Find the 13 t h term in the expansion of 9 x - 1 3 x 18 , x ≠ 0

We know that, In the expansion of ( a + b ) n , the ( r + 1 ) t h term is given by, T r + 1 = C r n a n - r b r . Here, a = 9 x , b = 1 3 x , r = 12 and n = 18 . Therefore, the 13 t h term is given by, T 13 = T 12 + 1 = C 12 18 . ( 9 x ) 18 - 12 - 1 3 x 12 = - 1 12 . C 12 18 . 9 x 6 1 3 x 12 = 18 ! 6 ! 12 ! × 9 6 × 1 3 12 × x 6 × 1 x 6 ∵ C r n = n ! r ! ( n - r ) ! = 18 ! 6 ! 12 ! 9 6 = 3 2 6 = 3 12 = 18 × 17 × 16 × 15 × 14 × 13 × 12 ! 6 × 5 × 4 × 3 × 2 × 1 × 12 ! = 18564

Find the middle terms in the expansions of 3 - x 3 6 7 .

It is known that the general term, T r + 1 , in the binomial expansion of ( a + b ) n is given by T r + 1 = n C r a n - r b r . Similarly, the general term of 3 - x 3 6 7 is given by T r + 1 = 7 C r ( 3 ) 7 - r - x 3 6 r . Here, n = 7 , which is odd. So, there are two middle terms, given by 4 th term = n + 1 2 t h term and 5 t h term = n + 1 2 + 1 t h term. So, T 4 = T 3 + 1 = 7 C 3 ( 3 ) 7 - 3 - x 3 6 3 = - 7 C 3 3 4 x 9 6 3 = - 7 ! 3 ! 4 ! × 81 × x 9 216 = - 105 8 x 9 T 5 = T 4 + 1 = 7 C 4 ( 3 ) 7 - 4 - x 3 6 4 = C 4 7 3 3 x 12 6 4 = 7 ! 4 ! 3 ! × 27 × x 12 1296 = 35 48 x 12

Find the middle term in the expansions of x 3 + 9 y 10 .

To find: The middle term in the expansions of x 3 + 9 y 10 . We know that, in the expansion of ( a + b ) n , number of terms = n + 1 . So, if n is even, the middle term is the n 2 + 1 t h term. Therefore, here the middle term = 10 2 + 1 = 6 t h term. We know that the ( r + 1 ) t h in the given expansion is T r + 1 = C r n a n - r b r . So, the 6 t h term is given by, T 6 = T 5 + 1 = C 5 10 x 3 10 - 5 9 y 5 = 10 ! 5 ! 5 ! × ( x 5 3 5 ) ( 9 5 y 5 ) ∵ C r n = n ! r ! ( n - r ) ! = 10 × 9 × 8 × 7 × 6 × 5 ! 1 × 2 × 3 × 4 × 5 × 5 ! ( x 5 3 5 ) ( 3 10 y 5 ) = 252 . x 5 . 3 5 . y 5 = 61236 x 5 y 5

E M B I B E

MATHEMATICS Textbook for Class XI>Chapter 8 - Binomial Theorem>EXERCISE 8.2

Mizoram Board Solutions for Chapter: Binomial Theorem, Exercise 2: EXERCISE 8.2

Author:Mizoram Board

Mizoram Board Mathematics Solutions for Exercise - Mizoram Board Solutions for Chapter: Binomial Theorem, Exercise 2: EXERCISE 8.2

Attempt the practice questions on Chapter 8: Binomial Theorem, Exercise 2: EXERCISE 8.2 with hints and solutions to strengthen your understanding. MATHEMATICS Textbook for Class XI solutions are prepared by Experienced Embibe Experts.

Questions from Mizoram Board Solutions for Chapter: Binomial Theorem, Exercise 2: EXERCISE 8.2 with Hints & Solutions

EASY

11th Mizoram Board

IMPORTANT

MATHEMATICS Textbook for Class XI>Chapter 8 - Binomial Theorem>EXERCISE 8.2>Q 4

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

EASY

11th Mizoram Board

IMPORTANT

MATHEMATICS Textbook for Class XI>Chapter 8 - Binomial Theorem>EXERCISE 8.2>Q 6

Find the $13^{t h}$ term in the expansion of ${(9 x - \frac{1}{3 \sqrt{x}})}^{18}, x \neq 0$

MEDIUM

11th Mizoram Board

IMPORTANT

MATHEMATICS Textbook for Class XI>Chapter 8 - Binomial Theorem>EXERCISE 8.2>Q 7

Find the middle terms in the expansions of ${(3 - \frac{x^{3}}{6})}^{7}$ .

MEDIUM

11th Mizoram Board

IMPORTANT

MATHEMATICS Textbook for Class XI>Chapter 8 - Binomial Theorem>EXERCISE 8.2>Q 8

Find the middle term in the expansions of ${(\frac{x}{3} + 9 y)}^{10} .$

HARD

11th Mizoram Board

IMPORTANT

MATHEMATICS Textbook for Class XI>Chapter 8 - Binomial Theorem>EXERCISE 8.2>Q 9

In the expansion of ${(1 + a)}^{m + n},$ prove that the coefficients of $a^{m}$ and $a^{n}$ are equal.

HARD

11th Mizoram Board

IMPORTANT

MATHEMATICS Textbook for Class XI>Chapter 8 - Binomial Theorem>EXERCISE 8.2>Q 10

The coefficients of the ${(r - 1)}^{t h}, r^{t h}$ and ${(r + 1)}^{t h}$ terms in the expansion of ${(x + 1)}^{n}$ are in the ratio $1 : 3 : 5 .$ Find $n$ and $r .$

HARD

11th Mizoram Board

IMPORTANT

MATHEMATICS Textbook for Class XI>Chapter 8 - Binomial Theorem>EXERCISE 8.2>Q 11

Prove that the coefficient of $x^{n}$ in the expansion of ${(1 + x)}^{2 n}$ is twice the coefficient of $x^{n}$ in the expansion of ${(1 + x)}^{2 n - 1} .$