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

done Critical Thinking Total Questions - 35

• question_answer1)
  The value of \[{{(\sqrt{2}+1)}^{6}}+{{(\sqrt{2}-1)}^{6}}\] will be [RPET 1997]
  A)
   - 198 done clear
  B)
  198 done clear
  C)
  99 done clear
  D)
  - 99 done clear
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• question_answer2)
  If \[{{(1+ax)}^{n}}=1+8x+24{{x}^{2}}+....,\]then the value of a and n is [IIT 1983; Pb. CET 1994, 99]
  A)
  2, 4 done clear
  B)
  2, 3 done clear
  C)
  3, 6 done clear
  D)
  1, 2 done clear
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• question_answer3)
  The coefficient of \[{{x}^{5}}\] in the expansion of \[{{(1+{{x}^{2}})}^{5}}{{(1+x)}^{4}}\] is            [EAMCET 1996; UPSEAT 2001; Pb. CET 2002]
  A)
  30 done clear
  B)
  60 done clear
  C)
  40 done clear
  D)
  None of these done clear
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• question_answer4)
  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)
  \[\left( 1,\frac{35}{24} \right)\] done clear
  B)
  \[\left( 1,-\frac{35}{24} \right)\] done clear
  C)
  \[\left( 2,\frac{35}{12} \right)\] done clear
  D)
  \[\left( 2,-\frac{35}{12} \right)\] done clear
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• question_answer5)
  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 done clear
  B)
  11 done clear
  C)
  12 done clear
  D)
  None of these done clear
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• question_answer6)
  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)
  \[p+q=r\] done clear
  B)
  \[p+r=q\] done clear
  C)
  \[p=q+r\] done clear
  D)
  \[p+q+r=0\] done clear
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• question_answer7)
   In the polynomial \[(x-1)(x-2)(x-3).............(x-100),\]the coefficient of \[{{x}^{99}}\] is [AMU 2002]
  A)
  5050 done clear
  B)
  - 5050 done clear
  C)
  100 done clear
  D)
  99 done clear
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• question_answer8)
  The coefficient of \[{{x}^{100}}\] in the expansion of  \[\sum\limits_{j=0}^{200}{{{(1+x)}^{j}}}\] is  [UPSEAT 2004]
  A)
  \[\left( \begin{align}   & 200 \\  & 100 \\ \end{align} \right)\] done clear
  B)
  \[\left( \begin{align}   & 201 \\  & 102 \\ \end{align} \right)\] done clear
  C)
  \[\left( \begin{align}   & 200 \\  & 101 \\ \end{align} \right)\] done clear
  D)
  \[\left( \begin{align}   & 201 \\  & 100 \\ \end{align} \right)\] done clear
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• question_answer9)
  If the coefficient of \[{{x}^{7}}\] in \[{{\left( a{{x}^{2}}+\frac{1}{bx} \right)}^{11}}\] is equal to the coefficient of \[{{x}^{-7}}\]in \[{{\left( ax-\frac{1}{b{{x}^{2}}} \right)}^{11}}\], then ab = [MP PET 1999; AMU 2001; Pb. CET 2002; AIEEE 2005]
  A)
  1 done clear
  B)
  1/2 done clear
  C)
  2 done clear
  D)
  3 done clear
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• question_answer10)
  If the coefficient of x in the expansion of \[{{\left( {{x}^{2}}+\frac{k}{x} \right)}^{5}}\] is 270, then k = [EAMCET 2002]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
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• question_answer11)
  The coefficients of three successive terms in the expansion of \[{{(1+x)}^{n}}\] are 165, 330 and 462 respectively, then the value of n will be [UPSEAT 1999]
  A)
  11 done clear
  B)
  10 done clear
  C)
  12 done clear
  D)
  8 done clear
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• question_answer12)
  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 done clear
  B)
  10 done clear
  C)
  8 done clear
  D)
  6 done clear
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• question_answer13)
  The middle term in the expansion of \[{{(1+x)}^{2n}}\] is  [Pb. CET 1998]
  A)
  \[\frac{1.3.5....(5n-1)}{n!}{{x}^{n}}\] done clear
  B)
  \[\frac{2.4.6....2n}{n!}{{x}^{2n+1}}\] done clear
  C)
  \[\frac{1.3.5....(2n-1)}{n!}{{x}^{n}}\] done clear
  D)
  \[\frac{1.3.5....(2n-1)}{n!}{{2}^{n}}{{x}^{n}}\] done clear
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• question_answer14)
  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)
  \[^{60}{{C}_{20}}\] done clear
  B)
  \[^{30}{{C}_{10}}\] done clear
  C)
  \[^{60}{{C}_{30}}\] done clear
  D)
  \[^{40}{{C}_{30}}\] done clear
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• question_answer15)
  Middle term in the expansion of \[{{(1+3x+3{{x}^{2}}+{{x}^{3}})}^{6}}\]is [MP PET 1997]
  A)
  \[{{4}^{th}}\] done clear
  B)
  \[{{3}^{rd}}\] done clear
  C)
  \[{{10}^{th}}\] done clear
  D)
  None of these done clear
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• question_answer16)
  Two middle terms in the expansion of \[{{\left( x-\frac{1}{x} \right)}^{11}}\] are
  A)
  231x and \[\frac{231}{x}\] done clear
  B)
  \[462x\] and \[\frac{462}{x}\] done clear
  C)
  \[-462x\] and \[\frac{462}{x}\] done clear
  D)
  None of these done clear
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• question_answer17)
  The term independent of y in the expansion of \[{{({{y}^{-1/6}}-{{y}^{1/3}})}^{9}}\] is [BIT Ranchi 1980]
  A)
  84 done clear
  B)
  8.4 done clear
  C)
  0.84 done clear
  D)
  - 84 done clear
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• question_answer18)
  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)
  \[\frac{1}{3}\] done clear
  B)
  \[\frac{19}{54}\] done clear
  C)
  \[\frac{17}{54}\] done clear
  D)
    \[\frac{1}{4}\] done clear
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• question_answer19)
  The term independent of x in \[{{\left[ \frac{\sqrt{x}}{3}+\frac{\sqrt{3}}{{{x}^{2}}} \right]}^{10}}\] is  [EAMCET 1984; RPET 2000]
  A)
  \[\frac{2}{3}\] done clear
  B)
  \[\frac{5}{3}\] done clear
  C)
  \[\frac{4}{3}\] done clear
  D)
  None of these done clear
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• question_answer20)
  The term independent of x in \[{{\left( \sqrt{x}-\frac{2}{x} \right)}^{18}}\]is  [EAMCET 1990]
  A)
  \[^{18}{{C}_{6}}{{2}^{6}}\] done clear
  B)
  \[^{18}{{C}_{6}}{{2}^{12}}\] done clear
  C)
  \[^{18}{{C}_{18}}{{2}^{18}}\] done clear
  D)
  None of these done clear
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• question_answer21)
  The largest term in the expansion of \[{{(3+2x)}^{50}}\] where \[x=\frac{1}{5}\] is [IIT Screening 1993]
  A)
  5th done clear
  B)
  51st done clear
  C)
  7th done clear
  D)
  6th done clear
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• question_answer22)
  \[\frac{{{C}_{1}}}{{{C}_{0}}}+2\frac{{{C}_{2}}}{{{C}_{1}}}+3\frac{{{C}_{3}}}{{{C}_{2}}}+....+15\frac{{{C}_{15}}}{{{C}_{14}}}=\] [IIT 1962]
  A)
   100 done clear
  B)
  120 done clear
  C)
  \[-120\] done clear
  D)
  None of these done clear
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• question_answer23)
  \[\left( \begin{matrix}    n  \\    0  \\ \end{matrix} \right)+2\,\left( \begin{matrix}    n  \\    1  \\ \end{matrix} \right)+{{2}^{2}}\left( \begin{matrix}    n  \\    2  \\ \end{matrix} \right)+.....+{{2}^{n}}\left( \begin{matrix}    n  \\    n  \\ \end{matrix} \right)\] is equal to   [AMU 2000]
  A)
  \[{{2}^{n}}\] done clear
  B)
  0 done clear
  C)
  \[{{3}^{n}}\] done clear
  D)
  None of these done clear
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• question_answer24)
  If \[{{C}_{r}}\] stands for \[^{n}{{C}_{r}}\], the sum of the given series\[\frac{2(n/2)!(n/2)!}{n!}[C_{0}^{2}-2C_{1}^{2}+3C_{2}^{2}-.....+{{(-1)}^{n}}(n+1)C_{n}^{2}]\], Where n is an even positive integer, is [IIT 1986]
  A)
  0 done clear
  B)
  \[{{(-1)}^{n/2}}(n+1)\] done clear
  C)
  \[{{(-1)}^{n}}(n+2)\] done clear
  D)
  \[{{(-1)}^{n/2}}(n+2)\] done clear
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• question_answer25)
  Sum of odd terms is A and sum of even terms is B in the expansion \[{{(x+a)}^{n}},\]then  [RPET 1987; UPSEAT 2004]
  A)
  \[AB=\frac{1}{4}{{(x-a)}^{2n}}-{{(x+a)}^{2n}}\] done clear
  B)
  \[2AB={{(x+a)}^{2n}}-{{(x-a)}^{2n}}\] done clear
  C)
  \[4AB={{(x+a)}^{2n}}-{{(x-a)}^{2n}}\] done clear
  D)
  None of these done clear
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• question_answer26)
  In the expansion of \[{{(x+a)}^{n}}\], the sum of odd terms is P and sum of even terms is Q, then the value of \[({{P}^{2}}-{{Q}^{2}})\] will be       [RPET 1997; Pb. CET 1998]
  A)
  \[{{({{x}^{2}}+{{a}^{2}})}^{n}}\] done clear
  B)
  \[{{({{x}^{2}}-{{a}^{2}})}^{n}}\] done clear
  C)
  \[{{(x-a)}^{2n}}\] done clear
  D)
  \[{{(x+a)}^{2n}}\] done clear
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• question_answer27)
  The sum of the coefficients in the expansion of \[{{(1+x-3{{x}^{2}})}^{2163}}\] will be [IIT 1982]
  A)
   0 done clear
  B)
  1 done clear
  C)
  \[-1\] done clear
  D)
  \[{{2}^{2163}}\] done clear
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• question_answer28)
  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)
  \[a=3b\] done clear
  B)
  \[a={{b}^{3}}\] done clear
  C)
  \[b={{a}^{3}}\] done clear
  D)
  None of these done clear
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• question_answer29)
  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 done clear
  B)
  924 done clear
  C)
  824 done clear
  D)
  724 done clear
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• question_answer30)
  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 done clear
  B)
  1 done clear
  C)
  May be any real number done clear
  D)
  No such value exist done clear
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• question_answer31)
  For every natural number n, \[{{3}^{2n+2}}-8n-9\] is divisible by [IIT 1977]
  A)
  16 done clear
  B)
  128 done clear
  C)
  256 done clear
  D)
  None of these done clear
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• question_answer32)
  The least remainder when \[{{17}^{30}}\] is divided by 5 is [Karnataka CET  2003]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
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• question_answer33)
  The value of the natural numbers n such that the inequality \[{{2}^{n}}>2n+1\] is valid is [MNR 1994]
  A)
  For \[n\ge 3\] done clear
  B)
  For n < 3 done clear
  C)
  For mn done clear
  D)
  For any n done clear
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• question_answer34)
  Let P(n) be a statement and let P(n) Þ p(n + 1) for all natural numbers n, then P(n) is true
  A)
  For all n done clear
  B)
  For all n > 1 done clear
  C)
  For all n > m, m being a fixed positive integer done clear
  D)
  Nothing can be said done clear
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• question_answer35)
  \[{{(1+x)}^{n}}-nx-1\] is divisible by (where \[n\in N\])
  A)
  \[2x\] done clear
  B)
  \[{{x}^{2}}\] done clear
  C)
  \[2{{x}^{3}}\] done clear
  D)
  All of these done clear
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