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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.
| 4151. |
The value of \[\int_{{}}^{{}}{x\sin kx\ dx}\]is |
| A. | \[\frac{\sin kx}{k}+c\] |
| B. | \[\frac{\cos kx}{k}+c\] |
| C. | \[\frac{\sin x}{k}+c\] |
| D. | \[-\frac{x\,\cos kx}{k}+\frac{\sin kx}{{{k}^{2}}}+c\] |
| Answer» E. | |
| 4152. |
\[\int_{{}}^{{}}{[f(x)g''(x)-f''(x)g(x)]}\ dx\] is equal to [MP PET 2001] |
| A. | \[\frac{f(x)}{g'(x)}\] |
| B. | \[f'(x)g(x)-f(x)g'(x)\] |
| C. | \[f(x)g'(x)-f'(x)g(x)\] |
| D. | \[f(x)g'(x)+f'(x)g(x)\] |
| Answer» D. \[f(x)g'(x)+f'(x)g(x)\] | |
| 4153. |
If \[\int_{{}}^{{}}{x\sin xdx=-x\cos x+A}\], then \[A=\] [MP PET 1992, 2000; RPET 1997] |
| A. | \[\sin x+\]Constant |
| B. | \[\cos x+\]Constant |
| C. | Constant |
| D. | None of these |
| Answer» B. \[\cos x+\]Constant | |
| 4154. |
\[\int_{{}}^{{}}{{{e}^{2x}}(-\sin x+2\cos x)\ dx=}\] [DSSE 1987] |
| A. | \[{{e}^{2x}}\sin x+c\] |
| B. | \[-{{e}^{2x}}\sin x+c\] |
| C. | \[-{{e}^{2x}}\cos x+c\] |
| D. | \[{{e}^{2x}}\cos x+c\] |
| Answer» E. | |
| 4155. |
\[\int_{{}}^{{}}{{{e}^{x}}\sin x\ dx=}\] [IIT 1978; AI CBSE 1980; MP PET 1999] |
| A. | \[\frac{1}{2}{{e}^{x}}(\sin x+\cos x)+c\] |
| B. | \[\frac{1}{2}{{e}^{x}}(\sin x-\cos x)+c\] |
| C. | \[{{e}^{x}}(\sin x+\cos x)+c\] |
| D. | \[{{e}^{x}}(\sin x-\cos x)+c\] |
| Answer» C. \[{{e}^{x}}(\sin x+\cos x)+c\] | |
| 4156. |
\[\int_{{}}^{{}}{(1-{{x}^{2}})\log x\ dx=}\] [DSSE 1982] |
| A. | \[\left( x-\frac{{{x}^{3}}}{3} \right)\log x-\left( x-\frac{{{x}^{3}}}{9} \right)+c\] |
| B. | \[\left( x-\frac{{{x}^{3}}}{3} \right)\log x+\left( x-\frac{{{x}^{3}}}{9} \right)+c\] |
| C. | \[\left( x+\frac{{{x}^{3}}}{3} \right)\log x+\left( x+\frac{{{x}^{3}}}{9} \right)+c\] |
| D. | None of these |
| Answer» B. \[\left( x-\frac{{{x}^{3}}}{3} \right)\log x+\left( x-\frac{{{x}^{3}}}{9} \right)+c\] | |
| 4157. |
\[\int_{{}}^{{}}{\frac{\log x}{{{(1+\log x)}^{2}}}dx=}\] |
| A. | \[\frac{1}{1+\log x}+c\] |
| B. | \[\frac{x}{{{(1+\log x)}^{2}}}+c\] |
| C. | \[\frac{x}{1+\log x}+c\] |
| D. | \[\frac{1}{{{(1+\log x)}^{2}}}+c\] |
| Answer» D. \[\frac{1}{{{(1+\log x)}^{2}}}+c\] | |
| 4158. |
\[\int_{{}}^{{}}{\left( \frac{2+\sin 2x}{1+\cos 2x} \right)\,\,{{e}^{x}}dx=}\] [AISSE 1982] |
| A. | \[{{e}^{x}}\cot x+c\] |
| B. | \[-{{e}^{x}}\cot x+c\] |
| C. | \[-{{e}^{x}}\tan x+c\] |
| D. | \[{{e}^{x}}\tan x+c\] |
| Answer» E. | |
| 4159. |
\[\int_{{}}^{{}}{{{x}^{n}}\log x\ dx=}\] |
| A. | \[\frac{{{x}^{n+1}}}{n+1}\left\{ \log x+\frac{1}{n+1} \right\}+c\] |
| B. | \[\frac{{{x}^{n+1}}}{n+1}\left\{ \log x+\frac{2}{n+1} \right\}+c\] |
| C. | \[\frac{{{x}^{n+1}}}{n+1}\left\{ 2\log x-\frac{1}{n+1} \right\}+c\] |
| D. | \[\frac{{{x}^{n+1}}}{n+1}\left\{ \log x-\frac{1}{n+1} \right\}+c\] |
| Answer» E. | |
| 4160. |
\[\int_{{}}^{{}}{\log x(\log x+2)\ dx=}\] |
| A. | \[x{{(\log x)}^{2}}+c\] |
| B. | \[x{{(1+\log x)}^{2}}+c\] |
| C. | \[x[1+{{(\log x)}^{2}}]+c\] |
| D. | None of these |
| Answer» B. \[x{{(1+\log x)}^{2}}+c\] | |
| 4161. |
If \[\int_{{}}^{{}}{{{e}^{x}}\sin x\ dx=\frac{1}{2}{{e}^{x}}\ .\ a+c}\], then \[a=\] [MP PET 1989] |
| A. | \[\sin x-\cos x\] |
| B. | \[\cos x-\sin x\] |
| C. | \[-\cos x-\sin x\] |
| D. | \[\cos x+\sin x\] |
| Answer» B. \[\cos x-\sin x\] | |
| 4162. |
\[\int_{{}}^{{}}{\frac{1}{{{\log }_{x}}e}dx=}\] [MP PET 1994] |
| A. | \[\log {{\log }_{x}}e+c\] |
| B. | \[\frac{1}{{{({{\log }_{x}}e)}^{2}}}+c\] |
| C. | \[x\log \left( \frac{x}{e} \right)+c\] |
| D. | None of these |
| Answer» D. None of these | |
| 4163. |
\[\int_{{}}^{{}}{\sin (\log x)dx=}\] |
| A. | \[\frac{1}{2}x[\cos (\log x)-\sin (\log x)]\] |
| B. | \[\cos (\log x)-x\] |
| C. | \[\frac{1}{2}x[\sin (\log x)-\cos (\log x)]\] |
| D. | \[-\cos \log x\] |
| Answer» D. \[-\cos \log x\] | |
| 4164. |
\[\int_{{}}^{{}}{{{x}^{2}}\sin 2x}\ dx=\] [IIT 1974] |
| A. | \[\frac{1}{2}{{x}^{2}}\cos 2x+\frac{1}{2}x\sin 2x+\frac{1}{4}\cos 2x+c\] |
| B. | \[-\frac{1}{2}{{x}^{2}}\cos 2x+\frac{1}{2}x\sin 2x+\frac{1}{4}\cos 2x+c\] |
| C. | \[\frac{1}{2}{{x}^{2}}\cos 2x-\frac{1}{2}x\sin 2x+\frac{1}{4}\cos 2x+c\] |
| D. | None of these |
| Answer» C. \[\frac{1}{2}{{x}^{2}}\cos 2x-\frac{1}{2}x\sin 2x+\frac{1}{4}\cos 2x+c\] | |
| 4165. |
\[\int_{{}}^{{}}{\log xdx=}\] [MNR 1979; BIT Ranchi 1992; SCRA 1996] |
| A. | \[x+x\log x+c\] |
| B. | \[x\log x-x+c\] |
| C. | \[{{x}^{2}}\log x+c\] |
| D. | \[\frac{1}{x}\log x+x+c\] |
| Answer» C. \[{{x}^{2}}\log x+c\] | |
| 4166. |
If \[\int_{{}}^{{}}{\ln ({{x}^{2}}+x)dx=x\ln ({{x}^{2}}+x)+A}\], then \[A=\] [MP PET 1992] |
| A. | \[2x+\ln (x+1)+\]constant |
| B. | \[2x-\ln (x+1)+\]constant |
| C. | Constant |
| D. | None of these |
| Answer» E. | |
| 4167. |
\[\int_{{}}^{{}}{\log (x+1)dx=}\] [Roorkee 1974] |
| A. | \[(x+1)\log (x+1)-x+c\] |
| B. | \[(x+1)\log (x+1)+x+c\] |
| C. | \[(x-1)\log (x+1)-x+c\] |
| D. | \[(x-1)\log (x+1)+x+c\] |
| Answer» B. \[(x+1)\log (x+1)+x+c\] | |
| 4168. |
\[\int_{{}}^{{}}{{{e}^{2x+\log x}}}dx=\] |
| A. | \[\frac{1}{4}(2x-1)+\frac{2}{x+1}+c\] |
| B. | \[\frac{1}{4}(2x+1)+\frac{2}{x+1}+c\] |
| C. | \[\frac{1}{2}(2x+1){{e}^{2x}}+c\] |
| D. | \[\frac{1}{2}(2x+1){{e}^{2x}}+c\] |
| Answer» B. \[\frac{1}{4}(2x+1)+\frac{2}{x+1}+c\] | |
| 4169. |
\[\int_{{}}^{{}}{x{{\sin }^{2}}x\ dx=}\] [BIT Ranchi 1977; IIT 1972] |
| A. | \[\frac{{{x}^{2}}}{4}+\frac{x}{4}\sin 2x+\frac{1}{8}\cos 2x+c\] |
| B. | \[\frac{{{x}^{2}}}{4}-\frac{x}{4}\sin 2x+\frac{1}{8}\cos 2x+c\] |
| C. | \[\frac{{{x}^{2}}}{4}+\frac{x}{4}\sin 2x-\frac{1}{8}\cos 2x+c\] |
| D. | \[\frac{{{x}^{2}}}{4}-\frac{x}{4}\sin 2x-\frac{1}{8}\cos 2x+c\] |
| Answer» E. | |
| 4170. |
\[\int_{{}}^{{}}{x\sin x{{\sec }^{3}}x\,dx=}\] |
| A. | \[\frac{1}{2}[{{\sec }^{2}}x-\tan x]+c\] |
| B. | \[\frac{1}{2}[x{{\sec }^{2}}x-\tan x]+c\] |
| C. | \[\frac{1}{2}[x{{\sec }^{2}}x+\tan x]+c\] |
| D. | \[\frac{1}{2}[{{\sec }^{2}}x+\tan x]+c\] |
| Answer» C. \[\frac{1}{2}[x{{\sec }^{2}}x+\tan x]+c\] | |
| 4171. |
\[\int_{{}}^{{}}{\frac{\log x\ dx}{{{x}^{3}}}=}\] [Roorkee 1986] |
| A. | \[\frac{1}{4{{x}^{2}}}(2\log x-1)+c\] |
| B. | \[-\frac{1}{4{{x}^{2}}}(2\log x+1)+c\] |
| C. | \[\frac{1}{4{{x}^{2}}}(2\log x+1)+c\] |
| D. | \[\frac{1}{4{{x}^{2}}}(1-2\log x)+c\] |
| Answer» C. \[\frac{1}{4{{x}^{2}}}(2\log x+1)+c\] | |
| 4172. |
\[\int_{{}}^{{}}{[\sin (\log x)+\cos (\log x)]}\ dx=\] [MP PET 1991] |
| A. | \[x\cos (\log x)+c\] |
| B. | \[\sin (\log x)+c\] |
| C. | \[\cos (\log x)+c\] |
| D. | \[x\sin (\log x)+c\] |
| Answer» E. | |
| 4173. |
\[\int_{{}}^{{}}{x{{\sec }^{2}}x\ dx}=\] [RPET 1996, 2003; MP PET 1987, 97; Pb. CET 2002] |
| A. | \[\tan x+\log \cos x+c\] |
| B. | \[\frac{{{x}^{2}}}{2}{{\sec }^{2}}x+\log \cos x+c\] |
| C. | \[x\tan x+\log \sec x+c\] |
| D. | \[x\tan x+\log \cos x+c\] |
| Answer» E. | |
| 4174. |
\[1+{{i}^{2}}+{{i}^{4}}+{{i}^{6}}+.....+{{i}^{2n}}\]is [EAMCET 1980] |
| A. | Positive |
| B. | Negative |
| C. | Zero |
| D. | Cannot be determined |
| Answer» E. | |
| 4175. |
The value of \[\frac{{{i}^{592}}+{{i}^{590}}+{{i}^{588}}+{{i}^{586}}+{{i}^{584}}}{{{i}^{582}}+{{i}^{580}}+{{i}^{578}}+{{i}^{576}}+{{i}^{574}}}-1=\] |
| A. | \[-1\] |
| B. | -2 |
| C. | \[-3\] |
| D. | -4 |
| Answer» C. \[-3\] | |
| 4176. |
\[{{\left( \frac{1+i}{1-i} \right)}^{2}}+{{\left( \frac{1-i}{1+i} \right)}^{2}}\]is equal to |
| A. | \[2i\] |
| B. | \[-2i\] |
| C. | \[-2\] |
| D. | \[2\] |
| Answer» D. \[2\] | |
| 4177. |
The multiplication inverse of a number is the number itself, then its initial value is [RPET 2003] |
| A. | i |
| B. | -1 |
| C. | 2 |
| D. | |
| Answer» C. 2 | |
| 4178. |
The statement \[(a+ib) |
| A. | \[{{a}^{2}}+{{b}^{2}}=0\] |
| B. | \[{{b}^{2}}+{{c}^{2}}=0\] |
| C. | \[{{a}^{2}}+{{c}^{2}}=0\] |
| D. | \[{{b}^{2}}+{{d}^{2}}=0\] |
| Answer» E. | |
| 4179. |
The real part of \[\frac{1}{1-\cos \theta +i\,\sin \theta }\] is equal to [Karnataka CET 2001, 05] |
| A. | 44287 |
| B. | 44228 |
| C. | tan q/2 |
| D. | 1/1- cos q |
| Answer» C. tan q/2 | |
| 4180. |
The complex number \[\frac{1+2i}{1-i}\] lies in which quadrant of the complex plane [MP PET 2001] |
| A. | First |
| B. | Second |
| C. | Third |
| D. | Fourth |
| Answer» C. Third | |
| 4181. |
Solving \[3-2yi={{9}^{x}}-7i\], where \[{{i}^{2}}=-1,\] for x and y real, we get [AMU 2000] |
| A. | \[x=0.5\,\,,\,\,y=3.5\] |
| B. | \[x=5\,\,,\,\,y=3\] |
| C. | \[x=\frac{1}{2}\,\,,\,\,y=7\] |
| D. | \[x=0,\,y=\frac{3+7i}{2i}\] |
| Answer» B. \[x=5\,\,,\,\,y=3\] | |
| 4182. |
If \[a=\cos \,\theta +i\,\sin \,\theta ,\] then \[\frac{1+a}{1-a}=\] [Karnataka CET 2000] |
| A. | \[\cot \theta \] |
| B. | \[\cot \frac{\theta }{2}\] |
| C. | \[i\,\cot \frac{\theta }{2}\] |
| D. | \[i\,\tan \frac{\theta }{2}\] |
| Answer» D. \[i\,\tan \frac{\theta }{2}\] | |
| 4183. |
The value of \[{{(1+i)}^{5}}\times {{(1-i)}^{5}}\] is [Karnataka CET 1992] |
| A. | -8 |
| B. | \[8i\] |
| C. | 8 |
| D. | 32 |
| Answer» E. | |
| 4184. |
If \[\,\left| \begin{align} & \,6i\,\,\,\,\,-3i\,\,\,\,\,\,\,\,\,1 \\ & \,\,4\,\,\,\,\,\,\,\,\,3i\,\,\,\,\,\,-1 \\ & \,20\,\,\,\,\,\,\,\,3\,\,\,\,\,\,\,\,\,\,i \\ \end{align} \right|\,\]=\[x+iy\], then (x, y) is [MP PET 2000] |
| A. | (3, 1) |
| B. | (1, 3) |
| C. | (0, 3) |
| D. | (0, 0) |
| Answer» E. | |
| 4185. |
If \[z=1+i,\] then the multiplicative inverse of z2 is (where i = \[\sqrt{-1}\]) [Karnataka CET 1999] |
| A. | 2 si |
| B. | 1 - i |
| C. | |
| D. | i/2 |
| Answer» D. i/2 | |
| 4186. |
If \[{{z}_{1}}=(4,5)\] and \[{{z}_{2}}=(-3,2)\]then \[\frac{{{z}_{1}}}{{{z}_{2}}}\] equals [RPET 1996] |
| A. | \[\left( \frac{-23}{12},\frac{-2}{13} \right)\] |
| B. | \[\left( \frac{2}{13},\frac{-23}{13} \right)\] |
| C. | \[\left( \frac{-2}{13},\frac{-23}{13} \right)\] |
| D. | \[\left( \frac{-2}{13},\frac{23}{13} \right)\] |
| Answer» D. \[\left( \frac{-2}{13},\frac{23}{13} \right)\] | |
| 4187. |
If \[{{\left( \frac{1-i}{1+i} \right)}^{100}}=a+ib\], then [MP PET 1998] |
| A. | \[a=2,b=-1\] |
| B. | \[a=1,b=0\] |
| C. | \[a=0,b=1\] |
| D. | \[a=-1,b=2\] |
| Answer» C. \[a=0,b=1\] | |
| 4188. |
If \[x,y\in R\]and \[(x+iy)(3+2i)=1+i\], then \[(x,\,y)\] is |
| A. | \[\left( 1,\frac{1}{5} \right)\] |
| B. | \[\left( \frac{1}{13},\frac{1}{13} \right)\] |
| C. | \[\left( \frac{5}{13},\frac{1}{13} \right)\] |
| D. | \[\left( \frac{1}{5},\frac{1}{5} \right)\] |
| Answer» D. \[\left( \frac{1}{5},\frac{1}{5} \right)\] | |
| 4189. |
\[A+iB\] form of \[\frac{(\cos x+i\sin x)(\cos y+i\sin y)}{(\cot u+i)(1+i\tan v)}\] is [Roorkee 1980] |
| A. | \[\sin u\cos v\,[\cos (x+y-u-v)+i\sin (x+y-u-v)]\] |
| B. | \[\sin u\cos v\,[\cos (x+y+u+v)+i\sin (x+y+u+v)]\] |
| C. | \[\sin u\cos v\,[\cos (x+y+u+v)-i\sin (x+y+u+v)]\] |
| D. | None of these |
| Answer» B. \[\sin u\cos v\,[\cos (x+y+u+v)+i\sin (x+y+u+v)]\] | |
| 4190. |
If \[(x+iy)(p+iq)=({{x}^{2}}+{{y}^{2}})i\], then |
| A. | \[p=x,q=y\] |
| B. | \[p={{x}^{2}},\,\,q={{y}^{2}}\] |
| C. | \[x=q,y=p\] |
| D. | None of these |
| Answer» D. None of these | |
| 4191. |
Let \[{{z}_{1}},{{z}_{2}}\] be two complex numbers such that \[{{z}_{1}}+{{z}_{2}}\] and \[{{z}_{1}}{{z}_{2}}\] both are real, then [RPET 1996] |
| A. | \[{{z}_{1}}=-{{z}_{2}}\] |
| B. | \[{{z}_{1}}={{\bar{z}}_{2}}\] |
| C. | \[{{z}_{1}}=-{{\bar{z}}_{2}}\] |
| D. | \[{{z}_{1}}={{z}_{2}}\] |
| Answer» C. \[{{z}_{1}}=-{{\bar{z}}_{2}}\] | |
| 4192. |
If \[z(1+a)=b+ic\] and \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=1\], then \[\frac{1+iz}{1-iz}=\] |
| A. | \[\frac{a+ib}{1+c}\] |
| B. | \[\frac{b-ic}{1+a}\] |
| C. | \[\frac{a+ic}{1+b}\] |
| D. | None of these |
| Answer» B. \[\frac{b-ic}{1+a}\] | |
| 4193. |
If \[\sum\limits_{k=0}^{100}{{{i}^{k}}}=x+iy\], then the values of \[x\] and \[y\]are |
| A. | \[x=-1,y=0\] |
| B. | \[x=1,y=1\] |
| C. | \[x=1,y=0\] |
| D. | \[x=0,y=1\] |
| Answer» D. \[x=0,y=1\] | |
| 4194. |
If \[\frac{3x+2iy}{5i-2}=\frac{15}{8x+3iy}\], then |
| A. | \[x=1,y=-3\] |
| B. | \[x=-1,y=3\] |
| C. | \[x=1,y=3\] |
| D. | \[x=-1,y=-3\]or \[x=1,\]\[y=3\] |
| Answer» E. | |
| 4195. |
If \[{{(1-i)}^{n}}={{2}^{n}},\]then \[n=\] [RPET 1990] |
| A. | 1 |
| B. | 0 |
| C. | \[-1\] |
| D. | None of these |
| Answer» C. \[-1\] | |
| 4196. |
If \[{{z}_{1}}=1-i\] and \[{{z}_{2}}=-2+4i\], then \[\operatorname{Im}\left( \frac{{{z}_{1}}{{z}_{2}}}{{{z}_{1}}} \right)=\] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» E. | |
| 4197. |
If \[z=3-4i\], then \[{{z}^{4}}-3{{z}^{3}}+3{{z}^{2}}+99z-95\]is equal to |
| A. | 5 |
| B. | 6 |
| C. | -5 |
| D. | -4 |
| Answer» B. 6 | |
| 4198. |
If \[\frac{{{(p+i)}^{2}}}{2p-i}=\mu +i\lambda ,\]then \[{{\mu }^{2}}+{{\lambda }^{2}}\] is equal to |
| A. | \[\frac{{{({{p}^{2}}+1)}^{2}}}{4{{p}^{2}}-1}\] |
| B. | \[\frac{{{({{p}^{2}}-1)}^{2}}}{4{{p}^{2}}-1}\] |
| C. | \[\frac{{{({{p}^{2}}-1)}^{2}}}{4{{p}^{2}}+1}\] |
| D. | \[\frac{{{({{p}^{2}}+1)}^{2}}}{4{{p}^{2}}+1}\] |
| Answer» E. | |
| 4199. |
If \[x+iy=\frac{3}{2+\cos \theta +i\sin \theta },\]then \[{{x}^{2}}+{{y}^{2}}\] is equal to |
| A. | \[3x-4\] |
| B. | \[4x-3\] |
| C. | \[4x+3\] |
| D. | None of these |
| Answer» C. \[4x+3\] | |
| 4200. |
\[a+ib>c+id\]can be explained only when |
| A. | \[b=0,c=0\] |
| B. | \[b=0,d=0\] |
| C. | \[a=0,c=0\] |
| D. | \[a=0,d=0\] |
| Answer» C. \[a=0,c=0\] | |