Processing math: 7%

Binomial Theorem - Class 11 Engineering Maths - Extra Questions

Find the middle terms in the expansion of (5x7y)7.



Find the 28th term of (5x+8y)30.



Show that the expansion of (x2+1x)12 does not contain any term involving x1.



Find the middle term in the expansion (xa+yb)6.



Write down and simplify:
The 4th term of (x5)13



Find the 13th term in the expansion of (9x13x)18.



The sum of the coefficients in the first three terms of the expansion of (x22x)m is equal to 97. Find the term of the expansion containing x4.



Write general term of this:-
2x(3+2x2)20



Find the middle term of (ax+xa)10.



Show that nC0 nC1+nC2 .....=0



Write down and simplify:
The 7th term of (4x552x)9



Find the sum nr=1rnCrnCr1.



Find the 7th term of (3x213)10.



If 4th term in the expansion of (ax+1x)n is 52, then find the value of a and n.



Find the coefficient of x5 in the expansion of the product (1+2x)6(1x)7.



15C3+15C5+.......+15C15



Show that the middle term in the expansion of (1+x)n is 6x2 if   n=4 



The sum of the rational terms in the expansion of (2+31/5)10 is _______.



The sum of the coefficient of the polynomial (1+x3x2)2163 is______



Find the coefficient of x2 in (x2+1x3)6.



If the middle term in the expansion of (x2+2)8 is 1120; then the sum of possible real values of x is 



Find the middle terms in the expansion of (x2+1x)7.



Find the coefficient of x3 in the expansion of (2+5x)(12x)2.



The number of integral terms in the expansion of (51/2+71/8)1024 is................



If the second term in the expansion (a1/13+a3/2)n  is 14a5/2, then the value of nC3nC2 is .........



If n is odd natural number, then nr=0(1)rnCr equals................



Coefficient of x^3 and x^4 in { \left( 1+x+{ x }^{ 2 }+{ x }^{ 3 }+{ x }^{ 4 } \right)  }^{ 199 }{ \left( x-1 \right)  }^{ 201 } are  .



2.



If the sum of the coefficients in the expansion of {(x+y)}^{n} is 4096, find the greatest coefficient in the expansion.



The greatest binomial coefficient in the expansion of \displaystyle \left ( a+b \right )^{n} is \displaystyle ^{k}C_{m}  given that the sum of all the coefficients is equal to 4096. Find k-m ?



Following question contains statements given in two columns which have to be matched The statements in Column-I are labelled as A, B, C and D while the statements in Column-II are labelled as p, q, r and s Any given statement in Column-I can have correct matching with ONE statement in Column-II
Column-IColumn-II



If \displaystyle \left ( 1\times x \right )\left ( 1 +x+x^{2} \right )\left ( 1+x+x^{2}+x^{3} \right ).....\left ( 1+x+x^{2}+x^{3}+......+x^{n} \right )\equiv
a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+.......a_{m}x^{m} then \displaystyle \sum_{r=0}^{m}a_{r} has the value equal to \displaystyle \left ( n+k \right )! .Find k ?



Find the negative of middle term in the expansion of
\displaystyle \left ( \frac{2x}{3}-\frac{3}{2x} \right )^{6}



If \displaystyle ^{n+1}C_{2}+2\left ( ^2C_{2}+^3C_{2}+^4C_{2}+.....+^nC_{2} \right )=1^{2}+2^{2}+3^{2}+....+100^{2} then find sum of digits of n ?



Number of terms in the expansion of (x^{1/3} + x^{2/5})^{40} with integral power of x is equal to



Find the middle term in the expansion of (5x\, -\, 7y)^7.



Find the 7th term of \left (\dfrac {4x}{5}-\dfrac {5}{2x}\right )^9.



Find the number of rational terms in the expansion of \left ( 9^{1/4}+8^{1/6} \right )^{1000}.



\displaystyle \left ( 3a-\frac{a^{3}}{6} \right )^{9}



Find the middle term in the expansion of (2x + 3y)^8.



Find the middle term(s) in the expansion of:
\left (1-\dfrac {x^2}{2}\right )^{14}



Find the middle term(s) in the expansion of
\displaystyle \left ( 1-\frac{x^{2}}{2} \right )^{14}



Find the 13^{th} term in the expansion of \left ( 9x - \dfrac{1}{3\sqrt x} \right )^{18}, x \neq 0



Write the general term in the expansion of (x^2 - y)^6



Find the 4^{th} term in the expansion of (x-2y)^{12}



Find the middle terms in the expansion of \left (3- \dfrac{x^3}{6}  \right )^7



Find the middle terms in the expansion of \left (\dfrac {x}{3}+ 9y\right)^{10}



Write the middle terms in the expansion of \left(\dfrac{3x}{7}-2y\right)^{10}.



If P and Q are the sum of odd terms and the sum of even terms respectively, in the expansion of (x + a)^{n} then prove that
(i) P^{2} - Q^{2} = (x^{2} - a^{2})^{n}
(ii) 4PQ = (x + a)^{2n} - (x - a)^{2n}



Prove that C_0 +C_1 +C_2 +C_3 + ...... + 2^n . C_n = 3^n.



Find the fifth term of { \left( a+{ 2x }^{ 3 } \right)  }^{ 17 }.



Write down and simplify:
The 10th term of { \left( 1-2x \right)  }^{ 12 }



Find the fourteenth term of { \left( 3-a \right)  }^{ 15 }.



Find the middle term of { \left( 1-\cfrac { { x }^{ 2 } }{ 2 }  \right)  }^{ 14 }.



Expand \cfrac { 7+x }{ \left( 1+x \right) \left( 1+{ x }^{ 2 } \right)  } in ascending powers of x and find the general term.



The 2nd, 3rd and 4th terms in the expansion of {(x+y)}^{n} are 240,720, 1080 respectively; find x,y,n.



Write down and simplify:
The 4th term of { { \left( \cfrac { a }{ 3 } +9b \right)  }^{ 10 } }.



Show that the middle term in the expansion of { \left( 1+x \right)  }^{ 2n } is
\cfrac { 1.3.5....(2n-1) }{ |\underline { n }  } { 2 }^{ n }{ x }^{ n }.



Find the(p+2)th term from the end in { \left( x-\cfrac { 1 }{ x }  \right)  }^{ 2n+1 }



Find the rth term from the end in  { \left( x+a \right)  }^{ n }.



Write down and simplify:
The 25th term of { \left( 5x+8y \right)  }^{ 30 }



Write down and simplify:
The 12th term of { \left( 2x-1 \right)  }^{ 13 }.



Write down and simplify:
The 5th term of { \left( 2a-\cfrac { b }{ 3 }  \right)  }^{ 8 }.



If (1+x)^n=c_0+c_1x+c_2x^2+....+c_nx^n, find the value of 1^2c_1+2^2c_2+3^2c_3+.....+n^2c_n.



2.C_0+5.C_1 +8.C_2 + ...+ (3n+2)C_n = (3n+4)2^{n-1}



\sum_{k=1}^{3n} 6 ^nC_{2\, k-1}(-3)^k is equal to:



Prove that ^{n}C_{r} + ^{n - 1}C_{r} + ^{n - 2}C_{r} + ..... + ^{r}C_{r} = ^{n + 1}C_{r + 1}.



Write down and simplify 6^{th} term in the expansion of \displaystyle \left( \frac{2x}{3} + \frac{3y}{2} \right)^9.



The last term in (2^{1/3} \, + \, 2^{-1/2})^n \, is \, \left(\dfrac{1}{3(9)^{1/3}}\right)^{log_33} then show that the fifth term is 210



If 'n' is a positive integer and 'x' is any non-zero number, then prove that
\displaystyle C_0 + C_1 . \frac{x}{2} + C_2 . \frac{x^2}{3} + ..... + C_n . \frac{x^n}{n+1} = \frac{(1+x)^{n+1} - 1}{(n+1)x}



Find the third term of the expansion of \displaystyle\, \left ( z^2 + \frac{1}{z} \sqrt[3]{z} \right )^n, if the sum of all the binomial coefficients is equal to 2048.



Find the middle term of the expansion of \displaystyle\, \left ( \sqrt{x} - \frac{1}{x} \right )^6



Find the second term of the binomial expansion of \displaystyle\, \left ( \sqrt[13]{a} + \frac{a}{\sqrt{a^{-1}}} \right )^m, if \displaystyle\, C_{3}^{m} : C_{2}^{m} = 4 : 1



if a_0 \, , \, a_1 \, , \, a_2....be the coefficients in the expansion of (1 + x + x^2)^n in ascending powers of x , then prove that 
a_{0}^{2} \, - \, a_{1}^{2} \, + \, a_{2}^{2} \, - \, a_{3}^{2} \, + \, ......+ \, (- 1)^{n \, - \, 1} \,  a_{n \, - \, 1}^{2}
= \, \dfrac{1}{2}a_n(1 \, -(- 1)^n \, a_n)



If the coefficient of a^{r-1}, a^r and a^{r + 1} in the expansion of (1 + a)^n are in arithmetic progression, prove that n^2 - n(4r + 1) + 4r^2 - 2 = 0.



In  the expansion  of (7^{1/3} \, + \, 11^{1/9})^{6561} prove that three will be only  730 term  which are free from radicals 



If a_0 \, , \, a_1 \, , \, a_2....be the coefficients in the expansion of (1 + x + x^2)^n in ascending powers x , then prove that :
a_r \, = \, a_{2n \, - \, r}



Find the middle term(s) of (\frac{x^{3/2}y}{2} + \frac{2}{xy^{3/2}})^{13}



{ C }_{ 0 }+\dfrac { 3 }{ 2 } .{ C }_{ 1 }+\dfrac { 9 }{ 3 } .{ C }_{ 2 }+\dfrac { 27 }{ 4 } .{ C }_{ 3 }+...........+\dfrac { { 3 }^{ n } }{ n+1 } .{ C }_{ n }=\dfrac { { 4 }^{ n+1 }-1 }{ 3\left( n+1 \right)  } .Is it true ?If true enter 1 else 0.



Find the coefficient of x^{25} in expansion of expression \displaystyle \sum_{r = 0}^{50} {^{50}C}_r (2x - 3)^r (2 - x)^{50 -r}.



Find the middle term:
(i) \left[3x - \dfrac{2}{x}\right]^{15}



Find the coefficient of a^4 in the product (1 + 2a)^4 (2 - a)^5 using binomial theorem.



Evaluate {(\sqrt{3}+\sqrt{2})}^{6} -{(\sqrt{3}-\sqrt{2})}^{6}.



In the expansion of {\left( {{x^3} - \dfrac{1}{{{x^2}}}} \right)^2}, where n is a positive integer, the sum of the coefficients of {x^6} is 1.



Find the sum \sum _{ r=1 }^{ n }{ \overset { n+r }{ \underset { \quad \quad r }{ C }}} .



Find the sum of \sum _{ 0\le i\le j\le n  } \sum { j\overset { n }{ \underset { \quad i }{ C }  }  } 



In the expansion of {\left( {\dfrac{3}{2} + \dfrac{7}{3}} \right)^n} when x = \dfrac{1}{2} if is known that {6^{th}} term is the greatest term the find the possible integral value of n.



Find the sum \sum _{ r=1 }^{ n }{ \dfrac { r\overset { n }{ \underset { \quad \quad r }{ C }  }  }{ \overset { n }{ \underset { \quad \quad r-1 }{ C }  }  }  } 



Find the middle term in the expansion of \left ( \dfrac{x}{y}-\dfrac{y}{x} \right )^7.



Write the following form in expanded form:-
(A) {\left( {2a + 3b} \right)^3}
(B) {\left( {5x - 3y} \right)^3}



Show that the middle term in the expansion of {(1 + x)^{2n}} is \frac{{1.3.5.....(2n - 1)}}{{n!}} {2^n}{x^n}; where n is a positive integer.



If sum of coefficient in expansion of (2+3cx+c^2x^2)^{12} is 0. Then find the value of c.



Evaluate:
\sum\limits_{r=0}^{20}\ ^{20}C_r.



^{14}C_1+ ^{14}C_2+ ^{14}C_3+...+ ^{14}C_{14}=?



Is x^5 in the expansion of \left[2x^3 - \dfrac{1}{3x^3}\right]^{10}.



Find the 24th term in the sequence whose nth term  {a_n} = \dfrac{{n\left( {n - 2} \right)}}{{n + 3}} . using binomial theorem , Evaluate {\left( {102} \right)^3} 



Find the 7th term in the expansion of \left(\dfrac{4x}{5}-\dfrac{5}{2x}\right)^9.



Find the coefficient of x^4 in (1 + 2x)^4 (2 -x)^5



Find the middle terms in the expansions of 
i) \left(3 - \dfrac{x^3}{6} \right)^7
ii) \left(\dfrac{x}{3} + 9y \right)^{10}



Find the coeifficient of
(i)  x^{6}y^{3}in (x+y)^{9}



Show that the middle term in the expansion of (1+x)^{2n} is \dfrac{1.3.5...(2n-1)}{n!} 2^nx^n, where n is a positive integer.



Find the coefficient of x^{15} in
(1 + x)^{15} + (1 + x)^{16} + .... (1 + x)^{30}.



If the coefficients of \left( r-5 \right) th and \left( 2r-1 \right)th terms in the expansion of { \left( 1+x \right)  }^{ 34 } are equal, then write the value of r



Find the 8th term in the expansion of { \left( { x }^{ 3/2 }{ y }^{ 1/2 }-{ x }^{ 1/2 }{ y }^{ 3/2 } \right)  }^{ 10 }



Prove that there is no term involving x^6 in the expansion of \left (2x^2-\dfrac{3}{x}\right)^{11}, where r\neq 0.



If the coefficients of x and {x}^{2} in the expansion of {(1+x)}^{m}{(1-x)}^{n} are 3 and -6 respectively. Find the values of m and n.



Find the coefficient of x^{13} in the expansion of (1-x)^{5}x(1+x+x^{2}+x^{3})^{4}



The sum of the coefficients of first three terms in the expansion of { \left( x-\cfrac { 3 }{ { x }^{ 2 } }  \right)  }^{ m },x\neq 0, m being a natural number, is 559. Find the term of the expansion containing {x}^{3}.



If {x}^{p} occurs in the expansion of \quad { \left( { x }^{ 2 }+\cfrac { 1 }{ x }  \right)  }^{ 2n }, prove that its coefficient is \quad \left[ \cfrac { \left( 2n \right) ! }{ \left( \cfrac { 4n-p }{ 3 }  \right) !\left( \cfrac { 2n+p }{ 3 }  \right) ! }  \right]



The value of \sum_{r=1}^{15}r^{2}(\frac{^{15}Cr}{^{15}Cr-1}) is equal to:



The coefficient of x^{10} in the expansion of (1+x)^{2}



If the coefficients of rth, (r+1)th and (r+2)nd terms in the expansion of {(1+x)}^{n} are in A.P, then show that {n}^{2}-(4r+1)n+4{r}^{2}-2=0.



If \displaystyle \sum_{r = 1}^{n} {^{n}C_{r}} 3^{r} is equal to 4095 then n equals.



The coefficent of x^{10} in the expansion (x+10)^{10}



Find the middle term in the expansion of \left (\dfrac{2}{3}x^2-\dfrac{3}{2x}\right )^{20}.



Find the middle term of \left(x-\dfrac {1}{2x}\right)^{10}



Find the middle term in term  expansion of {\left(1+x\right)}^{2n}



If the sum of the  coefficients of x^7 and  \ x^4 in the expansion of \left(\dfrac{x^2}{a} - \dfrac{b}{x}\right)^{11} is zero, then ab=



 Find general term in the expansion of  \left( x _ { 1 } + x _ { 2 } + \ldots . + x _ { p } \right) ^ { n }  



In the expansion of  \left( 5 ^ { \dfrac { 1 } { 2 } } + 2 ^ { \dfrac { 1 } { 8 } } \right) ^ { 1024 },  the number of integral terms is



Find the coefficient of  { x }^{ 3 } in the expansion of  { \left( { x }^{ 2 }+\dfrac { 3\sqrt { 2 }  }{ x }  \right)  }^{ 9 }



Find the coefficient { x }^{ 8 } in the product \left( 1+2x \right) ^{ 6 }\left( 1-x \right) ^{ 7 } using binomial therm.



If the middle term in the expansion of \left(x+\dfrac {b}{x}\right)^{6} is 160, find b.



Find the sum of coefficients of odd powers of x in the expansion {\left(1+x\right)}^{50}



The coefficient of  x ^ { 10 }  in the expansion of  ( 1 + x ) ^ { 2 } \left( 1 + x ^ { 2 } \right) ^ { 3 } \left( 1 + x ^ { 3 } \right) ^ { 4 }  is equal to :



The middle term in the expansion of (5x-7y)^{7}.



Fidn the { 7 }^{ th } term from the end in the expansion of { \left( 9x-\dfrac { 1 }{ 3\sqrt { x }  }  \right)  }^{ 18 },x\neq 0.



In expansion of (1+x)^3 , the second term is 240 , find x



Find the middle term in the expansion of
\left( \dfrac { 2x }{ 3 } -\dfrac { 3 }{ 2x }  \right) ^{ 6 }



Find the middle term in the expansion of \left( \dfrac { { 2x }^{ 2 } }{ 3 } -\dfrac { 3 }{ 2x }  \right) ^{ 12 }.



Find the 8th term of {\left( {1 - \frac{{5x}}{2}} \right)^{ - 3/5}}



Find the { 7 }^{ th } term in { \left( \dfrac { 4 }{ { x }^{ 3 } } +\dfrac { { x }^{ 2 } }{ 2 }  \right)  }^{ 14 }



Prove that the coefficient of {x^n} in the expression of {\left( {1 + x} \right)^{2n}} is twice the coefficient of {x^n} in the expression of {\left( {1 + x} \right)^{2n - 1}}.



Find the middle term in the expansion of :
\left(\dfrac{x}{a}-\dfrac{a}{x}\right)^{10}



Find the coefficient of x^3 in the expansion of (3x + 1)(2 - x)^6



Find the coefficient of x^2 and x^3 in the expansion of (1-2x)^3.



Show that the expansion of (2x^2 - \frac {1}{x} )^{20} does not contain any term involving x^9



Show that the term containing x^3 does not exist in the expansion of (3 x -\frac {1}{2x})^8 .



Find the coefficient of  x^{-15} in the expansion of (3x^2 - \frac {a}{3x^3} )^{10} .



Find the coefficient of x in the expansion of ( 1 -3x +7x^2)(1-x)^{16}.



Write the general term in the expansion of ( x^2 -y)^6



Find the coefficient of  x^5 in the expansion of (x+ 3)^8 .



Find the coefficient of x^6 in the expansion of (3x^2 - \frac {1}{3x})^9 .



Find the two middle terms in the expansion of ( x^2 +a^2)^5



Find the middle term in the expansion of :
(\frac {x}{a} - \frac {a}{x})^{10}



Find the middle term in the expansion of :
( x^2 - \frac {2}{x})^{10}



Find the 4th term from the end in expansion of ( \frac {4x}{5} - \frac {5}{2x})^9



Find the two middle terms in the expansion of ( 3x - \frac {x^3}{6})^9



Find the two middle terms in the expansion of  ( x^4 - \frac {1}{x^3})^{11}



Find the 5th term from the end in expansion of ( x- \frac {1}{x})^{12} .



Find the middle term in the expansion of ( 3+ x)^6



Find the middle term in the expansion of (\frac {x+ 3}2 +3y)^8



Find the two middle terms in the expansion of 
(\frac {p}{x} + \frac {x}{p})^9



Find the coefficient of x^4 in the expansion of ( 1+ x)^n(1 -x)^n. deduce that C_2 = C_0C_4 - C_1C_3 +C_2C_2 - C_3C_1 +C_4C_0, where C_r stands for ^nC_r.



Find a and b so that the n^{th} term in the expansion of \dfrac {a + bx}{(1 - x)^2} may be (3n - 2)x ^{n - 1}.



The larger of 99^{50}+100^{50} is 101^{50} is ________.



If (1+ax)^{n}=1+8x+24x^{2}+.... then a= _______ and n= _______



Find a , b , c so that the coefficient of x^{n} in the expansion of \dfrac {a + bx + cx^2}{(1 - x)^3} may be n^2 + 1.



Prove that the sum of the coefficient of the odd powers of x the expansion of (1+x+x^2+x^3+x^4)^{n-1}, when n is a prime number other than 5, is divisible by n.



Match the statements a,b,c,d  in column I  with statements p,q,r,s in column II. 



Shew that the middle term in the expansion of (1+x)^{2n} is \dfrac{1.3.5...(2n-1)}{(n)!}2^nx^n.



Find the coefficient of x^n in the expansion of 
(1 - 2x + 3x^2 - 4x^3 + \dots)^{-n}.



Find n, if the ratio of fifth term from the beginning to the fifth term from the end in the expansion of  \left ( \sqrt[4]{2} + \cfrac{1}{\sqrt[4]{3}} \right )^{n} \text {is} \sqrt{6} : 1



Find the 4^{th} term in the expansion of (x - 2y)^{12}.



In the following expansions, find the term as stated: 
9th term of  \Bigg(\dfrac{x}{y} - \dfrac{3y}{x^2}\Bigg)^{12}



In the following expansions, find the term as stated :
5th term of (a + 2x^3)^{17}



Find a, if the 17^{th} and 18^{th} terms of the expansion (2 + a)^{50} are equal.



Find the middle terms in the expansions of
(i) \left (3 - \cfrac{x^{3}}{6} \right )^{7}
(ii)\left (\cfrac{x}{3} + 9 y \right)^{10}



Expand the following Binomials upto fourth term: (1+x^2)^{-2}



Find the 13^{th} term in the expansion of
\left ( 9x - \dfrac{1}{3\sqrt{x}} \right )^{18}, x \neq 0



Find the general term of the following expansions:
(1 x)^{-p/q}



Find the coefficient of x^6 in the expansion of (a + 2bx^2)^{-3}.



Find the middle term in the following expansions.
\bigg(\dfrac{x}{2}+2y\bigg)^6



Find the required terms in the following expansions:
Eighth term of (1+2x)^{-1/2}



Expand the following Binomials upto fourth term \bigg(1-\dfrac{x}{2}\bigg)^{1/2}.



Expand the following Binomials upto fourth term: (3-2x^2)^{-2/3}



Expand the following Binomials upto fourth term: \dfrac{1}{\sqrt{5+4x}}.



Find the middle term in the following expansions.
\bigg(x+\dfrac{1}{x}\bigg)^{2n}



Find the required terms in the following expansions:
Seventh term of (1+x)^{5/2}



Find the required terms in the following expansions: Fourth term of (1 3x)^{-1/3}.



The ratio of fifth term from the beginning to the fifth term from the end in the expansion of \left(\sqrt[4]{2} + \dfrac{1}{\sqrt[4]{3}}\right)^{n} is \sqrt{6} : 1. If n = \dfrac{20}{\lambda}, find the value of  \lambda.



Find the coefficient of { x }^{ 50 } in the polynomials after parenthesis have been removed and like terms have been collected in the expansion
{ \left( 1+x \right)  }^{ 1000 }+x{ \left( 1+x \right)  }^{ 999 }+{ x }^{ 2 }{ \left( 1+x \right)  }^{ 998 }+....+{ x }^{ 1000 }



If { \left( 1+x \right)  }^{ n }={ C }_{ 0 }+{ C }_{ 1 }x+{ C }_{ 2 }{ x }^{ 2 }+....+{ C }_{ n }{ x }^{ n } , then find the sum of the series
\cfrac { { C }_{ 0 } }{ 2 } -\cfrac { { C }_{ 1 } }{ 6 } +\cfrac { { C }_{ 2 } }{ 10 } -\cfrac { { C }_{ 3 } }{ 14 } +........+\cfrac { { \left( -1 \right)  }^{ n }{ C }_{ n } }{ 4n+2 }



For n\ge 2, let { C }_{ r }=\begin{pmatrix} n \\ r \end{pmatrix} and { a }_{ n }=\sum _{ r=0 }^{ n }{ \cfrac { 1 }{ { C }_{ r } }  }



Match the entries in column I with entries in column II



If the 2nd,3rd and 4th terms in the expansion of {(a+x)}^{n} are respectively 240,720,1080, find a,x,n.



Find the middle term in the expansion of {\left(\dfrac{a}{x}+bx\right)}^{12}



If {c}_{0},{c}_{1},{c}_{2},.......{c}_{n} denote the coefficients in the expansion of {(1+x)}^{n}, prove that
\cfrac { { c }_{ 1 } }{ { c }_{ 0 } } +\cfrac { 2{ c }_{ 2 } }{ { c }_{ 1 } } +\cfrac { 3{ c }_{ 3 } }{ { c }_{ 2 } } +...\cfrac { n{ c }_{ n } }{ { c }_{ n-1 } } =\cfrac { n(n+1) }{ 2 } .



Find the two middle terms of { \left( 3a-\cfrac { { a }^{ 3 } }{ 6 }  \right)  }^{ 9 }.



If p is a prime number, show that the coefficients of the terms of (1 + x)^{p - 1} are alternately greater and less by unity than some multiple of p.



If c_0 , c_1 , c_2 , ...... c_n  are the coefficients in the expansion (1+x)^n ,  where n is a positive integer, shew that

c_1 - \dfrac {c_2}{2} + \dfrac{c_3}{3} - ...... + \dfrac { (-1)^{n-1} c_n }{n} = 1 + \dfrac {1}{2} + \dfrac {1}{3} + .... \dfrac {1}{n}



If {c}_{0},{c}_{1},{c}_{2},.......{c}_{n} denote the coefficients in the expansion of {(1+x)}^{n}, prove that
{ c }_{ 0 }+\cfrac { { c }_{ 1 } }{ 2 } +\cfrac { { c }_{ 2 } }{ 3 } +......+\cfrac { { c }_{ n } }{ n+1 } =\cfrac { { 2 }^{ n+1 }-1 }{ n+1 } .



If {x}^{p} occurs in the equation { \left( { x }^{ 2 }+\cfrac { 1 }{ x }  \right)  }^{ 2n }., prove that its coefficient is \cfrac { |\underline { 2n }  }{ |\underline { \cfrac { 1 }{ 3 } (4n-p) } \quad |\underline { \cfrac { 1 }{ 3 } (2n+p) }  } .



If {c}_{0},{c}_{1},{c}_{2},.......{c}_{n} denote the coefficients in the expansion of {(1+x)}^{n}, prove that
({c}_{0}+{c}_{1})({c}_{1}+{c}_{2}).......({c}_{n-1}+{c}_{n})=\cfrac { { c }_{ 1 }{ c }_{ 2 }...{ c }_{ n }{ (n+1) }^{ n } }{ |\underline { n }  } .



Find the middle term(s) in the expansion of (1+3x+3x^2+x^3)^{2n}.



Find the middle terms in the expansion of { \left( { x }^{ 2 }+\dfrac { 1 }{ x }  \right)  }^{ 11 }



Prove that: \sum_{r \, = \, 0}^{n} \, r(n \, - \, r)C_r^2 \, = \, n^2(^{2n \, - \, 2}C_n)



If (1+x)^n=C_0+C_1x+C_2x^2+...+C_nx^n, then the value of C_0+C_2+C_4+.... is?



If {\left(x+\dfrac{1}{x}+1\right)}^{6}={a}_{0}+\left({a}_{1}x+\dfrac{{b}_{1}}{x}\right)+\left({a}_{2}{x}^{2}+\dfrac{{b}_{2}}{{x}^{2}}\right)+...+\left({a}_{6}{x}^{6}+\dfrac{{b}_{6}}{{x}^{6}}\right) then find the value of {a}_{0}



Find the integral part of (3 + \sqrt {7})^{5}.



Let\,\,n \in N;{S_n} = \sum\limits_{r = 0}^{3n} {\left( {^{3n}{C_{3r}}} \right)} .\,\,Find\,\,\left| {{S_n} - 3{T_n}} \right|.



If log 1001= 3.000434, find the number of digits in {1001^{101}}



Find the sum of coefficient of the expression
{(x+2y+4z)}^{10}



In the expansion of { \left( { 7 }^{ 1/3 }+{ 11 }^{ 1/9 } \right)  }^{ 6561 }, prove that there will be only 730 terms which are free from radicals.



Find the term containing x^{2}, if any, in the expansion of (3x-\dfrac{1}{2x})^{8}



The coefficient of x^{4} in the expansion of (\frac{x}{2}-\frac{3}{x^{2}})^{10} is equal to:



Prove that
C_{1}+2 C_{2}+3 C_{3}+4 C_{4}+.+ ^nC_{n}=n.2^{n-1}



In the expansion of {\left({3}^{\tfrac{-x}{4}}+{3}^{\tfrac{5x}{4}}\right)}^{n},  if the sum of the binomial coefficeints is 64 and the term with the greatest binomial coefficient exceeds the third by \left(n-1\right),then the value of x is  



Show that, if the greatest term in the expansion of {\left(1+x\right)}^{2n} has also the greatest coefficient , then x lies between \dfrac{n}{n+1} and \dfrac{n+1}{n}



The numerically greatest term in the expansion of {\left(2+3x\right)}^{9} when x=\dfrac{3}{2} is 



Coefficient of {x}^{7} in the expansion of \left( {1+3x-{2x}^{3}} \right )^{10}



Find a if the coefficients of {x}^{2} and {x}^{3} in the expansion of {\left(3+ax\right)}^{3} are equal.



Find the coefficient of x^5 in the product {\left(1+2x\right)}^{6}{\left(1-x\right)}^{7} using binomial theorem.



Find a, b and n in the expression of (a+b)^n if the first three terms of the expansion are 729, 7290 and 30375  respectively. 



Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of \left( \sqrt [ 4 ] { 2 } + \frac { 1 } { \sqrt [ 4 ] { 3 } } \right) ^ { ( n ) } is \sqrt { 6 } : 1



Show that the middle term in the expansion of (x - \frac{1}{x})^2n is \frac{1.3.5.....(2n-1)}{n;} (-2)^n



Find the coefficient of x^5 in the expansion of ( 1+ x)^3(1-x)^6 .



Show that the coefficient of x^4 in the expansion of ( \frac {x}{2} - \frac {3}{x^2} )^{10} is   \frac {405}{256}



If the 17th  and 18th terms in the expansion of ( 2+a )^{50} are equal , find the value of a.



Find the 9th term in the expansion of ( \frac {a}{b} - \frac {b}{2a^2})^{12}



Find the term independent of x in the expansion of the following expressions:
(1+x+2x^{3})\left(\dfrac{3}{2}x^{2}-\dfrac{1}{3x}\right)^{9}

Enter 1 if answer is \dfrac {17}{54} otherwise enter 0.



If the middle term in the expansion of ( \frac {p}{2}  + 2 )^8 is 1120, find p.



Find the sixth term in the expansion (y^{1/2}+x^{1/3})^{n}, if the binomial coefficient of the third term from the ends is 45.



Find the 6th term if expansion ( y^{1/2} + x^{ 1/3} )^n , if the binominal coefficient of 3rd term form the end is 45.



Show that the middle term in the expansion of ( \frac {2x^3}{3} + \frac {3}{2x^2} )^{10} is 252



Show that the coefficient of x^{-3} in the expansion of ( x -\frac {1}{x} )^{11} is -330



Write the number of terms in the expansion of (\sqrt {2} + 1)^5 +( \sqrt {2} -1)^5 .



If the coefficient of (r-5)^{th} and (2r -1)^{th} terms in the expansion of (1+ x)^{34} are equal, find the value of r.



Write the coefficient of x^7y^2 in the expansion of (x +2y)^9 .



Prove that there is no term involving x^6 in the expansion (2x^2 - \frac {3}{x})^{11} .



Show that the coefficient of x^4 in the expansion of ( 1+ 2x+ x^2)^5 is 210.



Write the coefficient of the middle term in the expansion of ( 1 +x)^{2n} .



Write down and simplify:
The 4^{th} term of (x-5)^{13}.



Write down and simplify:
The 12^{th} term of (2x-1)^{13}.



Write down and simplify:
The 5^{th} term of \left(\frac{x^{\frac{3}{2}}}{a^{\frac{1}{2}}}-\frac{y^{\frac{5}{2}}}{b^{\frac{3}{2}}}\right)^8.



Write down and simplify:
The 5^{th} term of \left(2a-\dfrac{b}{3}\right)^8.



Find the 13^{th} term of \left(9x-\dfrac{1}{3\sqrt{x}}\right)^{18}.



Find the middle term of \left(\dfrac{a}{x}+\dfrac{x}{a}\right)^{10}.



Write down and simplify:
The 7^{th} term of \left(\dfrac{4x}{5}-\dfrac{5}{2x}\right)^9.



Write down and simplify:
The 10^{th} term of (1-2x)^{13}.



Write down and simplify:
The 28^{th} term of (5x+8y)^{30}.



Write down and simplify:
The 4^{th} term of \left(\dfrac{a}{3}+9b\right)^{10}.



Prove that the coefficient of x^n in the expansion \dfrac{1}{1 + x+ x^2} is 1,  0, -1 according as n is of the form 3m, \ 3m-1, or \ 3m+1.



Class 11 Engineering Maths Extra Questions