓
${{\cos }^{-1}}(\sqrt{\sin \theta })$
- B
${{\sinh }^{-1}}(\sqrt{\sin \theta })$
- C
${{\sin }^{-1}}(\sqrt{\sin \theta })$
- D
${{\sin }^{-1}}(\sqrt{\cos \theta })$
AnswerCorrect option: A. ${{\cos }^{-1}}(\sqrt{\sin \theta })$
View full question & answer→MCQ 141 Mark
If $\omega $ is a complex cube root of unity, then the value of ${{\omega }^{99}}+{{\omega }^{100}}+{{\omega }^{101}}$ is [Pb. CET 2004]
View full question & answer→MCQ 151 Mark
If $1,\,\omega ,\,{{\omega }^{2}}$ are the roots of unity, then ${{(1-2\omega +{{\omega }^{2}})}^{6}}$ is equal to [Pb. CET 2001]
View full question & answer→MCQ 161 Mark
If $\omega =\frac{-1+\sqrt{3}i}{2}$then ${{(3+\omega +3{{\omega }^{2}})}^{4}}$= [Karnataka CET 2004; Pb. CET 2000]
- A
- B
- ✓
16 $\omega $
- D
16${{\omega }^{2}}$
AnswerCorrect option: C. 16 $\omega $
View full question & answer→MCQ 171 Mark
If $1,\omega ,{{\omega }^{2}}$ are the cube roots of unity, then$\Delta =\left| \,\begin{matrix} 1\,\,\,\, & {{\omega }^{n}} & {{\omega }^{2n}} \\ {{\omega }^{n}}\,\, & \,\,\,{{\omega }^{2n}}\,\, & 1 \\ {{\omega }^{2n}}\, & 1\,\, & {{\omega }^{n}} \\ \end{matrix} \right|$= [AIEEE 2003]
- ✓
- B
- C
$\omega $
- D
${{\omega }^{2}}$
View full question & answer→MCQ 181 Mark
If $\omega $ is a complex cube root of unity, then$225+$${{(3\omega +8{{\omega }^{2}})}^{2}}$$+{{(3{{\omega }^{2}}+8\omega )}^{2}}=$ [EAMCET 2003]
View full question & answer→MCQ 191 Mark
The value of (8)1/3 is [RPET 2003]
- A
$-1+i\sqrt{3}$
- B
$-1-i\sqrt{3}$
- C
- ✓
View full question & answer→MCQ 201 Mark
. Which of the following is a fourth root of $\frac{1}{2}+\frac{i\sqrt{3}}{2}$ [Karnataka CET 2003]
- A
$cis\left( \frac{\pi }{2} \right)$
- ✓
$cis\left( \frac{\pi }{12} \right)$
- C
$cis\left( \frac{\pi }{6} \right)$
- D
$cis\left( \frac{\pi }{3} \right)$
AnswerCorrect option: B. $cis\left( \frac{\pi }{12} \right)$
View full question & answer→MCQ 211 Mark
If $\omega $ is a non real cube root of unity, then $(a+b)$ $(a+b\omega )$ $(a+b{{\omega }^{2}})$ is [Kerala (Engg.) 2002]
- ✓
${{a}^{3}}+{{b}^{3}}$
- B
${{a}^{3}}-{{b}^{3}}$
- C
${{a}^{2}}+{{b}^{2}}$
- D
${{a}^{2}}-{{b}^{2}}$
AnswerCorrect option: A. ${{a}^{3}}+{{b}^{3}}$
View full question & answer→MCQ 221 Mark
Find the value of ${{(1+2\omega +{{\omega }^{2}})}^{3n}}-{{(1+\omega +2{{\omega }^{2}})}^{3n}}=$ [UPSEAT 2002]
- ✓
- B
- C
$\omega $
- D
${{\omega }^{2}}$
View full question & answer→MCQ 231 Mark
If ${{\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)}^{n}}$ is an integer, then n is [UPSEAT 2002]
View full question & answer→MCQ 241 Mark
If $\frac{1+\sqrt{3}\,i}{2}$ is a root of equation ${{x}^{4}}-{{x}^{3}}+x-1=0$ then its real roots are [EAMCET 2002]
View full question & answer→MCQ 251 Mark
If $z+{{z}^{-1}}=1,\,\text{then }\,{{z}^{100}}+{{z}^{-100}}$ is equal to [UPSEAT 2001]
View full question & answer→MCQ 261 Mark
Let ${{\omega }_{n}}=\cos \left( \frac{2\pi }{n} \right)+i\,\sin \left( \frac{2\pi }{n} \right)\,,\,{{i}^{2}}=-1$, then $(x+y{{\omega }_{3}}+z{{\omega }_{3}}^{2})$ $(x+y{{\omega }_{3}}^{2}+z{{\omega }_{3}})$ is equal to [AMU 2001]
- A
- B
${{x}^{2}}+{{y}^{2}}+{{z}^{2}}$
- ✓
${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-yz-zx-xy$$$
- D
${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+yz+zx+xy$
AnswerCorrect option: C. ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-yz-zx-xy$$$
View full question & answer→MCQ 271 Mark
If $z=\frac{\sqrt{3}+i}{-2}$, then ${{z}^{69}}$ is equal to [RPET 2001]
View full question & answer→MCQ 281 Mark
If $1,\omega ,{{\omega }^{2}}$ are the cube roots of unity, then their product is [Karnataka CET 1999, 2001]
View full question & answer→MCQ 291 Mark
The value of $(1-\omega +{{\omega }^{2}})\,{{(1-{{\omega }^{2}}+\omega )}^{6}}$, where $\omega ,{{\omega }^{2}}$ are cube roots of unity [DCE 2001]
- A
128$\omega $
- B
$-128{{\omega }^{2}}$
- ✓
$-128\omega $
- D
$128{{\omega }^{2}}$
AnswerCorrect option: C. $-128\omega $
View full question & answer→MCQ 301 Mark
If cube root of 1 is $\omega $, then the value of ${{(3+\omega +3{{\omega }^{2}})}^{4}}$ is [MP PET 2001]
- A
- B
- ✓
$16\,\omega $
- D
$16\,{{\omega }^{2}}$
AnswerCorrect option: C. $16\,\omega $
View full question & answer→MCQ 311 Mark
If $\pi /3$ is a complex root of the equation ${{z}^{3}}=1$, then $\omega +{{\omega }^{\left( \frac{1}{2}\,+\,\frac{3}{8}\,+\,\frac{9}{32}\,+\,\frac{27}{128}\,+... \right)}}$ is equal to [Roorkee 2000; AMU 2005]
View full question & answer→MCQ 321 Mark
$\frac{{{(-1+i\sqrt{3})}^{15}}}{{{(1-i)}^{20}}}+\frac{{{(-1-i\sqrt{3})}^{15}}}{{{(1+i)}^{20}}}$ is equal to [AMU 2000]
View full question & answer→MCQ 331 Mark
If $\omega $ is an imaginary cube root of unity, ${{(1+\omega -{{\omega }^{2}})}^{7}}$equals [IIT 1998; MP PET 2000]
- A
$128\omega $
- B
$-128\omega $
- C
$128{{\omega }^{2}}$
- ✓
$-128{{\omega }^{2}}$
AnswerCorrect option: D. $-128{{\omega }^{2}}$
View full question & answer→MCQ 341 Mark
${{\left( \frac{\sqrt{3}+i}{2} \right)}^{6}}+{{\left( \frac{i-\sqrt{3}}{2} \right)}^{6}}$is equal to [RPET 1997]
View full question & answer→MCQ 351 Mark
If $\omega $ is an imaginary cube root of unity, then the value of $\sin \,\left[ ({{\omega }^{10}}+{{\omega }^{23}})\,\pi -\frac{\pi }{4} \right]$ is [IIT Screening 1994]
- A
$-\sqrt{3}/2$
- B
$-1/\sqrt{2}$
- ✓
$1/\sqrt{2}$
- D
$\sqrt{3}/2$
AnswerCorrect option: C. $1/\sqrt{2}$
View full question & answer→MCQ 361 Mark
If $\omega $ is the cube root of unity, then ${{(3+5\omega +3{{\omega }^{2}})}^{2}}$ + ${{(3+3\omega +5{{\omega }^{2}})}^{2}}$ = [MP PET 1999]
View full question & answer→MCQ 371 Mark
If $\alpha $ and $\beta $ are imaginary cube roots of unity, then the value of ${{\alpha }^{4}}+{{\beta }^{28}}+\frac{1}{\alpha \beta }$,is [MP PET 1998]
View full question & answer→MCQ 381 Mark
${{\left( \frac{-1+i\sqrt{3}}{2} \right)}^{20}}+{{\left( \frac{-1-i\sqrt{3}}{2} \right)}^{20}}=$
- A
$20\sqrt{3}i$
- B
- C
$\frac{1}{{{2}^{19}}}$
- ✓
$-1$
View full question & answer→MCQ 391 Mark
If $\alpha $ is an imaginary cube root of unity, then for $n\in N$, the value of ${{\alpha }^{3n+1}}+{{\alpha }^{3n+3}}+{{\alpha }^{3n+5}}$ is [MP PET 1996; Pb. CET 2000]
View full question & answer→MCQ 401 Mark
If ${{z}_{1}},{{z}_{2}}{{z}_{3}},{{z}_{4}}$are the roots of the equation ${{z}^{4}}=1$, then the value of $\sum\limits_{i=1}^{4}{z_{i}^{3}}$is [Kurukshetra CEE 1996]
View full question & answer→MCQ 411 Mark
The common roots of the equations ${{x}^{12}}-1=0$, ${{x}^{4}}+{{x}^{2}}+1=0$ are [EAMCET 1989]
AnswerCorrect option: C. $\pm \omega ,\,\pm {{\omega }^{2}}$
View full question & answer→MCQ 421 Mark
If $1,\omega ,{{\omega }^{2}}$ are three cube roots of unity, then ${{(a+b\omega +c{{\omega }^{2}})}^{3}}$ + ${{(a+b{{\omega }^{2}}+c\omega )}^{3}}$ is equal to, if $a+b+c=0$ [West Bengal JEE 1992]
AnswerCorrect option: A. $27\,abc$
View full question & answer→MCQ 431 Mark
If ${{z}_{1}},{{z}_{2}},{{z}_{3}}......{{n}_{n}}$ are nth, roots of unity, then for $k=1,\,2,.....,n$
- A
$|{{z}_{k}}|=k|{{z}_{k+1}}|$
- B
$|{{z}_{k+1}}|=k|{{z}_{k}}|$
- C
$|{{z}_{k+1}}|\,=\,|{{z}_{k}}|+|{{z}_{k+1}}|$
- ✓
$|{{z}_{k}}|=|{{z}_{k+1}}|$
AnswerCorrect option: D. $|{{z}_{k}}|=|{{z}_{k+1}}|$
View full question & answer→MCQ 441 Mark
If $\omega $ is an nth root of unity, other than unity, then the value of $1+\omega +{{\omega }^{2}}+...+{{\omega }^{n-1}}$ is [Karnataka CET 1999]
View full question & answer→MCQ 451 Mark
If $n$ is a positive integer greater than unity and $z$ is a complex number satisfying the equation ${{z}^{n}}={{(z+1)}^{n}}$, then
- ✓
$\operatorname{Re}(z)<0$
- B
$\operatorname{Re}(z)>0$
- C
$\operatorname{Re}(z)=0$
- D
AnswerCorrect option: A. $\operatorname{Re}(z)<0$
View full question & answer→MCQ 461 Mark
Let $\Delta =\left| \,\begin{matrix} 1 & \omega & 2{{\omega }^{2}} \\ 2 & 2{{\omega }^{2}} & 4{{\omega }^{3}} \\ 3 & 3{{\omega }^{3}} & 6{{\omega }^{4}} \\ \end{matrix}\, \right|$ where $\omega $ is the cube root of unity, then
- ✓
$\Delta =0$
- B
$\Delta =1$
- C
$\Delta =2$
- D
$\Delta =3$
AnswerCorrect option: A. $\Delta =0$
View full question & answer→MCQ 471 Mark
$(1-\omega +{{\omega }^{2}})(1-{{\omega }^{2}}+{{\omega }^{4}})(1-{{\omega }^{4}}+{{\omega }^{8}})...........$to $2n$ factors is [EAMCET 1988]
- A
${{2}^{n}}$
- ✓
${{2}^{2n}}$
- C
- D
AnswerCorrect option: B. ${{2}^{2n}}$
View full question & answer→MCQ 481 Mark
If $1,\omega ,{{\omega }^{2}}$ are the three cube roots of unity, then ${{(3+{{\omega }^{2}}+{{\omega }^{4}})}^{6}}=$ [MP PET 1995]
View full question & answer→MCQ 491 Mark
The ${{n}^{th}}$roots of unity are in [Orissa JEE 2004]
View full question & answer→MCQ 501 Mark
If $\omega (\ne 1)$ is a cube root of unity, then $\left| \begin{matrix} 1 & 1+i+{{\omega }^{2}} & {{\omega }^{2}} \\ 1-i & -1 & {{\omega }^{2}}-1 \\ -i & -i+\omega -1 & -1 \\ \end{matrix} \right|$ is equal to [IIT 1995]
View full question & answer→