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JEE (Main+Advanced), , Binomial Theorem, , Exercise – I, , 9., , Section (A) : Expansion of ( x  a) n , general term,, 1., 2., 3., , 4., , middle term and coefficient of xk, 15, C3r 15 Cr 3 then value of r is, , (a) 3, (c) 5, , (b) 4, (d) 8, 2n, n, C3 : C2  44 : 3 then value of n C3 is, , (a) 20, (c) 56, The value of, (a), (c), , 15, , C3, , C10, , 17, , 16, , C16, , C13, , (d), , 17, , 10., , 11, , 1, (b) 11 C5 a 5b 6, 2, 1, (d) 11 C6 a 6b5, 2, , 1, , In the expansion of  3x   coefficient of x 7, x, , (c), , 13, , C3 310, , (b), (d), , The coefficient of, , (2  3x)12 is -, , (a), (c), , 12, 12, , C5 25.37, , C5 27.35, , (b), , 13, 13, , The term containing x in the expansion of, ,  2 1,  x   is x, , , 13., , 14., , 15., , 5, , (a) 2nd, (c) 4th, , (b) 3rd, (d) 5th, , is :, , ( x  a)100  ( x  a )100 after simplification is :, ,  1, , 2,  x8/3  x log10 x  is 5600, then x =, 8, , (a) 10, (c) 11, , C10 33, , (d) None of these, , 2 m 1, , 12., , C4 39, , C6 26.36, , x y, The ( m  1) term of   , y x, th, , (a) 50, (b) 202, (c) 51, (d) None of these, If the 6th term in the expansion of the binomial, , x 5 in the expansion of, 12, , (b) 5/4, (d) None of these, , 11., , 13, , is (a) 13 C5 (3)8, , 10, , (a) independent of x, (b) a constant, (c) dependent on the ratio x/y and m, (d) None of these, The total number of terms in the expansion of,, , C8 (3 x)8 (4 y )9, , b, , In the expansion of  2a   the 6th term from, 2, , , (c) 211 C5 a 6 b5, , 8., , 30, , C10 (3 x)10 (4 y ) 7, , (a) 211 C5 a 5b 6, , 7., , (b), , (d), , the end is, , 6., , C3 15 C13 is, ,  x, 3 ,  2  is ,  3 2x , , (a) 3/2, (c) 5/2, , (b) 35, (d) 10, , In the expansion of (3x  4 y )17 , 7th term from, beginning is, (a) 17 C7 (3 x ) 7 (4 y )10, (b) 17 C6 (3 x )11 (4 y ) 6, (c), , 5., , 16, , 15, , The term independent of x in the expansion of, , 16., , (b) 8, (d) 9, , , , 17, th, In the expansion of  3 , , 3, 2,  the 11, , 4, , , 15, , term is a :, (a) positive integer, (b) positive irrational number, (c) negative integer, (d) negative irrational number., If the second term of the, , expansion, ,  1/13, a , 5/ 2,  a  1  is 14a , then the value of, a , , n, C3, is :, n, C2, n, , (a) 4, (c) 12, , (b) 3, (d) 6, , In the expansion of (71/3  111/9 )6561 , the number, of terms free from radicals is :, (a) 730, (b) 729, (c) 725, (d) 750, The middle term in the expansion of, , (1  3x  3x 2  x3 )6 is -, , (a), , 18, , C10 x10, , (b), , 18, , C9   x , , 9, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 1

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JEE (Main+Advanced), (c), , 17., , 18., , 18, , Binomial Theorem, , (d) 18 C10 x10, , C9 x9, , k, , If k  R and the middle term of   2  is, 2, , 8, , 1120, then value of k is :, (a) 3, (b) 2, (c) -3, (d) -4, The term with coefficient 6 C2 in the expansion, , 19., , of (1  x ) is (a) T1 and T3, (b) T2 and T4, (c) T3 and T5, (d) None of these, The co-efficient of x in the expansion of, , 20., , (a) 56, (c) 154, Given that, , 21., , 22., ,  1, (1  2 x 3  3 x 5 )  1   is :,  x, 8, , of, , the, , (a) 1100, (b) 1010, (c) 1001, (d) none, The term independent of x in the expansion of, , 1 , 1, ,  x    x   is :, x , x, , 4, , 3, , (a) -3, (c) 1, , (b) 0, (d) 3, , Let the co-efficients of, , (1  x) 2 n1 be, , P, , &, , (a) 9, (c) 81, , Q, , x n in, , (1  x) 2 n &, , respectively,, , then, , (b) 27, (d) none of these, , If (1  by) n  (1  8 y  24 y 2  ...) where n  N, then the value of b and n are respectively(a) 4,2, (b) 2,-4, (c) 2,4, (d) -2,4, The, value, of, ,, 18  7  3.18.7.25, 3  6.243.2  15.81.4  20.27.8  15.9.16  6.3.32  64, 3, , 3, , 6, , is :, (a) 1, (c) 3, , (b) 2, (d) none, , , 100, , (a), 26., , C47, , (b), , x 52 in, , Cm ( x  3)100 m .2m is :, 100, , the, , expansion, , C48, , (d) 100 C100, , (1  x) 21  (1  x) 22  ...  (1  x)30 is :, 51, 31, , C5, , C6  21 C6, , (b) 9 C5, (d), , 30, , C5  20 C5, , , , , The term independent of x in (1  x) m  1 , is, (a), (c), , 28., , 100, , of, , The co-efficient of x 5 in the expansion of, , (c), 27., , 100, , coefficient, , (c) 100 C52, , (a), , expansion, , ( x13  x 1/ 2 )15 which does not contain x is 5 m,, where m  N , then m =, ,  PQ , ,  ,  Q , , 24., , (b) 65, (d) 62, the term, , The, m0, , 6, , 5, , 23., , 25., , mn, m 1, , Cn, , Cn, , (b), , (d), , mn, mn, , 1, , x, , n, , Cn, , Cn 1, , (1 x)(1 x x2)(1 x  x2  x3)....(1 x x2 ...  x100), , when written in the ascending power of x then, the highest exponent of x is, (a) 5000, (b) 5030, (c) 5050, (d) 5040, Section (B) : Remainder and Divisibility problems, 29., The remainder when 1710 is divided by 9 is :, (a) 8, (b) 7, (c) 2, (d) 1, 30., The remainder when 798 is divided by 5 is, (a) 4, (b) 0, (c) 2, (d) 3, 31., The remainder when 22003 is divided by 17 is :, (a) 1, (b) 2, (c) 8, (d) none of these, 32., If {x} denotes the fractional part of ‘x’, then, ,  31001 , , ,  82 , , 33., 34., 35., , (a) 9/82, (b) 81/82, (c) 3/82, (d) 1/82, The last three digits of 350 will be, (a) 249, (b) 259, (c) 349, (d) 241, The last two digits of the number 3400 are :, (a) 81, (b) 43, (c) 29, (d) 01, The last three digits in 10 ! are :, (a) 800, (b) 700, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 2

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JEE (Main+Advanced), , Binomial Theorem, , (c) 500, (d) 600, n, 36., If n  N then 10  5 is divisible by (a) 3, (b) 7, (c) 9, (d) 2, Section (C) : Sum of series, Product and division of, binomial coefficients, Reverse Expansion, 37., If, , 44., , (1  x ) n  C0  C1 x  C2 x 2  C3 x 3  ...  Cn x n ,, , then the value of C1  C2  C3  ...  Cn is -, , (a) 2, (b) 2, n, (c) 2  1, (d) 2 n  1, If (1  x )n  C0  C1 x  C2 x 2  ...Cn .x n then the, n1, , 38., , n1, , value of C0  3C1  5C2  ...  (2n  1)Cn is -, , (a) n.2, , (c) (n  2).2, 39., , 40., , The value of, , b, n, , 42., , 43., , , , (a) 99, (c) 101, , , ,  r., 10, , r 1, , (d) (n  1).2, , n, , n, , n, , Cr, is equal to, Cr 1, , (b) 10 n, , (d) None of these, , r 0, , br  1 , , (101), , then n equals to :, 100!, 100, , ar, and, ar 1, , n, , 47., , 11, C0 11 C1 11 C2, C10, , ,  ... , , 1, 2, 3, 11, 211  1, 211  1, (a), (b), 11, 6, 11, 11, 3 1, 3 1, (c), (d), 11, 6, , 48., , value, , (c), , 52, , The, , C4, , value, , of, , C0 C1 C2 C3, Cn, , , ,  ...  (1) n, is :, 1.3 2.3 3.3 4.3, (n  1).3, , (d), , 52, , of, , 49, , C4  , 5, , j 1, , 52  j, , C3, , C5, , C4, , 10, , The, , the, , expression, , value, ,  50   50   50  50 ,  50   50 ,          ...      is,,  0   1   1  2 ,  49   50 , n, where n Cr   , r, , of, , 100 , ,  51 , , (b) , ,  50 ,  50 , (c)  , (d)  ,  25 ,  25 , n, If (1  x )  C0  C1 x  C2 x 2  ...  Cn .x n then, 2, , for n odd, C12  C32  C52  ...  Cn2 is equal to, , (c), 49., , (b), , 47, , CK ,  10 10   10, K,   Cr    (1) 2 K  is :,  r 0,   k 0, , 10, 20, (a) 2, (b) 2, (c) 1, (d) 25, , (a) 2 2 n 2, , 11, , The, , The value of the expression, , (a) , , n, , (d) None of these, , in the expansion of (1  x  x 2  x 3 )5 is (a) 512, (b) -512, (c) 215, (d) None of these, , 100 , ,  50 , , (b) 100, (d) None of these, , n 1, 3, , (b), , The sum of the coefficients of even powers of x, , is equal to :, (a) 47 C5, 46., , n, , Cr, , n, r  0 Cr  Cr, n, n 1, (a), (b), 2, 2, n(n  1), n, (c) ( n  1), (d), 2, 2( n  1), n 1, , 45., , n, , (1  x) n   ar x r and, , r, , r 1, , 41., , n 1, , (a) 5(2n  9), (c) 9( n  4), , If, , (b) (n  1).2, , n, , 3, n 1, 1, (c), 3(n  1), (a), , (2n)!, 2(n !) 2, , (b) 2 n, (d), , (2n)!, ( n !) 2, , 1, , then value of, , n, r  0 Cr, n, 1, (a) an, (b) an, 2, 4, (c) nan, (d) 0, If an , , n, , , , n  2r, is :, n, Cr, r 0, n, , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 3

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JEE (Main+Advanced), , 50., , Binomial Theorem, ,  1, In the expansion of (1  x) 1   , the term,  x, n, , n, , independent of x is (a) C02  2C12  ...  ( n  1)Cn2, (b) (C0  C1  ...  Cn ) 2, (c) C02  C12  ...  Cn2, , Answer Key, , (d) None of these, Section (D) : Binomial Theorem for negative and, fractional index, 51., If x  1 then 7th term in the expansion of, , (1  x) 2 is, , (a) 7x 7, (c) 7x 6, 52., ,  2 1, x  , x, , , 4/3, , theorem if, , (a) x  1, , (c) x  1, , (b) 6x 7, (d) 6x 6, can be expanded by binomial, (b) x  1, , 3, a, 13, b, 23, c, 33, a, 43, c, 53, d, , 4, b, 14, a, 24, a, 34, b, 44, a, 54, a, , 5, b, 15, b, 25, b, 35, a, 45, c, 55, b, , 6, c, 16, b, 26, c, 36, a, 46, c, 56, a, , 7, c, 17, b, 27, b, 37, d, 47, b, , 8, c, 18, c, 28, c, 38, d, 48, c, , 9, b, 19, c, 29, d, 39, a, 49, d, , 10, c, 20, c, 30, a, 40, b, 50, c, , x  1, then the co-efficient of x n in the, , If, , 54., , (a) n, (b) n  1, (c) n  2, (d) n  1, The coefficient of x 4 in the expression (, , 56., , 2, a, 12, a, 22, d, 32, c, 42, b, 52, d, , (d) x  1, , 53., , 55., , 1, a, 11, c, 21, b, 31, c, 41, a, 51, c, , expansion of (1  x  x 2  x 3  .....) 2 is, , 1  2 x  3 x 2  4 x 3  ... up to  ) (where x  1 ), , is, (a) 1, (c) 2, , (b) 3, (d) 5, , The coefficient of x 4 in (1  x ) 3 (where x  1 ), is, (a) 10, (b) 15, (c) 9, (d) 6, If x is so small such that x 3 and higher power of, x may be neglected, then, may be approximated as, , 3 2, x, 8, 3, (c) 1  x 2, 8, (a), , (1  x)3/ 2  (1 , (1  x)1/ 2, , 1 3, x), 2, , x 3 2,  x, 2 8, 3, (d) 3 x  x 2, 8, (b), , ADDRESS : NEAR CANARA BANK, JAIL BYPASS ROAD, PADRI BAZAR, GORAKHPUR,, MOB: 6386566032,7992166350, , 4