MCQOPTIONS
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MCQOPTIONS
This section includes 20 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
\[\int_{0}^{1}{\frac{{{x}^{7}}}{\sqrt{1-{{x}^{4}}}}dx}\] is equal to[AMU 2000] |
| A. | 1 |
| B. | \[\frac{1}{3}\] |
| C. | \[\frac{2}{3}\] |
| D. | \[\frac{\pi }{3}\] |
| Answer» C. \[\frac{2}{3}\] | |
| 2. |
On the interval \[\left[ \frac{5\pi }{3},\,\,\frac{7\pi }{4} \right],\] the greatest value of the function \[f(x)=\int_{5\pi /3}^{x}{(6\cos t-2\sin t)\,dt=}\] |
| A. | \[3\sqrt{3}+2\sqrt{2}+1\] |
| B. | \[3\sqrt{3}-2\sqrt{2}-1\] |
| C. | Does not exist |
| D. | None of these |
| Answer» C. Does not exist | |
| 3. |
\[\int_{-\,\pi }^{\,\pi }{\frac{2x(1+\sin x)}{1+{{\cos }^{2}}x}dx}\] is [AIEEE 2002] |
| A. | \[{{\pi }^{2}}/4\] |
| B. | \[{{\pi }^{2}}\] |
| C. | 0 |
| D. | \[\pi /2\] |
| Answer» C. 0 | |
| 4. |
\[\int\limits_{0}^{\pi }{\,\frac{\sin \left( n+\frac{1}{2} \right)\text{ }x}{\sin x}}\,dx\], \[(n\in N)\] equals [Kurukshetra CEE 1998] |
| A. | \[n\pi \] |
| B. | \[(2n+1)\frac{\pi }{2}\] |
| C. | \[\pi \] |
| D. | 0 |
| Answer» D. 0 | |
| 5. |
If \[f(x)=A\sin \left( \frac{\pi x}{2} \right)+B,\] \[{f}'\left( \frac{1}{2} \right)=\sqrt{2}\] and \[\int_{0}^{1}{f(x)\,dx=\frac{2A}{\pi },}\] then the constants \[A\] and \[B\] are respectively [IIT 1995] |
| A. | \[\frac{\pi }{2}\] and \[\frac{\pi }{2}\] |
| B. | \[\frac{2}{\pi }\] and \[\frac{3}{\pi }\] |
| C. | \[\frac{4}{\pi }\] and 0 |
| D. | 0 and \[-\frac{4}{\pi }\] |
| Answer» D. 0 and \[-\frac{4}{\pi }\] | |
| 6. |
For which of the following values of m, the area of the region bounded by the curve \[y=x-{{x}^{2}}\] and the line \[y=mx\] equals\[\frac{9}{2}\] [IIT 1999] |
| A. | \[-4\] |
| B. | \[-2\] |
| C. | 2 |
| D. | 4 |
| Answer» C. 2 | |
| 7. |
If \[\int_{0}^{{{t}^{2}}}{xf(x)dx=}\frac{2}{5}{{t}^{5}},\,\,t>0,\]then\[f\left( \frac{4}{25} \right)=\] [IIT Screening 2004] |
| A. | \[\frac{2}{5}\] |
| B. | \[\frac{5}{2}\] |
| C. | \[-\frac{2}{5}\] |
| D. | None of these |
| Answer» B. \[\frac{5}{2}\] | |
| 8. |
The numbers P, Q and \[R\] for which the function \[f(x)=P{{e}^{2x}}+Q{{e}^{x}}+Rx\] satisfies the conditions \[f(0)=-1,\] \[{f}'(\log 2)=31\] and \[\int_{0}^{\log 4}{[f(x)-Rx]\,dx=\frac{39}{2}}\] are given by |
| A. | \[P=2,\] \[Q=-3,\] \[R=4\] |
| B. | \[P=-5,\] \[Q=2,\] \[R=3\] |
| C. | \[P=5,\] \[Q=-2,\] \[R=3\] |
| D. | \[P=5,\] \[Q=-6,\] \[R=3\] |
| Answer» E. | |
| 9. |
If \[f(x)=\frac{{{e}^{x}}}{1+{{e}^{x}}},\,\,\,\ {{I}_{1}}=\int_{f(-a)}^{f(a)}{xg\{x(1-x)\}dx}\], and \[{{I}_{2}}=\int_{f(-a)}^{f(a)}{g\{x(1-x))\}dx}\], then the value of \[\frac{{{I}_{2}}}{{{I}_{1}}}\] is [AIEEE 2004] |
| A. | 1 |
| B. | -3 |
| C. | -1 |
| D. | 2 |
| Answer» E. | |
| 10. |
If \[\int_{0}^{x}{f(t)\,dt}=x+\int_{x}^{1}{t\,f(t)\,dt,}\] then the value of \[f(1)\] is [IIT 1998; AMU 2005] |
| A. | 44228 |
| B. | 0 |
| C. | 1 |
| D. | -0.5 |
| Answer» B. 0 | |
| 11. |
Let \[f\] be a positive function. Let \[{{I}_{1}}=\int_{1-k}^{k}{x\,f\left\{ x(1-x) \right\}}\,dx\],\[{{I}_{2}}=\int_{1-k}^{k}{\,f\left\{ x(1-x) \right\}}\,dx\] when \[2k-1>0.\] Then \[{{I}_{1}}/{{I}_{2}}\] is [IIT 1997 Cancelled] |
| A. | 2 |
| B. | \[k\] |
| C. | \[1/2\] |
| D. | 1 |
| Answer» D. 1 | |
| 12. |
\[\int_{\,\pi }^{\,10\pi }{\,|\sin x|dx}\] is [AIEEE 2002] |
| A. | 20 |
| B. | 8 |
| C. | 10 |
| D. | 18 |
| Answer» E. | |
| 13. |
The area bounded by the curves \[y=\ln x\], \[y=\ln |x|\], \[y=\,|\ln x|\] and \[y=\,|\ln |x||\] is [AIEEE 2002] |
| A. | 4 sq. unit |
| B. | 6 sq. unit |
| C. | 10 sq. unit |
| D. | None of these |
| Answer» B. 6 sq. unit | |
| 14. |
\[{{I}_{n}}=\int_{\,0}^{\,\pi /4}{{{\tan }^{n}}x\,dx}\], then \[\underset{n-\infty }{\mathop{\lim }}\,n\,[{{I}_{n}}+{{I}_{n-2}}]\] equals [AIEEE 2002] |
| A. | 44228 |
| B. | 1 |
| C. | \[\infty \] |
| D. | 0 |
| Answer» C. \[\infty \] | |
| 15. |
If \[{{I}_{n}}=\int_{0}^{\infty }{{{e}^{-x}}{{x}^{n-1}}dx,}\] then \[\int_{0}^{\infty }{{{e}^{-\lambda x}}{{x}^{n-1}}dx=}\] |
| A. | \[\lambda {{I}_{n}}\] |
| B. | \[\frac{1}{\lambda }{{I}_{n}}\] |
| C. | \[\frac{{{I}_{n}}}{{{\lambda }^{n}}}\] |
| D. | \[{{\lambda }^{n}}{{I}_{n}}\] |
| Answer» D. \[{{\lambda }^{n}}{{I}_{n}}\] | |
| 16. |
If for a real number \[y,\,\,[y]\] is the greatest integer less than or equal to \[y,\] then the value of the integral \[\int\limits_{\pi /2}^{3\pi /2}{[2\sin x]\,dx}\] is [IIT 1999] |
| A. | \[-\pi \] |
| B. | 0 |
| C. | \[-\frac{\pi }{2}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» D. \[\frac{\pi }{2}\] | |
| 17. |
The volume of spherical cap of height h cut off from a sphere of radius a is equal to [UPSEAT 2004] |
| A. | \[\frac{\pi }{3}{{h}^{2}}(3a-h)\] |
| B. | \[\pi (a-h)(2{{a}^{2}}-{{h}^{2}}-ah)\] |
| C. | \[\frac{4\pi }{3}{{h}^{3}}\] |
| D. | None of these |
| Answer» B. \[\pi (a-h)(2{{a}^{2}}-{{h}^{2}}-ah)\] | |
| 18. |
Area enclosed between the curve \[{{y}^{2}}(2a-x)={{x}^{3}}\] and line \[x=2a\] above x-axis is [MP PET 2001] |
| A. | \[\pi \,{{a}^{2}}\] |
| B. | \[\frac{3\pi \,{{a}^{2}}}{2}\] |
| C. | \[2\pi \,{{a}^{2}}\] |
| D. | \[3\pi \,{{a}^{2}}\] |
| Answer» C. \[2\pi \,{{a}^{2}}\] | |
| 19. |
If \[l(m,\,n)=\int_{0}^{1}{{{t}^{m}}{{(1+t)}^{n}}dt,}\] then the expression for \[l(m,\,n)\] in terms of \[l(m+1,\,\,n-1)\] is [IIT Screening 2003] |
| A. | \[\frac{{{2}^{n}}}{m+1}-\frac{n}{m+1}l(m+1,\,n-1)\] |
| B. | \[\frac{n}{m+1}l(m+1,\,n-1)\] |
| C. | \[\frac{{{2}^{n}}}{m+1}+\frac{n}{m+1}l(m+1,\,n-1)\] |
| D. | \[\frac{m}{n+1}l(m+1,\,n-1)\] |
| Answer» B. \[\frac{n}{m+1}l(m+1,\,n-1)\] | |
| 20. |
Let \[f:R\to R\] and \[g:R\to R\] be continuous functions, then the value of the integral \[\int_{-\pi /2}^{\pi /2}{[f(x)+f(-x)]\,\,[g(x)-g(-x)]\,dx=}\] [IIT 1990; DCE 2000; MP PET 2001] |
| A. | \[\pi \] |
| B. | 1 |
| C. | \[-1\] |
| D. | 0 |
| Answer» E. | |