RD Sharma solutions for Mathematics [English] Class 11 chapter 18 - Binomial Theorem [Latest edition]

Using binomial theorem, write down the expansions  . 

(i)  \[\left( 2x + 3y \right)^5\]

Using binomial theorem, write down the expansions  :

(ii)  \[\left( 2x - 3y \right)^4\]

Using binomial theorem, write down the expansions  .

(iii)  \[\left( x - \frac{1}{x} \right)^6\]

Using binomial theorem, write down the expansions  :

(iv)  \[\left( 1 - 3x \right)^7\]

Using binomial theorem, write down the expansions  :

(v) \[\left( ax - \frac{b}{x} \right)^6\]

Using binomial theorem, write down the expansions  :

(vi) \[\left( \frac{\sqrt{x}}{a} - \sqrt{\frac{a}{x}} \right)^6\]

Using binomial theorem, write down the expansions  :

(vii)  \[\left( \sqrt[3]{x} - \sqrt[3]{a} \right)^6\]

Using binomial theorem, write down the expansions  :

(viii)  \[\left( 1 + 2x - 3 x^2 \right)^5\]

Using binomial theorem, write down the expansions  :

(ix) \[\left( x + 1 - \frac{1}{x} \right)\]

Using binomial theorem, write down the expansions  :

(x)  \[\left( 1 - 2x + 3 x^2 \right)^3\]

Evaluate the 

(i)\[\left( \sqrt{x + 1} + \sqrt{x - 1} \right)^6 + \left( \sqrt{x + 1} - \sqrt{x - 1} \right)^6\]

Evaluate the 

(ii) \[\left( x + \sqrt{x^2 - 1} \right)^6 + \left( x - \sqrt{x^2 - 1} \right)^6\]

Evaluate the 

(iii)\[\left( 1 + 2 \sqrt{x} \right)^5 + \left( 1 - 2 \sqrt{x} \right)^5\]

Evaluate the

(iv)  \[\left( \sqrt{2} + 1 \right)^6 + \left( \sqrt{2} - 1 \right)^6\]

Evaluate the

(v)  \[\left( 3 + \sqrt{2} \right)^5 - \left( 3 - \sqrt{2} \right)^5\]

Evaluate the

(vi)  \[\left( 2 + \sqrt{3} \right)^7 + \left( 2 - \sqrt{3} \right)^7\]

Evaluate the

(vii) \[\left( \sqrt{3} + 1 \right)^5 - \left( \sqrt{3} - 1 \right)^5\]

Evaluate the

(viii)  \[\left( 0 . 99 \right)^5 + \left( 1 . 01 \right)^5\]

Evaluate the

(ix) \[\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6\]

Evaluate the

(x) \[\left\{ a^2 + \sqrt{a^2 - 1} \right\}^4 + \left\{ a^2 - \sqrt{a^2 - 1} \right\}^4\]

Find  \[\left( a + b \right)^4 - \left( a - b \right)^4\] . Hence, evaluate \[\left( \sqrt{3} + \sqrt{2} \right)^4 - \left( \sqrt{3} - \sqrt{2} \right)^4\] .

Find \[\left( x + 1 \right)^6 + \left( x - 1 \right)^6\] . Hence, or otherwise evaluate \[\left( \sqrt{2} + 1 \right)^6 + \sqrt{2} - 1^6\] .

Using binomial theorem evaluate :

(i) (96)³

Using binomial theorem evaluate  .

(ii) (102)⁵

Using binomial theorem evaluate .

(iii) (101)⁴

Using binomial theorem evaluate .

Using binomial theorem, prove that \[2^{3n} - 7n - 1\] is divisible by 49, where \[n \in N\] .

Using binomial theorem, prove that  \[3^{2n + 2} - 8n - 9\]  is divisible by 64, \[n \in N\] .

If n is a positive integer, prove that \[3^{3n} - 26n - 1\]  is divisible by 676.

Find the 11th term from the beginning and the 11th term from the end in the expansion of  \[\left( 2x - \frac{1}{x^2} \right)^{25}\] .

Find the 7th term in the expansion of \[\left( 3 x^2 - \frac{1}{x^3} \right)^{10}\] .

Find the 5th term from the end in the expansion of \[\left( 3x - \frac{1}{x^2} \right)^{10}\]

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

Find the 7th term in the expansion of \[\left( \frac{4x}{5} + \frac{5}{2x} \right)^8\]

Find the 4th term from the beginning and 4th term from the end in the expansion of \[\left( x + \frac{2}{x} \right)^9\] .

Find the 4th term from the end in the expansion of \[\left( \frac{4x}{5} - \frac{5}{2x} \right)^8\] .

Find the 7th term from the end in the expansion of \[\left( 2 x^2 - \frac{3}{2x} \right)^8\] .

Find the coefficient of: 

(i) x¹⁰ in the expansion of  \[\left( 2 x^2 - \frac{1}{x} \right)^{20}\]

Find the coefficient of: 

(ii) x⁷ in the expansion of  \[\left( x - \frac{1}{x^2} \right)^{40}\]

Find the coefficient of: 

(iii)  \[x^{- 15}\]  in the expansion of  \[\left( 3 x^2 - \frac{a}{3 x^3} \right)^{10}\]

Find the coefficient of: 

(iv)  \[x^9\]  in the expansion of  \[\left( x^2 - \frac{1}{3x} \right)^9\]

Find the coefficient of: 

(v)  \[x^m\]  in the expansion of  \[\left( x + \frac{1}{x} \right)^n\]

Find the coefficient of: 

(vi) x in the expansion of  \[\left( 1 - 2 x^3 + 3 x^5 \right) \left( 1 + \frac{1}{x} \right)^8\]

Find the coefficient of: 

(vii) \[a^5 b^7\]  in the expansion of  \[\left( a - 2b \right)^{12}\]

Find the coefficient of: 

(viii) x in the expansion of \[\left( 1 - 3x + 7 x^2 \right) \left( 1 - x \right)^{16}\]

Which term in the expansion of \[\left\{ \left( \frac{x}{\sqrt{y}} \right)^{1/3} + \left( \frac{y}{x^{1/3}} \right)^{1/2} \right\}^{21}\]  contains x and y to one and the same power?

Does the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)\] contain any term involving x⁹?

Show that the expansion of \[\left( x^2 + \frac{1}{x} \right)^{12}\]  does not contain any term involving x⁻¹.

Find the middle term in the expansion of: 

(i)  \[\left( \frac{2}{3}x - \frac{3}{2x} \right)^{20}\]

Find the middle term in the expansion of: 

(ii)  \[\left( \frac{a}{x} + bx \right)^{12}\]

Find the middle term in the expansion of: 

(iii) \[\left( x^2 - \frac{2}{x} \right)^{10}\]

Find the middle term in the expansion of: 

(iv)  \[\left( \frac{x}{a} - \frac{a}{x} \right)^{10}\]

Find the middle terms in the expansion of: 

(i)  \[\left( 3x - \frac{x^3}{6} \right)^9\]

Find the middle terms in the expansion of:

(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]

Find the middle terms in the expansion of: 

(iii) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]

Find the middle terms in the expansion of:

(iv)  \[\left( x^4 - \frac{1}{x^3} \right)^{11}\]

Find the middle terms(s) in the expansion of: 

(i) \[\left( x - \frac{1}{x} \right)^{10}\]

Find the middle terms(s) in the expansion of:

(ii)  \[\left( 1 - 2x + x^2 \right)^n\]

Find the middle terms(s) in the expansion of:

(iii)  \[\left( 1 + 3x + 3 x^2 + x^3 \right)^{2n}\]

Find the middle terms(s) in the expansion of:

(iv)  \[\left( 2x - \frac{x^2}{4} \right)^9\]

Find the middle terms(s) in the expansion of:

(v) \[\left( x - \frac{1}{x} \right)^{2n + 1}\]

Find the middle terms(s) in the expansion of: 

(vi)  \[\left( \frac{x}{3} + 9y \right)^{10}\]

Find the middle terms(s) in the expansion of: 

(vii) \[\left( 3 - \frac{x^3}{6} \right)^7\]

Find the middle terms(s) in the expansion of:

(viii)  \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]

Find the middle terms(s) in the expansion of:

(ix)  \[\left( \frac{p}{x} + \frac{x}{p} \right)^9\]

Find the middle terms(s) in the expansion of:

(x)  \[\left( \frac{x}{a} - \frac{a}{x} \right)^{10}\]

Find the term independent of x in the expansion of the expression: 

(i) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9\]

Find the term independent of x in the expansion of the expression:

(ii)  \[\left( 2x + \frac{1}{3 x^2} \right)^9\]

Find the term independent of x in the expansion of the expression: 

(iii)  \[\left( 2 x^2 - \frac{3}{x^3} \right)^{25}\]

Find the term independent of x in the expansion of the expression: 

(iv) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]

Find the term independent of x in the expansion of the expression: 

(v)  \[\left( \frac{\sqrt{x}}{3} + \frac{3}{2 x^2} \right)^{10}\]

Find the term independent of x in the expansion of the expression: 

(vi)  \[\left( x - \frac{1}{x^2} \right)^{3n}\]

Find the term independent of x in the expansion of the expression: 

(vii)  \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\]

Find the term independent of x in the expansion of the expression: 

(ix) \[\left( \sqrt[3]{x} + \frac{1}{2 \sqrt[3]{x}} \right)^{18} , x > 0\]

Find the term independent of x in the expansion of the expression: 

(x) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^6\]

If the coefficients of \[\left( 2r + 4 \right)\text{ th and } \left( r - 2 \right)\] th terms in the expansion of  \[\left( 1 + x \right)^{18}\]  are equal, find r.

If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)⁴³ are equal, find r.

Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)^{n + 1} is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)ⁿ.

Prove that the term independent of x in the expansion of \[\left( x + \frac{1}{x} \right)^{2n}\]  is \[\frac{1 \cdot 3 \cdot 5 . . . \left( 2n - 1 \right)}{n!} . 2^n .\]

If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)²ⁿ are in A.P., show that  \[2 n^2 - 9n + 7 = 0\]

If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)ⁿ are in A.P., then find the value of n.

If in the expansion of (1 + x)ⁿ, the coefficients of pth and qth terms are equal, prove that p + q = n + 2, where  \[p \neq q\]

In the expansion of (1 + x)ⁿ the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n.

If in the expansion of (1 + x)ⁿ, the coefficients of three consecutive terms are 56, 70 and 56, then find n and the position of the terms of these coefficients.

If 3rd, 4th 5th and 6th terms in the expansion of (x + a)ⁿ be respectively a, b, c and d, prove that `(b^2 - ac)/(c^2 - bd) = (5a)/(3c)`.

If a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that \[\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}\].

If the coefficients of three consecutive terms in the expansion of (1 + x)ⁿ be 76, 95 and 76, find n.

If the 6th, 7th and 8th terms in the expansion of (x + a)ⁿ are respectively 112, 7 and 1/4, find x, a, n.

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

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

If the term free from x in the expansion of  \[\left( \sqrt{x} - \frac{k}{x^2} \right)^{10}\]  is 405, find the value of k.

Find the sixth term in the expansion  \[\left( y^\frac{1}{2} + x^\frac{1}{3} \right)^n\] , if the binomial coefficient of the third term from the end is 45.

If p is a real number and if the middle term in the expansion of  \[\left( \frac{p}{2} + 2 \right)^8\] is 1120, find p.

Find n in the binomial \[\left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^n\] , if the ratio of 7^th term from the beginning to the 7^th term from the end is  \[\frac{1}{6}\]

if the seventh term from the beginning and end in the binomial expansion of  \[\left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^n\] are equal, find n.

Write the number of terms in the expansion of \[\left( 2 + \sqrt{3}x \right)^{10} + \left( 2 - \sqrt{3}x \right)^{10}\] . 

Write the sum of the coefficients in the expansion of \[\left( 1 - 3x + x^2 \right)^{111}\]

Write the number of terms in the expansion of \[\left( 1 - 3x + 3 x^2 - x^3 \right)^8\]

Write the middle term in the expansion of `((2x^2)/3 + 3/(2x)^2)^10`.

Which term is independent of x, in the expansion of \[\left( x - \frac{1}{3 x^2} \right)^9 ?\]

If a and b denote respectively the coefficients of x^m and xⁿ in the expansion of \[\left( 1 + x \right)^{m + n}\], then write the relation between a and b.

If a and b are coefficients of xⁿ in the expansions of \[\left( 1 + x \right)^{2n} \text{ and } \left( 1 + x \right)^{2n - 1}\] respectively, then write the relation between a and b.

Write the middle term in the expansion of  \[\left( x + \frac{1}{x} \right)^{10}\]

If a and b denote the sum of the coefficients in the expansions of \[\left( 1 - 3x + 10 x^2 \right)^n\]  and \[\left( 1 + x^2 \right)^n\]  respectively, then write the relation between a and b.

Write the coefficient of the middle term in the expansion of \[\left( 1 + x \right)^{2n}\] . 

Write the number of terms in the expansion of  \[\left[ \left( 2x + y^3 \right)^4 \right]^7\] .

Find the sum of the coefficients of two middle terms in the binomial expansion of  \[\left( 1 + x \right)^{2n - 1}\]

Find the ratio of the coefficients of x^p and x^q in the expansion of \[\left( 1 + x \right)^{p + q}\] .

Find the number of terms in the expansion of\[\left( a + b + c \right)^n\]

If a and b are the coefficients of xⁿ in the expansion of  \[\left( 1 + x \right)^{2n} \text{ and }  \left( 1 + x \right)^{2n - 1}\]  respectively, find  \[\frac{a}{b}\]

Write the total number of terms in the expansion of  \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\] .

If  \[\left( 1 - x + x^2 \right)^n = a_0 + a_1 x + a_2 x^2 + . . . + a_{2n} x^{2n}\] , find the value of  \[a_0 + a_2 + a_4 + . . . + a_{2n}\] .

If in the expansion of (1 + x)²⁰, the coefficients of rth and (r + 4)th terms are equal, then ris equal to

The term without x in the expansion of \[\left( 2x - \frac{1}{2 x^2} \right)^{12}\] is 

If rth term in the expansion of \[\left( 2 x^2 - \frac{1}{x} \right)^{12}\]  is without x, then r is equal to

If in the expansion of (a + b)ⁿ and (a + b)ⁿ + 3, the ratio of the coefficients of second and third terms, and third and fourth terms respectively are equal, then n is

If A and B are the sums of odd and even terms respectively in the expansion of (x + a)ⁿ, then (x + a)²ⁿ − (x − a)²ⁿ is equal to

The number of irrational terms in the expansion of \[\left( 4^{1/5} + 7^{1/10} \right)^{45}\]  is

The coefficient of  \[x^{- 17}\]  in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] is 

In the expansion of \[\left( x^2 - \frac{1}{3x} \right)^9\] , the term without x is equal to

If an the expansion of \[\left( 1 + x \right)^{15}\]   , the coefficients of \[\left( 2r + 3 \right)^{th}\text{  and  } \left( r - 1 \right)^{th}\]  terms are equal, then the value of r is

The middle term in the expansion of \[\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)^{10}\] is 

If in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] ,  \[x^{- 17}\]  occurs in rth term, then

In the expansion of \[\left( x - \frac{1}{3 x^2} \right)^9\]  , the term independent of x is

If in the expansion of (1 + y)ⁿ, the coefficients of 5th, 6th and 7th terms are in A.P., then nis equal to

In the expansion of \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\] , the term independent of x is

If the sum of odd numbered terms and the sum of even numbered terms in the expansion of  \[\left( x + a \right)^n\]  are A and B respectively, then the value of \[\left( x^2 - a^2 \right)^n\] is 

If the coefficient of x in \[\left( x^2 + \frac{\lambda}{x} \right)^5\]  is 270, then \[\lambda =\]

The coefficient of x⁴ in \[\left( \frac{x}{2} - \frac{3}{x^2} \right)^{10}\] is

The total number of terms in the expansion of \[\left( x + a \right)^{100} + \left( x - a \right)^{100}\]  after simplification is

If  \[T_2 / T_3\]  in the expansion of \[\left( a + b \right)^n \text{ and } T_3 / T_4\]  in the expansion of \[\left( a + b \right)^{n + 3}\]  are equal, then n =

The coefficient of  \[\frac{1}{x}\]  in the expansion of \[\left( 1 + x \right)^n \left( 1 + \frac{1}{x} \right)^n\] is 

If the sum of the binomial coefficients of the expansion \[\left( 2x + \frac{1}{x} \right)^n\]  is equal to 256, then the term independent of x is

If the fifth term of the expansion  \[\left( a^{2/3} + a^{- 1} \right)^n\]  does not contain 'a'. Then n is equal to

The coefficient of \[x^{- 3}\]  in the expansion of \[\left( x - \frac{m}{x} \right)^{11}\]  is

The coefficient of the term independent of x in the expansion of \[\left( ax + \frac{b}{x} \right)^{14}\] is 

The coefficient of x⁵ in the expansion of \[\left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . + \left( 1 + x \right)^{30}\]

The coefficient of x⁸ y¹⁰ in the expansion of (x + y)¹⁸ is

If the coefficients of the (n + 1)^th term and the (n + 3)^th term in the expansion of (1 + x)²⁰are equal, then the value of n is

If the coefficients of 2nd, 3rd and 4th terms in the expansion of \[\left( 1 + x \right)^n , n \in N\]  are in A.P., then n =

The middle term in the expansion of \[\left( \frac{2x}{3} - \frac{3}{2 x^2} \right)^{2n}\] is 

If r^th term is the middle term in the expansion of \[\left( x^2 - \frac{1}{2x} \right)^{20}\]  then \[\left( r + 3 \right)^{th}\]  term is

The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is

Constant term in the expansion of \[\left( x - \frac{1}{x} \right)^{10}\]  is

If the coefficients of x² and x³ in the expansion of (3 + ax)⁹ are the same, then the value of a is