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

done Self Evaluation Test - Binomial Theorem Total Questions - 70

• question_answer1)
  The sum of the rational terms in the expansion of \[{{(\sqrt{2}+{{3}^{1/5}})}^{10}}\] is equal to
  A)
  40 done clear
  B)
  41 done clear
  C)
  42 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer2)
  The positive integer just greater than \[{{(1+0.0001)}^{10000}}\]
  A)
  4 done clear
  B)
  5 done clear
  C)
  2 done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer3)
  The coefficient of \[{{x}^{53}}\] in the 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_answer4)
  The value of \[^{20}{{C}_{0}}+{{\,}^{20}}{{C}_{1}}+{{\,}^{20}}{{C}_{2}}+{{\,}^{20}}{{C}_{3}}+{{\,}^{20}}{{C}_{4}}\]\[+{{\,}^{20}}{{C}_{12}}+{{\,}^{20}}{{C}_{13}}+{{\,}^{20}}{{C}_{14}}+{{\,}^{20}}{{C}_{15}}\] is
  A)
  \[{{2}^{19}}-\frac{\left( ^{20}{{C}_{10}}+{{\,}^{20}}{{C}_{9}} \right)}{2}\] done clear
  B)
  \[{{2}^{19}}-\frac{\left( ^{20}{{C}_{10}}+\,2{{\times }^{20}}{{C}_{9}} \right)}{2}\] done clear
  C)
  \[{{2}^{19}}-\frac{^{20}{{C}_{10}}}{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer5)
  The sum \[1+\frac{1+a}{2!}+\frac{1+a+{{a}^{2}}}{3!}+.....\infty \] is equal to
  A)
  \[{{e}^{a}}\] done clear
  B)
  \[\frac{{{e}^{a}}-e}{a-1}\] done clear
  C)
  \[(a-1){{e}^{a}}\] done clear
  D)
  \[(a+1){{e}^{a}}\] done clear
  View Solution play_arrow

• question_answer6)
  If \[\sum\limits_{r=0}^{n}{\frac{r+2}{r+1}{{\,}^{n}}{{C}_{r}}=\frac{{{2}^{8}}-1}{6}}\], then n is
  A)
  8 done clear
  B)
  4 done clear
  C)
  6 done clear
  D)
  5 done clear
  View Solution play_arrow

• question_answer7)
  If \[{{7}^{9}}+{{9}^{7}}\] is divided by 64 then the remainder is
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  63 done clear
  View Solution play_arrow

• question_answer8)
  The greatest integer less than or equal to; \[{{(\sqrt{2}+1)}^{6}}\] is
  A)
  196 done clear
  B)
  197 done clear
  C)
  198 done clear
  D)
  199 done clear
  View Solution play_arrow

• question_answer9)
  The fractional part of \[\frac{{{2}^{4n}}}{15}\] is
  A)
  \[\frac{1}{15}\] done clear
  B)
  \[\frac{2}{15}\] done clear
  C)
  \[\frac{4}{15}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer10)
  The coefficient of \[{{a}^{3}}{{b}^{4}}c\] in the expansion of \[{{(1+a-b+c)}^{9}}\] is equal to
  A)
  \[\frac{9!}{3!6!}\] done clear
  B)
  \[\frac{9!}{4!5!}\] done clear
  C)
  \[\frac{9!}{3!5!}\] done clear
  D)
  \[\frac{9!}{3!4!}\] done clear
  View Solution play_arrow

• question_answer11)
  The sum of the series \[\frac{2}{1}.\frac{1}{3}+\frac{3}{2}.\frac{1}{9}+\frac{4}{3}.\frac{1}{27}+\frac{5}{4}.\frac{1}{81}+......\infty \] is equal to
  A)
  \[{{\log }_{e}}3-{{\log }_{e}}2\] done clear
  B)
  \[\frac{1}{2}+{{\log }_{e}}3-{{\log }_{e}}2\] done clear
  C)
  \[\frac{1}{2}+{{\log }_{e}}3+{{\log }_{e}}2\] done clear
  D)
  \[{{\log }_{e}}3+{{\log }_{e}}2\] done clear
  View Solution play_arrow

• question_answer12)
  \[\sqrt{5}[{{(\sqrt{5}+1)}^{50}}-{{(\sqrt{5}-1)}^{50}}]\] is
  A)
  An irrational number done clear
  B)
  0 done clear
  C)
  A natural number done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  The value of \[\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}\sin (rx)}\] is equal to
  A)
  \[{{2}^{n}}\cdot {{\cos }^{n}}\frac{x}{2}\cdot \sin \frac{nx}{2}\] done clear
  B)
  \[{{2}^{n}}\cdot si{{n}^{n}}\frac{x}{2}\cdot \cos \frac{nx}{2}\] done clear
  C)
  \[{{2}^{n+1}}\cdot {{\cos }^{n}}\frac{x}{2}\cdot \sin \frac{nx}{2}\] done clear
  D)
  \[{{2}^{n+1}}\cdot si{{n}^{n}}\frac{x}{2}\cdot \cos \frac{nx}{2}\] done clear
  View Solution play_arrow

• question_answer14)
  For natural numbers \[m,n\,\,if{{(1-y)}^{m}}{{(1+y)}^{n}}\] \[=1+{{a}_{1}}y+{{a}_{2}}{{y}^{2}}+...\] and \[{{a}_{1}}={{a}_{2}}=10\], then \[(m,n)\]  is
  A)
  (20, 45) done clear
  B)
  (35, 20) done clear
  C)
  (45, 35) done clear
  D)
  (35, 45) done clear
  View Solution play_arrow

• question_answer15)
  What are the values of k if the term independent of x in the expansion of \[{{\left( \sqrt{x}+\frac{k}{{{x}^{2}}} \right)}^{10}}\] is 405?
  A)
  \[\pm 3\] done clear
  B)
  \[\pm 6\] done clear
  C)
  \[\pm 5\] done clear
  D)
  \[\pm 4\] done clear
  View Solution play_arrow

• question_answer16)
  The value of \[{{(}^{10}}{{C}_{0}})+{{(}^{10}}{{C}_{0}}{{+}^{10}}{{C}_{1}})+{{(}^{10}}{{C}_{0}}{{+}^{10}}{{C}_{1}})\] \[{{+}^{10}}{{C}_{2}})+.....+{{(}^{10}}{{C}_{0}}{{+}^{10}}{{C}_{1}}{{\,}^{10}}{{C}_{2}}+....{{+}^{10}}{{C}_{9}})\] is
  A)
  \[{{2}^{10}}\] done clear
  B)
  \[{{10.2}^{9}}\] done clear
  C)
  \[{{10.2}^{10}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer17)
  The greatest value of the term independent of x in the expansion \[{{(x\,\,\sin \,\,p+{{x}^{-1}}\cos \,\,p)}^{10}},p\in R\] is
  A)
  \[{{2}^{5}}\] done clear
  B)
  \[\frac{10!}{{{2}^{5}}{{(5!)}^{2}}}\] done clear
  C)
  \[\frac{10!}{{{(5!)}^{2}}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer18)
  If x is very small in magnitude compared with a, then \[{{\left( \frac{a}{a+x} \right)}^{\frac{1}{2}}}+{{\left( \frac{a}{a-x} \right)}^{\frac{1}{2}}}\]  can be approximately equal to
  A)
  \[1+\frac{1}{2}\frac{x}{a}\] done clear
  B)
  \[\frac{x}{a}\] done clear
  C)
  \[1+\frac{3}{4}\frac{{{x}^{2}}}{{{a}^{2}}}\] done clear
  D)
  \[2+\frac{3}{4}\frac{{{x}^{2}}}{{{a}^{2}}}\] done clear
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• question_answer19)
  If the 7th term in the binomial expansion of \[{{\left( \frac{3}{\sqrt[3]{84}}+\sqrt{3}\,ln\,\,x \right)}^{9}},x>0\], is equal to 729, then x can be
  A)
  \[{{e}^{2}}\] done clear
  B)
  e done clear
  C)
  \[\frac{e}{2}\] done clear
  D)
  2e done clear
  View Solution play_arrow

• question_answer20)
  If \[x={{\left( 2+\sqrt{3} \right)}^{n}}\], then find the value of \[x\,\left( 1-\left\{ x \right\} \right)\] where {x} denotes the fractional part of x                     
  A)
  1 done clear
  B)
  2 done clear
  C)
  \[{{2}^{2n}}\] done clear
  D)
  \[{{2}^{n}}\] done clear
  View Solution play_arrow

• question_answer21)
  In the binomial expansion \[{{(a+bx)}^{-3}}\] \[=\frac{1}{8}+\frac{9}{8}x+....\], then the value of a and b are:
  A)
  a = 2, b = 3 done clear
  B)
  a = 2, b = -6 done clear
  C)
  a = 3, b = 2 done clear
  D)
  a = -3, b = 2 done clear
  View Solution play_arrow

• question_answer22)
  If the sum of odd numbered terms and the sum of even numbered terms in the expansion of \[{{(x+a)}^{n}}\]are A and B respectively, then the value of \[{{({{x}^{2}}-{{a}^{2}})}^{n}}\] is
  A)
  \[{{A}^{2}}-{{B}^{2}}\] done clear
  B)
  \[{{A}^{2}}+{{B}^{2}}\] done clear
  C)
  4AB done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer23)
  The number of term in the expansion of \[{{[{{(x+4y)}^{3}}{{(x-4y)}^{3}}]}^{2}}\] is
  A)
  6 done clear
  B)
  7 done clear
  C)
  8 done clear
  D)
  32 done clear
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• question_answer24)
  The coefficient of \[{{x}^{n}}\] in the polynomial\[(x+{{\,}^{n}}{{C}_{0}})(x+3.{{\,}^{n}}{{C}_{1}})(x+5.{{\,}^{n}}{{C}_{2}})...(x+{{(2n+1)}^{n}}{{C}_{n}})\] is
  A)
  \[n{{.2}^{n}}\] done clear
  B)
  \[~n{{.2}^{n+1}}\] done clear
  C)
  \[(n+1){{.2}^{n}}\] done clear
  D)
  \[n{{.2}^{n}}+1\] done clear
  View Solution play_arrow

• question_answer25)
  \[1+\frac{1}{3}+\frac{1}{3}.\frac{3}{6}+\frac{1}{3}.\frac{3}{6}.\frac{5}{9}+.....\infty =\]
  A)
  \[\sqrt{\frac{2}{3}}\] done clear
  B)
  \[\sqrt{2}\] done clear
  C)
  \[\sqrt{3}\] done clear
  D)
  \[\sqrt{\frac{3}{2}}\] done clear
  View Solution play_arrow

• question_answer26)
  Sum of coefficients in the exansion of \[{{(x+2y+3z)}^{10}}\] is
  A)
  \[{{2}^{10}}\] done clear
  B)
  \[{{3}^{10}}\] done clear
  C)
  1 done clear
  D)
  \[{{6}^{10}}\] done clear
  View Solution play_arrow

• question_answer27)
  The term independent of x in the expansion of \[{{[({{t}^{-1}}-1)x+{{({{t}^{-1}}+1)}^{-1}}{{x}^{-1}}]}^{8}}\] is
  A)
  \[56{{\left( \frac{1-t}{1+t} \right)}^{3}}\] done clear
  B)
  \[56{{\left( \frac{1+t}{1-t} \right)}^{3}}\] done clear
  C)
  \[70{{\left( \frac{1-t}{1+t} \right)}^{4}}\] done clear
  D)
  \[70{{\left( \frac{1+t}{1-t} \right)}^{4}}\] done clear
  View Solution play_arrow

• question_answer28)
  The expression\[\frac{1}{\sqrt{3x+1}}\left[ {{\left( \frac{1+\sqrt{3x+1}}{2} \right)}^{7}}-{{\left( \frac{1-\sqrt{3x+1}}{2} \right)}^{7}} \right]\]is a polynomial in x of degree equal to
  A)
  3 done clear
  B)
  4 done clear
  C)
  2 done clear
  D)
  5 done clear
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• question_answer29)
  The value of \[\frac{{{C}_{1}}}{2}+\frac{{{C}_{3}}}{4}+\frac{{{C}_{5}}}{6}+.....\] is equal to
  A)
  \[\frac{{{2}^{n}}+1}{n+1}\] done clear
  B)
  \[\frac{{{2}^{n}}}{n+1}\] done clear
  C)
  \[\frac{{{2}^{n}}+1}{n-1}\] done clear
  D)
  \[\frac{{{2}^{n}}-1}{n+1}\] done clear
  View Solution play_arrow

• question_answer30)
  If 'n' is positive integer and three consecutive coefficient in the expansion of \[{{(1+x)}^{n}}\] are in the ratio 6 : 33 : 110, then n is equal to:
  A)
  9 done clear
  B)
  6 done clear
  C)
  12 done clear
  D)
  16 done clear
  View Solution play_arrow

• question_answer31)
  If \[\pi (n)\] denotes product of all binomial coefficients in \[{{(1+x)}^{n}}\] then ratio of \[\pi (2002)\] to \[\pi (2001)\] is
  A)
  2002 done clear
  B)
  \[\frac{{{(2002)}^{2001}}}{(2001)!}\] done clear
  C)
  \[\frac{{{(2001)}^{2002}}}{(2002)!}\] done clear
  D)
  2001 done clear
  View Solution play_arrow

• question_answer32)
  If \[x+y=1\], then \[\sum\limits_{r=0}^{n}{{{r}^{n}}{{C}_{r}}{{x}^{r}}{{y}^{n-r}}}\] equals
  A)
  1 done clear
  B)
  n done clear
  C)
  nx done clear
  D)
  ny done clear
  View Solution play_arrow

• question_answer33)
  If \[y=3x+6{{x}^{2}}+10{{x}^{3}}+........\infty \], then\[\frac{1}{3}y-\frac{1.4}{{{3}^{2}}2}{{y}^{2}}+\frac{1.4.7}{{{3}^{2}}3}{{y}^{3}}-.....\,\infty \] is equal to
  A)
  x done clear
  B)
  \[1-x\] done clear
  C)
  \[1 + x\] done clear
  D)
  \[{{x}^{x}}\] done clear
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• question_answer34)
  The coefficient of \[{{x}^{100}}\] in the expansion of \[\sum\limits_{j=0}^{200}{{{(1+x)}^{j}}}\] is:
  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
  View Solution play_arrow

• question_answer35)
  If the second term in the expansion \[{{\left( \sqrt[13]{a}+\frac{a}{\sqrt{{{a}^{-1}}}} \right)}^{n}}\] is \[14{{a}^{5/2}}\], then \[\frac{^{n}{{C}_{3}}}{^{n}{{C}_{2}}}=\]
  A)
  4 done clear
  B)
  3 done clear
  C)
  12 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer36)
  The remainder when \[{{27}^{40}}\] is divided by 12 is
  A)
  3 done clear
  B)
  7 done clear
  C)
  9 done clear
  D)
  11 done clear
  View Solution play_arrow

• question_answer37)
  The sum of the series\[^{20}{{C}_{0}}-{{\,}^{20}}{{C}_{1}}+{{\,}^{20}}{{C}_{2}}-{{\,}^{20}}{{C}_{3}}+....\] \[-....+{{\,}^{20}}{{C}_{10}}\] is
  A)
  0 done clear
  B)
  \[^{20}{{C}_{10}}\] done clear
  C)
  \[{{-}^{20}}{{C}_{10}}\] done clear
  D)
  \[\frac{1}{2}{{\,}^{20}}{{C}_{10}}\] done clear
  View Solution play_arrow

• question_answer38)
  If the middle term in the expansion of \[{{\left( \frac{1}{x}+x\,\sin \,x \right)}^{10}}\] equals to \[7\frac{7}{8}\] then x is equal to; \[(n\in I)\]
  A)
  \[2n\pi \pm \frac{\pi }{6}\] done clear
  B)
  \[n\pi +\frac{\pi }{6}\] done clear
  C)
  \[n\pi +{{(-1)}^{n}}\frac{\pi }{6}\] done clear
  D)
  \[n\pi +{{(-1)}^{n}}\frac{5\pi }{6}\] done clear
  View Solution play_arrow

• question_answer39)
  If the ratio of the 7th term from the beginning to the 7th term from the end in \[{{\left( \sqrt[3]{2}+\frac{1}{\sqrt[3]{3}} \right)}^{n}}\] is \[\frac{1}{6}\] them n equals to
  A)
  10 done clear
  B)
  9 done clear
  C)
  8 done clear
  D)
  12 done clear
  View Solution play_arrow

• question_answer40)
  \[\frac{1}{2}{{x}^{2}}+\frac{2}{3}{{x}^{3}}+\frac{3}{4}{{x}^{4}}+\frac{4}{5}{{x}^{5}}+\]................. is
  A)
  \[\frac{x}{1+x}+\log (1+x)\] done clear
  B)
  \[\frac{x}{1-x}+\log (1+x)\] done clear
  C)
  \[-\frac{x}{1-x}+\log (1+x)\] done clear
  D)
  \[\frac{x}{1-x}+\log (1-x)\] done clear
  View Solution play_arrow

• question_answer41)
  If \[{{a}_{n}}=2n+1\] and \[{{C}_{r}}={{\,}^{n}}{{C}_{r}}\] then\[{{a}_{0}}C_{0}^{2}+{{a}_{1}}C_{1}^{2}+{{a}_{2}}C_{2}^{2}+........{{a}_{n}}C_{n}^{2}=\]
  A)
  \[(n-1){{(}^{2n}}{{C}_{n}})\] done clear
  B)
  \[n{{(}^{2n}}{{C}_{n}})\] done clear
  C)
  \[(n+1){{(}^{2n}}{{C}_{n}})\] done clear
  D)
  \[(n+1){{(}^{n}}{{C}_{n/2}})\]   done clear
  View Solution play_arrow

• question_answer42)
  The ninth term in the expansion of\[{{\left\{ {{3}^{{{\log }_{3}}\sqrt{{{25}^{x-1}}+7}}}+{{3}^{-1/8\,\,{{\log }_{3}}\left( {{5}^{x-1}}+1 \right)}} \right\}}^{10}}\]is equal to 180, then x is
  A)
  A prime number done clear
  B)
  An irrational number done clear
  C)
  Has non-zero fractional part done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer43)
  The minimum positive integral value of m such that \[{{(1073)}^{71}}-m\] may be divisible by 10, is
  A)
  1 done clear
  B)
  3 done clear
  C)
  7 done clear
  D)
  9 done clear
  View Solution play_arrow

• question_answer44)
  The approximate value of \[{{(1.0002)}^{3000}}\] is
  A)
  1.6 done clear
  B)
  1.4 done clear
  C)
  1.8 done clear
  D)
  1.2 done clear
  View Solution play_arrow

• question_answer45)
  The coefficient of \[{{x}^{83}}\] in \[{{(1+x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}})}^{n}}\]\[{{(1-x)}^{n+3}},is-{{\,}^{n}}{{C}_{2\lambda }}\], then find the value of \[\lambda \]
  A)
  12 done clear
  B)
  10 done clear
  C)
  9 done clear
  D)
  8 done clear
  View Solution play_arrow

• question_answer46)
  If \[{{(1+x)}^{15}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+...+{{C}_{15}}{{x}^{15}}\] then\[{{C}_{2}}+2{{C}_{3}}+3{{C}_{4}}+.....+14{{C}_{15}}\] is equal to
  A)
  \[{{14.2}^{14}}\] done clear
  B)
  \[{{13.2}^{14}}+1\] done clear
  C)
  \[{{13.2}^{14}}-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer47)
  The interval in which x must lies so that the numerically greatest term in the expansion of \[{{(1-x)}^{21}}\] has the greatest coefficient is, (x > 0).
  A)
  \[\left[ \frac{5}{6},\frac{6}{5} \right]\] done clear
  B)
  \[\left( \frac{5}{6},\frac{6}{5} \right)\] done clear
  C)
  \[\left( \frac{4}{5},\frac{5}{4} \right)\] done clear
  D)
  \[\left[ \frac{4}{5},\frac{5}{4} \right]\] done clear
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• question_answer48)
  The co-efficient of \[{{x}^{n}}\] in the expansion of\[\frac{{{e}^{7x}}+{{e}^{x}}}{{{e}^{3x}}}\] is
  A)
  \[\frac{{{4}^{n-1}}+{{(-2)}^{n}}}{n!}\] done clear
  B)
  \[\frac{{{4}^{n-1}}+{{2}^{n}}}{n!}\] done clear
  C)
  \[\frac{{{4}^{n}}+{{(-2)}^{n}}}{n!}\] done clear
  D)
  \[\frac{{{4}^{n-1}}+{{(-2)}^{n-1}}}{n!}\] done clear
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• question_answer49)
  If \[{{C}_{0}},{{C}_{1}},\,{{C}_{2}}{{,}^{.}}.......,\,\,{{C}_{15}}\] are binomial coefficients in \[{{(1+x)}^{15}}\], then\[\frac{{{C}_{1}}}{{{C}_{0}}}+2\frac{{{C}_{2}}}{{{C}_{1}}}+3\frac{{{C}_{3}}}{{{C}_{2}}}+....+15\frac{{{C}_{15}}}{{{C}_{14}}}=\]
  A)
  60 done clear
  B)
  120 done clear
  C)
  64 done clear
  D)
  124 done clear
  View Solution play_arrow

• question_answer50)
  The number of integral terms in the expansion of \[{{(\sqrt{3}+\sqrt[8]{5})}^{256}}\] is
  A)
  35 done clear
  B)
  32 done clear
  C)
  33 done clear
  D)
  34 done clear
  View Solution play_arrow

• question_answer51)
  If number of terms in the expansion of\[{{(x-2y+3z)}^{n}}\] is 45, then n=
  A)
  7 done clear
  B)
  8 done clear
  C)
  9 done clear
  D)
  \[{{6}^{10}}\] done clear
  View Solution play_arrow

• question_answer52)
  The value of\[\left( \begin{matrix}    30  \\    0  \\ \end{matrix} \right)\left( \begin{matrix}    30  \\    10  \\ \end{matrix} \right)-\left( \begin{matrix}    30  \\    1  \\ \end{matrix} \right)\left( \begin{matrix}    30  \\    11  \\ \end{matrix} \right)+\left( \begin{matrix}    30  \\    2  \\ \end{matrix} \right)\left( \begin{matrix}    30  \\    12  \\ \end{matrix} \right)...\]\[+\left( \begin{matrix}    30  \\    20  \\ \end{matrix} \right)\left( \begin{matrix}    30  \\    30  \\ \end{matrix} \right)\] is where \[\left( \begin{matrix}    n  \\    r  \\ \end{matrix} \right)={{\,}^{n}}{{C}_{r}}\]
  A)
  \[\left( \begin{matrix}    30  \\    10  \\ \end{matrix} \right)\] done clear
  B)
  \[\left( \begin{matrix}    30  \\    15  \\ \end{matrix} \right)\] done clear
  C)
  \[\left( \begin{matrix}    60  \\    30  \\ \end{matrix} \right)\] done clear
  D)
  \[\left( \begin{matrix}    31  \\    10  \\ \end{matrix} \right)\] done clear
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• question_answer53)
  If \[x\ne 0\], then the sum of the series\[1+\frac{x}{2!}+\frac{2{{x}^{2}}}{3!}+\frac{3{{x}^{3}}}{4!}+.......\infty \] is
  A)
  \[\frac{{{e}^{x}}+1}{x}\] done clear
  B)
  \[\frac{{{e}^{x}}\,(x-1)}{x}\] done clear
  C)
  \[\frac{{{e}^{x}}\,(x-1)+1}{x}\] done clear
  D)
  \[\frac{{{e}^{x}}\,(x-1)+1+x}{x}\] done clear
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• question_answer54)
  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
  A)
  \[a=3b\] done clear
  B)
  \[a={{b}^{3}}\] done clear
  C)
  \[b={{a}^{3}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer55)
  If \[\frac{{{e}^{x}}}{1-x}={{B}_{0}}+{{B}_{1}}x+{{B}_{2}}{{x}^{2}}+...+{{B}_{n}}{{x}^{n}}\] then \[{{B}_{n}}-{{B}_{n-1}}\] is
  A)
  \[\frac{1}{n!}-\frac{1}{(n-1)!}\] done clear
  B)
  \[\frac{1}{n!}\] done clear
  C)
  \[\frac{1}{(n-1)!}\] done clear
  D)
  \[\frac{1}{n!}+\frac{1}{(n-1)!}\] done clear
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• question_answer56)
  Which of the following is the greatest?
  A)
  \[^{31}{{C}_{0}}^{2}-{{\,}^{31}}{{C}_{1}}^{2}+{{\,}^{31}}{{C}_{2}}^{2}-...-{{\,}^{31}}{{C}_{31}}^{2}\] done clear
  B)
  \[^{32}{{C}_{0}}^{2}-{{\,}^{32}}{{C}_{1}}^{2}+{{\,}^{32}}{{C}_{1}}^{2}-...+{{\,}^{32}}{{C}_{32}}^{2}\] done clear
  C)
  \[^{32}{{C}_{0}}^{2}+{{\,}^{32}}{{C}_{1}}^{2}+{{\,}^{32}}{{C}_{2}}^{2}-..+{{\,}^{32}}{{C}_{32}}^{2}\] done clear
  D)
  \[^{34}{{C}_{0}}^{2}-{{\,}^{34}}{{C}_{1}}^{2}+{{\,}^{34}}{{C}_{2}}^{2}-...+{{\,}^{34}}{{C}_{32}}^{2}\] done clear
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• question_answer57)
  If the fourth term in the expansion of \[{{\left( \sqrt{{{x}^{\left( \frac{1}{\log \,x+1} \right)}}}+{{x}^{1/12}} \right)}^{6}}\] is equal to 200 and \[\operatorname{x} > 1\], then x is equal to \[(log=lo{{g}_{10}})\]
  A)
  \[{{10}^{\sqrt{2}}}\] done clear
  B)
  10 done clear
  C)
  \[{{10}^{4}}\] done clear
  D)
  None of these done clear
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• question_answer58)
  Find the 7th term from the end in the expansion of \[{{\left( x-\frac{2}{{{x}^{2}}} \right)}^{10}}\].
  A)
  \[^{10}{{C}_{4}}\] done clear
  B)
  \[^{10}{{C}_{4}}{{.2}^{4}}x\] done clear
  C)
  \[{{2}^{4}}.{{x}^{2}}\] done clear
  D)
  \[^{10}{{C}_{4}}{{.2}^{4}}\left( \frac{1}{{{x}^{2}}} \right)\] done clear
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• question_answer59)
  The coefficient of \[{{x}^{m}}\] in \[{{(1+x)}^{m}}+{{(1+x)}^{m+1}}+......+{{(1+x)}^{n}},m\le n\] is
  A)
  \[^{n+1}{{C}_{m+1}}\] done clear
  B)
  \[^{n-1}{{C}_{m-1}}\] done clear
  C)
  \[^{n}{{C}_{m}}\]      done clear
  D)
  \[^{n}{{C}_{m+1}}\] done clear
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• question_answer60)
  The coefficient of \[{{x}^{-7}}\] in the expansion of \[{{\left[ ax-\frac{1}{b{{x}^{2}}} \right]}^{11}}\] will be:
  A)
  \[\frac{462}{{{b}^{5}}}{{a}^{6}}\] 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
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• question_answer61)
  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_answer62)
  The value of \[{{(}^{7}}{{C}_{0}}+{{\,}^{7}}{{C}_{1}})+{{(}^{7}}{{C}_{1}}+{{\,}^{7}}{{C}_{2}})+...+\]\[{{(}^{7}}{{C}_{6}}+{{\,}^{7}}{{C}_{7}})\] is
  A)
  \[{{2}^{8}}-2\] done clear
  B)
  \[{{2}^{8}}-1\] done clear
  C)
  \[{{2}^{8}}+1\] done clear
  D)
  \[{{2}^{8}}\] done clear
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• question_answer63)
  \[\frac{1}{1.2}+\frac{1.3}{1.2.3.4}+\frac{1.3.5}{1.2.3.4.5.6}+....\infty =\]
  A)
  \[\sqrt{e}\]         done clear
  B)
  \[\sqrt{e}+1\] done clear
  C)
  \[\sqrt{e}-1\] done clear
  D)
  \[e-1\] done clear
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• question_answer64)
  \[\frac{{{C}_{0}}}{1}+\frac{{{C}_{2}}}{3}+\frac{{{C}_{4}}}{5}+\frac{{{C}_{6}}}{7}+....=\]
  A)
  \[\frac{{{2}^{n+1}}}{n+1}\] done clear
  B)
  \[\frac{{{2}^{n+1}}-1}{n+1}\] done clear
  C)
  \[\frac{{{2}^{n}}}{n+1}\] done clear
  D)
  None of these done clear
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• question_answer65)
  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
  A)
  \[1-\frac{3}{8}{{x}^{2}}\] done clear
  B)
  \[3x+\frac{3}{8}{{x}^{2}}\] done clear
  C)
  \[-\frac{3}{8}{{x}^{2}}\] done clear
  D)
  \[\frac{x}{2}-\frac{3}{8}{{x}^{2}}\] done clear
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• question_answer66)
  What is the coefficient of \[{{x}^{3}}\] in \[\frac{(3-2x)}{{{(1+3x)}^{3}}}?\]
  A)
  - 272 done clear
  B)
  - 540 done clear
  C)
  - 870 done clear
  D)
  -918 done clear
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• question_answer67)
  If the third term in the expansion of \[{{[x+{{x}^{{{\log }_{\,10}}\,x}}]}^{5}}\] is \[{{10}^{6}}\], then x may be
  A)
  1 done clear
  B)
  \[\sqrt{10}\] done clear
  C)
  10 done clear
  D)
  \[{{10}^{-2/5}}\] done clear
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• question_answer68)
  The coefficient of \[{{x}^{n}}\] in the expansion of \[{{e}^{{{e}^{x}}}}\] is
  A)
  \[\frac{{{e}^{x}}}{n!}\] done clear
  B)
  \[\frac{{{n}^{n}}}{n!}\] done clear
  C)
  \[\frac{1}{n!}\] done clear
  D)
  None of these done clear
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• question_answer69)
  If \[{{P}_{n}}\] denotes the product of the binomial coefficients in the expansion of \[{{(1+x)}^{n}}\], then \[\frac{{{P}_{n+1}}}{{{P}_{n}}}\]  equals
  A)
  \[\frac{n+1}{n!}\] done clear
  B)
  \[\frac{{{n}^{n}}}{n!}\] done clear
  C)
  \[\frac{{{(n+1)}^{n}}}{(n+1)!}\] done clear
  D)
  \[\frac{{{(n+1)}^{n+1}}}{(n+1)!}\] done clear
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• question_answer70)
   If \[A=\sum\limits_{n=1}^{\infty }{\frac{2n}{(2n-1)!}B=\sum\limits_{n=1}^{\infty }{\frac{2n}{(2n+1)!}}}\] then AB is equal to
  A)
  \[{{e}^{2}}\] done clear
  B)
  e    done clear
  C)
  \[e+{{e}^{2}}\] done clear
  D)
   1 done clear
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