# ML Aggarwal Binomial Theorem ISC Class-11 Maths Understanding

ML Aggarwal **Binomial Theorem** ISC Class-11 Maths Understanding Solutions Chapter-9. Step by step Solutions of ML Aggarwal ISC Class-11 Mathematics with Exe-1, Exe-2, Exe-3 and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-11 Mathematics.

## ML Aggarwal Binomial Theorem ISC Class-11 Maths Understanding Ch-9

Class: | 11th |

Subject: | Mathematics |

Chapter : | Ch-9 Binomial Theorem of Section -A |

Board | ISC |

Writer | ML Aggarwal |

Publications | APC Arya Publications 2020-21 |

**-: Select Topics :- **

### ML Aggarwal Binomial Theorem ISC Class-11 Maths Understanding Ch-9

**Introduction to the Binomial Theorem**

The Binomial Theorem is the method of expanding an expression which has been raised to any finite power. A binomial Theorem is a powerful tool of expansion, which has application in Algebra, probability, etc.

**Binomial Expression: **A binomial expression is an algebraic expression which contains two dissimilar terms. Ex: a + b, a^{3} + b^{3}, etc.

**Binomial Expansion**

Important points to remember:

- The total number of terms in the expansion of (x+y)
^{n}are (n+1) - The sum of exponents of x and y is always n.
- nC
_{0}, nC_{1}, nC_{2}, … .., nC_{n}are called binomial coefficients and also represented by C_{0}, C_{1}, C2, ….., C_{n} - The binomial coefficients which are equidistant from the beginning and from the ending are equal i.e. nC
_{0 }= nC_{n}, nC_{1 }= nC_{n-1 }, nC_{2 }= nC_{n-2},….. etc.

**Binomial theorem**

Binomial theorem, statement that for any positive integer *n*, the *n*th power of the sum of two numbers *a* and *b* may be expressed as the sum of *n* + 1 terms of the form

The theorem is useful in algebra as well as for determining permutations and combinations and probabilities. For positive integer exponents, *n*, the theorem was known to Islamic and Chinese mathematicians of the late medieval period.

#### Binomial Expansions and Series

The binomial formula in statistics is mostly used for counting and for calculating probabilities in experiments. A very similar technique, called binomial series expansion, is used in calculus for rewriting complicated functions into a simpler (binomial) form.

**Exe-9.1**

### ML Aggarwal Binomial Theorem ISC Class-11 Maths Understanding Ch-9

**Question 1:**

Find the number of terms in the expression of the following :

(i) ………..

(ii)…………

(iii)………….

**Question 2:**

…………………..

………………….

…………………..

**Question 15:**

Prove that \( \sum_{r=0}^{n}3^{r}C_{r}^{n} = 4^{n} \).

**Exe-9.2**

Binomial Theorem ISC Class-11 Maths Understanding Ch-9

**Question 1:**

Write the general term in the expansion of

(i) (x^{2} – y)^{6}

(ii) (x^{2} – xy)^{12}

(iii) ……………

**Question 2:**

…………………..

……………………

…………………..

**Question 30:**

If the coefficient of ………………….. prove that n² – n (4r + 1) + 4r² – 2 = 0

**Exe-9.3**

### ML Aggarwal Binomial Theorem ISC Class-11 Maths Understanding Ch-9

**Question 1:**

Find the coefficient of x^{7 }in ………………………. if these coefficient are equal , find the relation between a and b.

**Question 2:**

………………….

…………………

………………….

**Question 7:**

If n is any positive integer, show that the integer part of ………………….. show that (λ + f) (1 – f) = 1.

**Chapter Test**

ML Aggarwal Binomial Theorem ISC Class-11 Maths Understanding Ch-9

**Question 1:**

Using Binomial Theorem, expand the following :

(i) (3x – 2y)^{5}

(ii) …………….

(iii)……………

**Question 2:**

…………………..

…………………….

…………………….

**Question 15:**

If three consecutive coefficient in the expansion of (1 + x)^{n } …………………………. 33 : 110, find n.

**-: End of Binomial Theorem ISC Class-11 ML Aggarwal Maths Understanding Chapter-9 Solution :-**

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