Title 6.BINOMIAL THEOREM.pdf ✅

6. BINOMIAL THEOREM (1.) The remainder left out when 2 2 1 8 (62) + − n n is divided by 9 is [AIEEE-2009] (a.) 2 (b.) 7 (c.) 8 (d.) 0 (2.) Let 10 10 10 10 1 1 2 1 ( 1) , =  − =  j j j j = = S j j C S j C and 10 2 10 3 1 =  j j = S j C Statement-1: 9 3 S =  55 2 Statement-2: 8 S 90 2 1 =  and 8 S 10 2 2 =  [AIEEE-2010] (a.) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1 (b.) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 (c.) Statement-1 is true, Statement-2 is false (d.) Statement-1 is false, Statement- 2 is true (3.) Statement-1: For each natural number 7 n n ,( 1) + 7 − − n 1 is divisible by 7 . Statement-2: For each natural number 7 n n n , − is divisible by 7 . [AIEEE-2011] (a.) Statement-1 is true, statement-2 is false. (b.) Statement-1 is false, statement-2, is true. (c.) Statement-1 is true, statement-2 is true, statement-2 is a correct explanation for statement-1 (d.) Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for statement-1 (4.) If n is a positive integer, then 2 2 ( 3 1) ( 3 1) + − − n n is [AIEEE-2012] (a.) An odd positive integer (b.) An even positive integer (c.) A rational number other than positive integer (d.) An irrational number (5.) Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A B having 3 or more elements is [JEE (Main)-2013] (a.) 256 (b.) 220 (c.) 219 (d.) 211 (6.) The term independent of x in expansion of 10 2/3 1/3 1/2 1 1 1   + −   −   − + − x x x x x x is [JEE (Main)-2013] (a.) 4 (b.) 120 (c.) 210 (d.) 310


(24.) The sum of the co-efficients of all even degree terms in x in the expansion of ( ) 6 3 x x + − + 1 ( ) 6 3 x x x − −  1 ,( 1) is equal to : [JEE (Main)-2019] (a.) 24 (b.) 32 (c.) 26 (d.) 29 (25.) If the fourth term in the binomial expansion of 10 6 1 12 1 log 1 +     +   x x x is equal to 200 , and x 1 , then the value of x is : [JEE (Main)-2019] (a.) 10 (b.) 3 10 (c.) 100 (d.) 4 10 (26.) If the fourth term in the Binomial expansion of b 6 2 log ( 0)     +    x x x x is 7 20 8 , then a value of x is [JEE (Main)-2019] (a.) 3 8 (b.) 8 (c.) 2 8 − (d.) 2 8 (27.) If some three consecutive coefficients in the binomial expansion of ( 1) + n x in powers of x are in the ratio 2:15:70 , then the average of these three coefficients is [JEE (Main)-2019] (a.) 625 (b.) 964 (c.) 232 (d.) 227 (28.) If the coefficients of 2 x and 3 x are both zero, in the expansion of the expression ( ) 2 1+ + ax bx 15 (1 3 ) − x in powers of x , then the ordered pair (a b, ) is equal to : [JEE (Main)-2019] (a.) (−54,315) (b.) (28,861) (c.) (−21,714) (d.) 28,315 (29.) The smallest natural number n , such that the coefficient of x in the expansion of 2 3   1   +   n x x is 23 nC , is [JEE (Main)-2019] (a.) 58 (b.) 35 (c.) 38 (d.) 23 (30.) The coefficient of 18 x in the product (1+ x) ( ) 9 10 2 (1 ) 1 − + + x x x is [JEE (Main)- 2019] (a.) 84 (b.) -126 (c.) -84 (d.) 126 (31.) If ( ) ( ) ( ) 20 2 20 2 20 2 20 1 2 3 20 2 3 . 20 C C C C + + + + (2 )  = A , then the ordered pair ( A,  ) is equal to [JEE (Main)-2019] (a.) (420,19) (b.) (380,19) (c.) (420,18) (d.) (380,18)