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# RD Sharma solutions for Class 11 Mathematics chapter 18 - Binomial Theorem

## Chapter 18 : Binomial Theorem

#### Pages 11 - 12

Q 1.01 | Page 11

Using binomial theorem, write down the expansions  .

(i)  $\left( 2x + 3y \right)^5$

Q 1.02 | Page 11

Using binomial theorem, write down the expansions  :

(ii)  $\left( 2x - 3y \right)^4$

Q 1.03 | Page 11

Using binomial theorem, write down the expansions  .

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

Q 1.04 | Page 11

Using binomial theorem, write down the expansions  :

(iv)  $\left( 1 - 3x \right)^7$

Q 1.05 | Page 11

Using binomial theorem, write down the expansions  :

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

Q 1.06 | Page 11

Using binomial theorem, write down the expansions  :

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

Q 1.07 | Page 11

Using binomial theorem, write down the expansions  :

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

Q 1.08 | Page 11

Using binomial theorem, write down the expansions  :

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

Q 1.09 | Page 11

Using binomial theorem, write down the expansions  :

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

Q 1.1 | Page 11

Using binomial theorem, write down the expansions  :

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

Q 2.01 | Page 11

Evaluate the

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

Q 2.02 | Page 11

Evaluate the

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

Q 2.03 | Page 11

Evaluate the

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

Q 2.04 | Page 11

Evaluate the

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

Q 2.05 | Page 11

Evaluate the

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

Q 2.06 | Page 11

Evaluate the

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

Q 2.07 | Page 11

Evaluate the

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

Q 2.08 | Page 11

Evaluate the

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

Q 2.09 | Page 11

Evaluate the

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

Q 2.1 | Page 11

Evaluate the

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

Q 3 | Page 11

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

Q 4 | Page 11

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

Q 5.1 | Page 12

Using binomial theorem evaluate :

(i) (96)3

Q 5.2 | Page 12

Using binomial theorem evaluate  .

(ii) (102)5

Q 5.3 | Page 12

Using binomial theorem evaluate .

(iii) (101)4

Q 5.4 | Page 12

Using binomial theorem evaluate .

(iv) (98)5

Q 6 | Page 12

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

Q 7 | Page 12

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

Q 8 | Page 12

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

Q 9 | Page 12

Using binomial theorem, indicate which is larger (1.1)10000 or 1000.

Q 10 | Page 12

Using binomial theorem determine which number is larger (1.2)4000 or 800?

Q 11 | Page 12

Find the value of (1.01)10 + (1 − 0.01)10 correct to 7 places of decimal.

Q 12 | Page 12

Show that  $2^{4n + 4} - 15n - 16$  , where n ∈  $\mathbb{N}$  is divisible by 225.

#### Pages 0 - 40

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 $\frac{1 \cdot 3 \cdot 5 . . . \left( 2n - 1 \right)}{n!} . 2^n .$

The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)n are in A.P., find 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  $p \neq q$

Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.

Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.

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 abc and d, prove that $\frac{b^2 - ac}{c^2 - bd} = \frac{5a}{3c}$  .

If abc 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 xan.

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

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 7th term from the beginning to the 7th 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.

#### Pages 45 - 46

Q 1 | Page 45

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

Q 2 | Page 45

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

Q 3 | Page 45

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

Q 4 | Page 45

Write the middle term in the expansion of

$\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)$

Q 5 | Page 45

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

Q 6 | Page 45

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

Q 7 | Page 45

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.

Q 8 | Page 45

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

Q 9 | Page 45

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.

Q 10 | Page 45

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

Q 11 | Page 45

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

Q 12 | Page 45

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

Q 13 | Page 45

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

Q 14 | Page 45

Write last two digits of the number 3400.

Q 15 | Page 45

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

Q 16 | Page 45

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  $\frac{a}{b}$

Q 17 | Page 46

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

Q 18 | Page 46

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  $a_0 + a_2 + a_4 + . . . + a_{2n}$ .

#### Pages 46 - 48

Q 1 | Page 46

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

(a) 7

(b) 8

(c) 9

(d) 10

Q 2 | Page 46

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

(a) 495

(b) −495

(c) −7920

(d) 7920

Q 3 | Page 46

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

(a) 8

(b) 7

(c) 9

(d) 10

Q 4 | Page 46

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

(a) 3

(b) 4

(c) 5

(d) 6

Q 5 | Page 46

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

(a) 4 (A + B)

(b) 4 (A − B)

(c) AB

(d) 4 AB

Q 6 | Page 46

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

(a) 40

(b) 5

(c) 41

(d) none of these

Q 7 | Page 46

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

(a) 1365

(b) −1365

(c) 3003

(d) −3003

Q 8 | Page 46

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

(a) $\frac{28}{81}$

(b) $\frac{-28}{243}$

(c) $\frac{28}{243}$

(d) none of these

Q 9 | Page 46

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

(a) 5

(b) 6

(c) 4

(d) 3

Q 10 | Page 47

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

(a) 251

(b) 252

(c) 250

(d) none of these

Q 11 | Page 47

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

(a) r = 10

(b) r = 11

(c) r = 12

(d) r = 13

Q 12 | Page 47

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

(a) T3

(b) T4

(c) T5

(d) none of these

Q 13 | Page 47

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

(a) 7, 11

(b) 7, 14

(c) 8, 16

(d) none of these

Q 14 | Page 47

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

(a) T5

(b) T6

(c) T7

(d) T8

Q 15 | Page 47

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

(a) $A^2 - B^2$

(b) $A^2 + B^2$

(c) 4 AB

(d) none of these

Q 16 | Page 47

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

(a) 3

(b) 4

(c) 5

(d) none of these

Q 17 | Page 47

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

(a) $\frac{405}{256}$

(b) $\frac{504}{259}$

(c) $\frac{450}{263}$

(d) none of these

Q 18 | Page 47

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

(a) 202

(b) 51

(c) 50

(d) none of these

Q 19 | Page 47

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 =

(a) 3

(b) 4

(c) 5

(d) 6

Q 20 | Page 47

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

(a)  $\frac{n !}{\left[ \left( n - 1 \right) ! \left( n + 1 \right) ! \right]}$

(b)$\frac{\left( 2n \right) !}{\left[ \left( n - 1 \right) ! \left( n + 1 \right) ! \right]}$

(c) $\frac{\left( 2n \right) !}{\left( 2n - 1 \right) ! \left( 2n + 1 \right) !}$

(d) none of these

Q 21 | Page 47

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

(a) 1120

(b) 1020

(c) 512

(d) none of these

Q 22 | Page 48

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

(a) 2

(b) 5

(c) 10

(d) none of these

Q 23 | Page 48

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

(a)  $- 924 m^7$

(b) $- 792 m^5$

(c) $- 792 m^6$

(d)  $- 330 m^7$

Q 24 | Page 48

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

(a)

$14! a^7 b^7$

(b)  $\frac{14!}{7!} a^7 b^7$

(c)  $\frac{14!}{\left( 7! \right)^2} a^7 b^7$

(d)  $\frac{14!}{\left( 7! \right)^3} a^7 b^7$

Q 25 | Page 48

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

(a) 51C5

(b) 9C5

(c) 31C6 − 21C6

(d) 30C5 + 20C5

Q 26 | Page 48

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

(a) 18C8

(b) 18p10

(c) 218

(d) none of these

Q 27 | Page 48

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

(a) 10

(b) 8

(c) 9

(d) none of these

Q 28 | Page 48

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 =

(a) 7

(b) 14

(c) 2

(d) none of these

Q 29 | Page 48

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

(a) $^{2n}{}{C}_n$

(b) \left( - 1 \right)^n "^2 n C_n x^{- n}

(c) $^{2n}{}{C}_n x^{- n}$

(d) none of these

Q 30 | Page 48

If rth 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

(a) $^{20}{}{C}_{14} \left( \frac{x}{2^{14}} \right)$

(b)  $^{20}{}{C}_{12} x^2 2^{- 12}$

(c) $- ^t{20}{}{C}_7 x, 2^{- 13}$

(d) none of these

Q 31 | Page 48

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

(a) 100

(b) 50

(c) 150

(d) 101

Q 32 | Page 48

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

(a) 152

(b) −152

(c) −252

(d) 252

Q 33 | Page 48

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

(a) $- \frac{7}{9}$

(b) $- \frac{9}{7}$

(c) $\frac{7}{9}$

(d) $\frac{9}{7}$

## RD Sharma solutions for Class 11 Mathematics chapter 18 - Binomial Theorem

RD Sharma solutions for Class 11 Maths chapter 18 (Binomial Theorem) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 18 Binomial Theorem are Introduction of Binomial Theorem, Simple Applications of Binomial Theorem, Rth Term from End, Proof of Binomial Therom by Combination, Proof of Binomial Therom by Pattern, General and Middle Terms, Binomial Theorem for Positive Integral Indices.

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