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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.
| 2101. |
\[\int_{\,0}^{\,\pi }{{{\cos }^{3}}x\,dx=}\] [MP PET 1996; Pb. CET 2002] |
| A. | \[-1\] |
| B. | 0 |
| C. | 1 |
| D. | \[\pi \] |
| Answer» C. 1 | |
| 2102. |
The value of \[\int_{0}^{2\pi }{{{\cos }^{99}}x\,dx}\] is |
| A. | 1 |
| B. | \[-1\] |
| C. | 99 |
| D. | 0 |
| Answer» E. | |
| 2103. |
If \[[x]\] denotes the greatest integer less than or equal to \[x,\] then the value of the integral \[\int_{0}^{2}{{{x}^{2}}[x]\,dx}\] equals [Kurukshetra CEE 1996; Pb. CET 2001] |
| A. | 5/3 |
| B. | 7/3 |
| C. | 8/3 |
| D. | 4/3 |
| Answer» C. 8/3 | |
| 2104. |
The value of \[\int_{0}^{1}{{{\tan }^{-1}}\left( \frac{2x-1}{1+x-{{x}^{2}}} \right)}\,dx\] is |
| A. | 1 |
| B. | 0 |
| C. | \[-1\] |
| D. | None of these |
| Answer» C. \[-1\] | |
| 2105. |
\[\int_{0}^{\pi }{x\log \sin x}\,dx=\] |
| A. | \[\frac{\pi }{2}\log \frac{1}{2}\] |
| B. | \[\frac{{{\pi }^{2}}}{2}\log \frac{1}{2}\] |
| C. | \[\pi \log \frac{1}{2}\] |
| D. | \[{{\pi }^{2}}\log \frac{1}{2}\] |
| Answer» C. \[\pi \log \frac{1}{2}\] | |
| 2106. |
The value of \[\int_{0}^{\pi /2}{\log \,\left( \frac{4+3\sin x}{4+3\cos x} \right)}\,dx\] is |
| A. | 2 |
| B. | \[\frac{3}{4}\] |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 2107. |
The value of \[\int_{-1}^{1}{(\sqrt{1+x+{{x}^{2}}}-\sqrt{1-x+{{x}^{2}}})\,dx}\] is |
| A. | 0 |
| B. | 1 |
| C. | \[-1\] |
| D. | None of these |
| Answer» B. 1 | |
| 2108. |
\[\int_{-\pi /2}^{\pi /2}{\frac{\cos x}{1+{{e}^{x}}}\,dx=}\] [EAMCET 1992] |
| A. | 1 |
| B. | 0 |
| C. | \[-1\] |
| D. | None of these |
| Answer» B. 0 | |
| 2109. |
If \[\int_{-1}^{4}{f(x)\,dx}=4\] and \[\int_{2}^{4}{(3-f(x))\,dx=7,}\] then the value of \[\int_{2}^{-1}{f(x)\,dx}\] is |
| A. | 2 |
| B. | ? 3 |
| C. | ? 5 |
| D. | None of these |
| Answer» D. None of these | |
| 2110. |
\[\int_{-1}^{1}{\log \left( \frac{1+x}{1-x} \right)\,dx=}\] [MP PET 1995; Pb. CET 2000] |
| A. | 2 |
| B. | 1 |
| C. | 0 |
| D. | \[\pi \] |
| Answer» D. \[\pi \] | |
| 2111. |
If \[I=\int_{0}^{100\pi }{\sqrt{(1-\cos 2x)}\,dx,}\]then the value of \[I\] is |
| A. | \[100\sqrt{2}\] |
| B. | \[200\sqrt{2}\] |
| C. | \[50\sqrt{2}\] |
| D. | None of these |
| Answer» C. \[50\sqrt{2}\] | |
| 2112. |
To find the numerical value of \[\int_{-2}^{2}{(p{{x}^{2}}+qx+s)\,dx,}\] it is necessary to know the values of constants [IIT 1992] |
| A. | \[p\] |
| B. | \[q\] |
| C. | \[s\] |
| D. | \[p\] and \[s\] |
| Answer» E. | |
| 2113. |
\[\int_{-1}^{1}{{{\sin }^{11}}x\,dx}\] is equal to [MNR 1995] |
| A. | \[\frac{10}{11}.\frac{8}{9}.\frac{6}{7}.\frac{4}{5}.\frac{2}{3}\] |
| B. | \[\frac{10}{11}.\frac{8}{9}.\frac{6}{7}.\frac{4}{5}.\frac{2}{3}.\frac{\pi }{2}\] |
| C. | 1 |
| D. | 0 |
| Answer» E. | |
| 2114. |
\[\int_{-1}^{1}{x\,|x|\,}dx=\] [MP PET 1990; Pb. CET 2004] |
| A. | 1 |
| B. | 0 |
| C. | 2 |
| D. | \[-2\] |
| Answer» C. 2 | |
| 2115. |
The value of \[\int_{-1}^{1}{\frac{\sin x-{{x}^{2}}}{3-|x|}\,dx}\] is [Roorkee 1995] |
| A. | 0 |
| B. | \[2\int_{0}^{1}{\frac{\sin x}{3-|x|}\,dx}\] |
| C. | \[2\int_{0}^{1}{\frac{-{{x}^{2}}}{3-|x|}}\,dx\] |
| D. | \[2\int_{0}^{1}{\frac{\sin x-{{x}^{2}}}{3-|x|}\,dx}\] |
| Answer» D. \[2\int_{0}^{1}{\frac{\sin x-{{x}^{2}}}{3-|x|}\,dx}\] | |
| 2116. |
\[\int_{0}^{\pi /2}{\frac{1}{1+\sqrt{\tan x}}}\,dx=\] [RPET 1995; Kurukshetra CEE 1998] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{6}\] |
| D. | 1 |
| Answer» C. \[\frac{\pi }{6}\] | |
| 2117. |
The value of \[\int_{0}^{\pi }{{{e}^{{{\cos }^{2}}x}}{{\cos }^{5}}3x}\,dx\] is [Bihar CEE 1994] |
| A. | 1 |
| B. | \[-1\] |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 2118. |
The value of \[\int_{2}^{3}{\frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}}\,dx\] is [IIT 1994; Kurukshetra CEE 1998] |
| A. | 1 |
| B. | 0 |
| C. | \[-1\] |
| D. | \[\frac{1}{2}\] |
| Answer» E. | |
| 2119. |
The value of \[\int_{1}^{5}{(|x-3|+|1-x|)\,dx}\] is [IIT Screening] |
| A. | 10 |
| B. | \[\frac{5}{6}\] |
| C. | 21 |
| D. | 12 |
| Answer» E. | |
| 2120. |
If \[I=\int_{0}^{\pi /4}{\,{{\sin }^{2}}x\,dx}\]and\[J=\int_{0}^{\pi /4}{{{\cos }^{2}}x\,dx,}\] then \[I=\] [SCRA 1989] |
| A. | \[\frac{\pi }{4}-J\] |
| B. | \[2J\] |
| C. | \[J\] |
| D. | \[\frac{J}{2}\] |
| Answer» B. \[2J\] | |
| 2121. |
If \[\int_{0}^{2a}{f(x)\,dx=2\int_{0}^{a}{f(x)\,dx,}}\] then [SCRA 1986] |
| A. | \[f(2a-x)=-f(x)\] |
| B. | \[f(2a-x)=f(x)\] |
| C. | \[f(a-x)=-f(x)\] |
| D. | \[f(a-x)=f(x)\] |
| Answer» C. \[f(a-x)=-f(x)\] | |
| 2122. |
If \[\int_{-a}^{a}{\sqrt{\frac{a-x}{a+x}}\,dx=k\pi ,}\] then \[k=\] [AISSE 1986; SCRA 1986] |
| A. | \[-a\] |
| B. | \[-2a\] |
| C. | \[2a\] |
| D. | \[a\] |
| Answer» E. | |
| 2123. |
\[\int_{0}^{\pi }{x\sin x\,dx=}\] [SCRA 1980, 91] |
| A. | \[\pi \] |
| B. | 0 |
| C. | 1 |
| D. | \[{{\pi }^{2}}\] |
| Answer» B. 0 | |
| 2124. |
If \[f(a+b-x)=f(x),\] then \[\int_{a}^{b}{x\,f(x)\,dx=}\] [CEE 1993; AIEEE 2003] |
| A. | \[\frac{a+b}{2}\int_{a}^{b}{f(b-x)\,dx}\] |
| B. | \[\frac{a+b}{2}\int_{a}^{b}{f(x)\,dx}\] |
| C. | \[\frac{b-a}{2}\int_{a}^{b}{f(x)\,dx}\] |
| D. | None of these |
| Answer» C. \[\frac{b-a}{2}\int_{a}^{b}{f(x)\,dx}\] | |
| 2125. |
If \[f(x)=\int_{a}^{x}{{{t}^{3}}{{e}^{t}}\,dt\,,}\] then \[\frac{d}{dx}\,f(x)=\] [MP PET 1989] |
| A. | \[{{e}^{x}}({{x}^{3}}+3{{x}^{2}})\] |
| B. | \[{{x}^{3}}{{e}^{x}}\] |
| C. | \[{{a}^{3}}{{e}^{a}}\] |
| D. | None of these |
| Answer» C. \[{{a}^{3}}{{e}^{a}}\] | |
| 2126. |
The value of \[\int_{\pi /4}^{3\pi /4}{\frac{\varphi }{1+\sin \varphi }\,d\varphi ,}\] is [AI CBSE 1990; IIT 1993] |
| A. | \[\pi \tan \frac{\pi }{8}\] |
| B. | \[\log \tan \frac{\pi }{8}\] |
| C. | \[\tan \frac{\pi }{8}\] |
| D. | None of these |
| Answer» B. \[\log \tan \frac{\pi }{8}\] | |
| 2127. |
The value of \[\int_{0}^{\pi /2}{\frac{dx}{1+{{\tan }^{3}}x}}\] is [IIT 1993; DCE 2000, 01] |
| A. | 0 |
| B. | 1 |
| C. | \[\frac{\pi }{2}\] |
| D. | \[\frac{\pi }{4}\] |
| Answer» E. | |
| 2128. |
\[\int_{-3}^{3}{\frac{{{x}^{2}}\sin x}{1+{{x}^{6}}}\,dx=}\] |
| A. | 4 |
| B. | 2 |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 2129. |
If \[\int_{0}^{\pi }{x\,f({{\cos }^{2}}x+{{\tan }^{4}}x)\,dx}\] \[=k\int_{0}^{\pi /2}{f({{\cos }^{2}}x+{{\tan }^{4}}x)\,dx,}\] then the value of \[k\] is |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\pi \] |
| C. | \[-\frac{\pi }{2}\] |
| D. | None of these |
| Answer» C. \[-\frac{\pi }{2}\] | |
| 2130. |
The value of the integral \[I=\int_{\,0}^{\,1}{\,x{{(1-x)}^{n}}dx}\] is [AIEEE 2003] |
| A. | \[\frac{1}{n+1}\] |
| B. | \[\frac{1}{n+2}\] |
| C. | \[\frac{1}{n+1}-\frac{1}{n+2}\] |
| D. | \[\frac{1}{n+1}+\frac{1}{n+2}\] |
| Answer» D. \[\frac{1}{n+1}+\frac{1}{n+2}\] | |
| 2131. |
The value of \[\int_{\pi }^{2\pi }{[2\sin x]\,dx,}\] where \[[\,\,.\,\,]\] represents the greatest integer function, is [IIT 1995] |
| A. | \[-\pi \] |
| B. | \[-2\pi \] |
| C. | \[-\frac{5\pi }{3}\] |
| D. | \[\frac{5\pi }{3}\] |
| Answer» D. \[\frac{5\pi }{3}\] | |
| 2132. |
\[\int_{\,0}^{\,\pi /2}{\{x-[\sin x]\}\,dx}\] is equal to [AMU 1999] |
| A. | \[\frac{{{\pi }^{2}}}{8}\] |
| B. | \[\frac{{{\pi }^{2}}}{8}-1\] |
| C. | \[\frac{{{\pi }^{2}}}{8}-2\] |
| D. | None of these |
| Answer» B. \[\frac{{{\pi }^{2}}}{8}-1\] | |
| 2133. |
\[\int_{1/e}^{e}{|\log x|\,dx=}\] [UPSEAT 2001] |
| A. | \[1-\frac{1}{e}\] |
| B. | \[2\,\left( 1-\frac{1}{e} \right)\] |
| C. | \[{{e}^{-1}}-1\] |
| D. | None of these |
| Answer» C. \[{{e}^{-1}}-1\] | |
| 2134. |
For any integer \[n,\] the integral \[\int_{0}^{\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}(2n+1)x\,dx=}\] [MNR 1982] |
| A. | \[-1\] |
| B. | 0 |
| C. | 1 |
| D. | \[\pi \] |
| Answer» C. 1 | |
| 2135. |
\[\int_{0}^{\pi /2}{\frac{d\theta }{1+\tan \theta }}=\] [Roorkee 1980; MP PET 1996; DCE 1999] |
| A. | \[\pi \] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\frac{\pi }{3}\] |
| D. | \[\frac{\pi }{4}\] |
| Answer» E. | |
| 2136. |
\[\int_{-1}^{1}{{{\sin }^{3}}x{{\cos }^{2}}x\,dx=}\] [MNR 1991; UPSEAT 2000] |
| A. | 0 |
| B. | 1 |
| C. | \[\frac{1}{2}\] |
| D. | 2 |
| Answer» B. 1 | |
| 2137. |
\[\int_{0}^{\pi }{\frac{x\,\tan x}{\sec x+\cos x}}\,dx=\] [MNR 1985; BIT Ranchi 1986; UPSEAT 2002] |
| A. | \[\frac{{{\pi }^{2}}}{4}\] |
| B. | \[\frac{{{\pi }^{2}}}{2}\] |
| C. | \[\frac{3{{\pi }^{2}}}{2}\] |
| D. | \[\frac{{{\pi }^{2}}}{3}\] |
| Answer» B. \[\frac{{{\pi }^{2}}}{2}\] | |
| 2138. |
\[\int_{0}^{\pi }{\frac{x\tan x}{\sec x+\tan x}}\,dx=\] [MNR 1984] |
| A. | \[\frac{\pi }{2}-1\] |
| B. | \[\pi \left( \frac{\pi }{2}+1 \right)\] |
| C. | \[\frac{\pi }{2}+1\] |
| D. | \[\pi \left( \frac{\pi }{2}-1 \right)\] |
| Answer» E. | |
| 2139. |
\[\int_{0}^{1.5}{[{{x}^{2}}]\,dx}\], where \[[\,\,.\,\,]\]denotes the greatest integer function, equals [IIT 1988; DCE 2000, 01] |
| A. | \[2+\sqrt{2}\] |
| B. | \[2-\sqrt{2}\] |
| C. | \[-2+\sqrt{2}\] |
| D. | \[-2-\sqrt{2}\] |
| Answer» C. \[-2+\sqrt{2}\] | |
| 2140. |
The value of the integral \[\int_{-\pi /4}^{\pi /4}{{{\sin }^{-4}}x}\,dx\] is [IIT Screening; MP PET 2003] |
| A. | 3/2 |
| B. | ?8/3 |
| C. | 3/8 |
| D. | 8/3 |
| Answer» C. 3/8 | |
| 2141. |
\[\int_{0}^{\pi }{|\cos x|\,dx=}\] [MP PET 1998; Pb. CET 2001] |
| A. | \[\pi \] |
| B. | 0 |
| C. | 2 |
| D. | 1 |
| Answer» D. 1 | |
| 2142. |
\[\int_{0}^{\pi /2}{\,\,\,\,\,|\sin x-\cos x|\,dx=}\] [Roorkee 1990; MP PET 2001; UPSEAT 2001] |
| A. | 0 |
| B. | \[2(\sqrt{2}-1)\] |
| C. | \[\sqrt{2}-1\] |
| D. | \[2(\sqrt{2}+1)\] |
| Answer» C. \[\sqrt{2}-1\] | |
| 2143. |
\[\int_{0}^{a}{f(x)\,dx}=\] [BIT Ranchi 1992] |
| A. | \[\int_{0}^{a}{f(a+x)\,dx}\] |
| B. | \[\int_{0}^{a}{f(2a+x)\,dx}\] |
| C. | \[\int_{0}^{a}{f(x-a)\,dx}\] |
| D. | \[\int_{0}^{a}{f(a-x)\,dx}\] |
| Answer» E. | |
| 2144. |
The correct evaluation of \[\int_{0}^{\pi /2}{\left| \,\sin \left( x-\frac{\pi }{4} \right)\, \right|\,dx}\] is [MP PET 1993] |
| A. | \[2+\sqrt{2}\] |
| B. | \[2-\sqrt{2}\] |
| C. | \[-2+\sqrt{2}\] |
| D. | 0 |
| Answer» C. \[-2+\sqrt{2}\] | |
| 2145. |
\[\int_{0}^{\pi /2}{\frac{x\sin x\cos x}{{{\cos }^{4}}x+{{\sin }^{4}}x}}\,dx=\] [IIT 1985] |
| A. | 0 |
| B. | \[\frac{\pi }{8}\] |
| C. | \[\frac{{{\pi }^{2}}}{8}\] |
| D. | \[\frac{{{\pi }^{2}}}{16}\] |
| Answer» E. | |
| 2146. |
\[\int_{0}^{\pi /2}{\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx=}\] [MNR 1989; UPSEAT 2002] |
| A. | 0 |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\frac{\pi }{4}\] |
| D. | None of these |
| Answer» D. None of these | |
| 2147. |
\[\int_{0}^{\pi }{x{{\sin }^{3}}x\,dx}=\] [CEE 1993] |
| A. | \[\frac{4\pi }{3}\] |
| B. | \[\frac{2\pi }{3}\] |
| C. | 0 |
| D. | None of these |
| Answer» C. 0 | |
| 2148. |
If n is a positive integer and [x] is the greatest integer not exceeding x, then \[\int_{0}^{n}{\,\,\{x-[x]\}\,dx}\] equals |
| A. | \[{{n}^{2}}/2\] |
| B. | \[n(n-1)/2\] |
| C. | \[n\,/\,2\] |
| D. | \[\frac{{{n}^{2}}}{2}-n\] |
| Answer» D. \[\frac{{{n}^{2}}}{2}-n\] | |
| 2149. |
\[\int_{-1}^{1}{|1-x|dx}=\] [Karnataka CET 2004] |
| A. | ? 2 |
| B. | 0 |
| C. | 2 |
| D. | 4 |
| Answer» D. 4 | |
| 2150. |
If \[\int_{-1}^{1}{f(x)\,dx=0}\], then [SCRA 1990] |
| A. | \[f(x)=f(-x)\] |
| B. | \[f(-x)=-f(x)\] |
| C. | \[f(x)=2f(x)\] |
| D. | None of these |
| Answer» C. \[f(x)=2f(x)\] | |