MCQOPTIONS
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MCQOPTIONS
This section includes 24 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
\[\int_{0}^{1}{{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\,dx=}\] [Karnataka CET 1999] |
| A. | \[\frac{\pi }{2}-2\log \sqrt{2}\] |
| B. | \[\frac{\pi }{2}+2\log \sqrt{2}\] |
| C. | \[\frac{\pi }{4}-\log \sqrt{2}\] |
| D. | \[\frac{\pi }{4}+\log \sqrt{2}\] |
| Answer» B. \[\frac{\pi }{2}+2\log \sqrt{2}\] | |
| 2. |
\[\int_{\,0}^{\,3}{\,\frac{3x+1}{{{x}^{2}}+9}dx=}\] [EAMCET 2003] |
| A. | \[\log (2\sqrt{2})+\frac{\pi }{12}\] |
| B. | \[\log (2\sqrt{2})+\frac{\pi }{2}\] |
| C. | \[\log (2\sqrt{2})+\frac{\pi }{6}\] |
| D. | \[\log (2\sqrt{2})+\frac{\pi }{3}\] |
| Answer» B. \[\log (2\sqrt{2})+\frac{\pi }{2}\] | |
| 3. |
\[\int_{\,-\,1}^{\,0}{\frac{dx}{{{x}^{2}}+2x+2}=}\] [MP PET 2000] |
| A. | 0 |
| B. | \[\pi /4\] |
| C. | \[\pi /2\] |
| D. | \[-\pi /4\] |
| Answer» C. \[\pi /2\] | |
| 4. |
\[\int_{\,1}^{\,x}{\frac{\log {{x}^{2}}}{x}\,dx=}\] [DCE 1999] |
| A. | \[{{(\log x)}^{2}}\] |
| B. | \[\frac{1}{2}{{(\log x)}^{2}}\] |
| C. | \[\frac{\log {{x}^{2}}}{2}\] |
| D. | None of these |
| Answer» B. \[\frac{1}{2}{{(\log x)}^{2}}\] | |
| 5. |
\[\int_{0}^{\pi /4}{\frac{\sec x}{1+2{{\sin }^{2}}x}}\] is equal to [MNR 1994] |
| A. | \[\frac{1}{3}\left[ \log (\sqrt{2}+1)+\frac{\pi }{2\sqrt{2}} \right]\] |
| B. | \[\frac{1}{3}\left[ \log (\sqrt{2}+1)-\frac{\pi }{2\sqrt{2}} \right]\] |
| C. | \[3\left[ \log (\sqrt{2}+1)-\frac{\pi }{2\sqrt{2}} \right]\] |
| D. | \[3\left[ \log (\sqrt{2}+1)+\frac{\pi }{2\sqrt{2}} \right]\] |
| Answer» B. \[\frac{1}{3}\left[ \log (\sqrt{2}+1)-\frac{\pi }{2\sqrt{2}} \right]\] | |
| 6. |
The value of \[\int_{3}^{5}{\frac{{{x}^{2}}}{{{x}^{2}}-4}\,dx}\] is[Roorkee 1992] |
| A. | \[2-{{\log }_{e}}\left( \frac{15}{7} \right)\] |
| B. | \[2+{{\log }_{e}}\left( \frac{15}{7} \right)\] |
| C. | \[2+4{{\log }_{e}}3-4{{\log }_{e}}7+4{{\log }_{e}}5\] |
| D. | \[2-{{\tan }^{-1}}\left( \frac{15}{7} \right)\] |
| Answer» C. \[2+4{{\log }_{e}}3-4{{\log }_{e}}7+4{{\log }_{e}}5\] | |
| 7. |
\[\int_{0}^{1}{\frac{{{e}^{x}}(x-1)}{{{(x+1)}^{3}}}\,dx=}\] [SCRA 1986] |
| A. | \[\frac{e}{4}\] |
| B. | \[\frac{e}{4}-1\] |
| C. | \[\frac{e}{4}+1\] |
| D. | None of these |
| Answer» C. \[\frac{e}{4}+1\] | |
| 8. |
\[\int_{0}^{\pi /2}{\frac{\cos x}{1+\cos x+\sin x}}\,dx=\] [Roorkee 1989] |
| A. | \[\frac{\pi }{4}+\frac{1}{2}\log 2\] |
| B. | \[\frac{\pi }{4}+\log 2\] |
| C. | \[\frac{\pi }{4}-\frac{1}{2}\log 2\] |
| D. | \[\frac{\pi }{4}-\log 2\] |
| Answer» D. \[\frac{\pi }{4}-\log 2\] | |
| 9. |
\[\int_{0}^{2}{\sqrt{\frac{2+x}{2-x}}}\,dx=\][MNR 1984; CEE 1993] |
| A. | \[\pi +2\] |
| B. | \[\pi +\frac{3}{2}\] |
| C. | \[\pi +1\] |
| D. | None of these |
| Answer» B. \[\pi +\frac{3}{2}\] | |
| 10. |
\[\int_{0}^{a}{\frac{x\,dx}{\sqrt{{{a}^{2}}+{{x}^{2}}}}}=\] |
| A. | \[a\,(\sqrt{2}-1)\] |
| B. | \[a\,(1-\sqrt{2})\] |
| C. | \[a\,(1+\sqrt{2})\] |
| D. | \[2a\sqrt{3}\] |
| Answer» B. \[a\,(1-\sqrt{2})\] | |
| 11. |
The area bounded by the curve \[y={{(x+1)}^{2}},\,y={{(x-1)}^{2}}\] and the line \[y=\frac{1}{4}\] is [IIT Screening 2005] |
| A. | 1/6 |
| B. | 2/3 |
| C. | 1/4 |
| D. | 1/3 |
| Answer» E. | |
| 12. |
\[\int_{0}^{\pi /4}{{}}\sec x\log (\sec x+\tan x)\,dx=\] |
| A. | \[\frac{1}{2}{{[\log (1+\sqrt{2})]}^{2}}\] |
| B. | \[{{[\log (1+\sqrt{2})]}^{2}}\] |
| C. | \[\frac{1}{2}{{[\log (\sqrt{2}-1)]}^{2}}\] |
| D. | \[\frac{1}{2}{{[\log (\sqrt{2}-1)]}^{2}}\] |
| Answer» B. \[{{[\log (1+\sqrt{2})]}^{2}}\] | |
| 13. |
\[\int_{0}^{\pi /2}{{{e}^{x}}\sin x\,dx=}\] [Roorkee 1978] |
| A. | \[\frac{1}{2}({{e}^{\pi /2}}-1)\] |
| B. | \[\frac{1}{2}({{e}^{\pi /2}}+1)\] |
| C. | \[\frac{1}{2}(1-{{e}^{\pi /2}})\] |
| D. | \[2({{e}^{\pi /2}}+1)\] |
| Answer» C. \[\frac{1}{2}(1-{{e}^{\pi /2}})\] | |
| 14. |
\[\int_{0}^{\pi /8}{{{\cos }^{3}}4\theta d\theta }=\] [Karnataka CET 2004] |
| A. | \[\frac{2}{3}\] |
| B. | \[\frac{1}{4}\] |
| C. | \[\frac{1}{3}\] |
| D. | \[\frac{1}{6}\] |
| Answer» E. | |
| 15. |
The parabolas \[{{y}^{2}}=4x\] and \[{{x}^{2}}=4y\] divide the square region bounded by the lines \[x=4\], \[y=4\]and the coordinate axes. If \[{{S}_{1}},{{S}_{2}},{{S}_{3}}\] are respectively the areas of these parts numbered from top to bottom, then \[{{S}_{1}}:{{S}_{2}}:{{S}_{3}}\] is [AIEEE 2005] |
| A. | \[2:1:2\] |
| B. | \[1:1:1\] |
| C. | \[1:2:1\] |
| D. | \[1:2:3\] |
| Answer» C. \[1:2:1\] | |
| 16. |
The area of the region bounded by the curve \[y=x|x|\], x-axis and the ordinates \[x=1,\,\,x=-1\]is given by [Pb. CET 2004] |
| A. | Zero |
| B. | \[\frac{1}{3}\] |
| C. | \[\frac{2}{3}\] |
| D. | 1 |
| Answer» D. 1 | |
| 17. |
The area between the parabola \[{{y}^{2}}=4ax\]and \[{{x}^{2}}=8ay\] is [RPET 1997] |
| A. | \[\frac{8}{3}{{a}^{2}}\] |
| B. | \[\frac{4}{3}{{a}^{2}}\] |
| C. | \[\frac{32}{3}{{a}^{2}}\] |
| D. | \[\frac{16}{3}{{a}^{2}}\] |
| Answer» D. \[\frac{16}{3}{{a}^{2}}\] | |
| 18. |
Area enclosed by the parabola \[ay=3({{a}^{2}}-{{x}^{2}})\] and x-axis is |
| A. | \[4\,{{a}^{2}}\]sq. unit |
| B. | \[12\,{{a}^{2}}\]sq. unit |
| C. | \[4\,{{a}^{3}}\]sq. unit |
| D. | None of these |
| Answer» B. \[12\,{{a}^{2}}\]sq. unit | |
| 19. |
Area inside the parabola \[{{y}^{2}}=4ax,\]between the lines \[x=a\]and\[x=4a\]is equal to [Pb. CET 2002; Karnataka CET 2005] |
| A. | \[4{{a}^{2}}\] |
| B. | \[8{{a}^{2}}\] |
| C. | \[28\frac{{{a}^{2}}}{3}\] |
| D. | \[35\frac{{{a}^{2}}}{3}\] |
| Answer» D. \[35\frac{{{a}^{2}}}{3}\] | |
| 20. |
The measurement of the area bounded by the co-ordinate axes and the curve \[y={{\log }_{e}}x\] is [MP PET 1998] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | \[\infty \] |
| Answer» E. | |
| 21. |
The area of the region (in the square unit) bounded by the curve \[{{x}^{2}}=4y,\] line \[x=2\] and x-axis is [MP PET 2002] |
| A. | 1 |
| B. | \[\frac{2}{3}\] |
| C. | \[\frac{4}{3}\] |
| D. | \[\frac{8}{3}\] |
| Answer» C. \[\frac{4}{3}\] | |
| 22. |
The area bounded by the \[x-\]axis and the curve \[y=\sin x\] and \[x=0,\] \[x=\pi \] is [Kerala (Engg.) 2002] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» C. 3 | |
| 23. |
If area bounded by the curves \[{{y}^{2}}=4ax\] and \[y=mx\] is \[{{a}^{2}}/3,\],then the value of \[m\] is |
| A. | 2 |
| B. | \[-2\] |
| C. | \[\frac{1}{2}\] |
| D. | None of these |
| Answer» B. \[-2\] | |
| 24. |
Area bounded by the curve \[y=\log x\,,\] \[x-\]axis and the ordinates \[x=1,\,\,x=2\] is [MP PET 2004] |
| A. | \[\log 4\]sq. unit |
| B. | \[(\log 4+1)\]sq. unit |
| C. | \[(\log 4-1)\]sq. unit |
| D. | None of these |
| Answer» D. None of these | |