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Binomial Theorem, , EXERCISE-1, Q.1, , r  1, , th, , term in the expansion of  x  a  will be, n, , [1] n Cr x n an r, Q.2, , [2] n Cr x n r ar, , n, , [2] n : r, , Q.5, , [3] x : n, , [4] None of these, , If the coefficient of 7th and 13th term in the expansion of 1  x  are equal, then n , n, , [1] 10, Q.4, , [4] n Cr x r an r, , The ratio of the coefficient of terms xnr ar and xr anr in the binomial expansion of  x  a  will be, [1] x : a, , Q.3, , [3] n Cr x n r an, , [2] 15, , [3] 18, , [4] 20, , The coefficient of two consecutive terms in the expansion of 1  x  will be equal, if, n, , [1] n is any integer, , [2] n is an odd integer, , [3] n is an even integer, , [4] None of these, , 10, , C1  10 C3  10 C5  10 C7  10 C9 , , [1] 29, , [2] 210, , [3] 210  1, , [4] None of these, , 10, , Q.6, , 1, 6th term in expansion of  2x 2  2 , 3x , , [1], , Q.7, , 14, , 4580, 17, , [2] , , [3], , [2] 214  1, , x a, The middle term in the expansion of   , a x, [1], , Q.9, , 896, 27, , 5580, 17, , [4] None of these, , C1  14 C2  14 C3  ...  14 C14 , , [1] 214, , Q.8, , is, , 20, , C11, , x, a, , [2], , 20, , C11, , [3] 214  2, , [4] 214  2, , [3], , [4] None of these, , 20, , is, , a, x, , 20, , C10, , In the expansion of 1  x  , coefficient of x 5 will be, 5, , [1] 1, , [2] –1, , [3] 5, , [4] –5, , 12, , Q.10, , a, , In the expansion of   bx  , the coefficient of x 10 will be, x, , [1] 12a11, , Q.11, , [2] 12b11a, , [3] 12a11b, , If A and B are the coefficients of xn in the expansions of 1  x , [1] A  B, , [2] A  2B, , 2n, , [4] 12a11b11, and 1  x , , [3] 2A  B, , 2n 1, , respectively, then, [4] None of these, 127

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Binomial Theorem, , Q.12, , Sum of odd terms is A and sum of even terms is B in the expansion  x  a  , then, n, , [1] AB , , 1, 2n, 2n,  x  a   x  a, 4, , [3] 4AB   x  a    x  a , 2n, , Q.13, , Q. 16, , 1.3.5.......n, 2n, 2.4.6......(n  1), , [2], , 1.3.5.......(n  1) n, 2, 2.4.6......n, , 1, , , The coefficient of y in the expansion of  1  2y 3 , , , , [3], , 2.4.6......n, 2n, 1.3.5.......(n  1), , [4], , 1.3.5...  2n  1, n!, , 2n xn, , [4] None of these, , [2] 4, , 1, , will be, [3] 8, , [4] 16, , The sum of the coefficients in the expansion of 1  x  3x 2 , , 2163, , [2] 1, , will be, , [3] –1, , [4] 22163, , [3] 0, , [4] 2n 1, , C0  C1  C2  C3  ...   1 Cn is equal to, n, , [2] 2n  1, , , , In the expansion of 1  x  x3  x 4, [1], , Q. 20, , is, , 2n+2, , [1] 2n, Q. 19, , 2n, , If the coefficient of middle term in the expansion of (1 + x), is p and coefficients of middle terms in the, 2n+1, expansion (1 + x), are q and r, then [1] r = p + q, [2] q = p + r, [3] p = q + r, [4] p + q + r = 0, , [1] 0, Q. 18, , 2n, , [4] None of these, , 1.3.5...  5n  1, , [1] 1, Q. 17, , 2n, , 1.3.5...  2n  1 n, 2.4.6...2n 2n1, x, xn, x, [2], [3], n!, n!, n!, n, The coefficient of middle term in the expansion of (1 +x) , when n is even, is, , [1], Q.15, , 2n, , The middle term in the expansion of 1  x , [1], , Q.14, , [2] 2AB   x  a    x  a , , 40, , [2], , C4, , 10, , , , 10, , , the coefficient of x 4 is, [3] 210, , C4, , [4] 310, , The number of terms in the expansion of ( 1 + 5 2 x)9 + ( 1 – 5 2 x)9 is –, [1] 5, , [2] 7, , [3] 9, , [4] 10, , 10, , Q.21, , x 2 , If the rth term in the expansion of   2 , 3 x , [1] 2, , [2] 3, , contains x 4 , then r is equal to, [3] 4, , [4] 5, , n, , Q. 22, , 3 , 2, If the 4th term in expansion of  x ,  is independent of x, then n , 3, 2x, , , [1] 5, , Q. 23, , [2] 6, , To make the term, [1], , 3n, , Cr  0, , 3n, , [3] 9, , [4] None of these, , Cr  1 x 3n r free from x, necessary condition is, r, , [2] x3nr  0, , [3] 3n  r, , [4] None of these, 128

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Binomial Theorem, , Q. 24, , Q. 25, , If the coefficients of p , (p+1) and (p+2) terms in the expansion of 1  x  are in A.P., then, th, , [2] n2  n  4p  1  4p2  2  0, , [3] n2  n  4p  1  4p2  0, , [4] None of these, , If the coefficients of 5th, 6th and 7th terms in the expansion of 1  x  be in A.P., then n=, n, , [2] 14 only, ,  2n !, 2, n!, , [2], , 2n  2, ,  2n  2 !, 2,  n  1 !, , [4] None of these, , is, , [3], ,  2n  2 !, n!  n  1!, , [4], , The sum of all the coefficients in the binomial expansion of  x 2  x  3 , , 319, , [1] 1, Q. 28, , [3] 7 or 14, , The greatest coefficient in the expansion of 1  x , , [1], Q. 27, , n, , th, , [1] n2  2np  4p 2  0, , [1] 7 only, Q. 26, , th, , [2] 2, ,  2n !, n!  n  1!, , is, , [3] –1, , [4] 0, , If the sum of the coefficients in the expansion of 1  3x  10x 2  is a and if the sum of the coefficients in the, n, , expansion of 1  x 2  is b, then, n, , [1] a  3b, Q.29, , where x , , [2] 51th, , 1, is, 5, , [3] 6th & 7th, , [4] 6th, , The numerically greatest term of ( 2 + 3x) when x = 3/2 is, [2] T7, , [3] T8, 3/2, , If sum of all the coefficients in the expansion of (x, [1] 35, , Q. 32, , 50, , [4] None of these, , 9, , [1] T6, Q.31, , [3] b  a3, , The largest term in the expansion of  3  2x , [1] 5th, , Q.30, , [2] a  b3, , [2] 45, , [4] None of these, , –1/3 n, , +x, , 5, , ) is 128, then the coefficient of x is, , [3] 7, , [4] none of these, , n, 1, In  3 2  1  if the ratio of 7th term from the begining to the 7th term from the end is , then n =, 6, 3, 3, , , [1] 7, , [2] 8, , [3] 9, , [4] None of these, , 8, , Q.33, ,  1, , th, 2, If 6 term in the expansion of  8 / 3  x log10 x  is 5600 , then x is equal to, x, , [1] 8, , [2] 10, , [3] 9, , [4] none of these, , n, , Q.34, , 7, 8, x, The coefficient of x and x in the expansion of  2   are equal, then n is equal to 3, , , [1] 35, , [2] 45, , [3] 55, , [4] none of these, 129

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Binomial Theorem, n, , Q.35, , 1, , If in the expansion of  x 2   , the coefficient of third term is 31, then the value of n is 4, , , , [1] 30, , [2] 31, , [3] 29, , [4] 32, , 10, , Q.36, , , 3 3, The term independent of x in the expansion of  x 2  3 , , x , , [1] 153090, , Q.37, , 1  x , , n, , [2] 150000, , [2] by x 2, , [2] 2n  1, , If 1  x   C0  C1x  C2 x 2  ...  ...Cn xn , then, n, , n, , n  n  1, , [2], , 2, , C0 , , [2], , n n  2, 2, , [4] 2n 1  1, , [3] 2n, C1 2C2 3C3, nCn, , ,  ... , , C0, C1, C2, Cn 1, , [3], , n  n  1, , [4], , 2, , n  1n  2 , 2, , 1, n, , [3], , 1, n 1, , [4], , 1, n 1, , C1  2C2  3C3  4C4  ...  nCn , , [1] 2n, Q.42, , [4] All of these, , n, Cn, 1n, 1, n, C1  n C2  ...   1, , 2, 3, n 1, , [1] n, Q.41, , [3] by 2x 3, , n, , [1], , Q.40, , [4] 153180, , If 1  x   C0  C1x  C2 x 2  ...  Cn x n , then the value of C0  C2  C4  .... is, [1] 2n 1, , Q.39, , [3] 150090, ,  nx  1 is divisible (where n  N ), , [1] by 2x, Q.38, , is -, , [2] n.2n, , [3] n.2n 1, , If a and d are two complex numbers, then the sum to, , [4] n.2n 1, , n  1, , terms of the following series, , aC0   a  d C1   a  2d C2  ...  ... is, [1], Q.43, , a, 2n, , [2] na, , [4] None of these, , In the expansion of 1  x  the sum of coefficients of odd powers of x is, n, , [1] 2n  1, Q.44, , [3] 0, , [2] 2n  1, , [3] 2n, , [4] 2n 1, , If 1  x   C0  C1x  C2 x 2  ...  Cn x n , then the value of C0  2C1  3C2  ...   n  1 Cn will be, n, , [1]  n  2  2n1, , [2]  n  1 2n, , [3]  n  1 2n1, , [4]  n  2  2n, , 130

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Binomial Theorem, , Q.45, , n, n , 2 n, n n,    2    2    ...  2   is equal to, 0,  1, 2, n, , [1] 2n, ,  1, k 1 , , n, k 1 , n, , Q.46, , [4] None of these, , [2] n  n  1, , [3] n2, , [4]  n  1, , [3] 0, , [4] 51, , , , The value of, , 15, , 2, , 2, is, C02  15 C12  15 C 22  ....  15 C15, , [1] 15, Q.48, , [3] 3n, , k 1, , [1] n  n  1, , Q.47, , [2] 0, , [2] –15, , n, , If C0 , C1 , C2 .... Cn denote the binomial coefficients in the expansion of ( 1 + x) , then the value of, n, , , , (r  1) Cr is, , r 0, , [1] n 2, Q.49, , n, , If (1 + x), , [2] ( n + 1) 2, 15, , 2, , 14, , 1, , [2], , 2, , The value of, , n–2, , 14, , 14, , . 13 + 1, , 14, , [3] 2 . 14 – 1, , [4] 2 . 14 + 1, , 1, , [3], , [4], , 2, , 3, , 1, 3, , 30 upto three decimals is, , [2] 5.477, , [3] 5.475, , [4] None of these, , [3] , , [4], , If n is odd, then C20  C12  C22  C23  ...   1 Cn2 , n, , [1] 0, , Q.53, , [4] (n + 2) 2, , 15, , [2] 2, , [1] 5.476, Q.52, , n–1, , 1 1.3 1.3.5, , ,  ..... , 4 4.8 4.8.12, , [1], Q.51, , [3] (n + 2) 2, , = C0 + C1x + C2x + ....+ C15x , then the value of C2 + 2C3 + 3C4 + ....+ 14C15 is -, , [1] 2 13 – 1, Q. 50, , n–1, , The expansion of, , [1] x  1, , [2] 1, , 1, ,  4  3x , , 1/ 2, , n!, 2, , n, 2 !,  , , by binomial theorem will be valid, if, , [2] x  1, , [3] , , 2, 3, , x, , 2, 3, , [4] , , 4, 4, x, 3, 3, , 131

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Binomial Theorem, , Q.54, , 1, 3, , 6  3x, , , , 1,  x 2x 2, , [1] 6 3 1   2  ..., 6, 6, , , , Q.55, , 1, , Q.58, , Q.59, , 1, 3, ,  x 2x 2, , 1   2  ..., 6, 6, , , , 1, 2, , , [3] 6 3 1  x  2x  ..., 2, 6, 6, , , , 1, , 1, , [2] 1  1  y  3, , [4] 6, , , , 1, 3, ,  x 2x 2, , 1   2  ...,  6 6, , , 1, , [3] 1  1  y  3, , [4] 1  1  y  3, , If the value of x is so small that x 2 and greater powers can be neglected, then, [1] 1 , , Q.57, , , , If y  3x  6x 2  10x 3  .... , then the value of x in terms of y is, [1] 1  1  y  3, , Q.56, , [2] 6, , 5, x, 6, , [2] 1 , , 5, x, 6, , [3] 1 , , –1/3, , The general term in the expansion of ( 1–3x), , 2, x, 3, , 1  x  3 1  x , 1 x  1 x, , [4] 1 , , 1.4.7 .....(3r  2) r, x, r!, , [2], , 1.4.7 .....(3r  2) r r, 3 x, r!, , [3], , 1.4.7 .....(3r  2), ( 1)r xr, r!, , [4], , 1.4.7 .....(3r  2), ( 3)r xr, r!, , 2, , 1/2, , –1/3, , is equal to, , 2, x, 3, , is, , [1], , The coefficient of x in (1 + 3x) (1 – 2x), [1] 6/13, [2] 55/72, , 2, , is [3] 7/19, 1/3, , The term independent of y in the bionomial expansion of (1/2y, [1] sixth, [2] seventh, [3] fifth, , [4] 2/9, –1/5 8, , +y, , ) is [4] none of these, , 4, , Q.60, ,  3x , 16  3x, , 1 , 4 , If x is very small and , = P + Qx, then (8  x ) 2 / 3, , [1] P = 1, Q =, , 305, 96, , [2] P = 1, Q = –, , 305, 96, , [3] P = 2, Q =, , 305, 48, , [4] P = 2, Q = –, , 305, 48, , *****, 132

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Binomial Theorem, , EXERCISE-2, 1, , Q.1, , If x  1 , then in the expansion of 1  2x  3x 2  4x 3  ...  2 , the coefficient of xn is, [1] n, , Q.2, , ac  ab  bc, b 2  ac, , [2], ,  1 1 , [2]  , ,  2 2, , ab  ac, b 2  ac, , [4] None of these, , [3]  2,2, , r, , 2, , [4]  2,2, , 3, , 2, , The coefficient of x in the expansion of ( 1 + 3x + 6x + 10x + .... ) is –, , r  1 r  2  r  3 , , [2], , 5!, , r  1  r  2 (r  3)  r  4  r  5 , , r  2  r  3  (r  4), 5!, , [4] None of these, , 5!, , If (1+x)n = C0 + C1 x + C2x2 + .......+ Cnxn , then the value of C0 Cr  C1Cr 1  C2 Cr  2  ...Cn r Cn , ,  2n !, [1] n  r ! n  r !,   , , n!, [2]  r !  n  r  !, , n, , 2, , n!, [3] n  r !, , [4] None of these, , n, , If (1 +x) = C0 + C1x + C2 x + .... + Cnx , then C0C1 + C1 C2 + C2C3 + .... + Cn–1Cn is equal to, 2n!, [1] n! n!, , Q.7, , [3], , 1, , [3], , Q.6, , 2ac  ab  bc, b2  ac, , To expand 1  2x  2 as an infinite series, the range of x should be, , [1], , Q.5, , [4] –1, , n, ,  1 1 , [1]  , ,  2 2, Q.4, , [3] 1, , If in the expansion of 1  x  , a, b, c are three consecutive coefficients, then n =, , [1], , Q.3, , [2] n + 1, , 2n!, [2] n! (n  1)!, , 2n!, [3] (n  1)! (n  1)!, , 2n!, [4] (n  1)! n!, n, , If C0 , C1, C2 , ..... Cn are binomial coefficients of different terms in the expansion of ( 1+ x) then, n, , C0 – 2.C1 + 3.C2 – 4.C3 + .... + (–1) . (n + 1) Cn equals, [1] –n.2, Q.8, , The value of, [1] 252, , Q.9, , n–1, , [2] 0, , , ,  , , 5 1, , 5, , [3] 2, , n–1, , . (2–n), , [4] none of these, , , , 5, , 5  1 is, , [2] 352, , [3] 452, n, , [4] 532, 2, , 3, , 4, , 2, , If |x| < 1 , then the coefficient of x in the expansion of (1 + x + x + x + x + .... ) is, [1] n, , [2] n – 1, , [3] n + 2, , [4] n + 1, , 133

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Binomial Theorem, , Q.10, , 12, , 4, [1]  , 3, Q.11, , 12, , The sum of 12 terms of the series, , r  1, [1], , th, , C1 ., , 1 12, 1 12, 1, + C2 ., + C3 ., + .... is, 3, 9, 27, , 1, , 3, [3]  , 4, , 12, , 3, [2]  , 4, , 1, , 12, , term in the expansion of 1  x , , xr, r!, , [2], , 4, , 1, , [4] none of these, , r  2 r  3  xr, , [4] None of these, , will be, , r  1r  2 r  3 , 6, , xr, , [3], , 2, , 2, , Q.12, ,  1 x , In the expansion of ,  , the coefficient of xn will be,  1 x , [1] 4n, , [2] 4n–3, , [3] 4n+1, , [4] None of these, , 1024, , Q.13, , 1,  1, , In the expansion of  5 2  7 8 , , , , [1] 128, , Q.14, , , the number of integral terms is, , [2] 129, , [3] 130, , [4] 131, , The sum of C02  C12  C22  ...   1 Cn2 where n is an even integer, is, n, , [1], , 2n, , [2]  1, , n 2n, , Cn, , Cn, , [3], , 2n, , [4] None of these, , Cn 1, , 1  n  n  1 , 1, , The sum of 1  n  1   ,  1  x   ... , will be, x, 2!, , , , , 2, , Q.15, , [1] x, , [2] x, , n, , 2, , Q.16, , If y =, , 2 1.3  2 , 1.3.5, , , 5, 2!  5 , 3!, , [1] 1, Q.17, , n, , [4] None of these, , 3, , 2, 2,  5   ...... then value of ( y + 2y – 4) is,  , , [2] –1, , [3] 0, , n, , [4] none of these, , 2 –1, , The coefficient of x in the expansion of ( 1–9x + 20x ) is, n, , [1] 5 – 4, Q.18, ,  1, [3]  1  ,  x, , n, , n, , [2] 5, , n+1, , –4, , n+1, , n–1, , [3] 5, , –4, , n–1, , [4] none of these, , The interval in which x must lie so that the numerically greatest term in the expansion of 1  x , , 21, , has the, , numerically greatest coefficient is, , 5 6, [1]  , , 6 5, Q. 19, , 5 6, [2]  , , 6 5, , 4 5, [3]  , , 5 4, , 4 5, [4]  , , 5 4, , [3] 10150, , [4] None of these, , The larger of 9950  10050 and 10150 is, [1] 9950  10050, , [2] Both are equal, , 134

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Binomial Theorem, q, , Q.20, , If x is nearly equal to 1, then the approximate value of, [1], , pq, 1 x, , [2], , 1, 1 x, , px  qx, , p, , is –, , x q  xp, [3], , 1, 1 x, , [4], , Pq, 1 x, , n, , Q.21, , 1 ,  2/3,  1/ 3  is 27 more than the coefficient of second term,, If the coefficient of third term in the expansion of  x, x , , then the value of n is, [1] 8, , Q.22, , [2] 9, , , , [4] none of these, ,    equals –, , 2, 2, 2, 2, n, 2, If n = 10, then C0  C1  C2  C3  .....  ( 1) Cn, 5, , 10, , [1] (–1) . C5, , Q.23, , [3] 10, , [2] 0, , [3], , 10, , 6, , C5, , 9x 17 , ,  log 2, th, The value of x , for which the 6 term in the expansion of 2, , , [1] 4 , 3, , [2] 0 , 3, , 10, , [4] (–1) . C6, 7, , , , [3] 0 , 2, , ,  is 84, is equal to, x 1, 1/ 5  log 2 (3 1) , 2, , 1, , [4] 1 , 2, , 10, , Q.24, , The sum of the series, , , , 20, , Cr is -, , r 0, , [1] 2, , Q.25, , 20, , [2] 2, , , , ,  nx , , [2] 2 ,  1  nx , , n, , 19, , +, , 1, 2, , 20, , C10, , 19, , [4] 2, , –, , 1, 2, , 20, , C10, , , 1 x n, 1  2x n, 1  3x,  C2 .,  C3 .,  ....  is equal to 2, 3, 1  nx, (1 nx), (1  nx), , ,  2nx , , [3] ,  1  nx , , n, , [4] none of these, , C3, C1, C2, C15, C 0 + 2 C1 + 3 C 2 + ....+ 15 C14 =, [1] 100, , Q.27, , [3] 2, , If n is positive integer then the sum of  n C0 n C1., , [1] 0, , Q.26, , 19, , [2] 120, n, , 2, , [3] –120, n, , 2, , If (1 + x) = C0 + C1 x + C2. x + .....+ Cnx then 2.C0 + 2 ., , [1], , 3n1  1, n 1, , [2], , 2n1  1, n 1, , [3], , [4] none of these, , C1, Cn, 3 C2, n+1, +2., is equal to + ....+ 2 ., 3, n 1, 2, , 3n  1, n 1, , [4], , 3 n 1  1, n 1, 135

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Binomial Theorem, , Q.28, , 3n(3n  1), 3n(3n  1)(3n  2), +, + .... is 2!, 3 !, , The sum of the series 1 + 3n +, n, , [1] (1 + 3n), , [2] (1 – 3n), , n, , [3] 2, , 2n, , [4] 2, , 3n, , 9, , Q.29, , 1 , 3 3 2,  is, The coefficient of the term independent of x in expansion of (1 + x + 2x )  x , 3x , 2, , [1] 1/3, Q.30, , [2] 19/54, 2 6, , [2] 64, 2, , [4] 31, , 2, , n, , 2, , If n > 2 then the value of C0 – 2 . C1 + 3 . C2 – 4 . C3 + ....+ (– 1) . (n + 1) . Cn is n–1, , n, , + 2 – n(n – 1).2, , n, , [3] 2 –n.2, , n–2, , [2] 3n.2, , n–1, , n–1, , n, , + 2 + n (n – 1). 2, , n–2, , [4] none of these, , 2 n, , 2, , 2n, , If (1 + x + x ) = (a0 + a1 x + a2 x + ....+ a2n x ) then (a0 + a3 + a6 + .....) equals n, , [1] 3, Q.33, , [3] 32, , 2, , [1] n.2, , Q.32, , [4] 1/4, , 12, , If (1 + x – 2x ) = a0 + a1x + a2x + ....+ a12x , then the value of a2 + a4 + a6 + ...+ a12 is [1] 1024, , Q.31, , [3] 17/54, , 2, , [2] 3, , n–1, , [3] 2, , n–1, , [4] none of these, n, , If C0, C1.....C n are binomial coefficient of different terms in the expansion of (1 + x) . (n  N) then, , , C1  C2   C3 , Cn ,  .1 ,  ......  1 ,  equals 1,  1 , C1   C2 ,  Cn1 ,  C0  , ,  n  1, , [1] ,  n , , n, , [2], , ((n  1)! )n, n!, , [3], , (n  1)n, n!, , 2, , n, , Q.34, , If n is a positive integer and Ck  nCk ,then the value of, , [1], Q.35, , n(n  1)(n  2), 12, , [2], , n(n  1)2, 12, , k, k 1, , [3], , [4] none of these, , 3, ,  Ck , ,  =,  C k 1 , , n(n  2)2 (n  1), 12, , [4] none of these, , n, , If T0, T1, T2,......,Tn represent the terms in the expansion of (x + a) , then the value of, 2, , 2, , (T0 – T2 + T4 – T6....) + (T1 – T3 + T5 – ....) is 2, , 2 n, , [1] (x – a ), , 2, , 2 n, , [2] (x + a ), , 2, , 2 n, , [3] (a –x ), , [4] none of these, , *****, , 136

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Binomial Theorem, , EXERCISE-3, Q.1, , 2 2, , 5, , Q.2, , 5, , 1/ 2, 1/ 2, 3, 3, The expansion  x   x  1    x   x  1  is a polynomial of degree, ,  , , [1] 5, [2] 6, [3] 7, , 100, , Q.3, , 53, , The coefficient of x in the expansion of, [1], , Q.4, , 100, , [2], , C 47, , 100, , , , 100, , Cm  x  3 , , 100  m, , m 0, , [IIT- 1992], [4] 8, , 2m is, , [IIT- 1992], , [3] 100 C53, , C53, , [4] 100 C100, , C20  C12  C22 .......  1 Cn2 , where n is an even integer is, 2n, , [1], Q.5, , 51, , If the sum of the coefficients in the expansion of ( x – 2x + 1) vanishes,then the valu of is[IIT- 1991], [1] 2, [2] –1, [3] 1, [4] – 2, , [2]  1, , n 2n, , Cn, , [IIT 1998], [3]  1, , n 2n, , Cn, , [4] None of these, , Cn 1, , If the sum of odd numbered terms and the sum of even numbered terms in the expansion of  x  a  are P and, n, , 2, 2, Q respectively then the value of  P  Q  equal to, , [1]  x2  a2 , Q.6, , 2n, , [4]  x 2  a2 , , n, , [3]  x  a , , n, , If 1  x   C0  C1x  C2 x 2  .....  Cn x n , then C02  C12  C22  .....Cn2 is equal to, n, , 2n, , [1], Q.7, , [2]  x 2  a2 , , 2n, , [RPET 1998], , [2], , Cn 1, , 2n, , 2n, , [3], , Cn, , [RPET 1998], [4] None of these, , C n 1, , If the coefficients of r term and  r  4  term are equal in the expansion of 1  x  , then the value of r will be, th, , th, , 20, , [RPET 1998], [1] 7, Q.8, , [2] 8, [2] 1  x , , If in the expansion of 1  x , [1] 6, , Q.10, , [4] 10, , The square root of 1  2x  3x 2  4x 3  ..... is, [1] 1  x , , Q.9, , [3] 9, , m, , 1  x , , n, , [2] 9, , [RPET 1998], [3] 1  x , , 1, , , the coefficients of x and x 2 are 3 and 6 respectively, then m is, [IIT 1999], [3] 12, [4] 24, , If the coefficient of second, third and fourth terms in the expansion of 1  x , , 2n  9n , [1] 7, , [4] 1  x , , 1, , 2n, , are in A.P., then the value of, [RPET 1999], , 2, , Q.11, , [2] –7, , If 1  x   C0  C1x  C2 x 2  ....  Cn xn then, n, , [1] 2n, , [2], , 2n, n 1, , [3] 14, , [4] –14, , C0 C 2 C 4, , ,  ...... , 1, 3, 5, 2n, n 1, , [3], , [RPET 1999], [4], , 2n, n, , 12, , Q.12, , 1, In the expansion of  2x 2   , the term independent of x is, x, , [1] 7920, [2] –7920, [3] 495, , [RPET 1999], [4] –495, , 10, , Q. 13, ,  2 3 3, The term independent of x in the expansion of  x  3 , x , , , [1] 153090, , [2] 150000, , is, , [3] 150090, , [RPET 2000], [4] 153180, 137

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Binomial Theorem, , Q.14, , n,  n   n , For 2  r  n,    2 , , , r ,  r  1  r  2 ,  n  1, , [1] ,  r  1, ,  n  1, , [2] 2 ,  r  1, , [IIT 2000], n  2, [4] , ,  r , , n  2, [3] 2 , ,  r , 10, , Q.15, ,  x, 3,  2 , The term independent of x in the expansion of , ,  3 x , , [1], Q. 16, , [2], , 3, 4, , [3], , [RPET 2000], , 7, 4, , [4], , 5, 3, , In the expansion of the following expression 1  1  x   1  x   ....  1  x  , the coefficient of xk  0  k  n  is, [RPET 2000], 2, , [1], Q.17, , 5, 4, , is, , n 1, , [2] n Ck, , C k 1, , n, , [3] n Cn k 1, , [4] None of these, , In the binomial expansion of  a  b  ,n  5 , the sum of the 5th and 6th terms is zero. Then, n, , [1], , n5, 6, , [2], , n4, 5, , [3], , 5, n4, , [4], , a, equals, b, [IIT 2001], , 6, n5, , 10, , Q. 18, , x 3 , 4, In the expansion of   2  , the coefficient of x is, 2 x , [1], , Q.19, , 405, 256, , [2], , 504, 259, , [RPET 2001], [3], , 450, 263, , [4] None of these, , The coefficient of x in the expansion of 1  x  x 2  x 3  is, n, , 4, , [2] C 4  C2, n, , n, , [1] C 4, , n, , [RPET 2001], , [3] C 4  C2  C 4 . C2, n, , n, , n, , n, ,  10  20  , , p, ,  ,  where    0 if p  q  is maximum when m is, i, m, , i, i 0 , ,  ,  q, , [2] 10, [3] 15, , [4] C 4  C2  n C1. n C2, n, , n, , m, , Q.20, , The sum, [1] 5, , , , [IIT 2002], [4] 20, , 10, , Q.21, , 1, , In the expansion of  x  , x, , [1], , 10, , the coefficient of middle term is, [2], , C4, , 10, , C4, , [3], , 10, , [RPET 2002], [4], , C5, , 10, , C6, , Q.22, , If the coefficient of second, third and fourth term in the expansion of 1  x  are in AP then n=, [RPET 2002], [1] 7, 5, [2] 5, 4, [3] 7, 2, [4] 3, 6, , Q. 23, , The sum of the coefficients in the expansion of  x  y  is 4096. The greatest coefficient in the expansion is, [AIEEE 2002], [1] 1024, [2] 924, [3] 824, [4] 724, , n, , n, , 1, , 1, , Q. 24, ,  a 2  a 2, a x ax , ,  , , , Q. 25, , 3x 2, 3x 2, x 3x 2, x 3x 2,  ....., [2] 1  2  ......, [3] 2   2  ....., [4] 2   2  ...., 2, a 4a, a 4a, 4a, 8a, th, th, 2n, If for positive integers r > 1, n > 2 the coefficients of the (3r) and (r+2) powers of x in the expansion of (1+x), are equal, then[AIEEE 2002], [1] n = 2r, [2] n = 3r, [3] n = 2r + 1, [4] n = 2r – 1, , [AIEEE 2002], , [1] 2 , , 138

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Binomial Theorem, , Q.26, , Find coefficient of t 24 in the expansion of 1  t 2 , , 12, , [1], Q.27, , 12, , C6  2, , [2], , 12, , C6  1, , [3], , 24, , 12, , C6  3, , [4], , th, , 16, , [2] 5, , [3] 6, , The sum of the coefficients in the expansion of  x  2y  z , , 10, , [1] 2, , [2] 4, , 10, , [3] 3, , is, If x is positive, the first negative term in the expansion of 1  x , [1] 7th term, [2] 5th term, [3] 8th term, , 100, , 100, , is, , The number of integral terms in the expansion of, , [4] 202, , , , 3 5, , [2] 33, , 8, , , , [AIEEE 2003], [4] 6th term, , 256, , is, , [AIEEE 2003], , [3] 34, , [4] 35, , The coefficient of the middle term in the binomial expansion in powers of x of 1  x  and of 1  x  is the, [AIEEE 2004], same if  equals, 4, , [1] , , 5, 3, , [2], , 10, 3, , [3], , 3, 10, , [4], , 6, , 3, 5, , The coefficient of x in expansion of 1  x 1  x  is, n, , n, , n, , [1] (n – 1), , [2] (–1) (1–n), , [AIEEE 2004], [3]  1, , n 1, , n  1, , 2, , [4]  1, , n 1, , n, , If the coefficients of rth, (r + 1)th and (r + 2)th terms in the binomial expansion (1 + y)m are in A.P., then m and r, satisfy the equation, [AIEEE 2005], [1] m2 – m (4r –1) + 4r2 + 2 = 0, [2] m2 – m (4r + 1) + 4r2 – 2 = 0, [3] m2 – m (4r + 1) + 4r2 + 2 = 0, [4] m2 – m(4r - 1) + 4r2 – 2 = 0, 11, , Q.35, , [RPET 2003], , 27 / 5, , [1] 32, , Q.34, , are equal then r , [RPET 2003], [4] 7, , [4] 1, , Q. 30, , Q. 33, , C6, , [RPET 2003], , The total number of terms in the expansion of  x  a    x  a , [1] 50, [2] 101, [3] 51, , Q. 32, , 12, , is, , 10, , 10, , Q.29, , Q. 31, , [IIT 2003], , If the coefficients of  2r  1 term &  r  1 term in the expansion of 1  x , th, , [1] 4, Q.28, , 12, , 1  t 1  t  is, , 11, ,  2  1 , ,  1 , If the coefficient of x in  ax  ,   equals the coefficient of x-7 in ax   2   , then a and b satisfy the,  bx  ,  bx  , , , 7, , relation, [1] ab = 1, , [AIEEE 2005], [2], , a, 1, b, , [3] a + b = 1, , [4] a – b = 1, 3, , Q.36, , 3,  1 , (1  x) 2  1  x ,  2  may be approximated, If x is so small that x3 and higher powers of x may be neglected, then, 1, (1  x ) 2, , as, [1], Q.37, , [AIEEE 2005], , x 3 2,  x, 2 8, , 3, 8, , 2, [2]  x, , If the expansion in powers of x of the function, , [1], , an  bn, ba, , [2], , an1  bn1, ba, , [3] 3x , , 3 2, x, 8, , [4] 1 , , 3 2, x, 8, , 1, 2, 3, is a0 + a1x + a2x + a3x + .... then an is, (1  ax )(1  bx ), [AIEEE 2006], [3], , bn1  an1, ba, , [4], , bn  an, ba, , 139

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Binomial Theorem, , Q.38, , m, , n, , 2, , For natural numbers m, n if (1 – y) (1 + y) = 1 + a1y + a2y + ...., and a1 = a2 = 10, then (m, n) is, [AIEEE 2006], [1] (35, 20), , Q.39, , [2] (45, 35), , Sum of the series, [1], , 20, , 20, , C0 –, , 20, , C1 +, , C10, , [2] –, , 20, , C2 –, , 20, , [3] (35, 45), 20, , C3 + ....+, , 20, , C10, , [4] (20, 45), , C10 is :, [3], , 1, 2, , [AIEEE 2007], 20, , C10, , [4] 0, , n, , Q.40, , n, n 1, Statement -I :  (r  1) C r  (n  2)2, , [AIEEE 2008], , r 0, , n, , Statement -II :, ,  (r  1), , n, , C r x r  (1  x )n  n(1  x )n1, , r 0, , [1] Statement I is true, statement II is true and II is a correct explanation for I, [2] Statement I is true, statement II is true but II is not a correct explanation for I, [3] Statement I is true, statement II is false, [4] Statement I is false, statement II is true, Q.41, , 2n, , The remainder left out when 8 – (62), [1] 2, , Q.42, , 2n+1, , is divided by 9 is :, , [2] 7, , [AIEEE 2009], , [3] 8, , [4] 0, r, , For r = 0, 1, ......, 10 let Ar, Br and Cr denote, respectively, the coefficient of x in the expansions of, 10, , 10, 20, 30, (1 + x) , (1 + x) and (1 + x) . Then  A r (B10B r  C10 A r ) is equal to :, , [IIT-JEE 2010], , r 1, , 2, , [1] B10 – C10, Q.43, , [2] A10(B10 – C10A10), , [4] C10 – B10, , The coefficient of x7 in the expansion of (1 – x – x2 + x3)6 is :, [1] 144, , Q.44, , [3] 0, , [2] – 132, , If n is a positive integer, then, , [AIEEE 2011], , [3] – 144, ,  3  1   3  1, 2n, , 2n, , [4] 132, , is :, , [AIEEE – 2012], , (1) An irrational number, , (2) An odd positive integer, , (3) An even positive integer, , (4) A rational number other than positive integers, , Q.45, , The coefficients of three consecutive terms of (1  x ) n5 are in the ratio 5 : 10 : 14. Then n = [JEE Adv. – 13], , Q.46, , Coefficient of x11 in the expansion of (1  x 2 ) 4 (1  x 3 ) 7 (1  x 4 )12 is :, (A) 1051, , (B) 1106, , (C) 1113, , , , x 1, x 1, , Q.47. The term independent of x in expansion of  2, 1, 1,  3,  x  x3 1 x  x2, (1) 210, (2) 310, (3) 4, 3, , 4, , [JEE Adv. – 2014], (D) 1120, , 10, , , , , , , , 2, , is :, , [JEE MAIN. – 2013], (4) 120, 18, , Q.48. If the coefficients of x and x in the expansion of (1  ax  bx ) (1  2x ), (a, b) is equal to :, 251 , , , (1) 16,, 3 , , , in powers of x are both zero, then, [JEE MAIN. – 2014], , 251 , , (2) 14,, , 3 , , , 272 , , , (3) 14,, 3 , , , 272 , , , (4) 16,, 3 , , 140

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Binomial Theorem, , Q.49. If (10) 9  2(11)1 (10) 8  3(11) 2 (10) 7  ...  10(11) 9  k (10) 9 , then k is equal to :, (1), , 121, 10, , (2), , 441, 100, , [JEE MAIN. – 2014], , (3) 100, , (4) 110, , ANSWER-KEY, EXERCISE # 1, Que . 1, Ans. 2, Que . 26, Ans. 2, Que . 51, Ans. 1, , 2, 4, 27, 3, 52, 1, , 3, 3, 28, 2, 53, 3, , 4, 2, 29, 3, 54, 2, , 5, 1, 30, 2, 55, 4, , 6, 2, 31, 1, 56, 2, , 7, 2, 32, 3, 57, 1, , 8, 3, 33, 2, 58, 2, , 9, 2, 34, 3, 59, 1, , 10, 3, 35, 4, 60, 2, , 11, 2, 36, 1, , 12, 3, 37, 2, , 13, 4, 38, 1, , 14, 2, 39, 3, , 15, 3, 40, 1, , 16, 3, 41, 3, , 17, 3, 42, 3, , 18, 3, 43, 3, , 19, 4, 44, 4, , 20, 1, 45, 1, , 21, 2, 46, 3, , 22, 2, 47, 3, , 23, 3, 48, 3, , 24, 2, 49, 3, , 25, 3, 50, 2, , 15, 1, , 16, 3, , 17, 2, , 18, 2, , 19, 3, , 20, 3, , 21, 2, , 22, 1, , 23, 4, , 24, 3, , 25, 1, , EXERCISE # 2, Que . 1, An s. 3, Q u e . 26, An s. 2, , 2, 2, 27, 1, , 3, 2, 28, 4, , 4, 3, 29, 3, , 5, 1, 30, 4, , 6, 3, 31, 4, , 7, 2, 32, 2, , 8, 2, 33, 3, , 9, 4, 34, 4, , 10, 1, 35, 2, , 11, 2, , 12, 1, , 13, 2, , 14, 4, , EXERCISE # 3, , 141