MCQOPTIONS
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MCQOPTIONS
This section includes 31 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
The term independent of x in the expansion of \[{{(1+x)}^{n}}{{\left( 1+\frac{1}{x} \right)}^{n}}\] is [EAMCET 1989] |
| A. | \[C_{0}^{2}+2C_{1}^{2}+....+(n+1)C_{n}^{2}\] |
| B. | \[{{({{C}_{0}}+{{C}_{1}}+....+{{C}_{n}})}^{2}}\] |
| C. | \[C_{0}^{2}+C_{1}^{2}+.....+C_{n}^{2}\] |
| D. | None of these |
| Answer» D. None of these | |
| 2. |
The middle term in the expansion of \[{{\left( x+\frac{1}{x} \right)}^{10}}\] is [BIT Ranchi 1991; RPET 2002; Pb. CET 1991] |
| A. | \[^{10}{{C}_{4}}\frac{1}{x}\] |
| B. | \[^{10}{{C}_{5}}\] |
| C. | \[^{10}{{C}_{5}}x\] |
| D. | \[^{10}{{C}_{7}}{{x}^{4}}\] |
| Answer» C. \[^{10}{{C}_{5}}x\] | |
| 3. |
In the expansion of \[{{\left( 2{{x}^{2}}-\frac{1}{x} \right)}^{12}}\], the term independent of x is [MP PET 2001] |
| A. | 10th |
| B. | 9th |
| C. | 8th |
| D. | 7th |
| Answer» C. 8th | |
| 4. |
The term independent of x in the expansion of \[{{\left( \frac{1}{2}{{x}^{1/3}}+{{x}^{-1/5}} \right)}^{8}}\] will be [Roorkee 1985] |
| A. | 5 |
| B. | 6 |
| C. | 7 |
| D. | 8 |
| Answer» D. 8 | |
| 5. |
The coefficient of \[{{x}^{7}}\] in the expansion of \[{{\left( \frac{{{x}^{2}}}{2}-\frac{2}{x} \right)}^{8}}\] is [MNR 1975] |
| A. | -56 |
| B. | 56 |
| C. | -14 |
| D. | 14 |
| Answer» D. 14 | |
| 6. |
If \[\frac{{{(1-3x)}^{1/2}}+{{(1-x)}^{5/3}}}{\sqrt{4-x}}\]is approximately equal to \[a+bx\]for small values of x,then \[(a,b)\]= |
| A. | \[\left( 1,\frac{35}{24} \right)\] |
| B. | \[\left( 1,-\frac{35}{24} \right)\] |
| C. | \[\left( 2,\frac{35}{12} \right)\] |
| D. | \[\left( 2,-\frac{35}{12} \right)\] |
| Answer» C. \[\left( 2,\frac{35}{12} \right)\] | |
| 7. |
\[\frac{{{C}_{1}}}{{{C}_{0}}}+2\frac{{{C}_{2}}}{{{C}_{1}}}+3\frac{{{C}_{3}}}{{{C}_{2}}}+....+15\frac{{{C}_{15}}}{{{C}_{14}}}=\] [IIT 1962] |
| A. | 100 |
| B. | 120 |
| C. | \[-120\] |
| D. | None of these |
| Answer» C. \[-120\] | |
| 8. |
The largest term in the expansion of \[{{(3+2x)}^{50}}\] where \[x=\frac{1}{5}\] is [IIT Screening 1993] |
| A. | 5th |
| B. | 51st |
| C. | 7th |
| D. | 6th |
| Answer» D. 6th | |
| 9. |
The term independent of x in \[{{\left( \sqrt{x}-\frac{2}{x} \right)}^{18}}\]is [EAMCET 1990] |
| A. | \[^{18}{{C}_{6}}{{2}^{6}}\] |
| B. | \[^{18}{{C}_{6}}{{2}^{12}}\] |
| C. | \[^{18}{{C}_{18}}{{2}^{18}}\] |
| D. | None of these |
| Answer» B. \[^{18}{{C}_{6}}{{2}^{12}}\] | |
| 10. |
The term independent of x in \[{{\left[ \frac{\sqrt{x}}{3}+\frac{\sqrt{3}}{{{x}^{2}}} \right]}^{10}}\] is [EAMCET 1984; RPET 2000] |
| A. | \[\frac{2}{3}\] |
| B. | \[\frac{5}{3}\] |
| C. | \[\frac{4}{3}\] |
| D. | None of these |
| Answer» C. \[\frac{4}{3}\] | |
| 11. |
The coefficient of the term independent of x in the expansion of \[(1+x+2{{x}^{3}}){{\left( \frac{3}{2}{{x}^{2}}-\frac{1}{3x} \right)}^{9}}\] is [DCE 1994] |
| A. | \[\frac{1}{3}\] |
| B. | \[\frac{19}{54}\] |
| C. | \[\frac{17}{54}\] |
| D. | \[\frac{1}{4}\] |
| Answer» D. \[\frac{1}{4}\] | |
| 12. |
\[{{\sum\limits_{k=1}^{n}{k\left( 1+\frac{1}{n} \right)}}^{k-1}}=\] [EAMCET 2002; Pb. CET 2002] |
| A. | \[n(n-1)\] |
| B. | \[n(n+1)\] |
| C. | \[{{n}^{2}}\] |
| D. | \[{{(n+1)}^{2}}\] |
| Answer» D. \[{{(n+1)}^{2}}\] | |
| 13. |
The expansion of \[\frac{1}{{{(4-3x)}^{1/2}}}\]binomial theorem will be valid, if |
| A. | \[x<1\] |
| B. | \[|x|\,<1\] |
| C. | \[-\frac{2}{\sqrt{3}}<x<\frac{2}{\sqrt{3}}\] |
| D. | None of these |
| Answer» E. | |
| 14. |
\[1+\frac{1}{4}+\frac{1.3}{4.8}+\frac{1.3.5}{4.8.12}+...........=\] [RPET 1996; EAMCET 2001] |
| A. | \[\sqrt{2}\] |
| B. | \[\frac{1}{\sqrt{2}}\] |
| C. | \[\sqrt{3}\] |
| D. | \[\frac{1}{\sqrt{3}}\] |
| Answer» B. \[\frac{1}{\sqrt{2}}\] | |
| 15. |
If \[|x|<1\], then the value of\[1+n\left( \frac{2x}{1+x} \right)+\frac{n(n+1)}{2!}{{\left( \frac{2x}{1+x} \right)}^{2}}+.....\infty \]will be [AMU 1983] |
| A. | \[{{\left( \frac{1+x}{1-x} \right)}^{n}}\] |
| B. | \[{{\left( \frac{2x}{1+x} \right)}^{n}}\] |
| C. | \[{{\left( \frac{1+x}{2x} \right)}^{n}}\] |
| D. | \[{{\left( \frac{1-x}{1+x} \right)}^{n}}\] |
| Answer» B. \[{{\left( \frac{2x}{1+x} \right)}^{n}}\] | |
| 16. |
In the expansion of \[{{(1+3x+2{{x}^{2}})}^{6}}\]the coefficient of \[{{x}^{11}}\] is [Kerala (Engg.) 2005] |
| A. | 144 |
| B. | 288 |
| C. | 216 |
| D. | 576 |
| E. | (3)(211) |
| Answer» E. (3)(211) | |
| 17. |
If the coefficients of \[{{x}^{2}}\]and \[{{x}^{3}}\]in the expansion of \[{{(3+ax)}^{9}}\] are the same, then the value of a is [DCE 2001] |
| A. | \[-\frac{7}{9}\] |
| B. | \[-\frac{9}{7}\] |
| C. | \[\frac{7}{9}\] |
| D. | \[\frac{9}{7}\] |
| Answer» E. | |
| 18. |
If p and q be positive, then the coefficients of \[{{x}^{p}}\] and \[{{x}^{q}}\] in the expansion of \[{{(1+x)}^{p+q}}\]will be [MNR 1983; AIEEE 2002] |
| A. | Equal |
| B. | Equal in magnitude but opposite in sign |
| C. | Reciprocal to each other |
| D. | None of these |
| Answer» B. Equal in magnitude but opposite in sign | |
| 19. |
The larger of \[{{99}^{50}}+{{100}^{50}}\] and \[{{101}^{50}}\] is [IIT 1980] |
| A. | \[{{99}^{50}}+{{100}^{50}}\] |
| B. | Both are equal |
| C. | \[{{101}^{50}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 20. |
The last digit in \[{{7}^{300}}\] is [Karnataka CET 2004] |
| A. | 7 |
| B. | 9 |
| C. | 1 |
| D. | 3 |
| Answer» D. 3 | |
| 21. |
If \[{{T}_{0}},{{T}_{1}},{{T}_{2}},....{{T}_{n}}\] represent the terms in the expansion of \[{{(x+a)}^{n}}\], then \[{{({{T}_{0}}-{{T}_{2}}+{{T}_{4}}-....)}^{2}}\] \[+{{({{T}_{1}}-{{T}_{3}}+{{T}_{5}}-....)}^{2}}=\] |
| A. | \[({{x}^{2}}+{{a}^{2}})\] |
| B. | \[{{({{x}^{2}}+{{a}^{2}})}^{n}}\] |
| C. | \[{{({{x}^{2}}+{{a}^{2}})}^{1/n}}\] |
| D. | \[{{({{x}^{2}}+{{a}^{2}})}^{-1/n}}\] |
| Answer» C. \[{{({{x}^{2}}+{{a}^{2}})}^{1/n}}\] | |
| 22. |
If the coefficient of the middle term in the expansion of \[{{(1+x)}^{2n+2}}\]is p and the coefficients of middle terms in the expansion of \[{{(1+x)}^{2n+1}}\] are q and r, then |
| A. | \[p+q=r\] |
| B. | \[p+r=q\] |
| C. | \[p=q+r\] |
| D. | \[p+q+r=0\] |
| Answer» D. \[p+q+r=0\] | |
| 23. |
The value of x in the expression \[{{[x+{{x}^{{{\log }_{10}}}}^{(x)}]}^{5}}\], if the third term in the expansion is 10,00,000 [Roorkee 1992] |
| A. | 10 |
| B. | 11 |
| C. | 12 |
| D. | None of these |
| Answer» B. 11 | |
| 24. |
If the sum of the coefficients in the expansion of \[{{(\alpha {{x}^{2}}-2x+1)}^{35}}\] is equal to the sum of the coefficients in the expansion of \[{{(x-\alpha y)}^{35}}\], then \[\alpha \]= |
| A. | 0 |
| B. | 1 |
| C. | May be any real number |
| D. | No such value exist |
| Answer» C. May be any real number | |
| 25. |
The sum of the coefficients in the expansion of \[{{(x+y)}^{n}}\] is 4096. The greatest coefficient in the expansion is [Kurukshetra CEE 1998; AIEEE 2002] |
| A. | 1024 |
| B. | 924 |
| C. | 824 |
| D. | 724 |
| Answer» C. 824 | |
| 26. |
If the sum of the coefficients in the expansion of \[{{(1-3x+10{{x}^{2}})}^{n}}\] is a and if the sum of the coefficients in the expansion of\[{{(1+{{x}^{2}})}^{n}}\] is b, then [UPSEAT 2001] |
| A. | \[a=3b\] |
| B. | \[a={{b}^{3}}\] |
| C. | \[b={{a}^{3}}\] |
| D. | None of these |
| Answer» C. \[b={{a}^{3}}\] | |
| 27. |
The term independent of y in the expansion of \[{{({{y}^{-1/6}}-{{y}^{1/3}})}^{9}}\] is [BIT Ranchi 1980] |
| A. | 84 |
| B. | 8.4 |
| C. | 0.84 |
| D. | -84 |
| Answer» E. | |
| 28. |
The value of \[\left( \begin{matrix} 30\\ 0\\\end{matrix} \right)\left( \begin{matrix} 30\\ 10\\\end{matrix} \right)-\left( \begin{matrix} 30\\ 1\\\end{matrix} \right)\left( \begin{matrix} 30\\ 11\\\end{matrix} \right)+\left( \begin{matrix} 30\\ 2\\\end{matrix} \right)\left( \begin{matrix} 30\\ 12\\\end{matrix} \right)+......+\left( \begin{matrix} 30\\ 20\\\end{matrix} \right)\left( \begin{matrix} 30\\ 30\\\end{matrix} \right)\][IIT Screening 2005] |
| A. | \[^{60}{{C}_{20}}\] |
| B. | \[^{30}{{C}_{10}}\] |
| C. | \[^{60}{{C}_{30}}\] |
| D. | \[^{40}{{C}_{30}}\] |
| Answer» C. \[^{60}{{C}_{30}}\] | |
| 29. |
If the coefficient of \[{{(2r+4)}^{th}}\] and \[{{(r-2)}^{th}}\]terms in the expansion of \[{{(1+x)}^{18}}\]are equal, then r= [MP PET 1997; Pb. CET 2001] |
| A. | 12 |
| B. | 10 |
| C. | 8 |
| D. | 6 |
| Answer» E. | |
| 30. |
The middle term in the expansion of \[{{(1+x)}^{2n}}\] is [Pb. CET 1998] |
| A. | \[\frac{1.3.5....(5n-1)}{n!}{{x}^{n}}\] |
| B. | \[\frac{2.4.6....2n}{n!}{{x}^{2n+1}}\] |
| C. | \[\frac{1.3.5....(2n-1)}{n!}{{x}^{n}}\] |
| D. | \[\frac{1.3.5....(2n-1)}{n!}{{2}^{n}}{{x}^{n}}\] |
| Answer» E. | |
| 31. |
If \[|x|>1\], then \[{{(1+x)}^{-2}}\] = |
| A. | \[1-2x+3{{x}^{2}}-....\] |
| B. | \[1+2x+3{{x}^{2}}+\].... |
| C. | \[1-\frac{2}{x}+\frac{3}{{{x}^{2}}}-....\] |
| D. | \[\frac{1}{{{x}^{2}}}-\frac{2}{{{x}^{3}}}+\frac{3}{{{x}^{4}}}-\]... |
| Answer» E. | |