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Cengage Maths Solutions Class 12 Binomial Theorem - Algebra

Cengage Maths Solutions Class 12 Binomial Theorem PDfs provides accurate solutions to practice before your final exam. Students can study this comprehensive Cengage Maths PDf Solutions on Binomial Theorem and its various sub-topics for effective exam preparation. The Cengage 12 PDF solutions help the student to swiftly practice and revise the important concepts just before the annual exam to score high marks in maths.<br><br>For more such video solutions/Pdf solution, visit https://doubtnut.com and clear your doubt instantly.

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Cengage Maths Solutions Class 12 Binomial Theorem - Algebra

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  1. CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER BINOMIAL THEOREM || ALGEBRA  Download Doubtnut Today Ques No. Question CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction Simplify: x5+ 10x4a + 40x3a2+ 80x2a3+ 80xa4+ 32a5. x5+ 10x4a + 40x3a2+ 80x2a3+ 80xa4 1 + 32a5.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction 183+ 73+ 3 × 18 × 7 × 25 Find the value of 2 36+ 6 × 243 × 2 + 15 × 18 × 4 + 20 × 27 × 8 + 15 × 9 × 16 + 6 × 3 × 32 + 64  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction Find the approximation of (0. 99)5 using the first three terms of its expansion. 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction Using the binomial theorem, evaluate (102)5 . 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( ) Find the 6th term in expansion of 2x2- 1/3x210. 5  Watch Free Video Solution on Doubtnut

  2. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction Find a if the 7th and 18th terms of the expansion (2 + a)50 are equal. 6  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction Findn, if the ratio of the fifth term form the beginning to the fifth term from the end in the expansion of 7  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( ) Find the coefficient of x4 in the expansion of x/2 - 3/x210. 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( √ ) 21 a b which has the same power of aandb. Find the term in √b3 + 9 a3  Watch Free Video Solution on Doubtnut

  3. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( )[( 3x2/2 - (1/3x) ) ] 9 Find the term independent of x in the expansion of 1 + x + 2x3 10  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction Find the coefficient of xk∈ 1 + (1 + x) + (1 + x)2+ + (1 + x)n(0 ≤ k ≤ n). xk∈ 1 + (1 + x) + (1 + x)2+ 11 + (1 + x)n(0 ≤ k ≤ n).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( ) Find the coefficient of x4 in the expansion of 2 - x + 3x26. 12  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( ) Find the coefficient of x50 in the expansion of (1 + x)101× 1 - x + x2100. 13  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction

  4. ( Find the sum of the coefficients of the first, second, and third terms of the expansion of x2 14 term that does not contain x.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction 50 ∑ r= 0 Find the coefficient of x25 in expansion of expression ^ (50)Cr(2x - 3)r(2 - x)50-r . 15  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction Find the a, b,andn in th expansion of (a + b)n if the first three terms of the expansion are 729, 7290, and 30375, respectively. 16  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction If aandb are distinct integers, prove that a - b is a factor of an- bn , wherever n is a positive integer. 17  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction The number of terms in the expansion of (a + b + c)n,wheren ∈ N.

  5. 18  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( 1 + x240. )( 1 x2 -5 ) Find the coefficient of x20 in x2+ 2 + 19  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction 4 . ( ) Find the coefficient of x5 in the expansion of 1 + x25 1 + x is60. 20  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( ) Find the coefficient of x13 in the expansion of (1 - x)5× 1 + x + x2+ x24. 21  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( ) Find the coefficient of x4 in the expansion of 1 + x + x2+ x311. 22  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( Find the number of terms which are free from radical signs in the expansion of y1/5+ x1/10 23  Watch Free Video Solution on Doubtnut

  6. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction If the number of terms in the expansion of (x + y + z)n are 36, then find the value of n. 24  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction Find the coefficient of a3b4c in the expansion of (1 + a - b + c)9. 25  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction Find the coefficient of a3b4c5 in the expansion of (bc + ca + ab)6. 26  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction ( ) Find the coefficient of x7 in the expansion of 1 + 3x - 2x310. 27  Watch Free Video Solution on Doubtnut

  7. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion Find the sum ^ 10C1+10C3+10C5+10C7+10C9. 28  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion 1 1 1 Find the sum of 1!(n - 1)!+ 1 3!(n - 3)+ 5!(n - 5)+ , 1 1 + + 29 1!(n − 1)! 3!(n − 3) 5!(n − 5) + ,  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion 10 ∑ k=0 ^ (20)Ck. Find the sum 30  Watch Free Video Solution on Doubtnut

  8. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion Find the sum of the series ^ 15C0+15C1+15C2+ +15C7. 31  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion If the sum of coefficient of first half terms in the expansion of (x + y)nis256 , then find the greatest coefficient in the expansion. 32  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) 319. Find the sum of all the coefficients in the binomial expansion of x2+ x - 3 33  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) 51 vanishes, then find the value of If the sum of the coefficient in the expansion of α2x2- 2αx + 1 34  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) Find the sum of the coefficients of all the integral powers of x in the expansion of 1 + 2√x 35  Watch Free Video Solution on Doubtnut

  9. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) If 1 + x - 2x220= a0a1x = a2x2+ a3x3+ + a40x40, (1 + x − 2x2)20= a0a1x = a2x2 36 + a3x3+ + a40x40, then find the value of a1+ a3+ a5+ + a39.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion 10! α!β!γ!= 310. Prove that ∑ α+β+γ= 10 37  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) n is 924x6, then find the value of n. If the middle term in the expansion of x2+ 1/x 38  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion Show that the middle term in the expansion of (1 + x)2nis(1.3. 5(2n - 1)) 2nxn,wheren n! (1.3.5(2n − 1)) (1 + x)2nis 2nxn, 39 n! wheren

  10. is a positive integer.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion If the coefficient of the middle term in the expansion of (1 + x)2n+ 2isα and the coefficients of middle terms in the expansion of (1 + x)2n+1 are β and γ then relate α, βandγ. 40  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Ratio Of Consecutive Terms/Coefficients If the coefficients of three consecutive terms in the expansion of (1 + x)n are in the ratio 1:7:42, then find the value of 41  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Ratio Of Consecutive Terms/Coefficients In the coefficients of rth, (r + 1)th, and(r + 2)th terms in the binomial expansion of m2- m(4r + 1) + 4r2- 2 = 0. 42  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Ratio Of Consecutive Terms/Coefficients ( )( Cn- 1+ Cn )( = C1C2Cn(n + 1)n/n! ) ( ) If Cr= ∧ nCr , then prove that C0+ C1 C1+ C2 (C0+ C1)(C1+ C2)(Cn−1+ Cn) 43 = (C1C2Cn)(n + 1)n/n!

  11.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Ratio Of Consecutive Terms/Coefficients If a1,a2, a3, a4 are the coefficients of any of four consecutive terms in the expansion of a1 a1+ a2 a3+ a4 a2+ a3 a3 2a2 44 . + =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Ratio Of Consecutive Terms/Coefficients n rnCr ^ nCr- 1 ∑ r=1 Find the sum of . 45  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion Find the positive integer just greater than (1 + 0. 0001)10000. 46  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion Find (i) the last digit, (ii) the last two digits, and (iii) the last three digits of 17256. 47  Watch Free Video Solution on Doubtnut

  12. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion If 10m divides the number 101100- 1 then, find the greatest value of m. 48  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion Prove that for each n ∈ N,23n- 1 is divisible by 7. 49  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion Find the remainder when 6n- 5n is divided by 25. 50  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion Find the remainder when 7103 is divided by 25. 51  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion Find the remainder when 599 is divided by 13. 52  Watch Free Video Solution on Doubtnut

  13. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion Find the value of 32003/28 ,where{.} denotes the fractional part. { } 53  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion Find the remainder when 16902608+ 26081690 is divided by 7. 54  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion Find the remainder when x = 555^ (5) (24 times 5) is divided by 24. 55  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion [( 100 is an even integer . ) ] ) 100- √10 - 1 ( Prove that √10 √10 + 1 56

  14.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion If 97+ 79 is divisible b 2n, then find the greatest value of n,wheren ∈ N. 57  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion {( )} ) ( 1 +√4x + 1 1 +√4x + 1 7 7 1 Find the degree of the polynomial - 2 2 √4x + 1 7 1 1 + √4x + 1 {( ) 2 √4x + 1 7 58 1 + √4x + 1 } − ( ) 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion ( ) n= I + f, whare I and n are positive integers and `0 59 If 2 +√3  Watch Free Video Solution on Doubtnut

  15. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion Find the sum C0- C2+ C4- C6+ ,whereCr= ∧ 6Cr=nCr C0− C2+ C4− C6+ ,whereCr= 60 ∧ 6Cr =nCr .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Use Of Complex Numbers In Binomial Theorem ( ) 1 3 2n+ 2cos(nπ) Prove that ^ nC0+nC3+nC6+ = 3 ^ nC0+nC3+nC6+ 61 cos(nπ) 1 (2n+ 2 ) = 3 3 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Use Of Complex Numbers In Binomial Theorem ( If T0,T1, T2, ,Tn represent the terms in the expansion of (x + a)n, then find the value of T0 (T0− T2+ T4− )2+ (T1− T3+ T5 62 − )2n ∈ N.  Watch Free Video Solution on Doubtnut

  16. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Use Of Complex Numbers In Binomial TheoremCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Use Of Complex Numbers In Binomial Theorem ( ) If 1 + x + x2n= a0+ a1x + a2x2+ + a2nx2n, (1 + x + x2) n= a0+ a1x + a2x2+ 63 + a2nx2n, find the value of a0+ a3+ a6+ + , n ∈ N.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Use Of Complex Numbers In Binomial Theorem ( ) 1 20 Find the greatest term in the expansion of √3 1 + . 64 √3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Greatest Term In Binomial Expansion Find the numerically greatest term in the expansion of (3 - 5x)15whenx = 1/5. 65  Watch Free Video Solution on Doubtnut

  17. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Greatest Term In Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Greatest Term In Binomial Expansion Given that the 4th term in the expansion of [2 + (3/8x)]10 has the maximum numerical value. Then find the range of value of 66  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Greatest Term In Binomial Expansion Find the greatest coefficient in the expansion of (1 + 2x/3)15. . 67  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Greatest Term In Binomial Expansion Find the sum C0+ 3C1+ 32C2+ + 3nCn. 68  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series n ∑ r=0 If (1 + x)n= Crxr, then prove that C1+ 2c2+ 3C1+ + nCn= n2n-1. . 69  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series ( ) If 1 + x + x2+ + xpn= a0+ a1x + a2x2+ + anpxnp, (1 + x + x2+ + xp) n= a0+ a1x 70 + a2x2+ + anpxnp, .. + npanp. then find the value of a1+ 2a2+ 3a3+  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series Find the sum 1C0+ 2C1+ 3C2+ + (n + 1)Cn, whereCr=nCr. 1C0+ 2C1+ 3C2+ + (n + 1)Cn, 71 whereCr =nCr.

  18.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series Find the sum 1 × 2 × C1+ 2 × 3C2++n(n + 1)Cn, whereCr= ∧ nCr. 72  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series If n > 2, then prove that C1(a - 1) - C2× (a - 2) + + ( - 1)n- 1Cn(a - n) = a, whereCr=nCr. C1(a − 1) − C2× (a − 2) + + ( − 1)n−1Cn(a − n) = a,whereCr 73 =nCr.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series Find the sum 3nC0- 8nC1+ 13nC2×nC3+. 74  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series

  19. 75 n ∑ r=0 ^ nCrxryn-r. If x + y = 1, prove that  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series n C1 2 Cn n + 1= 2n+1- 1 n + 1 ∑ r=0 If (1 + x)n= ^ nCr , show that C0+ . + + 76  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series C1 2 C3 4 C5 6 2n+1- 1 n + 1 . Prove that + + + = 77  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series 23 2C1+ 23 3C2+ 24 24 4C3+ +211 11C10. Find the sum 2C0 23 23 2C0 C1+ C2+ C3 + 78 3 2 4 211 + C10. 11

  20.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series 2 ^ nC2 ( ) 3 ^ nC3 ( ) ( ) - + ( - 1)n+1n ^ nCn ^ nC1 2 1 Prove that = - + n + 1 ^ nC1 3 4 n + 1 2( ^ nC2) 1 = − n + 1 2 3 79 3( ^ nC3) + − 4 n( ^ nCn) + ( − 1)n+1 n + 1 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series m ∑ r=1 If kandn are positive integers and sk= 1k+ 2k+ 3k+ + nk, then prove that ^ (m + 1)Crsr m ^ (m + 1)Crsr= (n + 1)m+1 ∑ 80 r=1 − (n + 1).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series n ( ) 1 2+1 3+ +1 1 n n ∑ r= 1 ( - 1)r-11 + Prove that Cr= r n 1 1 ( − 1)r−1(1 + ∑ r=1 + + 81 3 2 n 1 1 ) + Cr= r n .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series C1 1 C2 2 C3 3 C4 4 ( - 1)n- 1 n Cn= 1 +1 2+1 3+ +1 . Prove that - + - + + n 82

  21. C1 C2 C3 C4 − + − + 3 1 2 4 ( − 1)n−1 1 1 + Cn = 1 + + + 3 2 n 1 + . n  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series n ∑ r= 0 ^ nCrsinrxcos(n - r)x = 2n-1sin(nx). Prove that n ∑ 83 ^ nCrsinrxcos(n − r)x r=0 = 2n−1sin(nx).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series There are two bags each of which contains n balls. A man has to select an equal number of balls from both the bags. Prove that the number of ways in which a man can choose at least one ball from each bag is2nCn- 1. 84  Watch Free Video Solution on Doubtnut

  22. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series r ∑ i= 0 .n1Cr-i.n2Ci . Find the sum 85  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series 2n ∑ r= 0 ( ) 2= n4nC2n . .2nCr Prove that 86  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series Using binomial theorem (without using the formula for ^ nCr ) , prove that nC4+mC2-mCn nC4 +mC2 −mCn 1C2 =mC4 −m+nCm 1C3+m+nCm 2C2 −m+nCm 3C1+m+nC4. 1C 87  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series Prove that (r + 1)nCr- rnCr+ (r - 1)nC2-nC3+ + ( - 1)r^ nCr= ( - 1)r^ (n - 2)Cr. (r + 1)nCr− rnCr+ (r − 1)nC2−nC3 + + ( − 1)r^ nCr= ( − 1)r 88 ^ (n − 2)Cr.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series n ∑ r=0 Find the sum ^ (n + r)Cr . 89  Watch Free Video Solution on Doubtnut

  23. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series Prove that nC2n 0Cn-nC2n- 1 Cn+nC2×2n-2Cn+ + ( - 1)n^ nCn Cn+nC2×2n−2Cn nCn= 1. 1 0Cn−nC2n−1 + + ( − 1)n^ nCn nC2n 1 90 nCn = 1.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series Prove that ^ mCn 1Cm-mC2n 2Cm+mC3n 2Cm+mC3n 3Cm≡ ( - 1)m -1nm. 1Cm−mC2n ≡ ( − 1)m−1nm. ^ mCn 3Cm 91  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series

  24. If 18x2+ 12x + 4 ( ) n= a0+ a1x+ a2x2 + + a2nx2n, n= a0+ a1x + a2x2 (18x2+ 12x + 4) + + a2nx2n, prove that ar= 2n3r ( ) 92 ^ (2n)Cr+nC2n-2 Cr+nC2n- 4 Cr+ 1 2 ar= 2n3r( ^ (2n)Cr+nC2n−2 Cr 1 +nC2n−4 Cr + ) 2 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series Find the sum ∑∑ 0≤i<j≤nnCinCj 93  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series 94 Find the value of `sumsum_(0lt=i  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series Find the value of ∑ ∑ 1≤i≤j≤n-1(ij)ncn icj. 95  Watch Free Video Solution on Doubtnut

  25. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series ( ) Find the value of ∑ ∑ 0≤i≤j≤n(i + j) nCi+ nCj . 96  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series Find the sum ∑∑ 0≤i<j≤nnCi 97  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series s n ∑ r= 0 ∑ s= 1 ^ nCn sCr= 3n- 1. Prove that 98  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series 99 Find the sum `sumsum_(0lt=i  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series m(m - 1) 1 × 2 Find the condition for which the formula (a + b)mam+ mam-1b + am-2b2+

  26. 100 (a + b)mam+ mam−1b m(m − 1) am−2b2+ + 1 × 2 holds.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series Find the value of x, for which 1/√5 + 4x can be expanded as infinite series. ( ) 101  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any Index Find the fourth term in the expansion of (1 - 2x)3/2. 102  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any Index [ ] Prove that the coefficient of xr in the expansion of (1 - 2x)1/2is(2r)!/ 2r(r!)2. 103  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any Index 104

  27. 1 8+1 3 1 × 3 × 5 8 × 16 × 24+ Find the sum 1 - 8× 16-  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any Index . . . . . . Show that √3 = 1 +1 3+1 . 3 3 6+1 3 6 5 9+1 3 6 5 9 7 12+ 3 3 . 3 3 . 5 1 1 1 √3 = 1 + + + 105 3 . 7 3 6 3 6 9 . . 1 3 5 + + 3 6 9 12  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any Index ( 1 + Assuming x to be so small that x2 and higher power of x can be neglected, prove that −4(16 − 3x) 1 3x (1 + ) 2 106 4 = 1 2 (8 + x) 3 305 − ( )x 96  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any Index . y2 2x2 . x x √ √ If x is very large as compare to y, then prove that x - y= 1 + 107 x + y  Watch Free Video Solution on Doubtnut

  28. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any IndexCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any Index ( ) -1. Find the coefficient of xn in the expansion of 1 - 9x + 20x2 108  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any Index Find the constant term in the expansion of (x - 1/x)6. 109  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any Index ( ) a x+ bx 12 Find the coefficient of x-10 in the expansion of . 110  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series ( ) x2 2!+ + xn n! 2 x Find the coefficient of xn in 1 + . 1!+ 111  Watch Free Video Solution on Doubtnut

  29. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series n ar(x - α + 2)r- br(α - x - 1)r { } ∑ r= 0 n ∑ If = 0, {ar(x − α + 2)r− br(α − x − 1)r} r=0 112 = 0, then prove that bn- ( - 1)nan= 0.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series n n 2+1cos(nπ/4),wherei =√-1. ^ nCr( - 1)ri + i2r+ i3r+ i4r [ ] ∑ r= 0 = 2n+ 2 Prove that n ^ nCr( − 1)r[i + i2r+ i3r+ i4r] ∑ 113 r=0 n +1cos(nπ/4),wherei = 2n+ 2 2 = √−1.  Watch Free Video Solution on Doubtnut

  30. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series . Let a = 21/401- 1 and for each n ≥ 2, letbn=nC1+nC2 ( ) a +nC3a2+ ...... +nCn⋅ an- 1 .a +nC3a2+ 114 n ≥ 2, letbn=nC1+nC2 ...... +nCn⋅ an−1 ( ) . . Find the value of b2006- b2005  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series If n is a positive integer, prove that 1 - 2n +2n(2n - 1) 2n(2n - 1)(2n - 2) 3! + + ( - 1)n-12n(2n - - 2! (n 2n(2n − 1) 1 − 2n + 2! 2n(2n − 1)(2n − 2) − + 115 3! 2n(2n − 1)(n + 2) + ( − 1)n−1 (n − 1)! = ( − 1)n+1(2n)!/2(n!)2 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series ^ nC0 x ^ nC0 ^ nC0 x + 1 ^ nC1 x + 2 - + ( - 1)n^ nCn n! Prove that - + = x(x + 1)(x - n), x + n ^ nC0 ^ nC1 − + − x + 1 x + 2 116 x n! ^ nCn + ( − 1)n = , x + n x(x + 1)(x − n) where n is any positive integer and x is not a negative integer.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series 1 2 [ ] Prove that ^ 100C100 C2+100C100 C4+100C100 C6+ +100C100 ^ (200)C98-100C49. 98C100= 2 2 4 2C2+100C100 C4 +100C100 ^ 100C100 C6 2 4 1 117 + +100C100 98C100= [ ^ (200)C98 2 −100C49].  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series

  31. m -1 ∑ r= 1 2r2- r(m - 2) + 1 (m - r)mCr 1 m . Prove that = - 118  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion Find the coefficients of x50 in the expression (1 + x)1000+ 2x(1 + x)999+ 3x2(1 + x)998+ + 1001 (1 + x)1000+ 2x(1 + x)999 + 3x2(1 + x)998+ + 1001x1000 119 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous . . . bn-b {( b1+nC2 b2+ +nCn If b1,b2bn are the nth roots of unity, then prove that ^ nC1 . b1 +nC2 b2+ +nCn 1 + b2 b . . ^ nC1 bn 120 b {(1 + b2)n− 1} . − b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series

  32. ( ) 2√2 ^ nC2 ^ nC4 ^ nC6 .. = n . If n = 12m(m ∈ N), prove that ^ nC0- 2+ 4- 6+ ( ) ( ) ( ) 1 +√3 2 +√3 2 +√3 2 +√3 ^ nC2 ^ nC4 ^ nC0− + 2 4 121 (2 + √3) (2 + √3) n 2√2 ^ nC6 .. = ( ) − + . 6 1 + √3 (2 + √3)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Ratio Of Consecutive Terms/Coefficients Prove that in the expansion of (1 + x)n(1 + y)n(1 + z)n , the sum of the coefficients of the terms of degree 122  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series ( ) 2n+ √3 - 1 ( ) 2n , then prove that Un+1= 8Un- 4Un-1. If Un= √3 + 1 123  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion (1 - x)(1 - 2x)(1 - 3x)is1 1 3n+2- 2n+3+ 1 ( ) Prove that the coefficient of xn in the expansion of 2 1 is (3n+ 2 1 2 124 (1 − x)(1 − 2x)(1 − 3x) − 2n+3+ 1).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction If the coefficients of 5th, 6th , and 7th terms in the expansion of (1 + x)n are in A.P., then n these 125  Watch Free Video Solution on Doubtnut

  33. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion (m)!(2n - m)! b. ( (2n)! If xm occurs in the expansion x + 1/x2n , then the coefficient of xm is ( ) 126 these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion The coefficient x5 in the expansion of (1 + x)21+ (1 + x)22+ + (1 + x)30 is a. 51C5 b. 9C5 c. 31 127  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion

  34. n! (2n)! 128 The coefficient of 1/x in the expansion of (1 + x)n(1 + 1/x)n is (n - 1)!(n + 1)! b. (n - 1)!(n +  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion In the expansion of 1 + 3x + 2x26 , the coefficient of x11 is a. 144 b. 288 c. 216 d. 576 ( ) 129  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion If the coefficients of rth and (r + 1)th terms in the expansion of (3 + 7x)29 are equal, then r equals a. 15 b. 21 c. 14 d. none of these 130  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) 1 x2 n In the expansion of x3- , n ∈ N , if the sum of the coefficients of x5andx10is 0 , then n 131  Watch Free Video Solution on Doubtnut

  35. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion 2n ∑ r= 0 . If 1 + 2x + x2n= ( ) ( ) arxr, thena = 2 b. ^ nCr ^ nCr+ 1 c. ^ 2nCr d. ^ 2nCr+1 ^ nC2 132  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Ratio Of Consecutive Terms/Coefficients If in the expansion of (1 + x)n,a, b,c are three consecutive coefficients, then n =ac + ab + b2+ a 133 these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion The coefficient of the middle term in the binomial expansion in powers of x of(1 + αx)4 and of 134 10 3 c. - 10 d. 3 3 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion The coefficient of xn in the expansion of (1 - x)(1 - x)n is n - 1 b. ( - 1)n(1 - n) c. ( - 1)n-1(n - 1) 135  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion If (1 + x)5= a0+ a1x + a2x2 + a3x3+ a4x4+ a5x5, (1 + x)5= a0+ a1x + a2x2 + a3x3 + a4x4+ a5x5, 136 ( ) 2+ a1- a3+ a5 ( ) 2 is equal to 243 b. 32 c. 1 d. 210 then the value of a0- a2+ a4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion [ ( )] [ 1 11 If the coefficient of x7∈ ax2- equal the coefficient of x-7 in satisfy the ax bx2 137 a + b = 1 b. a - b = 1 c. b = 1 d. a = 1 b  Watch Free Video Solution on Doubtnut

  36. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion 10! The coefficient of a8b4c9d9 in (abc + abd + acd + bcd)10 is 10! b. 8!4!9!9! c. 2520 d. none of these 138  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) 5 is -83 b. -82 c. -86 d. -81 The coefficient of x5 in the expansion of x2- x - 2 139  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion 20! 213! b. - 20! 213! c. 20! 5!2!3! d. none of these The coefficient of x2y3 in the expansion of (1 - x + y)20 is 140  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) k 10 If the term independent of x in the √x - is 405, then k equals 2, - 2 b. 3, - 3 c. 4, - 4 141 x2

  37.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) The coefficient of x10 in the expansion of 1 + x2- x38 is 476 b. 496 c. 506 d. 528 142  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion The coefficient of xn in (1 + x)1011 - x + x2100 is nonzero, then n cannot be of the form 3r + ( ) 143  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion The coefficient of x28 in the expansion of 1 + x3- x630 is 1 b. 0 c. ^ 30C6 d. ^ 30C3 ( ) 144  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) 1 -30 The term independent of a in the expansion of 1 +√a + is ^ 30C20 b. 0 c. ^ 30 145 √a - 1  Watch Free Video Solution on Doubtnut

  38. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion 100 ∑ m =0 The coefficient of x53 in the expansion ^ (100)Cm(x - 3)100-m2m is ^ 100C47 b. ^ 100C53 146  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( x + 1 x - 1 x - x1 The coefficient of the term independent of x in the exampansion of - 147 x2/3- x1/3+ 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion In the expansion of 1 + x + x3+ x410, the coefficient of x4 is ^ 40C4 b. ^ 10C4 c. 210 d. 310 ( ) 148  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion If coefficient of a2b3c4∈ (a + b + c)m(wherem ∈ N)isL(L ≠ 0) a2b3c4∈ (a + b + c)m(wherem 149 ∈ N)isL(L ≠ 0) , then in same expansion coefficient of a4b4c1 will be L b. L 3 c. mL d. L 4 2  Watch Free Video Solution on Doubtnut

  39. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion The last two digits of the number 3400 are 81 b. 43 c. 29 d. 01 150  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion )(( )) The expression √2x2+ 1 +√2x2- 1 ( 2 6 6+ □ √2x2+ 1 +√2x2- 1 6 (√2x2+ 1 +√2x2− 1) 151 6 ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ 2 + □ (√2x2+ 1 + √2x2− 1) is polynomial of degree 6 b. 8 c. 10 d. 12  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series The coefficient of xr[0 ≤ r ≤ (n - 1)] in lthe expansion of (x + 3)n-1+ (x + 3)n- 2(x + 2) + (x + 3 (x + 3)n−1+ (x + 3)n−2(x + 2) + (x + 3)n− 3(x + 2)2+ + (x + 2)n− 1 ( ) 152 is ^ nCr3r- 2n b. ^ nCr3n- r- 2n-r c. ^ nCr3r+ 2n-r d. none of these ( ) ( )  Watch Free Video Solution on Doubtnut

  40. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial ExpansionCENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion If 1 + 2x + 3x210= a0+ a1x + a2x2+ + a20x20, thena1 ( ) 10= a0+ a1x + a2x2+ (1 + 2x + 3x2) 153 + a20x20,thena1 equals 10 b. 20 c. 210 d. none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion In the expansion of 51/2+ 71/81024, the number of integral terms is 128 b. 129 c. 130 d. 131 ( ) 154  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion ( The total number of terms which are dependent on the value of x in the expansion of x2 155 n + 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion

  41. ( ) 1 8 + x2(log)10x 156 is 5600, then x equals 1 b. (log)e10 c. If the 6th term in the expansion of 8 3 x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Introduction If n is an integer between 0 and 21, then the minimum value of n!(21 - n)! is attained for n = 157  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Greatest Term In Binomial Expansion In the expansion of 3- x/4+ 35x/4n the sum of binomial coefficient is 64 and term with the greatest binomial coefficient exceeds the ( ) 158 third by (n - 1) , the value of x must be 0 b. 1 c. 2 d. 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion )( ) ( n 1 1 log38 1 3 - If the last tem in the binomial expansion of 2 , then 5th term from the beginning is is 5 159 √2 3 3 these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion . . If ^ n + 1Cr+ 1 ^ nCr . ˆ nCr ^ (n - 1)Cr-1= 11:6:3, . ˆ (n − 1)Cr−1 ^ n + 1Cr+1 160 = 11:6:3, then nr =20 b. 30 c. 40 d. 50  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion [ 1 The value of x for which the sixth term in the expansion of 2log_ 2√9( x- 1) +7+ ( 21 5(log)23 161

  42. ⎡ ⎢⎢ ⎣ 2log_ 2√9(x−1) +7 7 ⎤ ⎥⎥ ⎦ 1 + 21 (log)2(3(x−1) +1) 5 is 84 is 4 b. 1 or 2 c. 0 or 1 d. 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion The last two digits of the number (23)14 are 01 b. 03 c. 09 d. none of these 162  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Use Of Complex Numbers In Binomial Theorem The number of real negative terms in the binomial expansion of (1 + ix)4n-2,n ∈ N, x > 0 is a. 163  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion If in the expansion of (a - 2b)n, the sum of 5th and 6th terms is 0, then the values of a/b = 164

  43.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion ( ) The number of distinct terms in the expansion of x +1 1 x2 15 x+ x2+ is/are (with respect to different power of 165 none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion ( ) The sum of the coefficients of even power of x in the expansion of 1 + x + x2+ x35is 256 b. 166  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series ( ) Maximum sum of coefficient in the expansion of 1 - xsinθ + x2n is 1 b. 2n c. 3n d. 0 167  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Greatest Term In Binomial Expansion If the sum of the coefficients in the expansion of (a + b)n is 4096, then the greatest coefficient in the expansion is

  44. 168 d. none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Binomial Theorem For Any Index If the sum of the coefficients in the expansion of 1 - 3x + 10x2nisa and if the sum of the coefficients in the expansion of ( ) 169 then a = 3b b. a = b3 c. b = a3 d. none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion If 1 + x - 2x26= 1 + a1x + a2x2+ a2x3+ , ( ) 6= 1 + a1x + a2x2 (1 + x − 2x2) 170 + a2x3+ , then the value of a2+ a4+ a6+ ... + a12 will be 32 b. 31 c. 64 d. 1024  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Applications Of Binomial Expansion The fractional part of 24n/15is(n ∈ N)1 15 b. 2 15 c. 4 15 d. none of these 171  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series The value of ^ 15C02 -15C12 +15C22 -15C152 is 15 b. -15 c. 0 d. 51 172  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series ( )( ) ( )( ) 30 0 30 10 30 1 30 11 The value of - + (302)(3012) + + (3020)(3030) = (30 0)(30 10) − (30 1)(30 11) 173 + (302)(3012) + + (3020)(3030) = ^ 60C20 b. ^ 30C10 c. ^ 60C30 d. ^ 40C30  Watch Free Video Solution on Doubtnut

  45. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series f1(1) 1 f2(1) 2! fn(1) n! f2(1) If f(x) = xn,f(1) + , wherefr(x) + + f1(1) f(x) = xn,f(1) + + 174 1 2! fn(1) ,wherefr(x) + n! denotes the rth order derivative of f(x) with respect to x, is n b. 2n c. 2n-1 d. none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series The value of 20C0+20C1+20C2+20C3+20C4+20C12+20C13+20C14+20C15 20C0+20C1 +20C2+20C3+20C4 +20C12+20C13+20C14+20C15 ^(20)C_10 +20C9 b. 219- 175 ( ) ( ) ^ (20)C10 + 2 ×20C9 ^ (20)C10 2 is a. 219- c. 219- d. none of these 2 2

  46.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series nC0 n nC1 n + 1+ nC2 n + 2+....+ nCn 2n is equal to a.∫ 1 0xn-1(1 - x)ndx b. ∫ 2 1xn(x - 1)n-1dx c. ∫ The value of + 176  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series If ^ C0,C1,C2,Cn are the binomial coefficient, then 2 × C1+ 23× C3+ 25× C5+ equals 3n+ 177  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series n nCr r + 1 is equal to a.- n + 1 b. 1 1 1 n ∑ r= 1 ( - 1)r+1 The value of n c. n + 1 d. 178 n + 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series If (1 + x)n= C0+ C1x + C2x2+ + Cnxn,thenC0C2+ C1C3+ C2C4+ + Cn-2Cn=(2n)! (n!)2 b. (n 179 these  Watch Free Video Solution on Doubtnut

  47. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series The sum of series ∧ (20)C0 - ∧ (20)C1 + ∧ (20)C2 - ∧ (20)C3 + + ∧ (20)C10 is 1 2 ∧ (20)C10 180  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series ˆAˆA404C4ˆAˆ a′ˆAˆA4C1ˆAˆAˆA303C4ˆA +ˆAˆA4C2ˆAˆAˆA202C4ˆAˆ a′ˆAˆA4C3ˆAˆAˆA101C4ˆA is equal to a. (401)4 b. (101)4 c. 0 d. (201)4 181  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series If 3 + x2008+ x20092010= a0+ a1x + a2x2+ + anxn, then the value of a0-1 none of these 1 2a2+ a3 ( ) 2a1- 182  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series 10 ∑ r= 0 r10Cr3r( - 2)10-r is 20 b. 10 c. 300 d. 30 The value of 183  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series

  48. 40 ∑ r= 0 r40C30 rCr is 4069C29 b. 4070C30 c. ^ 69C29 d. ^ 70C30 184 The value of  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series 15 ∑ r= 1 (r + 2)! is (17)! - 216 r2r b. (18)! - 217 (18)! c. (16)! - 215 (16)! d. (15)! - 214 (15)! The value of 185 (17)!  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series (n + 2)C02n+1- (n + 1)C12n+ (n)C22n-1- .... is equal to ( ) ( ) ( ) 186  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series 50 ∑ r= 0 ( - 1)r^ (50)Cr 1 1 1 The value of is equal to 50 × 51 b. 52 × 50 c. 52 × 51 d. none of these r + 2 187  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion

  49. In the expansion of [(1 + x)/(1 - x)]2, the coefficient of xn will be 4n b. 4n - 3 c. 4n + 1 d. none of these 188  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series ( ) ( ) ( ) 1 x +n(n + 1) 2! 1 x + ∞ will be xn b. x-n c. 1 -1 2 n The sum of 1 + n 1 - d. none of these 1 - 189 x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Sum Of Series ∞ ∑ k=1 k 1 - ( ) 1 n k-1 = a.n(n - 1) b. n(n + 1) c. n2 d. (n + 1)2 190  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion { } - 1 in ascending powers of x, when |x The coefficient of x4 in the expansion of √1 + x2- x 191  Watch Free Video Solution on Doubtnut

  50. CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series 1 3x +1 × 4 1 × 4 × 7 3 × 6 × 9x3+ is equal to x b. (1 + x)1/3 c. (1 - x)1/3 d. (1 - x)-1/3 3 × 6x2+ 1 + 192  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series ( ) ( ) ( ) 1 4 1 ⋅ 3 4 ⋅ 8 1 ⋅ 4 ⋅ 7 4 ⋅ 8 ⋅ 12 + .... = 1 + + + 193  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Miscellaneous Series ( ) ( ) ( ) ( ) ( 2x +n(n + 1) 2x 2x 1 + x 2x 2 n n If |x| < 1, then1 + n + ...... is equal to a. b. c. 194 1 + x 2! 1 + x 1 + x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) -3/2is(|x| < 1)21 b. 25 c. 26 d. none of these The coefficient of x5∈ 1 + 2x + 3x2+ 195  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_ALGEBRA_BINOMIAL THEOREM_Analysis Of Binomial Expansion ( ) 2 is n b. n - 1 c. n + 2 d. If |x| < 1, then the coefficient of xn in expansion of 1 + x + x2+ x3+ 196  Watch Free Video Solution on Doubtnut

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