RESONANCE - Binomial Theorem

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Part I Part II Part III Part IV Part - V. Part - VI. DPP No. _____ Home Assignment : Home Assignment : Home Assignment : Home Assignment : Home Assignment : Home Assignment :

Signature HOD Signature Faculty

DAILY LECTURE SHEET : ACADEMIC SESSION 2008-09

Title/Sub title of the Topic : ____________________________________________________________ Lecture No. of the particular topic : _____________________________________Cumulative Lecture No. ______ Contents to be discussed :

DPP No. _______ Home Assignments : ________________________________________________________ _________________________________________________________________________________________________ Theory Contents to be Covered in the Lecture : Assignments to be given after the Lecture :

Discussion : 2 Lecture

Batch : R

Binomial Theorem for positive index Theorem + basic properties

General term, middle term

Coefficient of xk_{in (ax + b)}n

Num greatest coefficient Num greatest term

Remainder

1

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LECTURE – 1 1. Expand the following binomials :

(i) (x – 3)5 _{[ Ans. x}5_{– 15x}4_{+ 90x}3_{– 270x}2_{+ 405x – 243 ]}

(ii) (2x – y)5 _{[ Ans. 32x}5_{– 80x}4_{y + 80x}3_y2_{– 40x}2_y3_{+ 10xy}4_{– y}5 _]

2. Find

(i) 28th term of (5x + 8y)30 _{[ Ans.}

! 3 ! 27 ! 30 (5x)3_{. (8y)}27_] (ii) 7th term of 9 x 2 5 5 x 4        _[Ans. 3 x 10500 ]

3. Find middle term of (i)

14 2 2 x 1          _{& (ii)} 9 3 6 a a 3          _{[Ans. (i)} 16 429  x14_{, (ii)} 8 189 a17_, 16 21  a19_]

4. Find the term independent of x in

9 2 x 3 1 x 2 3        [Ans. 18 7 ]

D5. Find the coefficients of x32_{& x}–17_in

15 3 4 x 1 x        [Ans. 1365, –1365]

6. Find the coefficient of x–1_{in (1 + 3x}2_{+ x}4₎

8 x 1 1        _{[Ans. 232]}

7. Find the numerically greatest coefficient in the expansion of (3x – 2)21 _{[ Ans. T} 9 ]

8. Find the algebracally greatest and least coefficient in the expansion of (2 – 5x)48

[ Ans. T₃₅, T₃₆]

9. Find the greatest term in the expansion of (7 – 5x)11_{where x =}

3 2 [Ans. T₄ = 9 440 × 78_{× 5}3_]

D10. Find numerically greatest term in the expansion of (3 – 5x)15_{when x =}

5 1

. [ Ans. T₃, T₄ ] 11. If n is a positive integer, show that

(i) 9n_{+ 7 is divisible by 8.}

D(ii) 32n+1_{+ 2}n+2_{is divisible by 7}

D(iii) 62n_{– 35n – 1 is divisible by 1225.}

12. What is the remainder when 599_{is divided by 13} _{[Ans. 8]}

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Batch : R

Sum of series involves binomial coefficient upper index is constant

(i) By proper substitution (ii)



tk =



tn k (iii) By reduction r n_C r = n n–1_C r–1 (iv) Calculus



Cr ,



rCr ,



(1)Cr ,



1Cr r 1 ,



r 2 C r ... 2 (A + B)n Binomial Theorem

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LECTURE – 2

1. _{If n is +ve integer, prove that the integral part of (7 + 4 3 )}n_{is an odd number..}

D2. _{If n is +ve integer, prove that the integeral part of (7 + 2 5 )}2n + 1_{is an even number..}

D3. _{If (7 + 4 3 )}n_{= , where  is a +ve integer and  is a proper fraction, then prove that}

(1–) () = 1

4. _{Show that the integer just above ( 3 +1)}2n_{is divisible by 2}n + 1_{for all n  N,}

5. If (1 + x)n_{= C}

0 + C1x + C2x

2_{+ ...+ C} nx

n_{, then show that}

(i) C₀ + C₁ + C₂ + ...+C_n = 2n (ii) C₀ + C₂ + C₄ + ... = 2n–1 D(iii) C₁ + C₃ + C₅ + ... = 2n–1 (iv) C₀ + 3C₁ + 32_C 2 + ... = 4 n_. 6. If (1 + x)n_{= C} 0 + C1x + C2x 2_{+ ...+ C} nx

n_{, then show that}

(i) C₁ + 2C₂ + 3.C₃ + ... + nC_n = n.2n–1

(ii) C₀ + 2C₁ + 3.C₂ + ... + (n+1) . C_n = 2n–1_(n+2)

(iii) C₀ + 3C₁ + 5C₂ + ... + (2n+1) . C_n = 2n_(n+1)

D(iv) a.C₀ + (a – b).C₁ _ (a – 2b)C₂ + ...+(a – nb).C_n = 2n–1_{(2a – nb)}

(v) C₀ – 22_.C 1 + 3 2_C 2 + ...+ (–1) n_(n+1)2 _.C n = 0, n > 2. D(vi) a2_.C 0 – (a – 1) 2_.C 1 + (a – 2) 2_.C 2 – (a – 3) 2_.C 3 + ...+ (–1) n _{(a – n)}2_.C n = 0. (vii) 1 n 1 2 1 n C ... 3 C 2 C C 1 n n 2 1 0          (viii) 1 n 1 ... 4 C 3 C 2 C C 1 2 3 0       (ix)



      n 0 r r r 2 n 1 ) 2 r )( 1 r ( C . ) 1 ( . (x)



   n 0 r r 1 2 r C_r = 1 n 1 ) 3 n ( 2n    .

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Batch : R

3

Problem of double sigma Product of binomial coefficient Upper index is variable (i) n_C r + n_C r + 1 = n+1_C r+1

(ii) Using binomial theorem for negative index

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LECTURE – 3 1. (i) n_C n + n+1_C n+ n+2_C n+ ...+ n+k_C n = n+k_C n+1 . (ii) m_C 1 + m+1_C 2 + m+2_C 3 + ... + m+n–1_C n = n_C 1 + n+1_C 2 + n+2_C 2 + ...+ n+m–1_C m. 2. Prove that If (1 + x)n_{= C} 0 + C1x + C2x 2_{+...+ C} nx n_, (i) C₀2_{+ C} 1 2 _{+ ... + C} n 2₌ )! n ( ! ) n 2 ( (ii) C₀C₁ + C₁C₂+ ... + C_n–1 C_n=

 

)! 1 n ( )! 1 n ( ! n 2   (iii) C₀C_n + C_{1 .}C_n–1+ ... + C_n C₀= 2 ) ! n ( )! n 2 ( D(iv)



    n 0 r 3 r r 3)! -(n 3)! (n ! n 2 C C _. (v) m_C r. n_C 0 + m_C r–1. n_C 1 + m_C r–2. n_C 2 + ...+ m_C 1. n_C r–1 + m_C 0. n_C r = m+n_C r ,

where m,n,r are positive integers and r < m, r < n. D(vi) C₀.2n_C

n – C1. 2n–2_C

n + C2. 2n–4_C

n – ... = 2n, where Cr stands for n_C r. 3. Evaluate :

 

  n 0 m m 0 p p m m n_C _. _C Ans. 3n 4. If (1 + x)n_{= C} 0 + C1x + C2x 2_{+ ...+ C} nx n_{, prove that} (i)



(ij)nC_j nC_i = n.22n D(ii)  (C_i + C_j)2_{= (n – 1).}2n_C n + 2 2n_{0  i < j  n,} D5. If (1 + x)n_{= C} 0 + C1x + C2x 2_{+ ...+ C} nx n_{, show that}  i.j.C_iC_j=n2_{ 1 n 2 n 2 3 n 2 _. _C 2 1 2    _ } 0  i < j  n. D6. Prove that If (1 + x)n_{= C} 0 + C1x + C2x 2_{+...+ C} nx n_, (i) ₂ 2 n 2 2 2 1 2 0 ] ! 1) [(n ! ) 1 n 2 ( 1 n C ... 3 C 2 C C         (ii) C₁2_{– 2.C} 2 2_{+ 3.C} 3 2_{– ...– 2n.C}2 2n = (–1) n–1_.n.C

n where Cr stands for 2n_C r. (iii)



    n 0 r n 2 n 2 2 2 r n . C C ) r n ( r _{, where C} r stands for n_C r. (iv) C₀– C₁ + C₂ – ...+ (–1)m–1_.C m–1 = (–1) m–1 ! ) 1 m ( ) 1 m n )...( 2 n )( 1 n (      where C_r stands for n_C

r. D7. If (1 + x)n_{= C} 0 + C1x + C2x 2_{+ ...+ C} nx n_{, show that }



           n 0 r r 2 j i C 1 2 n C j C i , 0  i < j  n.

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Batch : R

Binomial theorem for negative and fractional index

Multinomial theorem

Binomial Theorem

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LECTURE – 4 1. Write the expansion of

(i) (1 – x)–1 _{(ii) (1 – x)}–2 _{(iii) (1 – x)}–n

2. Find the general term in the expansion of (2–3x2₎–2/3 _[Ans.

r 3 2 2 ! r ) 1 r 3 ...( 8 . 5 . 2 2   x2r_]

D3. Find the no. of term in the expansion of (a + b + c + d + e + f)n_. _[Ans.n+5_c n]

D4. Find the coeff. of a3_.b4_.c7_{in the expansion of (bc + ca + ab)}8_. _{[Ans. 280]}

5. Find the coefficient of x3 _y4 _z2_{in the expansion of (2x – 3y + 4z)}9_. _Ans.

! 2 ! 4 ! 3 ! 9 23₃4₄2

6. Find the coefficient of x4_{in (1 + x – 2x}2₎7 _{Ans. 14}

D7. The sum of rational terms in



₃ ₆



10

5 3