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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 6251. |
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 |
| A. | \[\Delta =0\] |
| B. | \[\Delta =1\] |
| C. | \[\Delta =2\] |
| D. | \[\Delta =3\] |
| Answer» B. \[\Delta =1\] | |
| 6252. |
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. | \[\frac{10}{\sqrt{2}}(1-i)\] |
| D. | \[\frac{10}{\sqrt{2}}(1+i)\] |
| Answer» E. | |
| 6253. |
\[(1-\omega +{{\omega }^{2}})(1-{{\omega }^{2}}+{{\omega }^{4}})(1-{{\omega }^{4}}+{{\omega }^{8}})...........\]to \[2n\] factors is [EAMCET 1988] |
| A. | \[{{2}^{n}}\] |
| B. | \[{{2}^{2n}}\] |
| C. | 0 |
| D. | 1 |
| Answer» C. 0 | |
| 6254. |
If \[1,\omega ,{{\omega }^{2}}\] are the three cube roots of unity, then \[{{(3+{{\omega }^{2}}+{{\omega }^{4}})}^{6}}=\] [MP PET 1995] |
| A. | 64 |
| B. | 729 |
| C. | 2 |
| D. | 0 |
| Answer» B. 729 | |
| 6255. |
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] |
| A. | 0, 1 |
| B. | 1, 0 |
| C. | 1, 1 |
| D. | \[-1,\ 1\] |
| Answer» D. \[-1,\ 1\] | |
| 6256. |
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] |
| A. | 0 |
| B. | 1 |
| C. | \[\omega \] |
| D. | \[i\] |
| Answer» B. 1 | |
| 6257. |
The \[{{n}^{th}}\]roots of unity are in [Orissa JEE 2004] |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» C. H.P. | |
| 6258. |
One of the cube roots of unity is [MP PET 1994, 2003] |
| A. | \[\frac{-1+i\sqrt{3}}{2}\] |
| B. | \[\frac{1+i\sqrt{3}}{2}\] |
| C. | \[\frac{1-i\sqrt{3}}{2}\] |
| D. | \[\frac{\sqrt{3}-i}{2}\] |
| Answer» B. \[\frac{1+i\sqrt{3}}{2}\] | |
| 6259. |
The roots of the equation \[{{x}^{4}}-1=0\], are [MP PET 1986] |
| A. | \[1,\,1,i,-i\] |
| B. | \[1,\,-1,i,-i\] |
| C. | \[1,-1,\omega ,{{\omega }^{2}}\] |
| D. | None of these |
| Answer» C. \[1,-1,\omega ,{{\omega }^{2}}\] | |
| 6260. |
The roots of \[{{(2-2i)}^{1/3}}\] are |
| A. | \[\sqrt{2}\left( \cos \frac{\pi }{12}-i\sin \frac{\pi }{12} \right),\sqrt{2}\left( -\sin \frac{\pi }{12}+i\cos \frac{\pi }{12} \right),-1-i\] |
| 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. | None of the above |
| Answer» B. \[\sqrt{2}\left( \cos \frac{\pi }{12}+i\sin \frac{\pi }{12} \right),\sqrt{2}\left( -\sin \frac{\pi }{12}-i\cos \frac{\pi }{12} \right)\,,\,1+i\] | |
| 6261. |
If \[z=\frac{\sqrt{3}+i}{2}\], then the value of \[{{z}^{69}}\] is [RPET 2002] |
| A. | \[-i\] |
| B. | \[i\] |
| C. | 1 |
| D. | \[-1\] |
| Answer» B. \[i\] | |
| 6262. |
If \[\alpha ,\beta ,\gamma \] are the cube roots of \[p(p |
| A. | \[\frac{1}{2}(-1+i\sqrt{3})\] |
| B. | \[\frac{1}{2}(1+i\sqrt{3})\] |
| C. | \[\frac{1}{2}(1-i\sqrt{3})\] |
| D. | None of these |
| Answer» B. \[\frac{1}{2}(1+i\sqrt{3})\] | |
| 6263. |
\[{{\left( -\frac{1}{2}+\frac{\sqrt{3}}{2}i \right)}^{1000}}=\] |
| A. | \[\frac{1}{2}+\frac{\sqrt{3}}{2}i\] |
| B. | \[\frac{1}{2}-\frac{\sqrt{3}}{2}i\] |
| C. | \[-\frac{1}{2}+\frac{\sqrt{3}}{2}i\] |
| D. | None of these |
| Answer» D. None of these | |
| 6264. |
The cube roots of unity when represented on the Argand plane form the vertices of an [IIT 1988; Pb. CET 2004] |
| A. | Equilateral triangle |
| B. | Isosceles triangle |
| C. | Right angled triangle |
| D. | None of these |
| Answer» B. Isosceles triangle | |
| 6265. |
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] |
| A. | 1 |
| B. | -1 |
| C. | 2 |
| D. | -2 |
| Answer» C. 2 | |
| 6266. |
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. | \[-1\] |
| B. | 1 |
| C. | \[\frac{3}{2}\] |
| D. | \[-\frac{1}{2}\] |
| Answer» C. \[\frac{3}{2}\] | |
| 6267. |
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. | \[{{a}^{3}}+{{b}^{3}}\] |
| B. | \[3({{a}^{3}}+{{b}^{3}})\] |
| C. | \[3({{a}^{2}}+{{b}^{2}})\] |
| D. | None of these |
| Answer» C. \[3({{a}^{2}}+{{b}^{2}})\] | |
| 6268. |
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. | \[x={{\omega }^{2}}\] |
| D. | \[x=0\] |
| Answer» E. | |
| 6269. |
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] |
| A. | 0 |
| B. | 1 |
| C. | \[-1\] |
| D. | None of these |
| Answer» C. \[-1\] | |
| 6270. |
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. | \[{{x}^{2}}-{{y}^{2}}\] |
| C. | \[{{x}^{3}}-{{y}^{3}}\] |
| D. | \[{{x}^{3}}+{{y}^{3}}\] |
| Answer» D. \[{{x}^{3}}+{{y}^{3}}\] | |
| 6271. |
If \[z={{\left( \frac{\sqrt{3}}{2}+\frac{i}{2} \right)}^{5}}+{{\left( \frac{\sqrt{3}}{2}-\frac{i}{2} \right)}^{5}}\], then [MP PET 1997] |
| A. | \[\operatorname{Re}(z)=0\] |
| B. | \[\operatorname{Im}(z)=0\] |
| C. | \[\operatorname{Re}(z)>0,\operatorname{Im}(z)>0\] |
| D. | \[\operatorname{Re}(z)>0,\operatorname{Im}(z)<0\] |
| Answer» C. \[\operatorname{Re}(z)>0,\operatorname{Im}(z)>0\] | |
| 6272. |
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] |
| A. | 3 |
| B. | 1 |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 6273. |
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] |
| A. | 16 |
| B. | 32 |
| C. | 48 |
| D. | -32 |
| Answer» C. 48 | |
| 6274. |
If w is a complex cube root of unity, then \[(1-\omega )(1-{{\omega }^{2}})\] \[(1-{{\omega }^{4}})(1-{{\omega }^{8}})=\] |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | 9 |
| Answer» E. | |
| 6275. |
Square of either of the two imaginary cube roots of unity will be |
| A. | Real root of unity |
| B. | Other imaginary cube root of unity |
| C. | Sum of two imaginary roots of unity |
| D. | None of these |
| Answer» C. Sum of two imaginary roots of unity | |
| 6276. |
If \[\omega \] is a cube root of unity, then \[{{(1+\omega )}^{3}}-{{(1+{{\omega }^{2}})}^{3}}=\] |
| A. | 0 |
| B. | \[\omega \] |
| C. | \[{{\omega }^{2}}\] |
| D. | None of these |
| Answer» B. \[\omega \] | |
| 6277. |
\[{{(27)}^{1/3}}=\] |
| A. | 3 |
| B. | \[3,\,\,3i,\,3{{i}^{2}}\] |
| C. | \[3,\,3\omega ,\,3{{\omega }^{2}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 6278. |
If \[\omega \] is a cube root of unity, then \[(1+\omega -{{\omega }^{2}})\] \[(1-\omega +{{\omega }^{2}})\] = [MNR 1990; MP PET 1993, 2002] |
| A. | 1 |
| B. | 0 |
| C. | 2 |
| D. | 4 |
| Answer» E. | |
| 6279. |
The two numbers such that each one is square of the other, are [MP PET 1987] |
| A. | \[\omega ,\,{{\omega }^{3}}\] |
| B. | \[-i,\,\,i\] |
| C. | \[-1,\,1\] |
| D. | \[\omega ,\,\,{{\omega }^{2}}\] |
| Answer» E. | |
| 6280. |
If \[i{{z}^{4}}+1=0\], then \[z\] can take the value [UPSEAT 2004] |
| A. | \[\frac{1+i}{\sqrt{2}}\] |
| B. | \[\cos \frac{\pi }{8}+i\,\sin \frac{\pi }{8}\] |
| C. | \[\frac{1}{4i}\] |
| D. | i |
| Answer» C. \[\frac{1}{4i}\] | |
| 6281. |
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. | \[{{(\sqrt{2})}^{n-2}}\sin \left( \frac{n\pi }{4} \right)\] |
| C. | \[{{(\sqrt{2})}^{n+2}}\cos \left( \frac{n\pi }{4} \right)\] |
| D. | \[{{(\sqrt{2})}^{n+2}}\sin \left( \frac{n\pi }{4} \right)\] |
| Answer» D. \[{{(\sqrt{2})}^{n+2}}\sin \left( \frac{n\pi }{4} \right)\] | |
| 6282. |
\[{{\left( \frac{1+\sin \theta +i\,\cos \theta }{1+\sin \theta -i\,\cos \theta } \right)}^{n}}\]= [Kerala (Engg.) 2002] |
| A. | \[\cos \left( \frac{n\pi }{2}-n\theta \right)+i\,\sin \left( \frac{n\pi }{2}-n\theta \right)\] |
| 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. | \[\cos \,n\left( \frac{\pi }{2}+2\theta \right)+i\,\sin \,n\left( \frac{\pi }{2}+2\theta \right)\] |
| Answer» B. \[\cos \left( \frac{n\pi }{2}+n\theta \right)+i\,\sin \left( \frac{n\pi }{2}+n\theta \right)\] | |
| 6283. |
The value of i1/3 is [UPSEAT 2002] |
| A. | \[\frac{\sqrt{3}\,+i}{2}\] |
| B. | \[\frac{\sqrt{3}\,-i}{2}\] |
| C. | \[\frac{1+i\sqrt{3}}{2}\] |
| D. | \[\frac{1-i\sqrt{3}}{2}\] |
| Answer» B. \[\frac{\sqrt{3}\,-i}{2}\] | |
| 6284. |
\[\frac{{{(\cos \alpha +i\,\sin \alpha )}^{4}}}{{{(\sin \beta +i\,\cos \beta )}^{5}}}=\] [RPET 2002] |
| A. | \[\cos (4\alpha +5\beta )+i\,\sin (4\alpha +5\beta )\] |
| B. | \[\cos (4\alpha +5\beta )-i\,\sin (4\alpha +5\beta )\] |
| C. | \[\sin (4\alpha +5\beta )-i\cos (4\alpha +5\beta )\] |
| D. | None of these |
| Answer» D. None of these | |
| 6285. |
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] |
| A. | \[\frac{1+i\sqrt{3}}{2}\] |
| B. | \[\frac{-1+i\sqrt{3}}{2}\] |
| C. | \[\frac{1-i\sqrt{3}}{2}\] |
| D. | \[\frac{-1-i\sqrt{3}}{2}\] |
| Answer» B. \[\frac{-1+i\sqrt{3}}{2}\] | |
| 6286. |
\[{{\left[ \frac{1+\cos (\pi /8)+i\,\sin (\pi /8)}{1+\cos (\pi /8)-i\,\sin (\pi /8)} \right]}^{8}}\] is equal to [RPET 2001] |
| A. | -1 |
| B. | 0 |
| C. | 1 |
| D. | 2 |
| Answer» B. 0 | |
| 6287. |
If \[{{x}_{r}}=\cos \left( \frac{\pi }{{{2}^{r}}} \right)+i\sin \left( \frac{\pi }{{{2}^{r}}} \right)\], then\[{{x}_{1}}.{{x}_{2}}......\infty \]is [RPET 1990, 2000; BIT Mesra 1996; Karnataka CET 2000] |
| A. | \[-3\] |
| B. | \[-2\] |
| C. | \[-1\] |
| D. | 0 |
| Answer» D. 0 | |
| 6288. |
\[{{(\sin \theta +i\,\cos \theta )}^{n}}\,\]is equal to [RPET 2001] |
| A. | \[\cos n\theta +i\,\sin n\theta \] |
| B. | \[\sin n\theta +i\,\cos n\theta \] |
| C. | \[\cos n\left( \frac{\pi }{2}-\theta \right)+i\,\sin n\left( \frac{\pi }{2}-\theta \right)\] |
| D. | None of these |
| Answer» D. None of these | |
| 6289. |
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] |
| A. | \[\cos 49\theta -i\,\sin 49\theta \] |
| B. | \[\cos 23\theta -i\,\sin 23\theta \] |
| C. | \[\cos 49\theta +i\,\sin 49\theta \] |
| D. | \[\cos 21\theta +i\,\sin 21\theta \] |
| Answer» B. \[\cos 23\theta -i\,\sin 23\theta \] | |
| 6290. |
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] |
| A. | 0 |
| B. | -1 |
| C. | 1 |
| D. | 2 |
| Answer» C. 1 | |
| 6291. |
\[{{(-\sqrt{3}+i)}^{53}}\] where \[{{i}^{2}}=-1\] is equal to [AMU 2000] |
| A. | \[{{2}^{53}}(\sqrt{3}+2i)\] |
| B. | \[{{2}^{52}}(\sqrt{3}-i)\] |
| C. | \[{{2}^{53}}\,\left( \frac{\sqrt{3}}{2}+\frac{1}{2}i \right)\] |
| D. | \[{{2}^{53}}(\sqrt{3}-i)\] |
| Answer» D. \[{{2}^{53}}(\sqrt{3}-i)\] | |
| 6292. |
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] |
| A. | \[2\cos (\alpha +\beta +\gamma )\] |
| B. | \[\cos 2(\alpha +\beta +\gamma )\] |
| C. | 0 |
| D. | 1 |
| Answer» D. 1 | |
| 6293. |
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] |
| A. | 44257 |
| B. | 44230 |
| C. | 44228 |
| D. | 1 |
| Answer» C. 44228 | |
| 6294. |
\[{{\left( \frac{\cos \theta +i\sin \theta }{\sin \theta +i\cos \theta } \right)}^{4}}\]equals [RPET 1996] |
| A. | \[\sin 8\theta -i\cos 8\theta \] |
| B. | \[\cos 8\theta -i\sin 8\theta \] |
| C. | \[\sin 8\theta +i\cos 8\theta \] |
| D. | \[\cos 8\theta +i\sin 8\theta \] |
| Answer» E. | |
| 6295. |
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] |
| A. | \[-1\] |
| B. | \[1\] |
| C. | 0 |
| D. | 2 |
| Answer» B. \[1\] | |
| 6296. |
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] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» E. | |
| 6297. |
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. | \[{{\alpha }^{3}}+{{\beta }^{3}}\] |
| D. | \[{{\alpha }^{3}}-{{\beta }^{3}}\] |
| Answer» E. | |
| 6298. |
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. | 0 |
| B. | \[\frac{\sqrt{3}}{2}\] |
| C. | 1 |
| D. | 2 |
| Answer» D. 2 | |
| 6299. |
If \[{{\tan }^{-1}}(\alpha +i\beta )=x+iy,\] then x = [RPET 2002] |
| A. | \[\frac{1}{2}{{\tan }^{-1}}\left( \frac{2\alpha }{1-{{\alpha }^{2}}-{{\beta }^{2}}} \right)\] |
| 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. | None of these |
| Answer» B. \[\frac{1}{2}{{\tan }^{-1}}\left( \frac{2\alpha }{1+{{\alpha }^{2}}+{{\beta }^{2}}} \right)\] | |
| 6300. |
\[{{\left( \frac{1+\cos \varphi +i\sin \varphi }{1+\cos \varphi -i\sin \varphi } \right)}^{n}}=\] |
| A. | \[\cos n\varphi -i\sin n\varphi \] |
| B. | \[\cos n\varphi +i\sin n\varphi \] |
| C. | \[\sin n\varphi +i\cos n\varphi \] |
| D. | \[\sin n\varphi -i\cos n\varphi \] |
| Answer» C. \[\sin n\varphi +i\cos n\varphi \] | |