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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.
| 2151. |
The value of \[\int_{\,0}^{\,1}{\,\frac{dx}{x+\sqrt{1-{{x}^{2}}}}}\] is [MP PET 2003] |
| A. | \[\frac{\pi }{3}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\frac{1}{2}\] |
| D. | \[\frac{\pi }{4}\] |
| Answer» E. | |
| 2152. |
\[\int_{\,-1/2}^{\,1/2}{(\cos x)\,\left[ \log \left( \frac{1-x}{1+x} \right) \right]\,dx=}\] [Karnataka CET 2002] |
| A. | 0 |
| B. | 1 |
| C. | \[{{e}^{1/2}}\] |
| D. | \[2{{e}^{1/2}}\] |
| Answer» B. 1 | |
| 2153. |
Assume that \[f\] is continuous everywhere, then \[\frac{1}{c}\int_{ac}^{bc}{f\left( \frac{x}{c} \right)}\,dx=\] |
| A. | \[\int_{a}^{b}{f\left( \frac{x}{c} \right)}\,dx\] |
| B. | \[\frac{1}{c}\int_{a}^{b}{f(x)\,dx}\] |
| C. | \[\int_{a}^{b}{f(x)\,dx}\] |
| D. | None of these |
| Answer» D. None of these | |
| 2154. |
The smallest interval \[[a,\,\,b]\] such that \[\int_{0}^{1}{\frac{dx}{\sqrt{1+{{x}^{4}}}}}\in [a,\,\,b]\] is given by |
| A. | \[\left[ \frac{1}{\sqrt{2}},\,\,1 \right]\] |
| B. | \[[0,\,\,1]\] |
| C. | \[\left[ \frac{1}{2},\,\,2 \right]\] |
| D. | \[\left[ \frac{3}{4},\,\,1 \right]\] |
| Answer» B. \[[0,\,\,1]\] | |
| 2155. |
\[\int_{-4}^{4}{|x+2|\,dx}=\] |
| A. | 50 |
| B. | 24 |
| C. | 20 |
| D. | None of these |
| Answer» D. None of these | |
| 2156. |
\[\int_{0}^{1}{f(1-x)\,dx}\] has the same value as the integral [SCRA 1990] |
| A. | \[\int_{0}^{1}{f(x)\,dx}\] |
| B. | \[\int_{0}^{1}{f(-x)\,dx}\] |
| C. | \[\int_{0}^{1}{f(x-1)\,dx}\] |
| D. | \[\int_{-1}^{1}{f(x)\,dx}\] |
| Answer» B. \[\int_{0}^{1}{f(-x)\,dx}\] | |
| 2157. |
\[\int_{0}^{2\pi }{\frac{\sin 2\theta }{a-b\,\cos \theta }\,d\theta =}\] [Roorkee 1988] |
| A. | 1 |
| B. | 2 |
| C. | \[\frac{\pi }{4}\] |
| D. | 0 |
| Answer» E. | |
| 2158. |
\[\int_{0}^{\pi /4}{\log (1+\tan \theta )\,d\theta =}\] [SCRA 1986; Karnataka CET 2000, 05] |
| A. | \[\frac{\pi }{4}\log 2\] |
| B. | \[\frac{\pi }{4}\log \frac{1}{2}\] |
| C. | \[\frac{\pi }{8}\log 2\] |
| D. | \[\frac{\pi }{8}\log \frac{1}{2}\] |
| Answer» D. \[\frac{\pi }{8}\log \frac{1}{2}\] | |
| 2159. |
If \[f(x)=\left\{ \begin{matrix} 4x+3\,, & \text{if} & 1\le x\le 2 \\ 3x+5\,, & \text{if} & 2 |
| A. | 80 |
| B. | 20 |
| C. | \[-20\] |
| D. | 37 |
| Answer» E. | |
| 2160. |
If f is continuous function, then [Kerala (Engg.) 2005] |
| A. | \[\int_{-2}^{2}{f(x)dx=\int_{0}^{2}{[f(x)-f(-x)]dx}}\] |
| B. | \[\int_{-3}^{5}{2f(x)dx=\int_{-6}^{10}{f(x-1)dx}}\] |
| C. | \[\int_{-3}^{5}{f(x)dx=\int_{-4}^{4}{f(x-1)dx}}\] |
| D. | \[\int_{-3}^{5}{f(x)dx=\int_{-2}^{6}{f(x-1)dx}}\] |
| E. | \[\int_{-3}^{5}{f(x)dx=\int_{-6}^{10}{f(x/2)]dx}}\] |
| Answer» E. \[\int_{-3}^{5}{f(x)dx=\int_{-6}^{10}{f(x/2)]dx}}\] | |
| 2161. |
\[\int_{\ -\pi }^{\pi }{\frac{{{\sin }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}\ dx}=\] [Kerala (Engg.) 2005] |
| A. | \[\pi /4\] |
| B. | \[\pi /2\] |
| C. | \[3\pi /2\] |
| D. | \[2\pi \] |
| E. | \[\pi \] |
| Answer» F. | |
| 2162. |
\[\int_{\pi /6}^{\pi /3}{\frac{dx}{1+\sqrt{\tan x}}=}\] [Kerala (Engg.) 2005] |
| A. | \[\pi /12\] |
| B. | \[\pi /2\] |
| C. | \[\pi /6\] |
| D. | \[\pi /4\] |
| E. | \[2\pi /3\] |
| Answer» B. \[\pi /2\] | |
| 2163. |
\[\int_{0}^{2n\pi }{\left( |\sin x|-\left. \left| \frac{1}{2}\sin x \right. \right| \right)}\ dx\] equals [Orissa JEE 2005] |
| A. | n |
| B. | 2n |
| C. | ?2n |
| D. | None of these |
| Answer» C. ?2n | |
| 2164. |
If \[\int_{\sin x}^{1}{{{t}^{2}}f(t)\ dt=1-\sin x}\],\[x\in \left( 0,\frac{\pi }{2} \right)\] then \[f\ \left( \frac{1}{\sqrt{3}} \right)\] equal to [IIT Screening 2005] |
| A. | 3 |
| B. | \[\frac{1}{3}\] |
| C. | \[\frac{1}{\sqrt{3}}\] |
| D. | \[\sqrt{3}\] |
| Answer» B. \[\frac{1}{3}\] | |
| 2165. |
\[\int_{-\pi /2}^{\pi /2}{\sqrt{\frac{1}{2}(1-\cos 2x)}}\,dx=\] |
| A. | 0 |
| B. | 2 |
| C. | \[\frac{1}{2}\] |
| D. | None of these |
| Answer» C. \[\frac{1}{2}\] | |
| 2166. |
\[\int_{0}^{\pi /2}{x\cot x\,dx}\] equals [RPET 1997] |
| A. | \[-\frac{\pi }{2}\log 2\] |
| B. | \[\frac{\pi }{2}\log 2\] |
| C. | \[\pi \log 2\] |
| D. | \[-\pi \log 2\] |
| Answer» C. \[\pi \log 2\] | |
| 2167. |
\[\int_{0}^{\pi /2}{{{\left( \frac{\theta }{\sin \theta } \right)}^{2}}d\theta =}\] |
| A. | \[\pi \log 2\] |
| B. | \[\frac{\pi }{\log 2}\] |
| C. | \[\pi \] |
| D. | None of these |
| Answer» B. \[\frac{\pi }{\log 2}\] | |
| 2168. |
\[\int_{0}^{1}{\frac{\log x}{\sqrt{1-{{x}^{2}}}}\,dx=}\] [BIT Ranchi 1984] |
| A. | \[\frac{\pi }{2}\log 2\] |
| B. | \[\pi \log 2\] |
| C. | \[-\frac{\pi }{2}\log 2\] |
| D. | \[-\pi \log 2\] |
| Answer» D. \[-\pi \log 2\] | |
| 2169. |
\[\int_{0}^{1}{\log \sin \left( \frac{\pi }{2}x \right)}\,dx=\] [RPET 1997] |
| A. | \[-\log 2\] |
| B. | \[\log 2\] |
| C. | \[\frac{\pi }{2}\log 2\] |
| D. | \[-\frac{\pi }{2}\log 2\] |
| Answer» B. \[\log 2\] | |
| 2170. |
The value of \[\int_{0}^{n\pi +v}{|\sin x|\,dx}\] is [IIT 1994] |
| A. | x\[2n+1+\cos v\] |
| B. | \[2n+1-\cos v\] |
| C. | \[2n+1\] |
| D. | \[2n+\cos v\] |
| Answer» C. \[2n+1\] | |
| 2171. |
\[\int_{0}^{2a}{\frac{f(x)}{f(x)+f(2a-x)}\,dx=}\] [IIT 1988; Karnataka CET 2000] |
| A. | \[a\] |
| B. | \[\frac{a}{2}\] |
| C. | \[2a\] |
| D. | 0 |
| Answer» B. \[\frac{a}{2}\] | |
| 2172. |
The value of \[\int_{0}^{1}{(1+{{e}^{-{{x}^{2}}}})}\,dx=\] [IIT 1981] |
| A. | \[-1\] |
| B. | 2 |
| C. | \[1+{{e}^{-1}}\] |
| D. | None of these |
| Answer» E. | |
| 2173. |
\[\int_{0}^{\pi /2}{\frac{{{\sin }^{3/2}}x\,dx}{{{\cos }^{3/2}}x+{{\sin }^{3/2}}x}}=\] [Roorkee 1989; BIT Ranchi 1989] |
| A. | 0 |
| B. | \[\pi \] |
| C. | \[\pi /2\] |
| D. | \[\pi /4\] |
| Answer» E. | |
| 2174. |
\[\int_{-\pi }^{\pi }{{{(\cos px-\sin qx)}^{2}}dx}\] is equal to (where \[p\] and \[q\] are integers) [IIT 1992] |
| A. | \[-\pi \] |
| B. | 0 |
| C. | \[\pi \] |
| D. | \[2\pi \] |
| Answer» E. | |
| 2175. |
Let \[a,\,\,b,\,\,c\] be non-zero real numbers such that \[\int_{0}^{3}{(3a{{x}^{2}}+2bx+c)\,dx}=\int_{1}^{3}{(3a{{x}^{2}}+2bx+c})\,dx\,,\] then [BIT Ranchi 1991] |
| A. | \[a+b+c=3\] |
| B. | \[a+b+c=1\] |
| C. | \[a+b+c=0\] |
| D. | \[a+b+c=2\] |
| Answer» D. \[a+b+c=2\] | |
| 2176. |
The function \[L(x)=\int_{1}^{x}{\frac{dt}{t}}\] satisfies the equation [IIT 1996; DCE 2001] |
| A. | \[L(x+y)=L(x)+L(y)\] |
| B. | \[L\left( \frac{x}{y} \right)=L(x)+L(y)\] |
| C. | \[L(xy)=L(x)+L(y)\] |
| D. | None of these |
| Answer» D. None of these | |
| 2177. |
If \[P=\int_{0}^{3\pi }{f({{\cos }^{2}}x)dx}\,\,\text{and}\,\,Q=\int_{0}^{\pi }{f({{\cos }^{2}}x)dx}\], then [Orissa JEE 2004] |
| A. | \[P-Q=0\] |
| B. | \[P-2Q=0\] |
| C. | \[P-3Q=0\] |
| D. | \[P-5Q=0\] |
| Answer» D. \[P-5Q=0\] | |
| 2178. |
\[\int_{0}^{\pi /2}{{}}(\sin x-\cos x)\log (\sin x+\cos x)\,dx=\] [SCRA 1986] |
| A. | \[-1\] |
| B. | 1 |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 2179. |
If \[\int_{0}^{\pi }{xf(\sin x)dx=A}\int_{0}^{\pi /2}{f(\sin x)dx}\], then A is [AIEEE 2004] |
| A. | \[2\pi \] |
| B. | \[\pi \] |
| C. | \[\frac{\pi }{4}\] |
| D. | 0 |
| Answer» C. \[\frac{\pi }{4}\] | |
| 2180. |
If \[f(x)=|x-1|\], then \[\int_{0}^{2}{f(x)dx}\]is [Orissa JEE 2004] |
| A. | 1 |
| B. | 0 |
| C. | 2 |
| D. | ?2 |
| Answer» B. 0 | |
| 2181. |
\[\int_{-1}^{1}{{{x}^{17}}{{\cos }^{4}}x}\,dx=\] [MP PET 1990] |
| A. | \[-2\] |
| B. | \[-1\] |
| C. | 0 |
| D. | 2 |
| Answer» D. 2 | |
| 2182. |
\[\int_{0}^{\pi }{\frac{xdx}{1+\sin x}}\]is equal to [UPSEAT 2004] |
| A. | \[-\pi \] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\pi \] |
| D. | None of these |
| Answer» D. None of these | |
| 2183. |
The value of \[\int_{\,-2}^{\,2}{\left[ p\ln \left( \frac{1+x}{1-x} \right)+q\ln {{\left( \frac{1-x}{1+x} \right)}^{-2}}+r \right]\,dx}\] depends on [Orissa JEE 2003] |
| A. | The value of p |
| B. | The value of q |
| C. | The value of r |
| D. | The value of p and q |
| Answer» D. The value of p and q | |
| 2184. |
\[\int_{\,0}^{\,1}{\,{{\tan }^{-1}}\left( \frac{1}{{{x}^{2}}-x+1} \right)\,dx}\] is [Orissa JEE 2003] |
| A. | ln 2 |
| B. | \[-\ln 2\] |
| C. | \[\frac{\pi }{2}+\ln 2\] |
| D. | \[\frac{\pi }{2}-\ln 2\] |
| Answer» E. | |
| 2185. |
\[\int_{\,-\,2}^{\,2}{\,\left| \,[x]\, \right|\,dx=}\] [EAMCET 2003] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» E. | |
| 2186. |
\[\int_{\,0}^{\,2}{\,|x-1|\,dx=}\] [SCRA 1990; RPET 2001; UPSEAT 2003] |
| A. | 0 |
| B. | 2 |
| C. | 1/2 |
| D. | 1 |
| Answer» E. | |
| 2187. |
The value of \[\int_{\,0}^{\,8}{\,|x-5|\,dx}\] is [UPSEAT 2003] |
| A. | 17 |
| B. | 12 |
| C. | 9 |
| D. | 18 |
| Answer» B. 12 | |
| 2188. |
The value of \[\int_{-\pi /2}^{\,\pi /2}{(3\sin x+{{\sin }^{3}}x)\,dx}\] is [MP PET 2003] |
| A. | 3 |
| B. | 2 |
| C. | 0 |
| D. | \[\frac{10}{3}\] |
| Answer» D. \[\frac{10}{3}\] | |
| 2189. |
\[\int_{\,0}^{\,2\pi }{(\sin x+|\sin x|)\,dx=}\] [Karnataka CET 2003] |
| A. | 0 |
| B. | 4 |
| C. | 8 |
| D. | 1 |
| Answer» C. 8 | |
| 2190. |
The integral \[\int_{\,-1/2}^{\,1/2}{\,\left\{ [x]+\log \left( \frac{1+x}{1-x} \right) \right\}}\,dx\] equal (where [.] is the greatest integer function) [IIT Screening 2002] |
| A. | \[-\frac{1}{2}\] |
| B. | 0 |
| C. | 1 |
| D. | \[2\log \frac{1}{2}\] |
| Answer» B. 0 | |
| 2191. |
\[\int_{\,0}^{\,\pi /2}{\sin 2x\log \tan x\,dx}\] is equal to [Kerala (Engg.) 2002; AI CBSE 1990; Karnataka CET 1996, 98] |
| A. | \[\pi \] |
| B. | \[\pi /2\] |
| C. | 0 |
| D. | \[2\pi \] |
| Answer» D. \[2\pi \] | |
| 2192. |
\[\int_{-1}^{1}{\log \frac{2-x}{2+x}\,dx}=\] [Roorkee 1986; Kurukshetra CEE 1998] |
| A. | 2 |
| B. | 1 |
| C. | \[-1\] |
| D. | 0 |
| Answer» E. | |
| 2193. |
The value of the integral \[\int_{\,\frac{1}{n}}^{\,\frac{an-1}{n}}{\frac{\sqrt{x}}{\sqrt{a-x}+\sqrt{x}}dx}\] is [AMU 2002] |
| A. | \[\frac{a}{2}\] |
| B. | \[\frac{na+2}{2n}\] |
| C. | \[\frac{na-2}{2n}\] |
| D. | None of these |
| Answer» D. None of these | |
| 2194. |
The value of \[\int_{\,0}^{\,\sqrt{2}}{[{{x}^{2}}]\,dx},\] where [.] is the greatest integer function [AIEEE 2002] |
| A. | \[2-\sqrt{2}\] |
| B. | \[2+\sqrt{2}\] |
| C. | \[\sqrt{2}-1\] |
| D. | \[\sqrt{2}-2\] |
| Answer» D. \[\sqrt{2}-2\] | |
| 2195. |
\[\int_{\,0}^{\,1000}{{{e}^{x-[x]}}dx}\] is [AMU 2002] |
| A. | \[{{e}^{1000}}-1\] |
| B. | \[\frac{{{e}^{1000}}-1}{e-1}\] |
| C. | \[1000(e-1)\] |
| D. | \[\frac{e-1}{1000}\] |
| Answer» D. \[\frac{e-1}{1000}\] | |
| 2196. |
\[\int_{\,0}^{\,\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}x\,dx}\] is equals to [MP PET 2002] |
| A. | \[-1\] |
| B. | 0 |
| C. | 1 |
| D. | \[\pi \] |
| Answer» C. 1 | |
| 2197. |
\[\int_{\,0}^{\,2a}{f(x)dx=}\] [RPET 2002] |
| A. | \[2\int_{\,0}^{\,a}{\,f(x)dx}\] |
| B. | 0 |
| C. | \[\int_{\,0}^{\,a}{\,f(x)dx+\int_{\,0}^{\,a}{\,f(2a-x)dx}}\] |
| D. | \[\int_{\,0}^{\,a}{f(x)dx+}\int_{\,0}^{\,2a}{\,f(2a-x)dx}\] |
| Answer» D. \[\int_{\,0}^{\,a}{f(x)dx+}\int_{\,0}^{\,2a}{\,f(2a-x)dx}\] | |
| 2198. |
\[\int_{\,0}^{\,\pi }{\sqrt{\frac{1+\cos 2x}{2}}\,dx}\] is equal to [AMU 2001] |
| A. | 0 |
| B. | 2 |
| C. | 1 |
| D. | \[-1\] |
| Answer» C. 1 | |
| 2199. |
The value of the integral \[\int_{-\pi }^{\pi }{{{(\cos ax-\sin bx)}^{2}}dx}\], (a and b are integer) is [UPSEAT 2001] |
| A. | \[-\pi \] |
| B. | 0 |
| C. | \[\pi \] |
| D. | \[2\pi \] |
| Answer» E. | |
| 2200. |
\[\int_{0}^{\pi }{x}\,f\,(\sin x)\,dx=\] [IIT 1982; Kurukshetra CEE 1993] |
| A. | \[\pi \int_{0}^{\pi }{f(\sin x)\,dx}\] |
| B. | \[\frac{\pi }{2}\int_{0}^{\pi }{f(\sin x)\,dx}\] |
| C. | \[\frac{\pi }{2}\int_{0}^{\pi /2}{f(\sin x)\,dx}\] |
| D. | None of these |
| Answer» C. \[\frac{\pi }{2}\int_{0}^{\pi /2}{f(\sin x)\,dx}\] | |