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Archibald Garrett
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1 Fial sep (Biomial Theorem)-by abhiji Kumar Jha LEVEL I 5 Show ha he middle erm i he expasio of ( + x) is! ieger x, where is a posiive Fid he erms idepede of x, x 0, i he expasio of (i) x x 6 (ii) x x I he biomial expasio of ( + x), he coefficies of he fifh, sixh ad seveh erms are i arihmeic progressio Fid all he values of for which his ca happe 4 Show ha he coefficie of he middle erm of ( + x) is equal o he sum of he coefficies of he wo middle erms of ( + x) 5 If a ad b are disic iegers, prove ha a b is a facor of a b, wheever is posiive ieger 6 Usig Biomial Theorem, prove ha 6 5 always leaves he remaider, whe divided by 5 7 Usig Biomial Theorem, prove each of he followig ideiies : (i) ( + ) = + (ii) 0 (iii) 0 + (iv) = (v) If he middle erm of ( + x) (x > 0, x N) is he greaes erm of he expasio, he show ha x 9 If is a posiive ieger ad if ( + x + x ) = 0 a a a a a r0 a x r r, he show ha

2 Fial sep (Biomial Theorem)-by abhiji Kumar Jha 0 Prove each of he followig ideiies : (i) ( 0 + )( + ) ( + ) =! (ii) m! +! 0 + m! + + m! = m! m! (iii) r r 0 r r

3 Fial sep (Biomial Theorem)-by abhiji Kumar Jha LEVEL II If he greaes erm has he greaes coefficie i he expasio of ( + x), N, he show ha x, if is eve ad < x <, if is odd Fid he coefficie of x i (x 0 ) (x ) (x ) (x - ) Prove ha : r si (r ) si r 4 Prove each of he followig ideiies : (i) ( ) = (ii) = 0, if is odd /,if is eve / (iii) ( + ) + = ( ) (iv) = For ay posiive iegers m, (wih m), le m = m Prove ha m m m m m m 6 Prove ha : = cos 7 Show ha : , I + is divisible by 85 8 Fid umerically he greaes erm i he expasio of : (a) ( + x) 9 whe x = (b) Fid he idex of he biomial he greaes coefficie ( N) x 5 if he 9h erm of he expasio has umerically 5 q 9 Give S = + q + q ++ q ad S = q q, q, prove ha s + + s s = S 0 Fid he erm idepede of x i he expasio of ( + x + x ) x 9 x

4 Fial sep (Biomial Theorem)-by abhiji Kumar Jha The value of x i he expressio x x SET I x log 0 ( ) 5 if he hird erms i he expasio i he expasio is 0,00,000 (A) 0 (B) () (D) oe of hese If T i he expasio of (a + b) ad T i he expasio of (a + b) + are equal, he is equal o T T4 (A) (B) 4 () 5 (D) 6 If umber of erms i he expasio of (x - y + z) are 45, he is equal o (A) 7 (B) 8 () 9 (D) oe of hese 4 I he biomial ( / + -/ ), if he raio of he seveh erm from he begiig of he expasio o he seveh erm from is ed is /6, he is equal o (A) 6 (B) 9 () (D) 5 5 If ( + ax) = + 8x + 4x +he he value of a ad is (A), 4 (B), (), 6 (D), 6 The coefficie of x 4 i he expasio of ( + x + x + x ) is (A) 4 (B) () (D) 4 7 The greaes coefficie i he expasio of ( + x) + is b g b g!! (A) (B) b g b g!! () b g b g!!! (D) bg!!! b g 8 The umber whe divided by 00 leaves he remaider (A) 4 (B) 45 () 50 (D) oe of hese 9 Number of erms i he expasio of ( - 4x) -0 are (A) 9 (B) 0 () (D) ifiiely may 0 If he sum of he coefficies i he expasio of (p x - px + ) 5 vaishes, he he value of p is (A) (B) - () (D) - If ( + x + x ) 0 = a 0 + a x + a x a 40 x 40 he a + a + a a 9 equals (A) 9 ( 0 ) () 0 ( 9 9) () 9 ( 0 + ) (D) oe of hese The las digi of ( P + ), where P = 4 + is (A) (B) 9 () 4 (D) 5

5 Fial sep (Biomial Theorem)-by abhiji Kumar Jha If N ad is eve, he ( )!! ( )! 5! ( 5)! ( )!! is equal o (A) (B)! ()! (D) oe of hese 4 If 0,,, are he biomial coefficies i he expasio of ( + x) 5, he (A)0 (B) 0 () 40 (D) 50 5 The expressio 4x 4x 4x is a polyomial i x of degree 7 7 (A) 7 (B) 5 () 4 (D) 6 r r r is equal o (A) - (B) - () - + (D) 7 If 0,,, deoe he coefficies i he expasio of ( + x), he he value of ( r ) r is r0 (A) (B) ( + ) - () ( + ) - (D) (+) - 8 If ( + x) = 0 + x + x + + x, he (4 + ) is equal o (A) (B) ( + ) () ( + ) (D) (4 + ) 9 The co-efficie of x 9 i he polyomial give by, (x + ) (x + ) (x + 0) + (x + ) (x + ) (x + ) + + (x + ) (x + ) (x + 0) is : (A) 55 (B) 55 () 55 (D) 55 0 If ( - x + x ) = a 0 + a x + a x + + a x, he a 0 + a + a 4 ++a equals (A) (B) () (D)

6 Fial sep (Biomial Theorem)-by abhiji Kumar Jha SET II For r, he value of r r r r r is (A) r (B) r () r (D) oe of hese The coefficie of x m m i he expasio m ( x ) is 00 m0 (A) (B) 00 5 () 00 5 (D) If be a posiive ieger such ha, he he value of he sum o erms of -! ( - ) +! ( - )! ( - ) + is (A) 0 (B) () - (D) oe of hese 4 The umber of iegral erms i he expasio of (5 / + 7 /6 ) 64 is (A) 06 (B) 08 () 0 (D) 09 5 The raio of he coefficie of x i he expasio of ( + x) ad ( + x) - will be (A) : (B) : () : (D) : 6 The umber of erms i he expasio of (x + ) + (x ) +, is (A) (B) () + (D) + 7 If he sum of he coefficies i he expasio of x is equal o x is 4, he he erm idepede of x, (A) (B) 7 () 5 (D) 40 8 The value of 4 { } is (A) 0 (B) 5 + () 5 (D) 5-9 If A ad B respecively deoe he sum of he odd erms ad sum of he eve erms i he expasio of (x + y), he he value of (x y ), is equal o (A) A + B (B) A B () 4AB (D) (A B) 0 The value of r0, is equal o r! r! (A)! (B)! ()! (D)!

7 Fial sep (Biomial Theorem)-by abhiji Kumar Jha The value of he sum of he series upo ( + ) erms, where r = r, is equal o (A) (B) 4 () 0 (D) 4 The value of he expressio , is equal o (A) (B) () (D) oe of he above o-efficie of i he expasio of, ( + p) m + ( + p) m ( + q) + ( + p) m ( + q) + ( + q) m, where q ad p q is (A) () m m p q p q p q p q (B) (D) m m m m p q p q m m p q 4 I he expasio of ( + x) ( + y) ( + z), he sum of he co-efficies of he erms of degree ' r ' is p q (A) r (B) () r (D) r r 5 The value of he expressio (A) r r r is r0 r (B) () (D) m 6 The coefficie of x m i he expasio of x r0 mr, m, is equal o (A) + m (B) + m + () m (D) m 7 The sum of he las coefficies i he expasio of ( + x) whe expadig i ascedig power of x, is equal o (A) (B) () (D) 8 The iegral par of, is of he form (A) k +, k I () k, k I (B) (k + ), k I (D) oe of hese 9 The fracioal par of 8, is equal o (A) 8 (B) 7 8 () 8 (D) oe of hese 0 If he ui digi of + 7, N, is 7, he he value of is of he form (A) 4k +, k I (B) 4k +, ki () 4k +, k I (D) 4k, k I

8 Fial sep (Biomial Theorem)-by abhiji Kumar Jha SET III Muliple choice quesios wih oe or more ha oe correc opio If x <, he he coefficie of x i he expasio of log e ( + x + x + ) is (A)! (B) e log e! () (D) e log e I he expasio of x ad x x x i powers of x have oe erm idepede of x, he is divisible by (A) (B) () 4 (D) 6 I he expasio of (a + b + c) 0 (A) oal umber of erms is 66 (B) coefficie of a 8 bc is 90 () coefficie of a 4 b 5 c is 0 (D) oe of hese 4 k 4 Le ( + x ) (a + x) = kx If a, a, a are i A P, he is equal o k0 (A) 6 (B) 4 () (D) 5 I he expasio of x, a 0, if o erm is idepede of x, he is x (A) 0 (B) () 6 (D) 0 Quesio based o wrie-up If O be he sum of erms a odd posiio ad E ha of erms a he eve posiio i he expasio (x + a) 6 The value of (x + a) i erms of O, E is (A) OE (B) O + E () O E (D) oe of hese 7 The value of (x a) i erms of (A) OE (B) O + E () O E (D) oe of hese 8 The value of (x a ) is equal o (A) O + E (B) O + E () O E (D) oe of hese 9 The value of (x + a) (x a) is equal o (A) OE (B) OE () OE (D) 4 OE 0 The value of (x + a) + (x a) is equal o (A) (O + E ) (B) (O + E ) () (O E ) (D) (O E )

9 Fial sep (Biomial Theorem)-by abhiji Kumar Jha Le be a posiive ieger such ha ( + x) = 0 + x + x + + x The value of s s s r (whe s r0 s r is) (A) (B) () (D) The value of ( r s) r s, is r0 s0 (A) (B) ( ) () ( + ) (D) The value of (r s) rs, is 0r s (A) [ ] (B) [ ] () [ ] (D) [ ] 4 The value of (r s) (r s ), is 0r s (A) (B) ( ) () (D) 5 The value of 0r s ( r s)(r s rs ), is (A) [ ] (B) [ ] () [ ] (D) [ ] 6 The value of ( ), is r0 s0 0 u0 (A) 4 (B) + 4 () 4+ 4 (D) ( + ) 4 7 The value of ( ), is 0r s u (A) 4 (B) + 4 () 4+ 4 (D) ( +) 4 8 True ad False : (i) (ii) (iii) The raio of he coefficie of x 0 i ( x ) 0 ad he erm idepede of x i x x is : The coefficie of (r + ) h erm i he expasio of ( + x) + is equal o he sum of he coefficies of r h ad (r + ) h erms i he expasio of ( + x) The coefficie of x i he expasio of ( + x) is double he coefficie of x i he expasio of ( + x) 0

10 Fial sep (Biomial Theorem)-by abhiji Kumar Jha 9 Fill I The Blaks : (i) The coefficie of x 5 i expasio of ( + x ) 5 ( + x) 4 is (ii) If ( + x + x ) 6 = a 0 + a x + a x +, he a 0 + a + a 6 += (iii) If he coefficies of (r + ) h ad (r +5) h erms i he expasio of ( + x) 5 are equal, he value of r is (iv) The larges erm i he expasio of ( + x) 50 where x = /5, is (v) If ( + x - x ) 6 = + a x + a x ++ a x, he he expressio a + a 4 + a 6 ++a has he value, is 0 Mach he colum olum I (a) If he coefficies of x 7 ad x 8 i F HG olum II xi K J are equal, he is (P) (b) If he coefficie of x i he expasio of x is 5, ax he he value of a, is (Q) 0 0 (c) The remaider whe 00 i divided by 7 is (R) 55 (d) The las wo digis of 400, are (S) 8

11 Fial sep (Biomial Theorem)-by abhiji Kumar Jha LEVEL I (i) 5 ANSWER (ii) 495 = 7, 4 LEVEL II 8 (a) T 7 = 7 (b) 5 0 x SET I A B 4 B 5 A 6 D 7 B 8 A 9 D 0 A D B 4 A 5 D 6 D D 0 A SET II A 4 B 5 B 6 7 D 8 D 9 B 0 B B B B 7 B A SET III D AB AB 4 BD 5 AD 6 B D 0 B D A B 4 D 5 B 6 D 7 B 8 (i) T (ii) T (iii) T 9 (i) 60 (ii) 6 (iii) 4 OR 7 (iv) 6 h, 7 h (v) 0 a-r, b-p, c-s, d-q

EXERCISE - BINOMIAL THEOREM

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