MCQOPTIONS
Saved Bookmarks
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.
| 4201. |
\[\frac{1-2i}{2+i}+\frac{4-i}{3+2i}=\] [RPET 1987] |
| A. | \[\frac{24}{13}+\frac{10}{13}i\] |
| B. | \[\frac{24}{13}-\frac{10}{13}i\] |
| C. | \[\frac{10}{13}+\frac{24}{13}i\] |
| D. | \[\frac{10}{13}-\frac{24}{13}i\] |
| Answer» E. | |
| 4202. |
The true statement is [Roorkee 1989] |
| A. | \[1-i<1+i\] |
| B. | \[2i+1>-2i+1\] |
| C. | \[2i>1\] |
| D. | None of these |
| Answer» E. | |
| 4203. |
If \[\frac{5(-8+6i)}{{{(1+i)}^{2}}}=a+ib\], then\[(a,\,b)\] equals [RPET 1986] |
| A. | (15, 20) |
| B. | (20, 15) |
| C. | \[(-15,\,20)\] |
| D. | None of these |
| Answer» B. (20, 15) | |
| 4204. |
If \[z\ne 0\] is a complex number, then |
| A. | \[\operatorname{Re}(z)=0\Rightarrow \operatorname{Im}({{z}^{2}})=0\] |
| B. | \[\operatorname{Re}({{z}^{2}})=0\Rightarrow \operatorname{Im}({{z}^{2}})=0\] |
| C. | \[\operatorname{Re}(z)=0\Rightarrow \operatorname{Re}({{z}^{2}})=0\] |
| D. | None of these |
| Answer» B. \[\operatorname{Re}({{z}^{2}})=0\Rightarrow \operatorname{Im}({{z}^{2}})=0\] | |
| 4205. |
If \[{{\left( \frac{1+i}{1-i} \right)}^{m}}=1,\]then the least integral value of \[m\] is [IIT 1982; MNR 1984; UPSEAT 2001; MP PET 2002] |
| A. | 2 |
| B. | 4 |
| C. | 8 |
| D. | None of these |
| Answer» C. 8 | |
| 4206. |
The real values of \[x\] and \[y\] for which the equation \[({{x}^{4}}+2xi)-(3{{x}^{2}}+yi)=\]\[(3-5i)+(1+2yi)\] is satisfied, are [Roorkee 1984] |
| A. | \[x=2,y=3\] |
| B. | \[x=-2,y=\frac{1}{3}\] |
| C. | Both (a) and (b) |
| D. | None of these |
| Answer» D. None of these | |
| 4207. |
The real values of \[x\]and\[y\]for which the equation is \[(x+iy)\] \[(2-3i)\]= \[4+i\] is satisfied, are [Roorkee 1978] |
| A. | \[x=\frac{5}{13},y=\frac{8}{13}\] |
| B. | \[x=\frac{8}{13},y=\frac{5}{13}\] |
| C. | \[x=\frac{5}{13},y=\frac{14}{13}\] |
| D. | None of these |
| Answer» D. None of these | |
| 4208. |
\[{{\left\{ \frac{2i}{1+i} \right\}}^{2}}=\] [BIT Ranchi 1992] |
| A. | 1 |
| B. | \[2i\] |
| C. | \[\alpha -i\beta \,(\alpha ,\beta \,\text{real),}\] |
| D. | \[\left( \frac{3-4ix}{3+4ix} \right)=\] |
| Answer» C. \[\alpha -i\beta \,(\alpha ,\beta \,\text{real),}\] | |
| 4209. |
If \[{{(x+iy)}^{1/3}}=a+ib,\]then \[\frac{x}{a}+\frac{y}{b}\]is equal to [IT 1982; Karnataka CET 2000] |
| A. | \[4({{a}^{2}}+{{b}^{2}})\] |
| B. | \[4({{a}^{2}}-{{b}^{2}})\] |
| C. | \[4({{b}^{2}}-{{a}^{2}})\] |
| D. | None of these |
| Answer» C. \[4({{b}^{2}}-{{a}^{2}})\] | |
| 4210. |
The real part of \[{{(1-\cos \theta +2i\sin \theta )}^{-1}}\]is [IIT 1978, 86] |
| A. | \[\frac{1}{3+5\cos \theta }\] |
| B. | \[\frac{1}{5-3\cos \theta }\] |
| C. | \[\frac{1}{3-5\cos \theta }\] |
| D. | \[\frac{1}{5+3\cos \theta }\] |
| Answer» E. | |
| 4211. |
\[\frac{3+2i\sin \theta }{1-2i\sin \theta }\] will be purely imaginary, if \[\theta =\] [IIT 1976; Pb. CET 2003] |
| A. | \[2n\pi \pm \frac{\pi }{3}\] |
| B. | \[n\pi +\frac{\pi }{3}\] |
| C. | \[n\pi \pm \frac{\pi }{3}\] |
| D. | None of these [Where \[n\] is an integer] |
| Answer» D. None of these [Where \[n\] is an integer] | |
| 4212. |
If \[{{a}^{2}}+{{b}^{2}}=1,\] then \[\frac{1+b+ia}{1+b-ia}=\] |
| A. | 1 |
| B. | 2 |
| C. | \[b+ia\] |
| D. | \[a+ib\] |
| Answer» D. \[a+ib\] | |
| 4213. |
\[\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}=\] |
| A. | \[-\frac{3}{2}i\] |
| B. | \[\frac{3}{2}i\] |
| C. | \[-\frac{3}{2}\] |
| D. | \[\frac{3}{2}\] |
| Answer» B. \[\frac{3}{2}i\] | |
| 4214. |
\[\frac{3+2i\sin \theta }{1-2i\sin \theta }\]will be real, if \[\theta \] = [IIT 1976; EAMCET 2002] |
| A. | \[2n\pi \] |
| B. | \[n\pi +\frac{\pi }{2}\] |
| C. | \[n\pi \] |
| D. | None of these [Where \[n\] is an integer] |
| Answer» D. None of these [Where \[n\] is an integer] | |
| 4215. |
If \[n\] is a positive integer, then \[{{\left( \frac{1+i}{1-i} \right)}^{4n+1}}\]= |
| A. | 1 |
| B. | -1 |
| C. | \[i\] |
| D. | \[-i\] |
| Answer» D. \[-i\] | |
| 4216. |
\[\operatorname{Re}\frac{{{(1+i)}^{2}}}{3-i}\] = |
| A. | \[-1/5\] |
| B. | 44317 |
| C. | 44470 |
| D. | -0.1 |
| Answer» B. 44317 | |
| 4217. |
Additive inverse of \[1-i\]is |
| A. | \[0+0i\] |
| B. | \[-1-i\] |
| C. | \[-1+i\] |
| D. | None of these |
| Answer» D. None of these | |
| 4218. |
\[\left( \frac{1}{1-2i}+\frac{3}{1+i} \right)\,\,\left( \frac{3+4i}{2-4i} \right)=\] [Roorkee 1979; RPET 1999; Pb. CET 2003] |
| A. | \[\frac{1}{2}+\frac{9}{2}i\] |
| B. | \[\frac{1}{2}-\frac{9}{2}i\] |
| C. | \[\frac{1}{4}-\frac{9}{4}i\] |
| D. | \[\frac{1}{4}+\frac{9}{4}i\] |
| Answer» E. | |
| 4219. |
If \[{{z}_{1}}\] and \[{{z}_{2}}\] be two complex number, then Re\[({{z}_{1}}{{z}_{2}})=\] |
| A. | Re \[({{z}_{1}}).\operatorname{Re}({{z}_{2}})\] |
| B. | Re \[({{z}_{1}})\]. Im \[({{z}_{2}})\] |
| C. | Im \[({{z}_{1}}).\operatorname{Re}\,({{z}_{2}})\] |
| D. | None of these |
| Answer» E. | |
| 4220. |
The values of \[x\] and \[y\] satisfying the equation \[\frac{(1+i)x-2i}{3+i}\] \[+\frac{(2-3i)\,y+i}{3-i}=i\] are [IIT 1980; MNR 1987] |
| A. | \[x=-1,\,y=3\] |
| B. | \[x=3,\,y=-1\] |
| C. | \[x=0,\,y=1\] |
| D. | \[x=1,y=0\] |
| Answer» C. \[x=0,\,y=1\] | |
| 4221. |
The smallest positive integer \[n\]for which \[{{(1+i)}^{2n}}={{(1-i)}^{2n}}\]is [Karnataka CET 2004] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» C. 3 | |
| 4222. |
If \[x=3+i\], then \[{{x}^{3}}-3{{x}^{2}}-8x+15=\] [UPSEAT 2003] |
| A. | 6 |
| B. | 10 |
| C. | -18 |
| D. | -15 |
| Answer» E. | |
| 4223. |
If \[{{i}^{2}}=-1\], then sum \[i+{{i}^{2}}+{{i}^{3}}+...\]to 1000 terms is equal to [Kerala (Engg.) 2002] |
| A. | 1 |
| B. | -1 |
| C. | i |
| D. | 0 |
| Answer» E. | |
| 4224. |
The value of \[{{(1+i)}^{6}}+{{(1-i)}^{6}}\] is [RPET 2002] |
| A. | 0 |
| B. | 27 |
| C. | 26 |
| D. | None of these |
| Answer» B. 27 | |
| 4225. |
If \[n\] is a positive integer, then which of the following relations is false |
| A. | \[{{i}^{4n}}=1\] |
| B. | \[{{i}^{4n-1}}=i\] |
| C. | \[{{i}^{4n+1}}=i\] |
| D. | \[{{i}^{-4n}}=1\] |
| Answer» C. \[{{i}^{4n+1}}=i\] | |
| 4226. |
\[{{(1+i)}^{10}}\], where \[{{i}^{2}}=-1,\] is equal to [AMU 2001] |
| A. | 32 i |
| B. | 64 + i |
| C. | 24 i - 32 |
| D. | None of these |
| Answer» B. 64 + i | |
| 4227. |
The value of \[{{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}\,,\,(n\in N)\] is [RPET 2001] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | None of these |
| Answer» B. 1 | |
| 4228. |
The value of \[{{(1+i)}^{8}}+{{(1-i)}^{8}}\] is [RPET 2001; KCET 2001] |
| A. | 16 |
| B. | -16 |
| C. | 32 |
| D. | -32 |
| Answer» D. -32 | |
| 4229. |
If \[\]\[x+\frac{1}{x}=2\cos \theta ,\] then x is equal to [RPET 2001] |
| A. | \[\cos \theta +i\,\sin \theta \] |
| B. | \[\cos \theta -i\,\sin \theta \] |
| C. | \[\cos \theta \pm i\,\sin \theta \] |
| D. | \[\sin \theta \pm i\,\cos \theta \] |
| Answer» D. \[\sin \theta \pm i\,\cos \theta \] | |
| 4230. |
The value of \[{{i}^{1+3+5+...+(2n+1)}}\] is [AMU 1999] |
| A. | i if n is even, - i if n is odd |
| B. | 1 if n is even, - 1 if n is odd |
| C. | 1 if n is odd, - 1 if n is even |
| D. | i if n is even, - 1 if n is odd |
| Answer» D. i if n is even, - 1 if n is odd | |
| 4231. |
The least positive integer \[n\] which will reduce \[{{\left( \frac{i-1}{i+1} \right)}^{n}}\] to a real number, is [Roorkee 1998] |
| A. | 2 |
| B. | 3 |
| C. | 4 |
| D. | 5 |
| Answer» B. 3 | |
| 4232. |
The value of the sum \[\sum\limits_{n=1}^{13}{({{i}^{n}}+{{i}^{n+1}})}\], where \[i=\sqrt{-1}\], equals [IIT 1998] |
| A. | \[i\] |
| B. | \[i-1\] |
| C. | \[-i\] |
| D. | 0 |
| Answer» C. \[-i\] | |
| 4233. |
If \[{{i}^{2}}=-1\], then the value of \[\sum\limits_{n=1}^{200}{{{i}^{n}}}\]is [MP PET 1996] |
| A. | \[50\] |
| B. | -50 |
| C. | 0 |
| D. | 100 |
| Answer» D. 100 | |
| 4234. |
If \[i=\sqrt{-1}\], then \[1+{{i}^{2}}+{{i}^{3}}-{{i}^{6}}+{{i}^{8}}\] is equal to [RPET 1995] |
| A. | \[2-i\] |
| B. | 1 |
| C. | 3 |
| D. | \[-1\] |
| Answer» B. 1 | |
| 4235. |
\[{{i}^{2}}+{{i}^{4}}+{{i}^{6}}+......\]upto \[(2n+1)\] terms = [EAMCET 1980; Kerala (Engg.) 2005] |
| A. | \[i\] |
| B. | \[-i\] |
| C. | 1 |
| D. | \[-1\] |
| Answer» E. | |
| 4236. |
\[\sqrt{-2}\,\sqrt{-3}=\] [Roorkee 1978] |
| A. | \[\sqrt{6}\] |
| B. | \[-\sqrt{6}\] |
| C. | \[i\sqrt{6}\] |
| D. | None of these |
| Answer» C. \[i\sqrt{6}\] | |
| 4237. |
The function \[\sin x-bx+c\]will be increasing in the interval \[(-\infty ,\,\,\infty )\], if |
| A. | \[b\le 1\] |
| B. | \[b\le 0\] |
| C. | \[b<-1\] |
| D. | \[b\ge 0\] |
| Answer» D. \[b\ge 0\] | |
| 4238. |
The function \[\sin x-\cos x\]is increasing in the interval |
| A. | \[\left[ \frac{3\pi }{4},\frac{7\pi }{4} \right]\] |
| B. | \[\left[ 0,\frac{3\pi }{4} \right)\] |
| C. | \[\left[ \frac{\pi }{4},\frac{3\pi }{4} \right]\] |
| D. | None of these |
| Answer» C. \[\left[ \frac{\pi }{4},\frac{3\pi }{4} \right]\] | |
| 4239. |
A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched? [AIEEE 2005] |
| A. | Interval Function \[\left( -\infty ,\,\frac{1}{3} \right]\] \[3{{x}^{2}}-2x+1\] |
| B. | (? ¥, ? 4] \[{{x}^{3}}+6{{x}^{2}}+6\] |
| C. | (? ¥, ¥) \[{{x}^{3}}-3{{x}^{2}}+3x+3\] |
| D. | [2, ¥) \[2{{x}^{3}}-3{{x}^{2}}-12x+6\] |
| Answer» B. (? ¥, ? 4] \[{{x}^{3}}+6{{x}^{2}}+6\] | |
| 4240. |
On the interval \[\left( 0,\frac{\pi }{2} \right)\], the function log sin x is |
| A. | Increasing |
| B. | Decreasing |
| C. | Neither increasing nor decreasing |
| D. | None of these |
| Answer» B. Decreasing | |
| 4241. |
Function \[f(x)=\frac{4{{x}^{2}}+1}{x}\] is decreasing for interval |
| A. | \[\left( \frac{-1}{2},\,\frac{1}{2} \right)\] |
| B. | \[\left[ \frac{1}{2},\,-\frac{1}{2} \right]\] |
| C. | (? 1, 1) |
| D. | [1, ?1] |
| Answer» C. (? 1, 1) | |
| 4242. |
Given function \[f(x)=\left( \frac{{{e}^{2x}}-1}{{{e}^{2x}}+1} \right)\] is [Orissa JEE 2005] |
| A. | Increasing |
| B. | Decreasing |
| C. | Even |
| D. | None of these |
| Answer» B. Decreasing | |
| 4243. |
The function \[f(x)={{\tan }^{-1}}(\sin x+\cos x)\], \[x>0\] is always an increasing function on the interval [Kerala (Engg.) 2005] |
| A. | \[(0,\,\pi )\] |
| B. | \[(0,\,\pi /2)\] |
| C. | \[(0,\pi /4)\] |
| D. | \[(0,\,3\pi /4)\] |
| E. | \[(0,\,5\pi /4)\] |
| Answer» D. \[(0,\,3\pi /4)\] | |
| 4244. |
The function \[f(x)=2{{x}^{3}}-3{{x}^{2}}+90x+174\] is increasing in the interval [J & K 2005] |
| A. | \[\frac{1}{2}<x<1\] |
| B. | \[\frac{1}{2}<x<2\] |
| C. | \[3<x<\frac{59}{4}\] |
| D. | \[-\infty <x<\infty \] |
| Answer» E. | |
| 4245. |
If \[f(x)=x,-1\le x\le 1\], then function \[f(x)\] is [SCRA 1996] |
| A. | Increasing |
| B. | Decreasing |
| C. | Stationary |
| D. | Discontinuous |
| Answer» B. Decreasing | |
| 4246. |
For all \[x\in (0,\,1)\] [IIT Screening 2000] |
| A. | \[{{e}^{x}}<1+x\] |
| B. | \[{{\log }_{e}}(1+x)<x\] |
| C. | \[\sin x>x\] |
| D. | \[{{\log }_{e}}x>x\] |
| Answer» C. \[\sin x>x\] | |
| 4247. |
The length of the longest interval, in which the function \[3\sin x-4\sin x\] is increasing, is [IIT Screening 2002] |
| A. | \[\frac{\pi }{3}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\frac{3\pi }{2}\] |
| D. | \[\pi \] |
| Answer» B. \[\frac{\pi }{2}\] | |
| 4248. |
The interval in which the function \[{{x}^{2}}{{e}^{-x}}\]is non decreasing, is |
| A. | \[(-\infty ,\,\,2]\] |
| B. | [0, 2] |
| C. | \[[2,\,\,\infty )\] |
| D. | None of these |
| Answer» C. \[[2,\,\,\infty )\] | |
| 4249. |
\[2{{x}^{3}}-6x+5\] is an increasing function if [UPSEAT 2003] |
| A. | \[0<x<1\] |
| B. | \[-1<x<1\] |
| C. | \[x<-1\] or \[x>1\] |
| D. | \[-1<x<-1/2\] |
| Answer» D. \[-1<x<-1/2\] | |
| 4250. |
The function \[f(x)=1-{{x}^{3}}-{{x}^{5}}\] is decreasing for [Kerala (Engg.) 2002] |
| A. | \[1\le x\le 5\] |
| B. | \[x\le 1\] |
| C. | \[x\ge 1\] |
| D. | All values of x |
| Answer» E. | |