Find the middle term in the expansion of: x 2 - 1 x 6 .

Given, x 2 - 1 x 6 Middle term in the expansion of ( a + b ) n = n 2 + 1 t h term, when n is even. = n + 1 2 t h and n + 3 2 t h terms, when n is odd. ∴ Middle term in the expansion of x 2 - 1 x 6 is 6 2 + 1 t h term = 4 t h term ( r + 1 ) t h term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ 4 t h term in the expansion of x 2 - 1 x 6 = C 3 6 x 2 3 - 1 x 3 = 6 ! 313 ! x 3 - 1 ∵ C r n = n ! r ! ( n - r ) ! = 6 × 5 × 4 3 × 2 × 1 - x 3 = - 20 x 3 ∴ Middle term in the expansion of x 2 - 1 x 6 = - 20 x 3 .

Find the term independent of x in the expansion of: 3 x - 2 x 2 15 .

Given, 3 x - 2 x 2 15 General term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ General term in the expansion of 3 x - 2 x 2 15 = C r 15 3 x 15 - r - 2 x 2 r = C r 15 3 15 - r x 15 - r ( - 2 ) r x - 2 r = C r 15 3 15 - r x 15 - 3 r - 2 r Term independent of x implies index of x = 0 . ⇒ 15 - 3 r = 0 ⇒ r = 5 ∴ Term independent of x in the expansion of 3 x - 2 x 2 15 = C 5 15 3 15 - 5 - 2 5 = 15 ! 5 ! 10 ! 3 10 - 32 ∵ C r n = n ! r ! ( r - r ) ! = 3003 × - 32 × 3 10 = - 96064 × 3 10

E M B I B E

Understanding ISC Mathematics Class 11 Volume 1>Chapter 9 - Binomial Theorem>EXERCISE 9.2

M L Aggarwal Solutions for Chapter: Binomial Theorem, Exercise 2: EXERCISE 9.2

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Binomial Theorem, Exercise 2: EXERCISE 9.2

Attempt the practice questions on Chapter 9: Binomial Theorem, Exercise 2: EXERCISE 9.2 with hints and solutions to strengthen your understanding. Understanding ISC Mathematics Class 11 Volume 1 solutions are prepared by Experienced Embibe Experts.

Questions from M L Aggarwal Solutions for Chapter: Binomial Theorem, Exercise 2: EXERCISE 9.2 with Hints & Solutions

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 9 - Binomial Theorem>EXERCISE 9.2>Q 4

Find the middle term in the expansion of: ${(x - \frac{1}{x})}^{10}$ .

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 9 - Binomial Theorem>EXERCISE 9.2>Q 4

Find the middle term in the expansion of: ${(x^{2} - \frac{1}{x})}^{6}$ .

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 9 - Binomial Theorem>EXERCISE 9.2>Q 10

What is the sum of the binomial coefficients in the expansion of $(1 + x)^{50}$ ?

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 9 - Binomial Theorem>EXERCISE 9.2>Q 15

Find the coefficient of $x^{10}$ in the expansion of ${(2 x^{2} - \frac{3}{x})}^{11}$ .

MEDIUM

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 9 - Binomial Theorem>EXERCISE 9.2>Q 19

Find the term independent of $x$ in the expansion of: ${(3 x - \frac{2}{x^{2}})}^{15}$ .

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 9 - Binomial Theorem>EXERCISE 9.2>Q 28

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 ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 9 - Binomial Theorem>EXERCISE 9.2>Q 29

Let $n$ be a positive integer. If the coefficients of $2^{n d}$ , $3^{r d}$ , $4^{t h}$ terms in the expansion of $(1 + x)^{n}$ are in $A . P .$ , then find the value of $n$ .

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 1>Chapter 9 - Binomial Theorem>EXERCISE 9.2>Q 30

If the coefficients of $x^{r - 1}$ , $x^{r}$ and $x^{r + 1}$ in the binomial expansion of $(1 + x)^{n}$ are in $A.P.$ Prove that $n^{2} - n (4 r + 1) + 4 r^{2} - 2 = 0$ .

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