JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank

done Binomial theorem for any index Total Questions - 34

• question_answer1)
  The fourth term in the expansion of \[{{(1-2x)}^{3/2}}\]will be [RPET 1989]
  A)
  \[-\frac{3}{4}{{x}^{4}}\] done clear
  B)
  \[\frac{{{x}^{3}}}{2}\] done clear
  C)
  \[-\frac{{{x}^{3}}}{2}\] done clear
  D)
  \[\frac{3}{4}{{x}^{4}}\] done clear
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• question_answer2)
  Cube root of 217 is
  A)
  6.01 done clear
  B)
  6.04 done clear
  C)
  6.02 done clear
  D)
  None of these done clear
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• question_answer3)
  The expansion of \[\frac{1}{{{(4-3x)}^{1/2}}}\]binomial theorem will be valid, if
  A)
  \[x<1\] done clear
  B)
  \[|x|\,<1\] done clear
  C)
  \[-\frac{2}{\sqrt{3}}<x<\frac{2}{\sqrt{3}}\] done clear
  D)
  None of these done clear
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• question_answer4)
  If \[{{(a+bx)}^{-2}}=\frac{1}{4}-3x+......\], then \[(a,b)\]=  [UPSEAT 2002]
  A)
  (2, 12) done clear
  B)
  \[(-2,12)\] done clear
  C)
  \[(2,\,\,-12)\] done clear
  D)
  None of these done clear
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• question_answer5)
  \[\frac{1}{\sqrt[3]{6-3x}}=\]
  A)
  \[{{6}^{1/3}}\left[ 1+\frac{x}{6}+\frac{2{{x}^{2}}}{{{6}^{2}}}+.... \right]\] done clear
  B)
  \[{{6}^{-1/3}}\left[ 1+\frac{x}{6}+\frac{2{{x}^{2}}}{{{6}^{2}}}+.... \right]\] done clear
  C)
  \[{{6}^{1/3}}\left[ 1-\frac{x}{6}+\frac{2{{x}^{2}}}{{{6}^{2}}}-.... \right]\] done clear
  D)
  \[{{6}^{-1/3}}\left[ 1-\frac{x}{6}+\frac{2{{x}^{2}}}{{{6}^{2}}}-.... \right]\] done clear
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• question_answer6)
  \[{{\left( \frac{a}{a+x} \right)}^{\frac{1}{2}}}+{{\left( \frac{a}{a-x} \right)}^{\frac{1}{2}}}=\]  [DCE 1994; Pb. CET 2002; AIEEE 2002]
  A)
  \[2+\frac{3{{x}^{2}}}{4{{a}^{2}}}+....\] done clear
  B)
  \[1+\frac{3{{x}^{2}}}{8{{a}^{2}}}+....\] done clear
  C)
  \[2+\frac{x}{a}+\frac{3{{x}^{2}}}{4{{a}^{2}}}+....\] done clear
  D)
  \[2-\frac{x}{a}+\frac{3{{x}^{2}}}{4{{a}^{2}}}\]+...... done clear
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• question_answer7)
  \[{{(r+1)}^{th}}\] term in the expansion of \[{{(1-x)}^{-4}}\]will be
  A)
  \[\frac{{{x}^{r}}}{r!}\] done clear
  B)
  \[\frac{(r+1)(r+2)(r+3)}{6}{{x}^{r}}\] done clear
  C)
  \[\frac{(r+2)(r+3)}{2}{{x}^{r}}\] done clear
  D)
  None of these done clear
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• question_answer8)
  In the expansion of \[{{\left( \frac{1+x}{1-x} \right)}^{2}}\], the coefficient of \[{{x}^{n}}\] will be
  A)
  \[4n\] done clear
  B)
  \[4n-3\] done clear
  C)
  \[4n+1\] done clear
  D)
  None of these done clear
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• question_answer9)
  \[\frac{1}{{{(2+x)}^{4}}}=\]
  A)
  \[\frac{1}{2}\left( 1-2x+\frac{5}{2}{{x}^{2}}-.... \right)\] done clear
  B)
   \[\frac{1}{16}\left( 1-2x+\frac{5}{2}{{x}^{2}}-.... \right)\] done clear
  C)
  \[\frac{1}{16}\left( 1+2x+\frac{5}{2}{{x}^{2}}+.... \right)\] done clear
  D)
  \[\frac{1}{2}\left( 1+2x+\frac{5}{2}{{x}^{2}}+.... \right)\] done clear
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• question_answer10)
  \[\frac{1}{{{\left( {{x}^{2}}+\frac{1}{x} \right)}^{\frac{4}{3}}}}\] can be expanded by binomial theorem, if
  A)
  \[x<1\] done clear
  B)
  \[|x|<1\] done clear
  C)
  \[x>1\] done clear
  D)
  \[|x|>1\] done clear
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• question_answer11)
  The coefficient of \[{{x}^{3}}\] in the expansion of \[\frac{{{(1+3x)}^{2}}}{1-2x}\] will be
  A)
  8 done clear
  B)
  32 done clear
  C)
  50 done clear
  D)
  None of these done clear
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• question_answer12)
  If \[|x|<1\]then the coefficient of \[{{x}^{n}}\] in the expansion of \[{{(1+x+{{x}^{2}}+....)}^{2}}\]will be [Pb. CET 1989]
  A)
  1 done clear
  B)
  n done clear
  C)
  \[n+1\] done clear
  D)
  None of these done clear
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• question_answer13)
  If \[|x|>1\], then \[{{(1+x)}^{-2}}\] =
  A)
  \[1-2x+3{{x}^{2}}-....\] done clear
  B)
  \[1+2x+3{{x}^{2}}+\].... done clear
  C)
  \[1-\frac{2}{x}+\frac{3}{{{x}^{2}}}-....\] done clear
  D)
  \[\frac{1}{{{x}^{2}}}-\frac{2}{{{x}^{3}}}+\frac{3}{{{x}^{4}}}-\]... done clear
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• question_answer14)
  If \[|x|<1\], then in the expansion of \[{{(1+2x+3{{x}^{2}}+4{{x}^{3}}+....)}^{1/2}},\] the coefficient of \[{{x}^{n}}\]is
  A)
  n done clear
  B)
  \[n+1\] done clear
  C)
  1 done clear
  D)
  - 1 done clear
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• question_answer15)
  The approximate value of \[{{(7.995)}^{1/3}}\]correct to four decimal places is                  [MNR 1991; UPSEAT 2000]
  A)
  1.9995 done clear
  B)
  1.9996 done clear
  C)
  1.9990 done clear
  D)
  1.9991 done clear
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• question_answer16)
  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)
  \[{{\left( \frac{1+x}{1-x} \right)}^{n}}\] done clear
  B)
  \[{{\left( \frac{2x}{1+x} \right)}^{n}}\] done clear
  C)
  \[{{\left( \frac{1+x}{2x} \right)}^{n}}\] done clear
  D)
  \[{{\left( \frac{1-x}{1+x} \right)}^{n}}\] done clear
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• question_answer17)
  The sum of \[1+n\left( 1-\frac{1}{x} \right)+\frac{n(n+1)}{2!}\text{  }{{\left( 1-\frac{1}{x} \right)}^{2}}+.....\infty ,\] will be   [Roorkee 1975]
  A)
  \[{{x}^{n}}\] done clear
  B)
  \[{{x}^{-n}}\] done clear
  C)
  \[{{\left( 1-\frac{1}{x} \right)}^{n}}\] done clear
  D)
  None of these done clear
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• question_answer18)
  The first four terms in the expansion of \[{{(1-x)}^{3/2}}\] are [RPET 1989]
  A)
  \[1-\frac{3}{2}x+\frac{3}{8}{{x}^{2}}-\frac{1}{16}{{x}^{3}}\] done clear
  B)
  \[1-\frac{3}{2}x-\frac{3}{8}{{x}^{2}}-\frac{{{x}^{3}}}{16}\] done clear
  C)
  \[1-\frac{3}{2}x+\frac{3}{8}{{x}^{2}}+\frac{{{x}^{3}}}{16}\] done clear
  D)
  None of these done clear
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• question_answer19)
  The coefficient of \[{{x}^{n}}\] in \[\frac{{{(1+x)}^{2}}}{{{(1-x)}^{3}}}\]is
  A)
  \[3{{n}^{2}}+2n+1\] done clear
  B)
  \[2{{n}^{2}}+2n+1\] done clear
  C)
  \[{{n}^{2}}+n+1\] done clear
  D)
  \[2{{n}^{2}}-2n+1\] done clear
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• question_answer20)
  \[1+\frac{1}{3}x+\frac{1.4}{3.6}{{x}^{2}}+\frac{1.4.7}{3.6.9}{{x}^{3}}+....\]is equal to
  A)
  x done clear
  B)
  \[{{(1+x)}^{1/3}}\] done clear
  C)
  \[{{(1-x)}^{1/3}}\] done clear
  D)
  \[{{(1-x)}^{-1/3}}\] done clear
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• question_answer21)
  \[1-\frac{1}{8}+\frac{1}{8}.\frac{3}{16}-\frac{1.3.5}{8.16.24}+.....\]= [EAMCET 1990]
  A)
  \[\frac{2}{5}\] done clear
  B)
  \[\frac{\sqrt{2}}{5}\] done clear
  C)
  \[\frac{2}{\sqrt{5}}\] done clear
  D)
  None of these done clear
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• question_answer22)
  If \[{{(r+1)}^{th}}\] term is the first negative term in the expansion of  \[{{(1+x)}^{7/2}}\], then the value of r is
  A)
  5 done clear
  B)
  6 done clear
  C)
  4 done clear
  D)
  7 done clear
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• question_answer23)
  The coefficient of \[{{x}^{n}}\] in the expansion of \[{{(1-2x+3{{x}^{2}}-4{{x}^{3}}+.....)}^{-n}}\]is
  A)
  \[\frac{(2n)!}{n!}\] done clear
  B)
  \[\frac{(2n)!}{{{(n!)}^{2}}}\] done clear
  C)
  \[\frac{1}{2}\frac{(2n)!}{{{(n!)}^{2}}}\] done clear
  D)
  None of these done clear
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• question_answer24)
  The coefficient of \[{{x}^{n}}\]in the expansion of \[{{(1-9x+20{{x}^{2}})}^{-1}}\] is
  A)
  \[{{5}^{n}}-{{4}^{n}}\] done clear
  B)
  \[{{5}^{n+1}}-{{4}^{n+1}}\] done clear
  C)
  \[{{5}^{n-1}}-{{4}^{n-1}}\] done clear
  D)
  None of these done clear
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• question_answer25)
  The coefficient of \[{{x}^{n}}\] in the expansion of \[\frac{1}{(1-x)(3-x)}\] is
  A)
  \[\frac{{{3}^{n+1}}-1}{{{2.3}^{n+1}}}\] done clear
  B)
  \[\frac{{{3}^{n+1}}-1}{{{3}^{n+1}}}\] done clear
  C)
  \[\left( \frac{{{3}^{n+1}}-1}{{{3}^{n+1}}} \right)\] done clear
  D)
  None of these done clear
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• question_answer26)
  The coefficient of \[{{x}^{n}}\] in the expansion of \[{{(1+x+{{x}^{2}}+....)}^{-n}}\] is
  A)
  1 done clear
  B)
  \[{{(-1)}^{n}}\] done clear
  C)
  n done clear
  D)
  \[n+1\] done clear
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• question_answer27)
  If \[y=3x+6{{x}^{2}}+10{{x}^{3}}+....,\]then the value of x in terms of y is
  A)
  \[1-{{(1-y)}^{-1/3}}\] done clear
  B)
  \[1-{{(1+y)}^{1/3}}\] done clear
  C)
  \[1+{{(1+y)}^{-1/3}}\] done clear
  D)
  \[1-{{(1+y)}^{-1/3}}\] done clear
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• question_answer28)
  The coefficient of \[x\] in the expansion of \[{{[\sqrt{1+{{x}^{2}}}-x]}^{-1}}\]in ascending powers of  x, when \[|x|<1\], is [MP PET 1996]
  A)
  0 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[-\frac{1}{2}\] done clear
  D)
  1 done clear
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• question_answer29)
   \[1+\frac{1}{4}+\frac{1.3}{4.8}+\frac{1.3.5}{4.8.12}+...........=\] [RPET 1996; EAMCET 2001]
  A)
  \[\sqrt{2}\] done clear
  B)
  \[\frac{1}{\sqrt{2}}\] done clear
  C)
  \[\sqrt{3}\] done clear
  D)
  \[\frac{1}{\sqrt{3}}\] done clear
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• question_answer30)
  If x is positive, the first negative term in the expansion of \[{{(1+x)}^{27\,/\,5}}\] is [AIEEE 2003]
  A)
  7th term done clear
  B)
  5th term done clear
  C)
  8th term done clear
  D)
  6th term done clear
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• question_answer31)
  Coefficient of \[{{x}^{r}}\] in the expansion of \[{{(1-2x)}^{-1/2}}\] is                                [Kurukshetra CEE 2001]
  A)
  \[\frac{(2r)\,!}{{{(r\,!)}^{2}}}\] done clear
  B)
  \[\frac{(2r)\,!}{{{2}^{r}}{{(r!)}^{2}}}\] done clear
  C)
  \[\frac{(2r)!}{{{(r!)}^{2}}{{2}^{2r}}}\] done clear
  D)
  \[\frac{(2r)!}{{{2}^{r}}.(r+1)!.(r-1)!}\] done clear
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• question_answer32)
  \[{{\sum\limits_{k=1}^{n}{k\left( 1+\frac{1}{n} \right)}}^{k-1}}=\] [EAMCET 2002; Pb. CET 2002]
  A)
  \[n(n-1)\] done clear
  B)
  \[n(n+1)\] done clear
  C)
  \[{{n}^{2}}\] done clear
  D)
  \[{{(n+1)}^{2}}\] done clear
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• question_answer33)
  The expression \[{{[x+{{({{x}^{3}}-1)}^{1/2}}]}^{5}}+{{[x-{{({{x}^{3}}-1)}^{1/2}}]}^{5}}\] is a polynomial of degree [Pb. CET 2000]
  A)
  5 done clear
  B)
  6 done clear
  C)
  7 done clear
  D)
  8 done clear
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• question_answer34)
  If x is so small that \[{{x}^{3}}\] and higher powers of x may be neglected, then \[\frac{{{(1+x)}^{\frac{3}{2}}}-{{\left( 1+\frac{1}{2}x \right)}^{3}}}{{{(1-x)}^{\frac{1}{2}}}}\] may be approximated as [AIEEE 2005]
  A)
  \[-\frac{3}{8}{{x}^{2}}\] done clear
  B)
  \[\frac{x}{2}-\frac{3}{8}{{x}^{2}}\] done clear
  C)
  \[1-\frac{3}{8}{{x}^{2}}\] done clear
  D)
  \[3x+\frac{3}{8}{{x}^{2}}\] done clear
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