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Binomial Theorem notes For Class 11 math Download PDF

Binomial Theorem Notes For Class 11 Math Download PDF

Chapter 8 : Binomial Theorem

NCERT Notes For Math Class 11

Chapter 8 :- Binomial Theorem

Binomial Theorem for Positive Integer

If n is any positive integer, then

CERT Notes For Math Class 11

This is called binomial theorem.

Binomial Theorem

Properties of Binomial Theorem for Positive Integer

(i) Total number of terms in the expansion of (x + a)n is (n + 1).
(ii) The sum of the indices of x and a in each term is n.
(iii) The above expansion is also true when x and a are complex numbers.
(iv) The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients and

This is called binomial theorem.

(v) General term in the expansion of (x + c)n is given by

General term in the expansion

(vi) The values of the binomial coefficients steadily increase to maximum and then steadily decrease .

values of the binomial

(xii) (a) If n is odd, then (x + a)n + (x – a)n and (x + a)n – (x – a)n both have the same number of terms equal to (n +1 / 2).

(b) If n is even, then (x + a)n + (x – a)n has (n +1 / 2) terms. and (x + a)n – (x – a)n has (n / 2) terms.

(xiii) In the binomial expansion of (x + a)n, the r th term from the end is (n – r + 2)th term

binomial expansion

(xiv) If n is a positive integer, then number of terms in (x + y + z)n is (n + l)(n + 2) / 2.

Middle term in the Expansion of (1 + x)n

(i) It n is even, then in the expansion of (x + a)n, the middle term is (n/2 + 1)th terms.

(ii) If n is odd, then in the expansion of (x + a)n, the middle terms are (n + 1) / 2 th term and (n + 3) / 2 th term.

Greatest Coefficient

(i) If n is even, then in (x + a)n, the greatest coefficient is nCn / 2

(ii) Ifn is odd, then in (x + a)n, the greatest coefficient is nCn – 1 / 2 or nCn+ 1 / 2 both being equal.

Greatest Term

In the expansion of (x + a)n

(i) If n + 1 / x/a + 1 is an integer = p (say), then greatest term is Tp == Tp + 1.

(ii) If n + 1 / x/a + 1 is not an integer with m as integral part of n + 1 / x/a + 1, then Tm + 1. is the greatest term.

Important Results on Binomial Coefficients

Greatest Term
Important Results on Binomial Coefficients

Divisibility Problems

Divisibility Problems

We can conclude that,

Divisibility Problems

(ii)Divisibility Problems

(iii)

Multinomial theorem

Multinomial theorem

(iii) The general term in the above expansion is

Multinomial theorem

(iv)The greatest coefficient in the expansion of (x1 + x2 + … +

greatest coefficient

where q and r are the quotient and remainder respectively, when n is divided by m.

(v) Number of non-negative integral solutions of x1 + x2 + … + xn = n is n+ r – 1Cr – 1

R-f Factor Relations

Here, we are going to discuss problem involving (√A + B)sup>n = I + f, Where I and n are positive integers.

0 le; f le; 1, |A – B2| = k and |√A – B| < 1

Binomial Theorem for any Index If n is any rational number, then

Binomial Theorem for any Index

(i) If in the above expansion, n is any positive integer, then the series in RHS is finite otherwise infinite.

(ii) General term in the expansion of (1 + x)n is Tr + 1 = n(n – 1)(n – 2)… [n – (r – 1)] / r! * x r

(iii) Expansion of (x + a)n for any rational index

Binomial Theorem for any Index
rational index

rational index

Important Results

(i) Coefficient of xm in the expansion of (axp + b / xq)n is the coefficient of Tr + l where r = np – m / p + q

(ii) The term independent of x in the expansion of axp + b / xq)n is the coefficient of Tr + l where r = np / p + q

(iii) If the coefficient of rth, (r + l)th and (r + 2)th term of (1 + x)n are in AP, then n2 – (4r+1) n+ 4r2 = 2

(iv) In the expansion of (x + a)n

Tr + 1 / Tr = n – r + 1 / r * a / x

Important Results

(vi) If the coefficient of pth and qth terms in the expansion of (1 + x)n are equal, then p + q = n + 2

(vii) If the coefficients of xr and xr + 1 in the expansion of a + x / b)n are equal, then n = (r + 1)(ab + 1) – 1

(viii) The number of term in the expansion of (x1 + x2 + … + xr)n is n + r – 1C r – 1.

(ix) If n is a positive integer and a1, a2, … , am ∈ C,then the coefficient of xr in the expansion of
Important Results

number of term

Important Points to be Remembered

(xi) Total number of terms in the expansion of (a + b + c + d)n is (n + l)(n + 2)(n + 3) / 6.

Important Points to be Remembered

(i) If n is a positive integer, then (1 + x)n contains (n +1) terms i.e., a finite number of terms. When n is general exponent, then the expansion of (1 + x)n contains infinitely many terms.

(ii) When n is a positive integer, the expansion of (l + x)n is valid for all values of x. If n is general exponent, the expansion of (i + x)n is valid for the values of x satisfying the condition |x| < 1.



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