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

done General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient Total Questions - 76

• question_answer1)
  \[{{6}^{th}}\]term in expansion of \[{{\left( 2{{x}^{2}}-\frac{1}{3{{x}^{2}}} \right)}^{10}}\] is 
  A)
  \[\frac{4580}{17}\] done clear
  B)
  \[-\frac{896}{27}\] done clear
  C)
  \[\frac{5580}{17}\] done clear
  D)
    None of these done clear
  View Solution play_arrow

• question_answer2)
  If the ratio of the coefficient of third and fourth term in the expansion of \[{{\left( x-\frac{1}{2x} \right)}^{n}}\] is 1 : 2, then the value of  n  will be
  A)
  18 done clear
  B)
  16 done clear
  C)
  12 done clear
  D)
  - 10 done clear
  View Solution play_arrow

• question_answer3)
  If the coefficients of  \[{{r}^{th}}\]term and \[{{(r+4)}^{th}}\]term are equal in the expansion of \[{{(1+x)}^{20}}\], then the value of r will be [RPET 1985, 97; Kerala (Engg.) 2001; MP PET 2002]
  A)
  7 done clear
  B)
  8 done clear
  C)
  9 done clear
  D)
  10 done clear
  View Solution play_arrow

• question_answer4)
  \[{{r}^{th}}\]term in the expansion of  \[{{(a+2x)}^{n}}\]is
  A)
  \[\frac{n(n+1)....(n-r+1)}{r!}{{a}^{n-r+1}}{{(2x)}^{r}}\] done clear
  B)
  \[\frac{n(n-1)....(n-r+2)}{(r-1)\,!}{{a}^{n-r+1}}{{(2x)}^{r-1}}\] done clear
  C)
  \[\frac{n(n+1)....(n-r)}{(r+1)!}{{a}^{n-r}}{{(x)}^{r}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer5)
  \[{{16}^{th}}\] term in the expansion of \[{{(\sqrt{x}-\sqrt{y})}^{17}}\] is
  A)
  \[136x{{y}^{7}}\] done clear
  B)
  \[136xy\] done clear
  C)
  \[-136x{{y}^{15/2}}\] done clear
  D)
  \[-136x{{y}^{2}}\] done clear
  View Solution play_arrow

• question_answer6)
  In \[{{\left( \sqrt[3]{2}+\frac{1}{\sqrt[3]{3}} \right)}^{n}}\]  if the ratio of \[{{7}^{th}}\] term from the beginning to the \[{{7}^{th}}\] term from the end is \[\frac{1}{6}\], then  \[n=\]
  A)
  7 done clear
  B)
  8 done clear
  C)
  9 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer7)
  If coefficient of \[{{(2r+3)}^{th}}\] and \[{{(r-1)}^{th}}\] terms in the expansion of \[{{(1+x)}^{15}}\] are equal, then value of r is [RPET 1995, 2003; UPSEAT 2001]
  A)
  5 done clear
  B)
  6 done clear
  C)
  4 done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer8)
  If \[{{x}^{4}}\] occurs in the \[{{r}^{th}}\] term in the expansion of  \[{{\left( {{x}^{4}}+\frac{1}{{{x}^{3}}} \right)}^{15}}\], then \[r=\]         [MP PET 1995; Pb. CET 2002]
  A)
  7 done clear
  B)
  8 done clear
  C)
  9 done clear
  D)
  10 done clear
  View Solution play_arrow

• question_answer9)
  If the \[{{(r+1)}^{th}}\] term in the expansion of \[{{\left( \sqrt[3]{\frac{a}{\sqrt{b}}}+\sqrt{\frac{b}{\sqrt[3]{a}}} \right)}^{21}}\] has the same power of a and b, then the value of r is
  A)
  9 done clear
  B)
  10 done clear
  C)
  8 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer10)
  If the third term in the binomial expansion of \[{{(1+x)}^{m}}\] is \[-\frac{1}{8}{{x}^{2}}\],  then the rational value of m is
  A)
  2 done clear
  B)
  \[1/2\] done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer11)
  The first 3 terms in the expansion of \[{{(1+ax)}^{n}}\] \[(n\ne 0)\] are 1, 6x and 16x2. Then the value of a and n are respectively [Kerala (Engg.) 2002]
  A)
  2 and 9 done clear
  B)
  3 and 2 done clear
  C)
  2/3 and 9 done clear
  D)
  3/2 and \[6\] done clear
  View Solution play_arrow

• question_answer12)
  If the coefficients of \[{{T}_{r}},\,{{T}_{r+1}},\,{{T}_{r+2}}\] terms of \[{{(1+x)}^{14}}\] are in A.P., then r = [Pb. CET 2002]
  A)
  6 done clear
  B)
  7 done clear
  C)
  8 done clear
  D)
  9 done clear
  View Solution play_arrow

• question_answer13)
  Coefficient of x in the expansion of \[{{\left( {{x}^{2}}+\frac{a}{x} \right)}^{5}}\] is  [Orissa JEE 2004]
  A)
  \[9{{a}^{2}}\] done clear
  B)
  \[10{{a}^{3}}\] done clear
  C)
  \[10{{a}^{2}}\] done clear
  D)
  \[10a\] done clear
  View Solution play_arrow

• question_answer14)
  If the coefficients of \[{{p}^{th}}\], \[{{(p+1)}^{th}}\] and \[{{(p+2)}^{th}}\]terms in the expansion of  \[{{(1+x)}^{n}}\]are in A.P., then [AIEEE 2005]
  A)
  \[{{n}^{2}}-2np+4{{p}^{2}}=0\] done clear
  B)
  \[{{n}^{2}}-n\,(4p+1)+4{{p}^{2}}-2=0\] done clear
  C)
  \[{{n}^{2}}-n\,(4p+1)+4{{p}^{2}}=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer15)
  In the expansion of \[{{\left( \frac{a}{x}+bx \right)}^{12}}\],the coefficient of x-10 will be
  A)
  \[12{{a}^{11}}\] done clear
  B)
  \[12{{b}^{11}}a\] done clear
  C)
  \[12{{a}^{11}}b\] done clear
  D)
  \[12{{a}^{11}}{{b}^{11}}\] done clear
  View Solution play_arrow

• question_answer16)
  The ratio of the coefficient of terms \[{{x}^{n-r}}{{a}^{r}}\]and \[{{x}^{r}}{{a}^{n-r}}\] in the binomial expansion of \[{{(x+a)}^{n}}\]will be
  A)
  \[x:a\] done clear
  B)
  \[n:r\] done clear
  C)
  \[x:n\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer17)
  If A and B are the coefficients of \[{{x}^{n}}\] in the expansions of \[{{(1+x)}^{2n}}\] and \[{{(1+x)}^{2n-1}}\]respectively, then [MP PET 1999; Pb. CET 2004]
  A)
  \[A=B\] done clear
  B)
  \[A=2B\] done clear
  C)
  \[2A=B\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer18)
  If the expansion of \[{{\left( {{y}^{2}}+\frac{c}{y} \right)}^{5}}\], the coefficient of y will be [MNR 1983]
  A)
  \[20c\] done clear
  B)
  \[10c\] done clear
  C)
  \[10{{c}^{3}}\] done clear
  D)
  \[20{{c}^{2}}\] done clear
  View Solution play_arrow

• question_answer19)
  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 done clear
  B)
  Equal in magnitude but opposite in sign done clear
  C)
  Reciprocal to each other done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer20)
  In the expansion of \[{{\left( x-\frac{1}{x} \right)}^{6}}\],  the constant term is [AMU 1982; MP PET 1984; MNR 1979]
  A)
  - 20 done clear
  B)
  20 done clear
  C)
  30 done clear
  D)
  - 30 done clear
  View Solution play_arrow

• question_answer21)
  In the expansion of \[{{\left( {{x}^{2}}-2x \right)}^{10}}\], the coefficient of \[{{x}^{16}}\] is [MP PET 1985]
  A)
  -1680 done clear
  B)
  1680 done clear
  C)
  3360 done clear
  D)
  6720 done clear
  View Solution play_arrow

• question_answer22)
  .In the expansion of \[{{\left( \frac{x}{2}-\frac{3}{{{x}^{2}}} \right)}^{10}}\], the coefficient of \[{{x}^{4}}\]is                         [IIT 1983; EAMCET 1985; DCE 2000; RPET 2001; UPSEAT 2002; J & K 2005]
  A)
  \[\frac{405}{256}\] done clear
  B)
  \[\frac{504}{259}\] done clear
  C)
  \[\frac{450}{263}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer23)
  If the coefficients of \[{{5}^{th}}\], \[{{6}^{th}}\]and \[{{7}^{th}}\] terms in the expansion of \[{{(1+x)}^{n}}\]be in A.P., then n = [Roorkee 1984; Pb. CET 1999]
  A)
  7 only done clear
  B)
  14 only done clear
  C)
  7 or 14 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer24)
  The coefficient of \[{{x}^{7}}\] in the expansion of \[{{\left( \frac{{{x}^{2}}}{2}-\frac{2}{x} \right)}^{8}}\] is [MNR 1975]
  A)
  - 56 done clear
  B)
  56 done clear
  C)
  - 14 done clear
  D)
  14 done clear
  View Solution play_arrow

• question_answer25)
  The coefficient of \[{{x}^{-7}}\] in the expansion of \[{{\left( ax-\frac{1}{b{{x}^{2}}} \right)}^{11}}\] will be                       [IIT 1967; RPET 1996; Pb. CET 2003]
  A)
  \[\frac{462{{a}^{6}}}{{{b}^{5}}}\] done clear
  B)
  \[\frac{462{{a}^{5}}}{{{b}^{6}}}\] done clear
  C)
  \[\frac{-462{{a}^{5}}}{{{b}^{6}}}\] done clear
  D)
  \[\frac{-462{{a}^{6}}}{{{b}^{5}}}\] done clear
  View Solution play_arrow

• question_answer26)
  The coefficient of \[{{x}^{53}}\] in the following expansion  \[\sum\limits_{m=0}^{100}{{{\,}^{100}}{{C}_{m}}{{(x-3)}^{100-m}}}{{.2}^{m}}\]is
  A)
  \[^{100}{{C}_{47}}\] done clear
  B)
  \[^{100}{{C}_{53}}\] done clear
  C)
  \[{{-}^{100}}{{C}_{53}}\] done clear
  D)
  \[{{-}^{100}}{{C}_{100}}\] done clear
  View Solution play_arrow

• question_answer27)
  The coefficient of \[{{x}^{32}}\]in the expansion of \[{{\left( {{x}^{4}}-\frac{1}{{{x}^{3}}} \right)}^{15}}\] is [MP PET 1994]
  A)
  \[^{15}{{C}_{5}}\] done clear
  B)
  \[^{15}{{C}_{6}}\] done clear
  C)
  \[^{15}{{C}_{4}}\] done clear
  D)
  \[^{15}{{C}_{7}}\] done clear
  View Solution play_arrow

• question_answer28)
  If the coefficients of \[{{x}^{7}}\] and \[{{x}^{8}}\]in \[{{\left( 2+\frac{x}{3} \right)}^{n}}\]are equal, then n is [EAMCET 1983; Kurukshetra CEE 1998; DCE 2000; RPET 2001; UPSEAT 2001]
  A)
  56 done clear
  B)
  55 done clear
  C)
  45 done clear
  D)
  15 done clear
  View Solution play_arrow

• question_answer29)
   The coefficient of \[{{x}^{3}}\] in the expansion of \[{{\left( x-\frac{1}{x} \right)}^{7}}\] is    [MP PET 1997; Pb. CET 2001]
  A)
  14 done clear
  B)
  21 done clear
  C)
  28 done clear
  D)
  35 done clear
  View Solution play_arrow

• question_answer30)
  If in the expansion of \[{{(1+x)}^{m}}{{(1-x)}^{n}}\], the coefficient of  x and  \[{{x}^{2}}\]are 3 and - 6 respectively, then m is [IIT 1999; MP PET 2000]
  A)
  6 done clear
  B)
  9 done clear
  C)
  12 done clear
  D)
  24 done clear
  View Solution play_arrow

• question_answer31)
  If \[{{x}^{m}}\]occurs in the expansion of \[{{\left( x+\frac{1}{{{x}^{2}}} \right)}^{2n}},\]then the coefficient of \[{{x}^{m}}\] is [UPSEAT 1999]
  A)
  \[\frac{(2n)!}{(m)!\,(2n-m)!}\] done clear
  B)
  \[\frac{(2n)!\,3!\,3!}{(2n-m)!}\] done clear
  C)
  \[\frac{(2n)!}{\left( \frac{2n-m}{3} \right)\,!\,\left( \frac{4n+m}{3} \right)\,!}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer32)
  If coefficients of 2nd, 3rd and 4th terms in the binomial expansion of \[{{(1+x)}^{n}}\] are in A.P., then \[{{n}^{2}}-9n\] is equal to [RPET 1999; UPSEAT 2002]
  A)
  - 7 done clear
  B)
  7 done clear
  C)
  14 done clear
  D)
  - 14 done clear
  View Solution play_arrow

• question_answer33)
  In the expansion of \[{{(1+x+{{x}^{3}}+{{x}^{4}})}^{10}},\] the coefficient of \[{{x}^{4}}\] is [MP PET 2000]
  A)
  \[^{40}{{C}_{4}}\] done clear
  B)
  \[^{10}{{C}_{4}}\] done clear
  C)
  210 done clear
  D)
  310 done clear
  View Solution play_arrow

• question_answer34)
  If coefficients of \[{{(2r+1)}^{th}}\] term and \[{{(r+2)}^{th}}\] term are equal in the expansion of \[{{(1+x)}^{43}},\] then the value of  r  will be   [UPSEAT 1999]
  A)
  14 done clear
  B)
  15 done clear
  C)
  13 done clear
  D)
  16 done clear
  View Solution play_arrow

• question_answer35)
  If the coefficient of 4th term in the expansion of \[{{(a+b)}^{n}}\] is 56, then  n is [AMU 2000]
  A)
  12 done clear
  B)
  10 done clear
  C)
  8 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer36)
  The coefficient of \[{{x}^{39}}\] in the expansion of \[{{\left( {{x}^{4}}-\frac{1}{{{x}^{3}}} \right)}^{15}}\] is   [MP PET 2001]
  A)
  - 455 done clear
  B)
  - 105 done clear
  C)
  105 done clear
  D)
  455 done clear
  View Solution play_arrow

• question_answer37)
  The coefficient of \[{{x}^{5}}\] in the expansion of \[{{(1+x)}^{21}}+{{(1+x)}^{22}}+..........+{{(1+x)}^{30}}\] is [UPSEAT 2001]
  A)
  \[^{51}{{C}_{5}}\] done clear
  B)
  \[^{9}{{C}_{5}}\] done clear
  C)
  \[^{31}{{C}_{6}}{{-}^{21}}{{C}_{6}}\] done clear
  D)
  \[^{30}{{C}_{5}}{{+}^{20}}{{C}_{5}}\] done clear
  View Solution play_arrow

• question_answer38)
  If the coefficients of second, third and fourth term in the expansion of \[{{(1+x)}^{2n}}\] are in A.P., then \[2{{n}^{2}}-9n+7\] is equal to    [AMU 2001; MP PET 2004]
  A)
  - 1 done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  3/2 done clear
  View Solution play_arrow

• question_answer39)
  The coefficient of \[{{x}^{-9}}\] in the expansion of \[{{\left( \frac{{{x}^{2}}}{2}-\frac{2}{x} \right)}^{9}}\] is   [Kerala (Engg.) 2001]
  A)
  512 done clear
  B)
  - 512 done clear
  C)
  521 done clear
  D)
  251 done clear
  View Solution play_arrow

• question_answer40)
  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)
  \[-\frac{7}{9}\] done clear
  B)
  \[-\frac{9}{7}\] done clear
  C)
  \[\frac{7}{9}\] done clear
  D)
  \[\frac{9}{7}\] done clear
  View Solution play_arrow

• question_answer41)
  If the second, third and fourth term in the expansion of \[{{(x+a)}^{n}}\] are 240, 720 and 1080 respectively, then the value of n is                [Kurukshetra CEE 1991; DCE 1995, 2001]
  A)
  15 done clear
  B)
  20 done clear
  C)
  10 done clear
  D)
  5 done clear
  View Solution play_arrow

• question_answer42)
  In the expansion of \[{{(1+x)}^{n}}\]the coefficient of pth and \[{{(p+1)}^{th}}\] terms are respectively p and q. Then \[p+q=\] [EAMCET 2002]
  A)
  \[n+3\] done clear
  B)
  \[n+1\] done clear
  C)
  \[n+2\] done clear
  D)
  \[n\] done clear
  View Solution play_arrow

• question_answer43)
  Coefficient of \[{{x}^{2}}\] in the expansion of \[{{\left( x-\frac{1}{2x} \right)}^{8}}\] is [UPSEAT 2002]
  A)
  \[\frac{1}{7}\] done clear
  B)
  \[\frac{-1}{7}\] done clear
  C)
  - 7 done clear
  D)
  7 done clear
  View Solution play_arrow

• question_answer44)
  The coefficient of \[{{x}^{5}}\] in the expansion of \[{{(x+3)}^{6}}\] is  [DCE 2002]
  A)
  18 done clear
  B)
  6 done clear
  C)
  12 done clear
  D)
  10 done clear
  View Solution play_arrow

• question_answer45)
  The coefficient of \[{{x}^{32}}\] in the expansion of \[{{\left( {{x}^{4}}-\frac{1}{{{x}^{3}}} \right)}^{15}}\] is [Karnataka CET 2003; Pb. CET 2000]
  A)
  \[^{15}{{C}_{4}}\] done clear
  B)
  \[^{15}{{C}_{3}}\] done clear
  C)
  \[^{15}{{C}_{2}}\] done clear
  D)
  \[^{15}{{C}_{5}}\] done clear
  View Solution play_arrow

• question_answer46)
  If in the expansion of \[{{(1+x)}^{21}}\], the coefficients of \[{{x}^{r}}\] and \[{{x}^{r+1}}\] be equal, then r is equal to [UPSEAT 2004]
  A)
  9 done clear
  B)
  10 done clear
  C)
  11 done clear
  D)
  12 done clear
  View Solution play_arrow

• question_answer47)
  The term independent of x in the expansion of  \[{{\left( \sqrt{\frac{x}{3}}+\frac{3}{2{{x}^{2}}} \right)}^{10}}\] will be  [IIT 1965; BIT Ranchi 1993; KCET 2000; UPSEAT 2001]
  A)
  3/2 done clear
  B)
  5/4 done clear
  C)
  5/2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer48)
  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 done clear
  B)
  6 done clear
  C)
  7 done clear
  D)
  8 done clear
  View Solution play_arrow

• question_answer49)
  In the expansion of \[{{\left( \frac{3{{x}^{2}}}{2}-\frac{1}{3x} \right)}^{9}}\],the term independent of x  is                        [MNR 1981; AMU 1983; JMI EEE 2001]
  A)
  \[^{9}{{C}_{3}}.\frac{1}{{{6}^{3}}}\] done clear
  B)
  \[^{9}{{C}_{3}}{{\left( \frac{3}{2} \right)}^{3}}\] done clear
  C)
  \[^{9}{{C}_{3}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer50)
  The term independent of x in \[{{\left( 2x-\frac{1}{2{{x}^{2}}} \right)}^{12}}\]is  [RPET 1985]
  A)
  - 7930 done clear
  B)
  - 495 done clear
  C)
  495 done clear
  D)
  7920 done clear
  View Solution play_arrow

• question_answer51)
  In the expansion of \[{{\left( x+\frac{2}{{{x}^{2}}} \right)}^{15}}\], the term independent of \[x\] is                                           [MP PET 1993; Pb. CET 2002]
  A)
  \[^{15}{{C}_{6}}{{2}^{6}}\] done clear
  B)
  \[^{15}{{C}_{5}}{{2}^{5}}\] done clear
  C)
  \[^{15}{{C}_{4}}{{2}^{4}}\] done clear
  D)
  \[^{15}{{C}_{8}}{{2}^{8}}\] done clear
  View Solution play_arrow

• question_answer52)
  The term independent of x in the expansion of \[{{\left( {{x}^{2}}-\frac{1}{x} \right)}^{9}}\] is   [EAMCET 1982; MP PET 2003]
  A)
  1 done clear
  B)
  -1 done clear
  C)
  - 48 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer53)
  The term independent of x in the expansion of \[{{\left( 2x+\frac{1}{3x} \right)}^{6}}\]is  [MNR 1995]
  A)
  \[\frac{160}{9}\] done clear
  B)
    \[\frac{80}{9}\] done clear
  C)
  \[\frac{160}{27}\] done clear
  D)
    \[\frac{80}{3}\] done clear
  View Solution play_arrow

• question_answer54)
  The term independent of x in the expansion \[{{\left( {{x}^{2}}-\frac{1}{3x} \right)}^{9}}\]is  [Roorkee 1981; RPET 1990, 95; Pb. CET 2000]
  A)
  \[\frac{28}{81}\] done clear
  B)
  \[\frac{28}{243}\] done clear
  C)
  \[-\frac{28}{243}\] done clear
  D)
  \[-\frac{28}{81}\] done clear
  View Solution play_arrow

• question_answer55)
  The term independent of x in the expansion of \[{{\left( 2x-\frac{3}{x} \right)}^{6}}\] is [Pb. CET 1999]
  A)
  4320 done clear
  B)
  216 done clear
  C)
  - 216 done clear
  D)
  - 4320 done clear
  View Solution play_arrow

• question_answer56)
  In the expansion of \[{{\left( 2{{x}^{2}}-\frac{1}{x} \right)}^{12}}\], the term independent of x  is  [MP PET 2001]
  A)
  10th done clear
  B)
  9th done clear
  C)
  8th done clear
  D)
  7th done clear
  View Solution play_arrow

• question_answer57)
  In the expansion of \[{{\left( x-\frac{3}{{{x}^{2}}} \right)}^{9}},\] the term independent of x is   [Karnataka CET  2001]
  A)
  Non existent done clear
  B)
  \[^{9}{{C}_{2}}\] done clear
  C)
  2268 done clear
  D)
  - 2268 done clear
  View Solution play_arrow

• question_answer58)
  If the middle term in the expansion of \[{{\left( {{x}^{2}}+\frac{1}{x} \right)}^{n}}\]is \[924{{x}^{6}}\], then \[n=\]
  A)
  10 done clear
  B)
  12 done clear
  C)
  14 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer59)
  The middle term in the expansion of \[{{\left( x+\frac{1}{x} \right)}^{10}}\] is  [BIT Ranchi 1991; RPET 2002; Pb. CET 1991]
  A)
  \[^{10}{{C}_{4}}\frac{1}{x}\] done clear
  B)
  \[^{10}{{C}_{5}}\] done clear
  C)
  \[^{10}{{C}_{5}}x\] done clear
  D)
  \[^{10}{{C}_{7}}{{x}^{4}}\] done clear
  View Solution play_arrow

• question_answer60)
  The term independent of x in the expansion of  \[{{\left( {{x}^{2}}-\frac{3\sqrt{3}}{{{x}^{3}}} \right)}^{10}}\] is [RPET 1999]
  A)
  153090 done clear
  B)
  150000 done clear
  C)
  150090 done clear
  D)
  153180 done clear
  View Solution play_arrow

• question_answer61)
  The coefficient of middle term in the expansion of \[{{(1+x)}^{10}}\] is [UPSEAT 2001]
  A)
  \[\frac{10!}{5!\,6!}\] done clear
  B)
  \[\frac{10\,!}{{{(5\,!)}^{2}}}\] done clear
  C)
  \[\frac{10\,!}{5\,!\,7\,!}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer62)
   The middle term in the expansion of \[{{(1+x)}^{2n}}\] is [DCE 2002]
  A)
  \[\frac{(2n)!}{n!}{{x}^{2}}\] done clear
  B)
  \[\frac{(2n)!}{n!(n-1)!}{{x}^{n+1}}\] done clear
  C)
  \[\frac{(2n)!}{{{(n!)}^{2}}}{{x}^{n}}\] done clear
  D)
  \[\frac{(2n)!}{(n+1)!(n-1)!}\,{{x}^{n}}\] done clear
  View Solution play_arrow

• question_answer63)
  The greatest coefficient in the expansion of \[{{(1+x)}^{2n+2}}\]  is [BIT Ranchi 1992]
  A)
  \[\frac{(2n)!}{{{(n!)}^{2}}}\] done clear
  B)
  \[\frac{(2n+2)!}{{{\{(n+1)!\}}^{2}}}\] done clear
  C)
  \[\frac{(2n+2)!}{n!(n+1)!}\] done clear
  D)
  \[\frac{(2n)!}{n!(n+1)!}\] done clear
  View Solution play_arrow

• question_answer64)
  The greatest term in the expansion of \[\sqrt{3}{{\left( 1+\frac{1}{\sqrt{3}} \right)}^{20}}\]is
  A)
  \[\frac{25840}{9}\] done clear
  B)
  \[\frac{24840}{9}\] done clear
  C)
  \[\frac{26840}{9}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer65)
  If n is even positive integer, then the condition that the greatest term in the expansion of \[{{(1+x)}^{n}}\]may have the greatest coefficient also, is
  A)
  \[\frac{n}{n+2}<x<\frac{n+2}{n}\] done clear
  B)
  \[\frac{n+1}{n}<x<\frac{n}{n+1}\] done clear
  C)
  \[\frac{n}{n+4}<x<\frac{n+4}{4}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer66)
  The interval in which  x must lie so that the greatest term in the expansion of \[{{(1+x)}^{2n}}\]has the greatest coefficient, is
  A)
  \[\left( \frac{n-1}{n},\frac{n}{n-1} \right)\] done clear
  B)
  \[\left( \frac{n}{n+1},\frac{n+1}{n} \right)\] done clear
  C)
  \[\left( \frac{n}{n+2},\frac{n+2}{n} \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer67)
  The greatest coefficient in the expansion of \[{{(1+x)}^{2n+1}}\] is [RPET 1997]
  A)
  \[\frac{(2n+1)\,!}{n!(n+1)!}\] done clear
  B)
    \[\frac{(2n+2)!}{n!(n+1)!}\] done clear
  C)
    \[\frac{(2n+1)!}{{{[(n+1)!]}^{2}}}\] done clear
  D)
    \[\frac{(2n)!}{{{(n!)}^{2}}}\] done clear
  View Solution play_arrow

• question_answer68)
  The coefficient of \[{{x}^{4}}\] in the expansion of \[{{(1+x+{{x}^{2}}+{{x}^{3}})}^{n}}\] is                                [MNR 1993; RPET 2001; DCE 1998]
  A)
  \[^{n}{{C}_{4}}\] done clear
  B)
  \[^{n}{{C}_{4}}{{+}^{n}}{{C}_{2}}\] done clear
  C)
  \[^{n}{{C}_{4}}+{{\,}^{n}}{{C}_{2}}+\,{{\,}^{n}}{{C}_{4}}{{.}^{n}}{{C}_{2}}\] done clear
  D)
  \[^{n}{{C}_{4}}+{{\,}^{n}}{{C}_{2}}+{{\,}^{n}}{{C}_{1}}.{{\,}^{n}}{{C}_{2}}\] done clear
  View Solution play_arrow

• question_answer69)
  The coefficient of \[\frac{1}{x}\] in the expansion of \[{{(1+x)}^{n}}{{\left( 1+\frac{1}{x} \right)}^{n}}\]is
  A)
  \[\frac{n!}{(n-1)!(n+1)!}\] done clear
  B)
  \[\frac{(2n)\,!}{(n-1)!(n+1)!}\] done clear
  C)
  \[\frac{n!}{(n-1)!(n+1)!}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer70)
  The term independent of x in the expansion of \[{{(1+x)}^{n}}{{\left( 1+\frac{1}{x} \right)}^{n}}\] is [EAMCET 1989]
  A)
  \[C_{0}^{2}+2C_{1}^{2}+....+(n+1)C_{n}^{2}\] done clear
  B)
  \[{{({{C}_{0}}+{{C}_{1}}+....+{{C}_{n}})}^{2}}\] done clear
  C)
  \[C_{0}^{2}+C_{1}^{2}+.....+C_{n}^{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer71)
  The coefficient of \[{{t}^{24}}\] in the expansion of \[{{(1+{{t}^{2}})}^{12}}(1+{{t}^{12}})\,(1+{{t}^{24}})\] is [IIT Screening 2003]
  A)
  \[^{12}{{C}_{6}}+2\] done clear
  B)
  \[^{12}{{C}_{5}}\] done clear
  C)
  \[^{12}{{C}_{6}}\] done clear
  D)
  \[^{12}{{C}_{7}}\] done clear
  View Solution play_arrow

• question_answer72)
  The coefficient of \[{{x}^{5}}\] in the expansion of \[{{({{x}^{2}}-x-2)}^{5}}\] is  [EAMCET 2003]
  A)
  - 83 done clear
  B)
  - 82 done clear
  C)
  - 81 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer73)
  The coefficient of \[{{x}^{n}}\]in expansion of \[(1+x)\,{{(1-x)}^{n}}\] is  [AIEEE 2004]
  A)
  \[{{(-1)}^{n-1}}n\] done clear
  B)
  \[{{(-1)}^{n}}(1-n)\] done clear
  C)
  \[{{(-1)}^{n-1}}{{(n-1)}^{2}}\] done clear
  D)
  \[(n-1)\] done clear
  View Solution play_arrow

• question_answer74)
  The middle term in the expansion of \[{{\left( x+\frac{1}{2x} \right)}^{2n}}\], is  [MP PET 1995]
  A)
  \[\frac{1.3.5....(2n-3)}{n!}\] done clear
  B)
  \[\frac{1.3.5....(2n-1)}{n!}\] done clear
  C)
  \[\frac{1.3.5....(2n+1)}{n!}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer75)
  In the expansion of \[{{(1+3x+2{{x}^{2}})}^{6}}\]the coefficient of \[{{x}^{11}}\] is  [Kerala (Engg.) 2005]
  A)
    144 done clear
  B)
  288 done clear
  C)
  216 done clear
  D)
  576 done clear
  E)
  (3)(211) done clear
  View Solution play_arrow

• question_answer76)
  The middle term in the expression of \[{{\left( x-\frac{1}{x} \right)}^{18}}\] is   [Karnataka CET 2005]
  A)
  \[^{18}{{C}_{9}}\] done clear
  B)
  \[{{-}^{18}}{{C}_{9}}\] done clear
  C)
  \[^{18}{{C}_{0}}\] done clear
  D)
  \[{{-}^{18}}{{C}_{10}}\] done clear
  View Solution play_arrow