BI NOMI AL D
E F I N I T I O N SA
N DR
E S U LT S 1. STATEMENT OF BINOMIAL THEOREM :If x, y R and n N, then ; (x + y)n = nC 0 x n + nC 1 x n1 y + nC 2 x n2y2 + ... + nC r x nryr + ... + nC ny n = r n
0 nC r x nryr.This theorem can be proved by Induction .
2. PROPERTIES OF BINOMIAL THEOREM :
(i) The number of terms in the expansion is (n + 1) .
(ii) The sum of the indices of x & y in each term is n .
(iii) The binomial coefficients of the terms nC 0 ,
nC
1 .... equidistant from the beginning and
the end are equal ( nC r =
nC n r ) .
(iv) General term :
The general term or the (r + 1)th term in the expansion of (x + y)n is T r+1 =
nC r x
nr . yr .
(v) Middle term(s) :
(a) If n is even , there is only one middle term which is given by ;
T(n+2)/2 = nC n/2 . x
n/2 . yn/2
(b) If n is odd , there are two middle terms which are :
T(n+1)/2 & T[(n+1)/2]+1
3. BINOMIAL THEOREM FOR NEGATIVE OR FRACTIONAL INDICES :
If n Q , then (1 + x)n = 1 + nx + n n( ) ! 1 2 x 2 + n n( ) (n ) ! 1 2 3 x 3 + ... provided x < 1 . 4. EXPONENTIAL SERIES : (i) ex = 1 + x x x 1 2 3 2 3
! ! ! ... ; where x may be any real or complex & e = Limit n 1 1 n n (ii) ax = 1 + x l n a x l n a x l n a 1 2 3 2 2 3 3 ! ! ! ... where a > 0 . 5. LOGARITHMIC SERIES : (i) ln (1+ x) = x x x x 2 3 4 2 3 4 ... where 1 < x 1 . (ii) ln (1 x) = x x x x 2 3 4 2 3 4 ... where 1 x < 1 . (iii) ln ( ) ( ) 1 1 x x = 2 x x x 3 5 3 5 ... x < 1 .
IIT – ian’s P A C E
Note :
1. Term independent of x coefficient of x0 ..
2. Numerically greatest term in the expansion of (1 x)n , x > 0 , n N is the same as the
greatest term in (1 + x)n .
3. PROPERTIES OF BINOMIAL COEFFICIENTS : (i) C0 + C1 + C2 + ... + Cn = 2n (ii) C0 + C2 + C4 + ... = C1 + C3 + C5 + ... = 2n1 (iii) C0² + C1² + C2² + .... + Cn² = 2nC n = ( ) ! ! ! 2 n n n (iv) C0.Cr + C1.Cr+1 + C2.Cr+2 + ... + Cnr.Cn = ( ) ! ( ) ( ) ! 2 n nr n r
4. When the index n is a positive integer the number of terms in the expansion of
(1 + x)n is finite i.e. (n + 1) & the coefficient of successive terms are : nC 0 , nC 1 , nC 2 , nC 3 ... nC n
5. When the index is other than a positive integer such as negative integer or fraction,
the number of terms in the expansion of (1 + x)n is infinite and the symbol nC r
cannot be used to denote the Coefficient of the general term .
6. If
(
x < 1)
. (a) (1 + x)1 = 1 x + x2 x3 + x4 .... (b) (1 x)1 = 1 + x + x2 + x3 + x4 + .... (c) (1 + x)2 = 1 2x + 3x2 4x3 + .... (d) (1 x)2 = 1 + 2x + 3x2 + 4x3 + ... 7. PROPERTIES OF 'e' : (a) e = 1 + 1 1 1 2 1 3 ! ! !...(b) 'e' is an irrational number approximately equal to 2.72 ..
(c) e + e1 = 2 1 1 2 1 4 1 6 ! ! ! ... (d) e e 1 = 2 1 1 3 1 5 1 7 ! ! ! ...
(e) Logarithms to the base ‘e’ are also called Natural Logarithm .
EXERCISE I
1. If ‘a’ be the sum of the odd terms & ‘b’ the sum of the even terms of the expansion of
(1+x)n , then (1x²)n =
(A) a²b² (B) a²+b² (C) b²a² (D) none
2. If the sum of the coefficients in the expansion of (1 + 2x)n is 6561, then the greatest
coefficient in the expansion is
(A) 1592 (B) 1492 (C) 1792 (D) 1700
3. The term independent of x in 3
2 1 3 2 9 x x is :
(A) 27/5 (B) 7/18 (C) 8/81 (D) none of these
4. Given that the term of the expansion (x1/3 x1/2)15 which does not contain x is 5m
where m N, then m =
5. If the second , third and fourth terms in the expansion of (a + b)n are 135, 30 and 10/3
respectively , then :
(A) a = 3 (B) b = 1/3 (C) n = 5 (D) n = 7
6. In the binomial (21/3 +31/3)n, if the ratio of the seventh term from the beginning of the
expansion to the seventh term from its end is 1/6, then n =
(A) 6 (B) 9 (C) 12 (D) 15
7. Let R = (6 6 + 14)2n + 1 and f = R [R] , where [ ] denotes the greatest integer function.
Then Rf =
(A) 202n + 1 (B) 202n 1 (C) 202n (D) none
8. The term independent of x in the expansion of x
x x x 1 4 1 3 is : (A) 3 (B) 0 (C) 1 (D) 3
9. The sum of the coefficients in the expansion of (1 2x + 5x2)n is a and the sum of the
coefficients in the expansion of (1 + x)2n is b . Then :
(A) a = b (B) a = b2 (C) a2 = b (D) ab = 1
10. If k R and the middle term of (k
2 + 2)
8 is 1120 , then value of k is
(A) 3 (B) 2 (C) 3 (D) 4
11. The coefficient of x4 in the expansion of (1 x + 2x2)12 is :
(A) 12C 3 (B) 13C 3 (C) 14C 4 (D) 12C 3 + 3 13C 3 + 14C 3 12. If (1+x+x²)25 = a 0+ a1x + a2x² +...+ a50 .x 50 then a 0 + a2 + a4 + ... + a50 is :
(A) even (B) odd & of the form 3n
(C) odd & of the form (3n1) (D) odd & of the form (3n+1)
13. The expression 2 2 1 2 2 1 6 x x + 2 2 2 1 2 2 1 6 x x is a polynomial of degree (A) 5 (B) 6 (C) 7 (D) 8 14. If (1 + 2x + 3x2)10 = a0 + a1x + a2x2 + .... + a 20x 20 , then (A) a1 = 20 (B) a2 = 210 (C) a4 = 8085 (D) a20 = 22. 37. 7
15. If x is so small that x2 and higher powers of x can be neglected , then value of the
expression 1 3 1 4 5 3 1 2 x x x ( ) ( ) / / is (A) 1 + 35 24x (B) 1 35 24x (C) 1 + 8 9x (D) 1 8 9x 16. C0 C1 C2 C10 1 2 3 ... 11 = (A) 2 11 11 (B) 2 1 11 11 (C) 3 11 11 (D) 3 1 11 11 17. The (m + 1)th term of x y y x m 2 1 is
(A) independent of x (B) a constant
18. If (C0+C1) (C1+C2) (C2+C3) ... (Cn1+Cn) = m . C1C2C3 ....Cn1 , then m = (A)
n n n 1 1 1 ! (B) ( ) ! n n n 1 (C)
n
n n 1 1 ! (D)
n n n 1 1 !19. The number of positive terms in the sequence xn = 1
4 195 3 3 1 1 n n n n n P P P is :
(A) 2 (B) 3 (C) 4 (D) none of these
20. The numerically greatest terms in the expansion of (2x+5y)34 when x = 3 & y = 2 is
(A) T21 (B) T22 (C) T23 (D) T24
21. If the second term of the expansion a a
a n 1 13 1 / is 14a
5/2 then the value of n n C C 3 2 is : (A) 4 (B) 3 (C) 12 (D) 6 22. The value of 4 {nC 1 + 4 . nC 2 + 4 2 . nC 3 + ... + 4 n 1} is : (A) 0 (B) 5n + 1 (C) 5n (D) 5n 1
23. The greatest integer less than or equal to ( 2 + 1)6 is
(A) 196 (B) 197 (C) 198 (D) 199
24. The coefficient of x10 in the expansion of (1 + x2 x3)8 is
(A) 476 (B) 496 (C) 506 (D) 528 25. In the expansion of x x x x x x 1 1 1 2 3 1 3 1 2 10
/ / / , the term which does not contain x is :
(A) 10C 0 (B) 10C 7 (C) 10C 4 (D) none 26. If x = (7 + 4 3)2n = [x] + f , then x(1 f) = (A) 2 (B) 0 (C) 1 (D) 2520
27. Coefficient of xn 1 in the expansion of ,
(x + 3)n + (x + 3)n 1 (x + 2) + (x + 3)n 2 (x + 2)2 + ... + (x + 2)n is (A) n+1C 2(3) (B) n1C 2(5) (C) n+1C 2(5) (D) nC 2(5) 28. Let (1 + x2)2 (1 + x)n = A 0 + A1 x + A2 x 2 + ... If A
0, A1, A2 are in A.P. then the value of
n is :
(A) 2 (B) 3 (C) 5 (D) 7
29. Set of value of r for which, 18C
r 2 + 2 . 18C r 1 + 18C r 20C 13 contains :
(A) 4 elements (B) 5 elements (C) 7 elements (D) 10 elements
30. If (1 + x + x2)n = a0 + a1x + a2x2 +...+ a 2nx 2n then value of a0 + 2a1 + 3a2 + ...+ (2n + 1)a2n is (A) 3n (1 + n) (B) n.3n (C) 3n (D) none 31. If Cr stands for 4C r ,then C0C4 C1C3+ C2C2 C3C1 + C4C0
=
(A) C2 (B) C3 (D) 24 (D) 1832. The value of the expression 47C
4 + j
1 5 52 jC 3 is equal to : (A) 47C 5 (B) 52C 5 (C) 52C 4 (D) 49C 433. The greatest terms of the expansion (2x + 5y)13 when x = 10, y = 2 is : (A) 13C 5 . 20 8 . 105 (B) 13C 6 . 20 7 . 104 (C) 13C 4 . 20 9 . 104 (D) none of these 34. (nC 0) 2 + (nC 1) 2 + (nC 2) 2 + ... + (nC n) 2 = (A) 22n (B) 2nC n (C) ( 2nC n) 2 (D) (n!)2
35. The remainder when 22003 is divided by 17 is
(A) 1 (B) 2 (C) 8 (D) none of these
36. If n is a natural number which is not a multiple of 3 and (1 + x + x2)n =
a xr r r n
0 2 , then value of ( ) ( )( )
1 0 r r n r r n a C is (A) 1 (B) 2 (C) 0 (D) (1)nEXERCISE II
1. Show that the coefficient of x10 in the expansion of (1 + x2 x3)9 is 882 .
2. Show that the term independent of x in the expansion of 1 6
10 x x is , 1 + r 1 5 10C 2 r 2 r C r 6 r .
3. Show that there are 32 integer terms in the expansion of ,
345
124 .4. Find numerically the greatest term in the expansion of :
(i) (2 + 3x)9 when x = 3
2 (ii) (3 5x)
15 when x = 1
5
5. In the binomial expansion of , y
y n 1 24
(
)
the first three coefficient form an A.P.in the order . Find other terms in the expansion of which the power of ' y ' is a natural number . 6. Given sn= 1 + q + q² + ... + qn & S n = 1 + q 1 2 + q 1 2 2 + .... + q n 1 2 , q 1 , prove that n+1C 1 + n+1C 2.s1 + n+1C 3.s2 +....+ n+1C n+1.sn = 2 n . S n . 7. If x = 1 3 1 3 3 6 1 3 5 3 6 9 1 3 5 7 3 6 9 12 . . . . . . . . .
. . . ... then prove that x² + 2x 2 = 0 .
8. Given that , (1 + x + x2 + x3)5 = a 0 + a1 x + a2 x 2 + ... + a 15 x 15 . Find a 10 . 9. Prove that
( )! ! 72 36 2 1 is divisible by 73 . 10. Show that , lnx = x x x x x x 1 1 1 2 1 1 1 3 1 1 2 2 3 3 . ( ) .( ) +... (x > 0) 11. If
3 3
5 2n 1= p+f where p is an integer and f is a proper fraction then find the value of f (p+f) . n N .
12. Prove that if 'p' is a prime number greater than 2, then the difference
2 5
p 2p+1is divisible by p, where [ ] denotes greatest integer .
13. If 'n' & ' r ' are coprime , prove that nC
r is divisible by n .
14. If C0 , C1 , C2 , ... , Cn are the combinatorial coefficients in the expansion of (1 + x)n,
n N , then prove that ,
1 . Co² + 3 . C1² + 5 . C2² + ... + (2n+1) Cn² = ( ) ( )!
! ! n n
n n
1 2
15. If n is an integer greater than 1 , show that ; a nC 1(a1) + nC 2(a2) ... + (1) n (a n) = 0 16. C C C Cn n n 1 2 3 1 2 1 2 ... 17. If (1+x)n = C 0 + C1x + C2x² + .... + Cn x
n , then show that the sum of the products of the
Ci’s taken two at a time , represented by C C
i j n i j 0 is equal to 2 2n1 2 2 2 n n ! ( !) . 18. If (1 + 2x + 2 x2)n = k 0 + k1 x + k2 x 2 + ... + k 2 n x 2 n . Prove that , k2 = 4 n2 22 n n
(
1 + 3 . n 1C 2 + 5 . n 1C 3 + ... + (2n 1) n 1C n 1)
. 19. If (1 + x)n = C x r r r n .
0then prove that ;
2 1 2 2 2 3 2 3 4 2 1 2 3 2 5 1 2 2 0 3 1 4 2 2 2 . . . . . . ... . ( ) ( ) ( ) ( ) C C C C n n n n n n n n
20. Prove that the sum to (n+1) terms of C
n n C n n C n n 0 1 2 1 1 2 2 3 ( ) ( ) ( ) ( ) ( )... equals 0 1
xn1. (1 x)n+1 . dx & evaluate the integral .21. Prove that , k n 0 3 6 nC 2 k 1 ( 3) k = 0 . 22. Prove that , 2 1 12 2 28 3 50 4 78 5 ! ! ! ! ! + ... = 5 e + 2 23. If the series 1 + x x 3 6 3! 6! + ... ; x + x4 x7 4! 7! + ... ; x2 x5 x8 2! 5! 8! + ... are denoted
by a , b , c respectively, show that : a3 + b3 + c3 3abc = 1 .
24. Prove that the coefficient of xn in the expansion of log
e (1 + x + x²) is
2 n or
1 n
according as n is/or is not a multiple of 3 .
25. Find the sum of the following infinite series , x x x x
2 3 4 5 2 2 3 3 4 4 5 ...given x < 1 26. Prove that , 1 1 2 1 3 4 1 5 6 . . . ... = 1 2 1 1 2 3 1 3 4 5 1 5 6 7 . . . .... = ln 2
27. If , are the roots of the equation ax² + bx + c = 0 , then show that : loge (a + bx + cx²) = loge a () ... x x x 2 2 2 3 3 3 2 3 28. Show that : 1 1 1 2 1 1 3 1 2 3 n (n ) (n ) + .... = 1 1 2 1 3 2 3 n n n ... 29. Prove that , 1 2 3. .5 3 4 5. .7 5 6 7. .9 +... = 3 ln2 1 30. If x denotes
2 3
n, n N & [x] the integral part of x then find the value of ; x x² + x[x] .
31. Find the cube root of 1001 correct to 5 decimal places without using calculator or
tables . Also evaluate (0.99)15 correct to 4 places .
32. Find the index 'n' of the binomial x
n 5 2 5
if the 9th term of the expansion has numerically
the greatest coefficient (n N) .
If C0 , C1 , C2 , ... , Cn are the combinatorial coefficients in the expansion of (1 + x)n , n N , then prove the following :
33. Co² C1² + C2² C3² + ... + (1)n C n² = 0 or (1) n/2 C n/2 according as n is odd or even . 34. (n1)² . C1 + (n3)² . C3 + (n5)² . C5 +... = n (n + 1)2n3 .
35. If a0 , a1 , a2 , ... be the coefficients in the expansion of (1 + x + x²)n in ascending
powers of x , then prove that :
(i) a0 a1 a1 a2 + a2 a3 .... = 0 (ii) a0a2 a1a3 +a2a4 ...+ a2n 2 a2n = an + 1 . (iii) E1 = E2 = E3 = 3n1 ; where E 1= a0 + a3 + a6 + ... ; E2 = a1 + a4 + a7 + ... & E3 = a2 + a5 + a8 + ... 36. Prove that : C C C C C n n n n n n n 0 1 2 3 1 5 9 13 1 4 1 4 1 5 9 13 4 3 4 1 ...( ) . ! . . . ... ( ) ( ) . 37. If (1+x)n = C 0 + C1x + C2x² + ... + Cn x
n , then show that :
C1(1x) C2 2 (1x)²+ C3 3 (1x) 3 ....+ (1)n11 n(1x) n = (1x)+ 1 2(1x²)+ 1 3(1x 3) +...+1 n(1x n) 38. Prove that , 1 2 nC 1 2 3 nC 2+ 3 4 nC 3 4 5 nC 4 + ... + ( ) 1 1 1 n n n . nC n= 1 1 n . 39. Prove that , (2nC 1)²+ 2 . ( 2nC 2)² + 3 . ( 2nC 3)² + ... + 2n . ( 2nC 2n)² =
( )! ( ) ! 4 1 2 1 2 n n 40. If (1+x+x2)n = r n
0 2ar xr , n N, then prove that
EXERCISE III
1. Find the coefficient of x50 in the expression :
(1 + x)1000 + 2x . (1 + x)999 + 3x² (1 + x)998 + ... + 1001 x1000
[ REE ’90 , 6 ]
2. [ REE ’91 , 6 + 6 ]
(a) If n is a positive integer & Ck = nC
k , find the value of k
C C k n k k 3 1 1 2
.(b) Find the sum of the following series upto infinity .
1 2 1 2 2 3 2 2 12 5 2 7 24 2 17 12 2 80 ... 3. [ JEE ’92 , 6 + 2 ] (a) If ar x b x r r n r r r n ( ) ( )
2
3 0 2 0 2& ak = 1 for all k n, then show that bn = 2n+1C n+1.
(b) The expression [x + (x31)1/2]5 + [x (x31)1/2]5 is a polynomial of degree :
(A) 5 (B) 6 (C) 7 (D) 8
4. [ REE '92 , 6 + 2 ]
(a) Determine the value of 'x' in the expression (x + xt)5 , if the third term in the expression
is 10,00,000 . Where t = log10 x .
(b) Sum the following series : 9 16
2 27 3 42 4 ! ! ! ... 5. [ REE '93 , 6 + 6 ]
(a) Find the value of 'x' for which the sixth term of 2 10 3
2
1/ 2 2 31/ 5 log log x x m is equal to 21 & binomial coefficients of second , third & fourth terms are the first, third & fifth terms of an arithmetic progression . [ Take every where base of log as 10 ]
(b) Find the sum of a x
x a x x a x x 2 2 2 4 4 3 6 6 1 2 1 3 1
... and determine the values of a & x for which it is valid .
6. Let n be a positive integer . If the coefficients of 2nd , 3rd & 4th terms in the
expansion of (1+x)n are in AP , then the value of n is ______ .
[ JEE ’94 , 2 ]
7. Given that the 4th term in the expansion of 2 3
8 10 x
has the maximum numerical value , find the range of values of 'x' for which this will be true .
[ REE ’94 , 6 ]
8. If a0 , a1 , a2 , ... be the coefficients in the expansion of (1+x+x²)n in ascending powers
of x , then prove that a0² a1²+ a2² a3²+ ... + a2n² = an .
[ JEE ’94 , 5 ]
9. Find the sum of the infinite series a1 + a2 + a3 + ... , where an = (loge3)n
k n
1 2k 1 k n k [ REE ’95 , 6 ] 10. Let (1+x²)² . (1+x)n = a x K K K n .
0 4. If a1 , a2 & a3 are in AP, find n .
[ REE ’96 , 6 ]
11. In the expansion of the expression (x + a)15 , if the eleventh term is the geometric mean of
the eighth and twelfth terms , which term in the expansion is the greatest ?
[ REE ’96 , 6 ]
12. In the binomial expansion of (a b)n, n 5, the sum of the 5th and 6th terms is zero. Then
a/b equals : (A) n 5 6 (B) n 4 5 (C) 5 4 n (D) 6 5 n [ JEE '2001 , 1 ]