EXERCISE - BINOMIAL THEOREM

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1 BINOMIAL THOEREM / EXERCISE - BINOMIAL THEOREM LEVEL I SUBJECTIVE QUESTIONS. Expad the followig expressios ad fid the umber of term i the expasio of the expressios. (a) (x + y) 99 (b) ( + a) 9 + ( a) 9 (c) ( p + q) ( p q) (d) ( + x + x ) (e) ( x + x ) (f) (ax y) (ax + y) (g) ( + x + x ) 7. Write all the terms of the expasio of the followig expressio usig biomial theorem. (a) ( + x x )4, x (b) (x ax + a ) (c) (a + b )4 (d) (x + x )9. Expad (x + y) 5 (x y) 5 ad fid the value of ( + ) 5 ( ) If O is the sum of odd terms ad E is the sum of eve term i the expasio of (x + a), the prove that; (a) O E = (x a ) (b) 4OE = (x + a) (x a) (c) (O + E ) = (x + a) + (x a) 5. Prove that r= C r r = Evaluate the followig : (a) (99) 5 (b) () 6 (c) (.) 5 (d) (96) (e) (.99) 5 + (.) 5 (f) ( + ) 7 + ( ) 7 (g) (a + a ) 5 + (a + a ) 5 7. State biomial approximatio ad approximatio of (.99) 5 usig the first three terms of its expasio. 8. Which is larger (a) (.) or (b) (.) or (c) (.) 4 or Prove that () 5 > () 5 + (99) 5.. Usig biomial theorem, prove that 6 5 always leaves remaider whe divided by 5.. Usig biomial theorem, prove that 7 is divisible by 49, where N.. Show that is divisible by 64, wheever is a positive iteger.. If is a positive iteger, show that 6 is divisible by Write the geeral term i the expasio of (a) (x y) 6, (b) (x xy) 99, x. 5. Write the th term i the followig expasio: (a) (9x x )8, x (b) ( ex +e x ). 6. Fid the middle term(s) i the followig expasios: (a) (x x ) (b) ( x 7 )7 (c) (a + b ) (d) ( x + x ) (e) ( + x + x + x ) 7. Fid the middle term(s) i the expasio of ( eix +e ix ) whe (a) is odd (b) is eve. 8. Fid the th term from the ed i the expasio of (x y x y ). 9. Fid the coefficiet of a 5 b 7 i the expasio of (a b).. I the expasio of (x + y) coefficiets of seveth ad thirteeth terms are equal. Fid the value of.. If the coefficiets of (r + 4) th term ad (r + ) th term i the expasio of ( + x) 4 are equal, fid r.. I the expasio of (a + b) coefficiets of a ad b are equal. Fid the value of.. I the expasio of ( + a) m+, prove that the coefficiets of a m ad a are equal. 4. Prove that the term idepedet of a i the expasio of (a + a ) is..5 ( ).! 5. Fid the umber of real egative term i the expasio of ( ix) 4, N ad x >. 6. If i the expasio of ( + x), the coefficiets of p th ad q th term are equal, prove that p + q = +, where p q. LEVEL I OBJECTIVE QUESTIONS 7. The coefficiet of x 7 i the expasio of (x 4 x )5 is (A) 65 (B) 65 (C) 495 (D) The coefficiet of i the biomial expasio of ( + x) ( + x ) is (A) (C)! ( )!(+)! ()! ( )!(+)! x (B) ()! ( )!(+)! 9. If r th term i the expasio of (x x ) is without x, the r is equal to (A) 8 (B) 7 (C) 9 (D). I the expasio of (x x )9, the term without x is equal to (A) 8 8 (B) 8 4 (C) 8 4 (D) 8 8. I the expasio of (x 4 x )5, x 7 is r th term, the r =

2 (A) (B) (C) (D). I the expasio of ( x + x 5 ) 8 the term idepedet of x is (A) 5 th term (B) 6 th term (C) 7 th term (D) 8 th term. The total umber of term i the expasio of (x + a) + (x a) after simplificatio is, (A) (B) 5 (C) 5 (D) 4. If the coefficiet of x i (x + λ x )5 is 7, the λ = (A) (B) 4 (C) 5 (D) 8 5. The coefficiet of term idepedet of x i the BINOMIAL THOEREM / 4. Two middle terms i the expasio of (x x ) are (A) x ad x (C) 46x ad 46 x (B) 46x ad 46 x 4. The largest term i the expasio of ( + x) 5 where x = / is (A) 5 th (B) 5 st (C) 7 th (D)6 th 4. The value of the atural umbers such that the iequality > + is valid is (A) For all (B) For all < (C) For all values of expasio of (ax + b x )6 is 4. C + C + C C 5 = C C C C 4 (C) 6! a 8 b 8 (B) 6! 8! a8 b 8 6! (8!) a8 b 8 6! (D) (8!) a8 b 8 (A) (B) (C) 6. The term idepedet of x i the expasio 44. If the coefficiet of x i the expasio of (x + k x )5 is of( x + x ) 7, the k = will be (A) (B) (C) (D) 4 (A) (B) 5 4 (C) The sum of the coefficiets i the expasio 7. If i the expasio of (a + b) ad (a + b) +, the ratio of the coefficiets of the secod ad third of (x + y) is 496. The greatest coefficiet i the expasio is terms, ad third ad fourth terms respectively are (A) 4 (B) 94 (C) 84 (D) 74 equal, the the value of is whe divided by 7 leaves the remaider (A) (B) 4 (C) 5 (D) 6 (A) (B) 6 (C) 5 (D) 8. If A ad B are the sums of odd ad eve terms respectively i the expasio of (x + a), the 47. If T /T i the expasio of(x + y) ad T /T 4 i the expasio of(x + y) + are equal, the = (x + a) (x a) is equal to (A) (B) 4 (C) 5 (D) 6 (A) 4(A + B) (B) 4(A B) (C) AB (D) 4AB 48. The coefficiet of middle term i the biomial 9. If A ad B are the sums of odd ad eve terms expasio i powers of x of ( + αx) 4 ad ( αx) 6 respectively i the expasio of (x + a), the is same if α equals [AIEEE 4] (x + a) (x a) is equal to (A) /5 (B) / (C) / (D) / (A) A B (B) A + B (C) AB (D) 4AB LEVEL II SUBJECTIVE QUESTIONS 49. Fid the term idepedet of x i the expasio of ( x + ) 8, x >. x 5. Fid the coefficiet of x i the biomial expasio of(x x ), x. 5. Fid a positive value of m for which the coefficiet of x i the expasio ( + x) m is Which term i the expasio of {( x ) + ( y ) } cotais x ad y to oe ad y x the same power. 5. Prove that the coefficiet of (r + ) th term i the expasio of ( + x) + is equal to the sum of the coefficiets of the r th ad (r + ) th terms i the expasio of ( + x). 54. Prove that the coefficiet of the middle term i the expasio of ( + x) is sum of the coefficiets of the middle terms i the expasio of ( + x) ad is equal to..5 ( ).! 55. The sum of the coefficiets of the first three terms i the expasio of (x x )m, x, m beig a atural umber is 559. Fid the term of the expasio cotaiig x. 56. If the coefficiets of (r 5) th ad (r ) th terms i the expasio of ( + x) 4 are equal, fid r. 57. Fid, if the ratio of the fifth term from the begiig to the fifth term from the ed i the 4 expasio of ( + 4 ) is 6:. 58. If the fourth term i the expasio of (ax + x ) is 5, the fid the value of a ad. 59. Fid the coefficiet of x 5 i the product ( + x) 6 ( x) 7 usig biomial theorem. 6. If the third term i the expasio of ( x + xlog x ) 5 is. Fid the value of x. 6. If the fourth term i the expasio of ( x + xlog x ) is 65. Fid the value of x.

3 6. I the expasio of ( xlog x+ + x), the fourth term is ad x >, the fid x. 6. For what value of x is the ith term i the expasio of ( log 5 x +7 + ( /8)log (5 x +) ) is equal to Fid the value of λ, λ for which the coefficiets of the middle term i the expasio of ( + λx) 4 ad ( λx) 6 are equal. 65. If the coefficiets of x 7 i (ax + bx ) ad x 7 i (ax bx ) are equal, fid the relatio betwee a ad b. 66. Fid a, if the coefficiet of x ad x are equal i the expasio of ( + ax) If the coefficiet of d, rd ad 4 th terms i the expasio of ( + x) are i A.P., the prove that =. 68. If the coefficiets of a r, a r ad a r+ i the expasio of ( + a) are i arithmetic progressio, prove that (4r + ) + 4r =. 69. Show that the coefficiet of α 5 i the product of ( + α) 4 ( α + α ) 4 is zero. 7. Fid the coefficiet of x 5 i the expasio of ( + x) + ( + x) + + ( + x) I the expasio of ( + x) the biomial coefficiets of three cosecutive terms are respectively, 495 ad 79, fid the value of ad the positio of the terms of these coefficiets. 7. Fid the greatest value of the term idepedet of x i expasio of (x cos θ + cos θ x 6 ), where θ R. 7. If i the expasio of ( x), the coefficiet of x r are deoted by a r, the show that a r + a r =. BINOMIAL THOEREM / 74. I the biomial expasio of ( + x), the coefficiets of ith, teth ad eleveth terms are i A.P. Fid all the values of. 75. The coefficiets of the (r ) th, r th ad (r + ) th terms i the expasio of (x + ) are i the ratio : : 5. Fid ad r. 76. If a, a, a, a 4 be the coefficiets of four cosecutive terms i the expasio of ( + x), the prove that a + a = a. a +a a +a 4 a +a 77. If the rd, 4 th ad 5 th terms i the expasio of (x + a) are respectively 84, 8 ad 56. Fid the value of x, a ad. 78. How may terms are free from radical sigs i the expasio of (x 5 + y ) Fid the umber of itegral terms i the expasio of ( ) If the biomial coefficiet of d, rd, 4 th ad 5 th terms i the expasio of (α + β) be respectively a, b, c ad d, the prove that b ac c bd = a c. 8. If the biomial coefficiet of th, th, th, ad 4 th terms i the expasio of (α + β) be respectively a, b, c, ad d the, prove that b ac c bd = a c. 8. If a ad b are the coefficiets of x i the expasio of ( + x) ad ( + x) respectively, the fid a as a fuctio of b. 8. If a ad b are the coefficiets of x m ad x i the expasio of ( + x) m. ( + x) respectively, the fid the relatio betwee a ad b. 84. Prove that the greatest term i the expasio of ( + x) has also the greatest coefficiet, the x ( +, + ). 85. Fid the last two digits of 4. LEVEL II OBJECTIVE QUESTIONS 86. If the coefficiet of x ad x i the expasio of ( + ax) 9 are the same, the a is, (A) 7 9 (B) 9 7 (C) 9 7 (D) The umber of terms with itegral coefficiets i the expasio of (7 + 5 ) 6 is (A) (B) 5 (C) 5 (D) 88. The umber of irratioal terms i the expasio of ( ) 45 is (A) 4 (B) 5 (C) If ( + x) 5 = C + C x + C x C 5 x 5, the C + C + C C 5 = (A) 4. 4 (B). 4 + (C) The uit digit of is, (A) (B) (C) (D) 9. If x + = ad x x4 + x4 = λ ad μ be the digit at uit place i the umber +, N, ad > the λ + μ is equal to (A) 8 (B) 7 (C) 6 (D) 5 9. I the expasio of ( x x ), coefficiet of x 4 is (A) 45/56 (B) 54/59 (C) 45/6 9. If ( r+ ) r+. C r =, the is equal to r= 6 (A) 6 (B) (C) 8 (D) Number of terms i the expasio of ( x ++x 6 x ) where N is, (A) + (B) Σ++ C (C) + (D) + +,

4 95. For a positive iteger, let a() = (A) a() (B) a() > (C) a() 96. If ( + ax) = + 8x + 4x +, the the value of a ad is (A),4 (B), (C),6 (D), 97. (5) 96 (96) 5 is divisible by, (A) 5 (B) 7 (C) 9 (D) 98. If N, the 7 +. is always divisible by (A) 5 (B) 5 (C) If 5 4 is divided by, the remaider is α ad whe is divided by 7, the remaider is β. The the value of β α is, (A) (B) 5 (C) 7 (D) 8. The value of is (A) (B) 5 (C) 5 (D). The rage of the values of the term idepedet of x i the expasio of (x si θ + cos θ x), θ [,] is (A) [ C 5 π, C 5 π ] (B) [ C 5 π, C 5 π ] 5 5 (C) [ C 5 π 5, C 5 π 5 ] (D) [ C 5 π 5, C 5 π 5 ] 5 5. If ( + x) = C + C x + C x + + C x, the the value of C + C + C + + ( + )C = (A) ( + ) (B) ( + ) (C) ( + ) (D) ( + ). If ( + x) = r= a r x r ad b r = + a r ad r= b r = ()!, the the value of is, a r (A) 99 (B) (C) (D) 4. The value of x i the expressio ( x + xlog x ) 5, if the third term i the expasio is 6, (A) (B) (C) 5. If is a iteger greater tha, the a C (a )+ C (a ) + + ( ) (a ) is equal to (A) a (B) (C) a (D) 6. The term idepedet of t i the expasio of (t 6 t ) 9 is (A) 84 (B) 8.4 (C).84 (D) The sum of + ( ) + (+) ( x! x ) + will be 8. If (A) x (B) x (C) ( x ) ( x) +( x) 5 4 x is approximately equal to a + bx for small values of x, the (a, b)= (A) (, ) (B) (, ) (C) (, ) (D) (, 4 4 ) BINOMIAL THOEREM / 4 9. Let P() deote the statemet that + is odd. It is see that P() P( + ), P is true for all (A) > (B) > m, m beig a fixed positive iteger (C) Nothig ca be said. The least remaider whe 7 is divided by 5 is, (A) (B) (C) (D) 4. If N, the is divisible by (A) (B) (C). If the sum of the coefficiets i the expasio of (α x αx + ) 5 vaishes, the value of α is (A) (B) (C) (D). I the polyomial (x )(x )(x ) (x ) the coefficiet of x 99 is (A) 55 (B) 55 (C) (D) The sum of the coefficiets of the polyomial ( + x x ) 6 is (A) (B) (C) (D) 6 5. If C r stads for C r ad >, the the value of k ( C k C k ) k= (A) (+)(+) (C) (+)(+) is (B) (+) 6. The sum of the series is equal to (A) / 5 (B) / (C) 5/ (D) 5 7. If the value of x is so small that x ad greater powers ca be eglected, the +x+ ( x) +x+ +x equal to (A) + 5 x 6 (B) 5 x (C) + x 6 (D) x 8. The coefficiet of x i the expasio of r= ( + x) r is (A) ( ) (B) () (C) () (D) ( ) 9. If 6 th term i the expasio of the biomial [ log( x) 5 + (x )log ] is equal to ad it is kow that the biomial coefficiets of the d, rd ad 4 th terms i the expasio represet respectively the first, third ad fifth terms of a A.P. (the symbol log stads for logarithm to the base ), the x = (A) (B) (C). I the biomial expasio of (a b), 5, the sum of the 5 th ad 6 th terms is zero. The a/b is equal to [IIT-JEE ] (A) 6 ( 5) (B) 5 ( 4) (C) 5 4 (D) 6 5. If ( x + x ) = a + a x + a x + a x, the a + a + a a = (A) + (B) (C) (D) +. If ( + x) = C r x r, the fid the value of r= is

5 ( + C C ) ( + C C ) ( + C C )= (A) ( )! (B) (+) ( )! (C) (+)! (D) (+)+!. For ay atural umber m ad, if ( t) m ( + t) = + a y + a y +, ad a = a =, the (m, ) is, (A) (5,) (B) (45,5) (C) (5,45) (D) (,45) 4. The umber of ratioal term i the expasio of ( + + ) 6 is, (A) 5 (B) 6 (C) 7 (D) 8 5. The umber of irratioal terms i the expasio 8 6 of ( 5 + ) is (A) 97 (B) 98 (C) 96 (D) 99 BINOMIAL THOEREM / 5 6. The value of the expressio { + C + [ C + C + 4 C + + C ]} is (A) (B) (C) (D) (+)/ 7. The coefficiet of x 5 i the expasio of ( + x) + x( + x) x ( + x) x is (A) C 5 (B) C 5 (C) C 5 (D) C 5 8. The value of C 4 5 (A) + (B) + (C) The value of x, for which the 6 th term of the expasio { log 9 x +7 + ( 5 )log ( x +) }7 is 84, is equal to (A) 4 (B) (C) (D) ENGINEERING ENTERANCE EXAM QUESTIONS OBJECTIVE QUESTIONS. The coefficiet of x i the expasio of ( + x)( x) is [AIEEE 4] (A) ( ) (B) ( ) ( ) (C) ( ) ( ) (D) ( ). Uit digit of. 7. is [AIEEE 7] (A) (B) (C) 7 (D) 9. The remaider left out whe 8 (6) + is divided by 9 is, [AIEEE 9] (A) (B) 7 (C) 8 (D). The term idepedet of x i the expasio of { x+ x x x + x x } is, [JEE Mais ] (A) (B) (C) (D) 4 4. Give positive itegers r >, > ad the coefficiet of the (r) th ad (r + ) th terms i the biomial expasio of ( + x) are equal, the [IIT-JEE 98] (A) = r (B) = r (C) = r + (A) ( ) ( + ) (B) ( ) ( + ) (C) ( ) / ( + ) (D) 7. If i the expasio of ( + x) m ( x) m, the coefficiet of x ad x are ad 6 respectively, the m is [IIT-JEE 999] (A) 6 (B) 9 (C) (D) 4 m 8. The sum ( i= )( i m i ), (where ( p ) = if p < q) q is maximum whe m is, [IIT-JEE ] (A) 5 (B) (C) 5 (D) 9. The coefficiet of t 4 i the expasio of ( + t ) ( + t )( + t 4 ) is [IIT-JEE ] (A) + C 6 (B) C 5 (C) C 6 (D)+ C 6 4. If C r = (k ). C r+, the k belogs to [IIT-JEE 4] (A) (, ] (B) [, ) (C) [, ] (D) (, ] 4. ( )( ) ( )( ) + ( )( ) + ( )( ) is 5. The expressio [x + (x ) / ] 5 equal to [IIT-JEE 5] + [x (A) C (x ) / ] 5 (B) 6 C (C) C (D) 65 C 55 is a degree of polyomial 4. For r =,,, ; let A r, B r ad C r deotes [IIT-JEE 99] respectively the coefficiet of x r i the expasios (A) 5 (B) 6 (C) 7 (D) 8 of ( + x), ( + x) ad ( + x). The 6. If C r stads for C r, the the sum of the give r= A r (B B r C C r ) is equal to series ( )!( )! [C! C + C [IIT-JEE ] + (A) B C (B) A (B C A ) ( ) ( + )C ], where is a eve positive iteger, is equal to [IIT-JEE 986] (C) (D) C B ANALYTICAL AND DISCRIPTIVE QUESTIONS 4. Prove that ( C ) ( C ) + ( C ) + ( C ) = ( ). C [IIT-JEE 978] 44. Prove that C C + C. C = ( ). C. [IIT-JEE 979] 45. If ( + x) = C + C x + C x + + C x, the, show that the sum of the products of the C i s take two at a time represeted by C i C j, ( i j ) is equal to ()!. [IIT-JEE 98] 46. Give, s = + q + q + q + + q ad S = + (q+) (!) + ( q+ ) + ( q+ ) + + ( q+ ), q. Prove that

6 BINOMIAL THOEREM / 6 ( + C + + C s + + C s C + s ) = S [IIT-JEE 984] 47. If r= a r (x ) r = r= b r (x ) r ad a k = for all k, the show that b = + C +. [IIT-JEE 99] 48. Prove that k r= ( ) r. C r =, where k = / ad is a eve positive iteger. [IIT-JEE 99] 49. Let be a positive iteger ad ( + x + x ) = a + a x + a x + + a x. Show that a a + + = a. [IIT-JEE 994] a 5. Fid the sum of ratioal terms i the expasio of ( + 5 ). [IIT-JEE 997]! 5. Prove that = (+) r= ( )r ( C r / r+ C r ). [IIT-JEE 997C] 5. For ay positive iteger m, (with m ), let ( m ) = C m. Prove that ( m ) + ( m ) + ( m ) + + (m m ) = ( + m+ ). Hece or otherwise, prove that ( m ) + ( m ) + ( m ) + + ( m + )(m m ) = ( + m+ ). [IIT-JEE ] 5. Prove that k ( )( k ) k ( )( k ) + k ( )( k ) + ( )k ( k )( k ) = ( ). [IIT-JEE ] k ASSERTION AND REASON This sectio cotais reasoig type questio ad has 4 choices, A, B, C ad D, out of which oly ONE is correct. (A) Statemet is True, Statemet is True; Statemet is a correct explaatio of Statemet. (B) Statemet is True, Statemet is True; Statemet is NOT a correct explaatio of Statemet. (C) Statemet is True, Statemet is False; (D) Statemet is False, Statemet is True 54. Statemet : r= (r + ). C r = ( + ) [AIEEE 8] Statemet : r=(r + ). C r x r = ( + x) + x( + x) 55. Let S = {j(j ). j= C j }, S = (j. j= C j ) ad S = j (j. C j ) [AIEEE ] Statemet : S = 55 9 Statemet : S = 9 8 ad S = 8 LEVEL III FOOD FOR THOUGHT 56. If {x} deotes the fractioal part of x, the { }, 8 N, is equal to (A) /8 (B) 7/8 (C) /8 57. If S be the sum of coefficiets i the expasio of (αx + βy γz), where (α, β, γ) >, the the S value of lim (A) e (αβ γ ) is [S / +] (B) e ( α+β γ α+β γ+ ) (C) αβ γ (D) 58. The umber of ratioal terms i the expasio of ( / + / + 5 /6 ) is (A) (B) (C) (D) Let R = (5 5 + ) + ad f = R [R], where [. ] deotes greatest iteger fuctio. The value of R. f is (A) 4 + (B) 4 (C) 4 (D) 4 6. If ( + x) = C + C x + C x + + C x ; the C C + 5C 5 is equal to (A). ( ). cos [( ) π] 4 (B). ( ). si [( 4 ) π] (C) ( ). ( ). cos [( ) π] 4 6. Let be a odd iteger. If si θ = r= b r si r θ for every value of θ, the (A) b =, b = (B) b =, b = + (C) b =, b = (D) b =, b = 6. The sum of the series ( ) r. C r ( 7 r 5r r + (A) m m ( ) 4r + m terms) is (B) m r= r r + + r (C) m + + ) is equal to 6. ( ) + ( ) + ( ) + + ( (A) (B) (C) 64. If ( + x) = C + C x + C x + + C x, the (C i + C j ) i j equals (A). C + (B) ( ). C + (C) ( + ). C The coefficiet of x i the expasio of ( x + x 4x + to ) is (A) ()!!( )! (B) ()! [( )!] ()! (C) (!) 66. The value of a ad b so that coefficiet x i the expasio of a+bx ( x) is +, are (A), (B), (C), 67. Let ad k be positive itegers such that k(k+). The umber of solutios (x, x. x k ), x, x,, x k k, all itegers, satisfyig x + x + + x k =, is (A) m C k (B) m C k+ (C) m C k {Where m = ( k + k )}

7 68. If the sum of the coefficiets i the expasio of (ax x + ) 5 is equal to the sum of the coefficiets i the expasio of(x ay) 5, the a = (A) (B) May be ay real umber (C) (D) No such value exist 69. Whe P is a atural umber, the P + + (P + ) is divisible by (A) P (B) P + P (C) P + P + (D) P 7. Algebraically greatest term i the expasio of ( 5x), whe x = /5 is, (A) 55 8 (B) 55 9 (C) 9 (D)N.O.T. 7. To expad ( + x) as a ifiite series, the rage of x should be (A) [, ] (B) (, ) (C) [,] (D) (,) BINOMIAL THOEREM / 7 7. Give S + = S S ad S =, S =, the the value of S for all N is, (A (B) + (C) 7. If a, b, c, d be four cosecutive coefficiets i the biomial expasio of ( + x), the the value of the expressio {( b b+c ) ac (a+b)(c+d) }, where x>, is (A) < (B) > (C) = (D) 74. The coefficiets of x y, yzt ad xyzt i the expasio of (x + y + z + t) 4 are i ratio, (A) 4: : (B) : : 4 (C) : 4: (D) : 4: 75. If the expasio of a x +, the a is (A) a b b a (C) b+ a + b a ( ax)( bx) (B) a+ b + b a (D) b a b a is a + a x + COMPREHENSION 76. PASSAGE: Let be a positive iteger, such that ( + x + x ) =a + a x + a x + + a x, the (I) The value of a r is, for (all r [, ]) (A) a r (B) a r (C) a (D). a (II) The value of a + a + a + + a is (A) (B) ( a ) (C) a (D). a (III) The value of a a + a a + + a is (A) a (B) ( + ) a i= i (C) a (D) a 77. PASSAGE: Numerically greatest term i the expasio of (x + a). Let T r+ be r th term i the biomial expasio of (x + a) ad T r+ be the umerically greatest term. The, T r ad also T r+. I the biomial expasio T T r+ T r = C r x r+ a r, T r+ = C r x r a r ad r+ T r+ = C r+ x r a r+. So T r = ( r+ ) a + r (= k say). T r+ r x (x/a)+ If k is a iteger, the both T k ad T k+ are umerically greatest term. If k is ot a iteger, the T [k] is umerically greatest term, where [k] deotes the greatest iteger fuctio. (I) Numerically greatest term i the expasio of ( x) 8, whe x = is (A) 7. 4 (B) 5. 9 (C) 7 (II) Magitude wise the greatest term i the expasio of ( x) 9, whe x = is (A) 9 C 7 (B) 9 C 6 (C) 9 C (D) both B ad C (III) Give that 4 th term i the expasio of ( + 8 x) has the maximum umerical value, the rage of value of x is (A) x [ , ] [, ] (B) < x < (C) < x < 4 (IV) If is eve positive iteger, the the coditio that the umerically greatest term i the expasio of ( + x) may also have the greatest coefficiet is (A) + < x < (B) + < x < + + (C) +4 < x < +4 (D) + < x < + (V) The iterval i which x (x > ) lies so that umerically greatest term also have the greatest coefficiet i the expasio of ( x) is, (A) [ 5 6, 6 5 ] (B) (5 6, 6 5 ) (C) (4 5, 5 4 ) (D) [4 5, 5 4 ] 78. PASSAGE: If α ad β are positive itegers ad t is a positive iteger, which is ot a perfect square, the the umber α + β t ad its positive itegral powers are essetially irratioal. If is a positive iteger, the it is oticed that E = (α + β t) + (α β t), where E I +. Further if < (α β t) < the E is the iteger just ext to (α + β t). From the equality (α + β t) + (α β t) = E, it ca be deduced that sum of fractioal parts is equal to. Similarly, (α + β t) (α β t) is also a positive iteger, where < (α β t) <.

8 (I) If R = (5 5 + ) + ad f = R [R], the Rf must be equal to (A) 5 + (B) + (C) 4 + (D) BINOMIAL THOEREM / 8 (II) Let S = ( + ) + ( ), the (A) 4S + = S S (B) S + = 4S S (C) S + = 8S 4S (D) S + = S (III) The iteger just greater tha ( + ) is always divisible by, (A) + (B) + (C) + (D) PASSAGE: If C, C, C,,C are biomial coefficiets, the ( + x) = C + C x + C x + + C x. Various relatios amog biomial coefficiets ca be derived by puttig x =,, x = i, x = ω, (i =, ω = + i). Some other idetities ca be derived by addig ad subtractig two idetities. The expressio (a + ib) ca be evaluated by usig de Moivre s idetity by puttig a = r cos θ ad b = r si θ. (I) The value of the expressio ( C C + C 4 C 6 + ) +( C C + C 5 ) must be (A) (B) (C) (II) The value of C + C 4 + C 8 + is (A) / cos π (B) / si π (C) / cos π (D) + cos π 4 4 (III) The value of k r= ( ) r C r, where is a eve positive iteger ad k = is (A) (B) (C) 6 (D) LEVEL III MULTIPLE CORRECT CHOICE QUESTIONS 8. Which of the followig is correct? (A) () > () (B) (6) 5 > (5) 6 (C) () 99 < (99) (D) (7) 7 = (7) 7 8. I the expasio of ( ) 6 (A) the umber of irratioal terms = 9 (B) middle term is irratioal (C) the umber of ratioal term = (D) 9 th term is ratioal 8. If C, C, C,, C have their usual meaigs, the + to ( + ) C C + C (+) (+)(+) (+)(+) terms is equal to (A) x + ( x) + dx (B) x ( + x) + dx (C) x ( x) + dx (D) x + ( x) dx 8. The series C to (A) x ( x) dx + C + + C C (B) x ( + x) equal (C) x ( x) dx (D) x (x ) dx 84. If the sum of the coefficiets i the expasio of ( + cx + c x ) vaishes the c equals (A) (B) (C) (D) 85. If f = ( + ) [( + ) ], where [ ] is dx greatest iteger fuctio x, the f is f (A) a positive iteger (B) a egative iteger (C) a ratioal umber 86. Let R = (8 + 7) ad [R] = the greatest iteger less tha or equal to R, the (A) [R] is eve (B) [R] is odd (B) R [R] = (D) R + [R]R = + R (8+ 7) 87. The value of C + + C + + C + + +k C k is equal to, (A) +k+ C k (B) +k+ C + (C) +k C Number of values of r satisfyig the equatio 69 C r 69 C r = 69 C r 69 C r is, (A) (B) (C) (D) If ( + x + x ) = a + a x + a x + + a x, the (A) a = (B) a = (C) a 4 = 885 (D) a = If ( + x + x ) = a + a x + a x + + a x, the the value of a + a + a 6 + is (A) a + a 4 + a 7 + (B) a + a 5 + a 8 + (C) 9. The largest coefficiet i the expasio of (4 + x) 5 is, (A) 5 C 5 ( 4 )4 (B) 5 C 4 5 ( 4 ) (C) 5 C (D) 5 C If ac > b, the the sum of the coefficiets i the expasio of (aα x + bαx + c) is, where a, b, c, α R ad R (A) positive if a > (B) positive if c >

9 (C) egative if a < ad is odd (D) positive if c < ad is eve 9. If is a positive iteger, the i the triomial expasio of (x + x + ), the coefficiet of (A) x is. (B) x is. (C) x is. + C (D) All of the above 94. Suppose x, x, x,,x (>) are real umbers, such that x i = x i+ for i. Cosider the sum S = x i x j x k, i, j, k ad i, j, k are distict. The, which of the followig is ot true? (A) S =! x. x. x x (B) S = ( )( 4) (C) S = ( )( 4)( 5) (D) S =, N 95. If is a positive iteger ad ( + 5) + = P + f, where P I ad <f<, the BINOMIAL THOEREM / 9 (A) P is a eve iteger (B) (P + f) is divisible by + (C) the iteger just below ( + 5) + is divisible by (D) P is divisible by 96. The umber of distict term of the expasio of (x + y + z 5u + 7w) is (A) + (B) +4 C 4 (C) +4 C (D) (+)(+)(+)(+4) The coefficiet of a 8 b 6 c 4 i the expasio of (a + b + c) 8 is (A) 8 C 4. 4 C 8 (B) 8 C. C 6 (C) 8 C 6. C 8 (D) 8 C 4. 4 C 6 LEVEL III SUBJECTIVE QUESTIONS 98. Fid a, b ad i the expasio of (a + b) if the first three terms of the expasio are 79, 79 ad 75 respectively. 99. If a ad b are distict itegers, prove that a b is a factor of a b, wheever is a positive iteger.. Prove that the coefficiets of x m i the expasio ()! of (x + x ) is ( 4 m )!( +m. Does the expasio (x x ) cotai ay term ivolvig x 9.. Prove that there is o term ivolvig x 6 i the expasio of (x x ).. I the expasio of ( + x), for all <5, fid all the values of for which coefficiets of three cosecutive terms are i A.P. Also fid cosecutive terms for the correspodig values of. 4. If the coefficiet of (r + ) th, (r + ) th, (r + ) rd, ad (r + 4) th terms i the expasio of ( + x) be respectively a, b, c ad d, the prove that b ac = a(r+). c bd c(r+) 5. If a ad b deote the sum of the coefficiets i the expasios of ( x + x ) ad ( + x ) respectively, the fid relatio betwee a ad b. 6. Fid the coefficiet of x 5 i the expressio C m (x ) m. m. m= )!. 7. Fid the sum of the series. C +. C +. C + +. C. 8. If C, C,, C are biomial coefficiets i the expasio of ( + x), the prove that ( +x ) C +x + ( +x (+x) ) C ( +x (+x) ) C + + ( ) ( +x (+x) ) C =. r+ 9. Evaluate r=. C r +6r ++r+6 r.. If C, C,, C are biomial coefficiets i the expasio of ( + x) ad p + q =, the prove that (A) r= r C r p r q r = p (B) r= r C r p r q r = p + pq. Show that the greatest coefficiet i the expasio of (x + x ) is [..5 ( )].!.. Prove that whe () divided by 7, the remaider is 4.. Fid the remaider, whe (5 + ) is divided by The term idepedet of x i the expasio of (9x x )8, x > is k times the correspodig biomial coefficiet. Fid the value of k. 5. Fid the coefficiet of x i ( + x x 4 ) ( + x )8. 6. ( + x)( + x + x )( + x + x + x ) ( + x + x + + x ) whe writte i ascedig powers of x, the the highest expoet of x is λ(). Fid the value of λ. ANSWERS. DIY. DIY. x 4 y + x y + y 5, 5 4. P.T. 5. P.T. 6. DIY a. (.) b. (.) c. (.) 4 9. P.T.. P.T.

10 . P.T.. P.T.. P.T. 4. a. 6 C r ( ) r x ( r) y r b. 99 C r ( ) r x 98 r y r 49. T 5. C m = 4 5. a. 8 C 6 x 5. No Term b. C e x( ) / 6. a. C 5 5. P.T. 54. P.T. b. 7 C 4 x 9 7,7 C x 55. T r = = 58. a = / c. C 6 ( ab )6 d. C ( ) x f. 6 C x 7. DIY 8. ( )(xy)49/ 9. ( 7 )7/ a 5 b 7. = 8. r =. =. P.T. 4. P.T. 5. DIY 6. P.T. 7. B 8. B 9. C. C. C. B. B 4. A 5. C 6. B 7. C 8. D 9. A 4. C 4. A 4. A 4. B 44. C 45. B 46. A 47. C 48. C 6. x=, 6. (4± 9 ) 6. 4, 6. log λ = / 65. ab + = 66. a = 9/7 67. P.T. 68. P.T. 69. P.T = ( C 5 ) 7. =, r= 7. C 5 7. P.T. 74. = 4, 75. =, r = 76. P.T. 77. x=, a=, = P.T. 8. P.T. 8. a = b 8. a = b 84. P.T D 87. D 88. C 89. C 9. B 9. C 9. A 9. D 94. D 95. A 96. A 97. C 98. A 99. C. A. B. A. B 4. A 5. B 6. D 7. A 8. B 9. D. D. C. C. A 4. C 5. D 6. C 7. B 8. A 9. A. B. A. C. C 4. B 5. A 6. B 7. C 8. A 9. C. B. B. A. B BINOMIAL THOEREM / 4. A 5. C 6. A 7. C 8. C 9. A 4. D 4. C 4. D 4. P.T. 44. P.T. 45. P.T. 46. P.T. 47. P.T. 48. P.T. 49. P.T P.T. 5. P.T. 5. P.T. 54. A 55. C 56. C 57. D 58. C 59. A 6. A 6. D 6. A 6. C 64. B 65. C 66. A 67. A 68. C 69. C 7. B 7. B 7. B 7. B 74. B 75. C 76. (I)A (II) B (III) A 77. (I)B (II) D(III) A (IV) A (V)B 78. (I) C(II) C(III)C 79. (I) B(II) D(III)D 8. B,C 8. A,B,C,D 8. C,D 8. B,D 84. A,D 85. A,C 86. B,C,D 87. A,B 88. C,D 89. A,B,C 9. A,B,C 9. A,B,C 9. A,B,C,D 9. A,B,C,D 94. A,B,C,D 95. A,B,D 96. B,C,D 97. A,B,C,D 98. a=,b=5,=6 99. P.T.. P.T.. P.T.. P.T.. DIY* 4. P.T. 5. DIY 6. C 5 7. ( + ) 8. P.T. 9. DIY. P.T.. P.T.. P.T