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Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce PDF Download

Binomial Theorem

A. Binomial Theorem

The formula by which any positive integral power of a binomial expression can be expanded in the form of a series is known as Binomial Theorem. If Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce then

(x + y)n = nC0xn + nC1xn – 1y + nC2xn – 2 y2 + ........+ nCrxn–ryr + ......+ nCnyn  Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

This theorem can be proved by induction.

Observations : 

(a) The number of terms in the expansion is (n + 1) i.e. one more than the index.

(b) The sum of the indices of x & y in each term is n.

(c) The binomial coefficients of the terms (nC0, nC1.........) equidistant from the beginning and the end are equal.

Ex.1 The value of Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Sol. The numerator is of the form a3 + b3 + 3ab (a + b) = (a + b)3 where a = 18, and b = 7

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce Nr = (18 + 7)3 = (25)3. Denominator can be written as

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

= (3 + 2)= 56 = (25)3 

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

B. IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE 

(a) General term : The general term or the (r + 1)th term in the expansion of (x + y)n is given by

Tr+1 = nCrxn-r. yr 

Ex.2 Find : (a) The coefficient of xin the expansion of Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

(b) The coefficient of x-7 in the expansion of Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Also, find the relation between a and b, so that these coefficients are equal.

Sol. 

 (a) In the expansion of   Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce  

the general terms is Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

putting 22 – 3r = 7 ⇒  3r = 15 ⇒  r = 5

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce  

Hence the coefficient of Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

(b) In the expansion of   Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce   ,

general terms is   Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

putting 11 – 3r = – 7 ⇒ 3r = 18  ⇒  r = 6

 Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Hence the coefficient of Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Also given coefficient of     Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce  

= coefficient of   Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce 

⇒  ab = 1 ( ∴ 11C5 = 11C6). Which is a required relation between a and b.

Hence the coefficient of x–7 in   Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Also given coefficient of x7 in Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

11C5 a6 b-5 = 11C6 a5b-6 ⇒  ab = 1 (∵11C5 = 11C6). Which is a required relation between a and b.

 Ex.3 Find the number of rational terms in the expansion of (91/4 + 81/6)1000. 

Sol. 

The general term in the expansion of (91/4 + 81/6)1000 is Tr + 1Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

The above term will be rational if exponent of 3 and 2 are integers. i.e 1000-r/2 and r/2 must be integers

The possible set of values of r is {0, 2, 4, ..........1000}. Hence, number of rational terms is 501

(b) Middle term : The middle term(s) in the expansion of (x + y)n is (are)

(i) If n is even, there is only one middle term which is given by T(n + 2)/2 = nCn/2. xn/2. yn/2

(ii) If n is odd, there are two middle terms which are Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce & Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Ex.4 Find the middle term in the expansion of Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Sol. 

The number of terms in the expansion of  Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce is 10 (even). So there are two middle terms. 

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce two middle terms. They are given by T5 and T6

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

(c) TERM INDEPENDENT OF x : Term independent of x contains no x ; Hence find the value of r for which the exponent of x is zero.

Ex.5 The term independent of x in Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Sol. General term in the expansion is Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

For constant term, 3r/2 = 10, r = 20/3 which is not an integer. Therefore, there will be no constant term.

(d) NUMERICALLY GREATEST TERM : To find the greatest term in the expansion of (x + a)n.

We have  Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce therefore, since xn multiplies every term in  Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce  , it will be sufficient to find the greatest term in this later expansion. Let the Tr and Tr + 1 be the rth and (r +1)th terms in the expansion of  Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce   Let numerically, Tr + 1 be the greatest term in the above expansion. Then

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce  Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce 

Substituting values of n and x, we get r < m + f or r < m where m is a positive integer and f is fraction such that 0 < f < 1. In the first case Tm + 1 is the greatest term, while in the second case Tm and Tm + 1 are the greatest terms and both are equal.

Ex.6 Find numerically the greatest term in the expansion of (3 – 5x)11 when x = 1/5

Sol. Since  Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

so, the greatest terms are T2 + 1 and T3+1.  Greatest term (when r = 2)

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

 Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

and greatest term (when r = 3)  Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce  Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

From above we say that the value of both greatest terms are equal.

C. If ( √A +B)n = I + f , where I & n are positive integers, n being odd and 0 < f < 1, then (I + f). f = Kn where A – B2 = K > 0 & A – B <1 . If n is an even integer, then (I + f) (1 – f) = kn

Ex.7 If (6√6 + 14)2n+1 = [N] + F and F = N – [N] ; where [*] denotes greatest integer, then NF is equal to

Sol. Since (66 + 14)2n+1 = [N] + F. Let us assume that f = (66 – 14)2n+1 ; where 0 ≤ f < 1.

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce

[N] + F – f = even integer.

Now 0 < F < 1 and 0 < f < 1 so –1 < F – f < 1 and F – f is an integer so it can only be zero

Thus NF = (6√6 + 14)2n+1 (6√6 – 14)2n+1 = 202n  + 1.

The document Binomial Theorem and Important Terms of Binomial Expansion | Mathematics (Maths) Class 11 - Commerce is a part of the Commerce Course Mathematics (Maths) Class 11.
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FAQs on Binomial Theorem and Important Terms of Binomial Expansion - Mathematics (Maths) Class 11 - Commerce

1. What is the Binomial Theorem?
Ans. The Binomial Theorem is a mathematical formula that allows us to expand a binomial expression raised to a positive integer power. It states that for any positive integer n, the expansion of (a + b)^n can be expressed as the sum of terms in the form of a^k * b^(n-k), where k takes on values from 0 to n.
2. What are the important terms used in binomial expansion?
Ans. In binomial expansion, there are several important terms: - Binomial Coefficient: The coefficient of each term in the expansion is given by the binomial coefficient, which is calculated using combinations. - Exponents: The exponents of the variables 'a' and 'b' in each term determine the power to which they are raised. - Powers of 'a' and 'b': The powers of 'a' and 'b' in each term depend on the binomial coefficient and the exponents. - Term: Each individual expression in the expansion is called a term. - Constant term: The term that does not have any variable is called the constant term.
3. How can the Binomial Theorem be applied to expand binomial expressions?
Ans. To apply the Binomial Theorem, follow these steps: 1. Identify the values of 'a', 'b', and 'n' in the binomial expression (a + b)^n. 2. Determine the powers of 'a' and 'b' in each term by using the binomial coefficient and the exponents. 3. Write down each term in the expansion, using the calculated powers of 'a' and 'b'. 4. Simplify and combine like terms, if necessary.
4. What is the significance of the Binomial Theorem in mathematics?
Ans. The Binomial Theorem is highly significant in mathematics as it allows us to efficiently expand binomial expressions without having to perform repetitive multiplication. It is used in various branches of mathematics, such as algebra, calculus, and probability theory. The theorem also has applications in fields like physics, engineering, and computer science, where it helps in solving complex equations and analyzing patterns.
5. Can the Binomial Theorem be used for negative exponents?
Ans. No, the Binomial Theorem cannot be directly applied to negative exponents. The theorem is specifically designed for expanding binomial expressions raised to positive integer powers. However, there are extensions of the Binomial Theorem, such as the generalized binomial theorem, which can be used to expand binomial expressions with negative exponents or fractional powers. These extensions involve using concepts like combinatorics, series, and mathematical functions to handle more complex cases.
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