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

Chapter 18: Binomial Theorem

Exercise 18.1Exercise 18.2Exercise 18.3Exercise 18.4
Exercise 18.1 [Pages 11 - 12]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 18 Binomial Theorem Exercise 18.1 [Pages 11 - 12]

Exercise 18.1 | Q 1.01 | Page 11

Using binomial theorem, write down the expansions  .

(i)  $\left( 2x + 3y \right)^5$

Exercise 18.1 | Q 1.02 | Page 11

Using binomial theorem, write down the expansions  :

(ii)  $\left( 2x - 3y \right)^4$

Exercise 18.1 | Q 1.03 | Page 11

Using binomial theorem, write down the expansions  .

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

Exercise 18.1 | Q 1.04 | Page 11

Using binomial theorem, write down the expansions  :

(iv)  $\left( 1 - 3x \right)^7$

Exercise 18.1 | Q 1.05 | Page 11

Using binomial theorem, write down the expansions  :

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

Exercise 18.1 | Q 1.06 | Page 11

Using binomial theorem, write down the expansions  :

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

Exercise 18.1 | Q 1.07 | Page 11

Using binomial theorem, write down the expansions  :

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

Exercise 18.1 | Q 1.08 | Page 11

Using binomial theorem, write down the expansions  :

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

Exercise 18.1 | Q 1.09 | Page 11

Using binomial theorem, write down the expansions  :

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

Exercise 18.1 | Q 1.1 | Page 11

Using binomial theorem, write down the expansions  :

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

Exercise 18.1 | Q 2.01 | Page 11

Evaluate the

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

Exercise 18.1 | Q 2.02 | Page 11

Evaluate the

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

Exercise 18.1 | Q 2.03 | Page 11

Evaluate the

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

Exercise 18.1 | Q 2.04 | Page 11

Evaluate the

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

Exercise 18.1 | Q 2.05 | Page 11

Evaluate the

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

Exercise 18.1 | Q 2.06 | Page 11

Evaluate the

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

Exercise 18.1 | Q 2.07 | Page 11

Evaluate the

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

Exercise 18.1 | Q 2.08 | Page 11

Evaluate the

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

Exercise 18.1 | Q 2.09 | Page 11

Evaluate the

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

Exercise 18.1 | Q 2.1 | Page 11

Evaluate the

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

Exercise 18.1 | Q 3 | Page 11

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

Exercise 18.1 | Q 4 | Page 11

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

Exercise 18.1 | Q 5.1 | Page 12

Using binomial theorem evaluate :

(i) (96)3

Exercise 18.1 | Q 5.2 | Page 12

Using binomial theorem evaluate  .

(ii) (102)5

Exercise 18.1 | Q 5.3 | Page 12

Using binomial theorem evaluate .

(iii) (101)4

Exercise 18.1 | Q 5.4 | Page 12

Using binomial theorem evaluate .

(iv) (98)5

Exercise 18.1 | Q 6 | Page 12

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

Exercise 18.1 | Q 7 | Page 12

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

Exercise 18.1 | Q 8 | Page 12

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

Exercise 18.1 | Q 9 | Page 12

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

Exercise 18.1 | Q 10 | Page 12

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

Exercise 18.1 | Q 11 | Page 12

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

Exercise 18.1 | Q 12 | Page 12

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

Exercise 18.2 [Pages 37 - 40]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 18 Binomial Theorem Exercise 18.2 [Pages 37 - 40]

Exercise 18.2 | Q 1 | Page 37

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

Exercise 18.2 | Q 2 | Page 37

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

Exercise 18.2 | Q 3 | Page 37

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

Exercise 18.2 | Q 4 | Page 37

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

Exercise 18.2 | Q 5 | Page 37

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

Exercise 18.2 | Q 6 | Page 37

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

Exercise 18.2 | Q 7 | Page 37

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

Exercise 18.2 | Q 8 | Page 37

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

Exercise 18.2 | Q 9.1 | Page 37

Find the coefficient of:

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

Exercise 18.2 | Q 9.2 | Page 37

Find the coefficient of:

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

Exercise 18.2 | Q 9.3 | Page 37

Find the coefficient of:

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

Exercise 18.2 | Q 9.4 | Page 37

Find the coefficient of:

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

Exercise 18.2 | Q 9.5 | Page 37

Find the coefficient of:

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

Exercise 18.2 | Q 9.6 | Page 37

Find the coefficient of:

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

Exercise 18.2 | Q 9.7 | Page 37

Find the coefficient of:

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

Exercise 18.2 | Q 9.8 | Page 37

Find the coefficient of:

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

Exercise 18.2 | Q 10 | Page 38

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?

Exercise 18.2 | Q 11 | Page 38

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

Exercise 18.2 | Q 12 | Page 38

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

Exercise 18.2 | Q 13.1 | Page 38

Find the middle term in the expansion of:

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

Exercise 18.2 | Q 13.2 | Page 38

Find the middle term in the expansion of:

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

Exercise 18.2 | Q 13.3 | Page 38

Find the middle term in the expansion of:

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

Exercise 18.2 | Q 13.4 | Page 38

Find the middle term in the expansion of:

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

Exercise 18.2 | Q 14.1 | Page 38

Find the middle terms in the expansion of:

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

Exercise 18.2 | Q 14.2 | Page 38

Find the middle terms in the expansion of:

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

Exercise 18.2 | Q 14.3 | Page 38

Find the middle terms in the expansion of:

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

Exercise 18.2 | Q 14.4 | Page 38

Find the middle terms in the expansion of:

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

Exercise 18.2 | Q 15.01 | Page 38

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

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

Exercise 18.2 | Q 15.02 | Page 38

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

(ii)  $\left( 1 - 2x + x^2 \right)^n$

Exercise 18.2 | Q 15.03 | Page 38

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

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

Exercise 18.2 | Q 15.04 | Page 38

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

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

Exercise 18.2 | Q 15.05 | Page 38

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

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

Exercise 18.2 | Q 15.06 | Page 38

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

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

Exercise 18.2 | Q 15.07 | Page 38

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

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

Exercise 18.2 | Q 15.08 | Page 38

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

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

Exercise 18.2 | Q 15.09 | Page 38

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

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

Exercise 18.2 | Q 15.1 | Page 38

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

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

Exercise 18.2 | Q 16.01 | Page 39

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

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

Exercise 18.2 | Q 16.02 | Page 39

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

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

Exercise 18.2 | Q 16.03 | Page 39

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

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

Exercise 18.2 | Q 16.04 | Page 39

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

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

Exercise 18.2 | Q 16.05 | Page 39

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

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

Exercise 18.2 | Q 16.06 | Page 39

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

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

Exercise 18.2 | Q 16.07

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

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

Exercise 18.2 | Q 16.09 | Page 39

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

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

Exercise 18.2 | Q 16.1 | Page 39

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

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

Exercise 18.2 | Q 17 | Page 39

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.

Exercise 18.2 | Q 18 | Page 39

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

Exercise 18.2 | Q 19 | Page 39

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.

Exercise 18.2 | Q 20 | Page 39

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

Exercise 18.2 | Q 21 | Page 39

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

Exercise 18.2 | Q 22 | Page 39

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

Exercise 18.2 | Q 23 | Page 39

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.

Exercise 18.2 | Q 24 | Page 39

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

Exercise 18.2 | Q 25 | Page 39

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

Exercise 18.2 | Q 26 | Page 39

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

Exercise 18.2 | Q 27 | Page 40

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.

Exercise 18.2 | Q 28 | Page 40

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.

Exercise 18.2 | Q 29 | Page 40

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

Exercise 18.2 | Q 30 | Page 40

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

Exercise 18.2 | Q 31 | Page 40

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

Exercise 18.2 | Q 32 | Page 40

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

Exercise 18.2 | Q 33 | Page 40

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

Exercise 18.2 | Q 34 | Page 40

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.

Exercise 18.2 | Q 35 | Page 40

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.

Exercise 18.2 | Q 36 | Page 40

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.

Exercise 18.2 | Q 37 | Page 40

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.

Exercise 18.2 | Q 38 | Page 40

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  $\frac{1}{6}$

Exercise 18.2 | Q 39 | Page 40

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.

Exercise 18.3 [Pages 45 - 46]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 18 Binomial Theorem Exercise 18.3 [Pages 45 - 46]

Exercise 18.3 | Q 1 | Page 45

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

Exercise 18.3 | Q 2 | Page 45

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

Exercise 18.3 | Q 3 | Page 45

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

Exercise 18.3 | Q 4 | Page 45

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

Exercise 18.3 | Q 5 | Page 45

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

Exercise 18.3 | 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.

Exercise 18.3 | 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.

Exercise 18.3 | Q 8 | Page 45

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

Exercise 18.3 | 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.

Exercise 18.3 | Q 10 | Page 45

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

Exercise 18.3 | Q 11 | Page 45

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

Exercise 18.3 | Q 12 | Page 45

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

Exercise 18.3 | Q 13 | Page 45

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

Exercise 18.3 | Q 14 | Page 45

Write last two digits of the number 3400.

Exercise 18.3 | Q 15 | Page 45

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

Exercise 18.3 | 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  $\frac{a}{b}$

Exercise 18.3 | Q 17 | Page 46

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

Exercise 18.3 | 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  $a_0 + a_2 + a_4 + . . . + a_{2n}$ .

Exercise 18.4 [Pages 46 - 48]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 18 Binomial Theorem Exercise 18.4 [Pages 46 - 48]

Exercise 18.4 | 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

•  7

• 8

•  9

• 10

Exercise 18.4 | Q 2 | Page 46

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

• 495

• −495

• −7920

•  7920

Exercise 18.4 | 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

• 8

•  7

• 9

•  10

Exercise 18.4 | 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

• 3

• 4

•  5

• 6

Exercise 18.4 | 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

•  4 (A + B)

•  4 (A − B)

•  AB

• 4 AB

Exercise 18.4 | Q 6 | Page 46

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

•  40

•  5

• 41

• none of these

Exercise 18.4 | Q 7 | Page 46

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

•  1365

• −1365

• 3003

• −3003

Exercise 18.4 | Q 8 | Page 46

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

•  $\frac{28}{81}$

• $\frac{-28}{243}$

• $\frac{28}{243}$

•  none of these

Exercise 18.4 | 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

• 5

•  6

•  4

•  3

Exercise 18.4 | 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

•  251

• 252

•  250

•  none of these

Exercise 18.4 | 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

•  r = 10

•  r = 11

•  r = 12

• r = 13

Exercise 18.4 | Q 12 | Page 47

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

•  T3

• T4

• T5

• none of these

Exercise 18.4 | 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

• 7, 11

•  7, 14

•  8, 16

•  none of these

Exercise 18.4 | 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

• T5

•  T6

•  T7

• T8

Exercise 18.4 | 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

•  $A^2 - B^2$

• $A^2 + B^2$

•  4 AB

•  none of these

Exercise 18.4 | Q 16 | Page 47

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

• 3

• 4

•  5

•  none of these

Exercise 18.4 | Q 17 | Page 47

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

•  $\frac{405}{256}$

•  $\frac{504}{259}$

•  $\frac{450}{263}$

• none of these

Exercise 18.4 | 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

• 202

• 51

•  50

•  none of these

Exercise 18.4 | 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 =

• 3

•  4

•  5

•  6

Exercise 18.4 | 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

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

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

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

•  none of these

Exercise 18.4 | 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

•  1120

•  1020

• 512

•  none of these

Exercise 18.4 | 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

• 2

• 5

•  10

•  none of these

Exercise 18.4 | Q 23 | Page 48

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

• $- 924 m^7$

•  $- 792 m^5$

• $- 792 m^6$

•   $- 330 m^7$

Exercise 18.4 | Q 24 | Page 48

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

• $14! a^7 b^7$

• $\frac{14!}{7!} a^7 b^7$

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

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

Exercise 18.4 | Q 25 | Page 48

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

• 51C5

•  9C5

•  31C6 − 21C6

•  30C5 + 20C5

Exercise 18.4 | Q 26 | Page 48

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

• 18C8

•  18p10

• 218

•  none of these

Exercise 18.4 | 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

• 10

• 8

• 9

• none of these

Exercise 18.4 | 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 =

• 7

•  14

• 2

•  none of these

Exercise 18.4 | Q 29 | Page 48

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

• $^{2n}{}{C}_n$

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

•  $^{2n}{}{C}_n x^{- n}$

•  none of these

Exercise 18.4 | 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

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

•   $^{20}{}{C}_{12} x^2 2^{- 12}$

• $- ^t{20}{}{C}_7 x, 2^{- 13}$

•  none of these

Exercise 18.4 | 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

• 100

•  50

•  150

• 101

Exercise 18.4 | Q 32 | Page 48

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

• 152

•  −152

• −252

•  252

Exercise 18.4 | 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

• $- \frac{7}{9}$

• $- \frac{9}{7}$

• $\frac{7}{9}$

• $\frac{9}{7}$

Chapter 18: Binomial Theorem

Exercise 18.1Exercise 18.2Exercise 18.3Exercise 18.4

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

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

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