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1.Sets, Relation and Functions
12-
Lecture1.1
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Lecture1.2
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Lecture1.3
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Lecture1.4
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Lecture1.5
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Lecture1.6
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Lecture1.7
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Lecture1.8
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Lecture1.9
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Lecture1.10
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Lecture1.11
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Lecture1.12
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2.Trigonometric Functions
28-
Lecture2.1
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Lecture2.2
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Lecture2.3
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Lecture2.4
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Lecture2.5
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Lecture2.6
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Lecture2.7
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Lecture2.8
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Lecture2.9
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Lecture2.10
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Lecture2.11
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Lecture2.12
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Lecture2.13
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Lecture2.14
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Lecture2.15
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Lecture2.16
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Lecture2.17
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Lecture2.18
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Lecture2.19
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Lecture2.20
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Lecture2.21
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Lecture2.22
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Lecture2.23
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Lecture2.24
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Lecture2.25
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Lecture2.26
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Lecture2.27
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Lecture2.28
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3.Mathematical Induction
5-
Lecture3.1
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Lecture3.2
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Lecture3.3
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Lecture3.4
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Lecture3.5
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4.Complex Numbers and Quadratic Equation
15-
Lecture4.1
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Lecture4.2
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Lecture4.3
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Lecture4.4
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Lecture4.5
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Lecture4.6
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Lecture4.7
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Lecture4.8
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Lecture4.9
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Lecture4.10
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Lecture4.11
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Lecture4.12
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Lecture4.13
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Lecture4.14
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Lecture4.15
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5.Linear Inequalities
4-
Lecture5.1
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Lecture5.2
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Lecture5.3
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Lecture5.4
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6.Permutations and Combinations
5-
Lecture6.1
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Lecture6.2
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Lecture6.3
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Lecture6.4
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Lecture6.5
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7.Binomial Theorem
19-
Lecture7.1
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Lecture7.2
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Lecture7.3
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Lecture7.4
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Lecture7.5
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Lecture7.6
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Lecture7.7
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Lecture7.8
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Lecture7.9
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Lecture7.10
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Lecture7.11
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Lecture7.12
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Lecture7.13
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Lecture7.14
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Lecture7.15
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Lecture7.16
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Lecture7.17
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Lecture7.18
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Lecture7.19
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8.Sequences and Series
14-
Lecture8.1
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Lecture8.2
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Lecture8.3
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Lecture8.4
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Lecture8.5
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Lecture8.6
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Lecture8.7
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Lecture8.8
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Lecture8.9
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Lecture8.10
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Lecture8.11
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Lecture8.12
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Lecture8.13
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Lecture8.14
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9.Properties of Triangles
2-
Lecture9.1
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Lecture9.2
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10.Straight Lines
30-
Lecture10.1
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Lecture10.2
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Lecture10.3
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Lecture10.4
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Lecture10.5
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Lecture10.6
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Lecture10.7
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Lecture10.8
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Lecture10.9
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Lecture10.10
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Lecture10.11
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Lecture10.12
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Lecture10.13
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Lecture10.14
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Lecture10.15
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Lecture10.16
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Lecture10.17
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Lecture10.18
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Lecture10.19
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Lecture10.20
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Lecture10.21
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Lecture10.22
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Lecture10.23
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Lecture10.24
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Lecture10.25
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Lecture10.26
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Lecture10.27
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Lecture10.28
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Lecture10.29
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Lecture10.30
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11.Conic Sections
21-
Lecture11.1
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Lecture11.2
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Lecture11.3
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Lecture11.4
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Lecture11.5
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Lecture11.6
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Lecture11.7
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Lecture11.8
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Lecture11.9
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Lecture11.10
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Lecture11.11
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Lecture11.12
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Lecture11.13
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Lecture11.14
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Lecture11.15
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Lecture11.16
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Lecture11.17
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Lecture11.18
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Lecture11.19
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Lecture11.20
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Lecture11.21
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12.Coordinate Geometry
8-
Lecture12.1
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Lecture12.2
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Lecture12.3
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Lecture12.4
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Lecture12.5
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Lecture12.6
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Lecture12.7
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Lecture12.8
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13.Three Dimensional Geometry
3-
Lecture13.1
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Lecture13.2
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Lecture13.3
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14.Limits And Derivatives
12-
Lecture14.1
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Lecture14.2
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Lecture14.3
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Lecture14.4
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Lecture14.5
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Lecture14.6
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Lecture14.7
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Lecture14.8
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Lecture14.9
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Lecture14.10
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Lecture14.11
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Lecture14.12
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15.Mathematical Reasoning
3-
Lecture15.1
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Lecture15.2
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Lecture15.3
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16.Statistics
5-
Lecture16.1
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Lecture16.2
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Lecture16.3
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Lecture16.4
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Lecture16.5
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17.Probability
3-
Lecture17.1
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Lecture17.2
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Lecture17.3
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18.Binary Number
2-
Lecture18.1
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Lecture18.2
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Chapter Notes – Binomial Theorem
Binomial Theorem for Positive Integer
If n is any positive integer, then
This is called binomial theorem.
Here, nC0, nC1, nC2, … , nno are called binomial coefficients and
nCr = n! / r!(n – r)! for 0 ≤ r ≤ n.
Properties of Binomial Theorem for Positive Integer
- Total number of terms in the expansion of (x + a)n is (n + 1).
- The sum of the indices of x and a in each term is n.
- The above expansion is also true when x and a are complex numbers.
- The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients and
nCr = nCn – r, r = 0,1,2,…,n.
- General term in the expansion of (x + c)n is given by Tr + 1 = nCrxn – r ar.
- The values of the binomial coefficients steadily increase to maximum and then steadily decrease .
- (vii)
- The coefficient of xr in the expansion of (1+ x)n is nCr. (x)
- (a)
(b)
- (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.
- In the binomial expansion of (x + a)n, the r th term from the end is (n – r + 2)th term from the
beginning.
- 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
- It n is even, then in the expansion of (x + a)n, the middle term is (n/2 + 1)th terms.
- 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
- If n is even, then in (x + a)n, the greatest coefficient is nCn / 2
- 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
- If n + 1 / x/a + 1 is an integer = p (say), then greatest term is Tp == Tp + 1.
- 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
Divisibility Problems
From the expansion, (1+ x)n = 1+ nC1x + nC1x2+ … +nCnxn We can conclude that,
(i) (1+ x)n – 1 = nC1x + nC1x2+ … +nCnxn is divisible by x i.e., it is multiple of x. (1+ x)n – 1 = M(x)
(ii)
(iii)
Multinomial theorem
For any n ∈ N,
(i)
(ii)
- The general term in the above expansion is
- The greatest coefficient in the expansion of (x1 + x2 + … +
xm)n is where q and r are the quotient and remainder respectively, when n is divided by m.
- 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
- If in the above expansion, n is any positive integer, then the series in RHS is finite otherwise infinite.
- General term in the expansion of (1 + x)n is Tr + 1 = n(n – 1)(n – 2)… [n – (r – 1)] / r! * xr
- Expansion of (x + a)n for any rational index
(vii) (1 + x)– 1 = 1 – x + x2 – x3 + …∞
(viii) (1 – x)– 1 = 1 + x + x2 + x3 + …∞ (ix) (1 + x)– 2 = 1 – 2x + 3x2 – 4x3 + …∞ (x) (1 – x)– 2 = 1 + 2x + 3x2 – 4x3 + …∞ (xi) (1 + x)– 3 = 1 – 3x + 6x2 – …∞
(xii) (1 – x)– 3 = 1 + 3x + 6x2 – …∞
(xiii) (1 + x)n = 1 + nx, if x2, x3,… are all very small as compared to x.
Important Results
- Coefficient of xm in the expansion of (axp + b / xq)n is the coefficient of Tr + l where r = np – m / p + q
- The term independent of x in the expansion of axp + b / xq)n is the coefficient of Tr + l where r = np / p + q
- 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
- In the expansion of (x + a)n Tr + 1 / Tr = n – r + 1 / r * a / x
- (a) The coefficient of xn – 1 in the expansion of (x – l)(x – 2) ….(x – n) = – n (n + l) / 2
(b) The coefficient of xn – 1 in the expansion of (x + l)(x + 2) ….(x + n) = n (n + l) / 2
- If the coefficient of pth and qth terms in the expansion of (1 + x)n are equal, then p + q = n + 2
- 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
- The number of term in the expansion of (x1 + x2 + … + xr)n is n + r – 1C r – 1.
- If n is a positive integer and a1, a2, … , am ∈ C, then the coefficient of xr in the expansion of (a1 + a2x + a3x2 +… + amxm – 1)n is
- For |x| < 1,
(a) 1 + x + x2 + x3+ … + ∞ = 1 / 1 – x (b) 1 + 2x + 3x2 + … + ∞ = 1 / (1 – x)2
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
- 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.
- 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.