Binomial Theorem And Its Simple Applications — practice questions

105 curated practice questions from Binomial Theorem And Its Simple Applications (Mathematics JEE Main, NTA Exams) — 15 medium, 70 easy, 20 hard. Every question includes the final answer free; step-by-step worked solutions with a free account.

• The approximate value of (7.995)^ 1 / 3 correct to four decimal places is
• Sum of the last 30 coefficients in the expansion of (1 + x) 59 , when expanded in ascending powers of x is
• If n N and the coefficients of x –7 and x –8 to the expansion of (2+ 1 3 x )^ n are equal then n =
• If the fourth term in the expansion of ( ax + 1 x )^ n is 5/2. Then find the value of a and n ?
• If (1+x)^ n = ^ n C_ n + ^ n C_ 1 x+ ^ n C_ 2 x^ 2 + + ^ n C_ n x^ n where, ^ n C_ n , ^ n C_ 1 , ^ n C_ 2 , are…
• If the coefficients of (r + 4)th term and (2r + 1)th term in the expansion of (1 + x) 18 are equal, then r =
• a, b, c, d are any four consecutive co-efficients of any binomial expansion, then a + b a , ~b + c ~b , c + d c are in :
• The positive value of for which the co-efficient of x 2 in the expression x^ 2 ( x + x^ 2 )^ 10 is 720, is-
• If the term independent of x in the expansion of ( x - x ^ 2 )^ 10 is 405, then λ equals
• If 1 18 + 1 9 = x 10 , then x is equal to
• The term void of x in the expanion of (x- 3 x^ 2 )^ 8 is
• The coefficient of x^ 4 in the expansion of (1+x+x^ 2 +x^ 3 )^ 11 is :
• Find the number of terms in the expansion ( x + y )^ 1! +( x - y )^ 1! .
• The coefficient of the term independent of x in the expansion of [ x 3 + 3 2 x^ 2 ]^ 1 is :
• The sum r C r + r+1 C r + r+2 C r + ... + n C r (n > r) equals
• In the expansion of ( 1 2 x^ 1 3 +x^ - 1 5 )^ 8 the term independent of x is
• C_ 1 1 + C_ 2 3 + C_ 4 5 + C_ 5 7 + is equal to
• The greatest integer in the expansion of (1+x)^ 2 n+2 is
• The sum of the coefficients of all odd degree terms in the expansion of (x+ x^ 3 -1 )^ 5 + (x- x^ 3 -1 )^ 5 ,(x>1) is:
• If (1-x+x^ 2 )^ 7 =a_ n +a_ 1 x+ +a_ 2 n x^ 2 n then the value of a_ n +a_ 2 +a_ 4 + +a_ 2 n i 5
• The coefficient of x 17 in the expansion of (x – 1) (x – 2) ........ (x – 18) is
• If the expansion in powers of x of the function 1 (1-a x)(1-b x) is a_ n +a_ 1 x+a_ 2 x^ 2 +a_ 3 x^ 3 + , then a_ n is
• The expansion (x + a) n = n C 0 x n + n C 1 x n–1 a 1 + ..... + n C n a n is valid when n is
• The total number of terms in the expansion of (x+y)^ 10 +(x-y)^ 10 after simplification is :
• If a_ n = ( _ e 3 )^ n _ k=0 ^ n 1 k!(n-k)! , for n ≥ 0 then a_ n +a_ 1 +a_ 2 + =
• The co-efficient of x 4 in the expansion of (1 + x + x 2 + x 3 ) n is
• Given below are two statements. STATEMENT - 1: _ r=n ^ n 1 r+1 ^ n C_ r x ^ r = 1 (n+1) x (1+ x )^ n+1 -1 STATEMENT …
• In the expansion of (2 x^ 2 - 1 x )^ 12 the term independent of x is :
• Let (x+10)^ 5 +(x-10)^ 5 a =a_ a +a_ 1 x+a_ 2 x^ 2 + +a_ 5 a x^ 5 a , for all x R; then a_ 2 a_ n is equal to:
• If the sum of the coefficients in the expansion of (1+2 x)^ n is 6561, then the greatest term in the expansion when x…
• The number of integral terms in the expansion of ( 3 + 2 )^ 500 is :
• The sum of coefficient in the expansion of (1 + x – 3x 2 ) 3148 is
• The ninth term of the expansion (3 x- 1 2 x )^ is
• Find the value of x for which the expansion 2+x (3-2 x)^ 2 is valid
• The coefficient of x^ 5 in (1+2 x+3 x^ 2 + )^ - 3 2 is
• 5th term from the end in the expansion of ( x^ 3 2 - 2 x^ 2 )^ 2 is
• If the last term in the expansion of (z^ 1 / 3 +z^ -1 / 2 )^ n is ( 1 3(9)^ 1 / 3 )^ _ 3 ~A C , then the value of n is
• Let n be a positive integer. If the binomial coefficients of 2 nd , 3 rd and 4 th term in the expansion of (1+x)^ n are…
• Coefficient of x 5 in the expansion of (1 + x 2 ) 5 (1 + x) 4 is
• If (1+x)^ =C_ n +C_ 1 x+C_ 2 x^ 2 + +C_ n x^ n , then for odd n, c_ 1 ^ 2 +c_ 3 ^ 2 +c_ 5 ^ 2 + +c_ n ^ 2 is equal to
• The term independent of x in expansion of ( x+1 x^ 2 / 3 -x^ 1 / 3 +1 - x-1 x-x^ 1 / 2 )^ 1 is
• If (2-x-x^ 2 )^ n =a_ n +a_ 1 x+a_ 2 x^ 2 + , then the value of a _ a + a _ 2 + a _ 4 +
• The total number of terms in the expansion of (x+y)^ 10 +(x-y)^ 10 after simplification is :
• The value of 1 81^ n - 10 81^ n ^ 2 n c_ 1 + 10^ 2 81^ n ^ 2 n c_ 2 - 10^ 3 81^ n ^ 2 n c_ 3 + + 10^ 2 n 81^ n is
• 10 C 1 + 10 C 2 + 10 C 3 + .............+ 10 C 10 =
• The number of terms in the expansions of (1+5 2 x)^ 9 +(1-5 2 x)^ 9 is
• If the sum of the coefficients in the expansion of (1+2 x)^ n is 6561. The greatest term in the expansion for x= 1 2 is
• If r and n are positive integers such that r>1, n>2 and coefficients of (r+2) th term and 3r th term in the expansion…
• If | x |<1 , then the coefficient of x ^ in expansion of (1+ x + x ^ 2 + x ^ 3 + )^ 2 is
• In the expansion of (x^ 3 - 1 x^ 2 )^ 15 , the constant term is :
• The number of ordered pairs (r, k) for which 6 ^ 35 c_ r = (k^ 2 -3 )^ 36 c_ r+1 , where k is an integer, is :
• If n N then 49 n + 16 n – 1 is divisible by
• If s_ n = _ r=n ^ n 1 ^ n C_ r and t_ n = _ r=n ^ n r ^ n C_ r then t_ n 5 _ n is equal to
• The sum of coefficients of integral powers of x in the binomial expansion of (1-2 x )^ 5 t is:
• The sum of the series 1- 1 2 ( 1 2 )+ 1 2 3 4 ( 1 2 )^ 2 - ( 1 2 ) ( 3 4 ) ( 5 6 ) ( 1 2 )^ 3 + is :
• The sum of the numerical coefficients in the expansion of (1+ x 3 + 2 y 3 )^ 12 , is
• If the ninth term in the expansion of [3^ _ 3 25^ x-1 +7 +3^ - 1 4 _ 3 [5^ x-1 +1 ] ] is equal to 180 and x>1 , then x…
• Co-efficient of x 5 in the expansion of (x 2 – x – 2) 5 is :
• The coefficient of a^ 4 b^ 5 c^ 4 in the expansion of (a+b+c)^ 1 is
• If (10)^ 9 +2(11)^ 1 (10)^ 1 +3(11)^ 2 (10)^ 7 + +10(11)^ 9 =k(10)^ 9 , then k is equal to :
• The coefficient of x 18 in the product (1+x)(1-x)^ 1 (1+x+x^ 2 )^ 3 is:
• If n N and (1 – x + x 2 ) n = a 0 + a 1 x + a 2 x 2 + ...... + a 2n x 2n , then a 0 + a 2 + a 4 + ... +a 2n is equal to…
• If the sum of the coefficients in the expansion of (a 2 x 2 – 6ax + 11) 10 , where a is constant, is 1024, then the…
• The value of ( ^ 7 C_ 0 + ^ 7 C_ 1 )+ ( ^ 7 C_ 1 + ^ 7 C_ 2 )+ + ( ^ 7 C_ 6 + ^ 7 C_ 7 ) is
• The greatest value of the term independent of x, as α varies over R, in the expansion of (x + x )^ 20 is :
• If n is not a multiple of 3, then the coefficient of x n in the expansion of _ r [1+x+x^ 2 ] is :
• The coefficients of x^ p and x^ 4 in the expansion of (1+x)^ p+q are
• (x+ x^ 3 -1 )^ 5 + (x- x^ 3 -1 )^ 5 is a polynomial of degree
• The coefficient of x 8 y 10 in the expansion of (x + y) 18 is
• If x is so small that x^ 3 and higher powers of x may be neglected, then (1+x)^ 3 / 2 - (1+ 1 2 x )^ 3 (1-x)^ 1 / 2 may…
• If n+1 C 4 = 9 n C 2 , then n =
• The coefficient of x 99 in (x + 1) (x + 3) (x + 5) ..... (x + 199) is
• If (2+3 x)^ -5 expands in ascending powers of x , then x belongs to
• In the expansion of (1 + x) 50 , the sum of the coefficient of odd powers of x is
• If n C 3 = n C 2 , then n is equal to
• Let S_ 1 = _ j=1 ^ 10 j(j-1) ^ 10 C_ j , S_ 2 = _ j=1 ^ 10 j ^ 10 C_ j and S_ 3 = _ j=1 ^ 10 j^ 2 ^ 10 C_ j Statement I…
• If f(x)=(1+x)^ then the value of f(0)+f^ (0)+ f^ (0) 2! + + f^ (0) n! is
• Given below are two statements. STATEMENT - 1: If n ∈ N and 'n' is not a multiple of 3 and (1+x+x^ 2 )^ n = _ r =0 ^ 2…
• Find the coefficient of x ^ in the series 1+(a+b x)+ (a+b x)^ 2 2! + (a+b x)^ 3 3! +
• The total number of irrational terms in the binomial expansion of (7^ 1 / 5 -3^ 1 / 16 )^ 61 is:
• Coefficient of x^ 2 in the expansion of ( x - 1 2 x )^ 1 is equal to :
• If n>1 then (1+ x )^ n - n x -1 is divisible by
• The term independent of x in the expansion of [ (t^ -1 -1 ) x+ (t^ -1 +1 )^ -1 x^ -1 ]^ 8 is :
• In the expansion of ( x + 1 x )^ 15 , if I 1 is the least value of the term independent of x when 8 4 and I 2 is the…
• The number of terms which are free from radical signs in the expansion of (y^ 1 5 +x^ 1 10 )^ 55 is
• The co-efficient of x in the expansion of (x^ 2 + c x )^ 5 is equal to :
• If the fourth term in the binomial expansion of ( 1 x^ 1+ g 10^ x +x^ 1 12 )^ 6 is equal to 200, and x > 1, then the…
• The term independent of x in the expansion of (2 x- 3 x^ 2 )^ 9 is
• If the coefficients of r th , (r+1) th and (r+2) th terms in the binomial expansion of (1+y)^ m are in AP., then m and…
• The number of non-zero terms in the expansion of [(1+3 2 x)^ 9 +(1-3 2 x)^ 9 ] is
• Let n 5 and b 0 . In the binomial expansion of (a-b)^ n , the sum of the 5^ th and 6^ th term is zero then a b equals :
• Given below are two statements. STATEMENT - 1: The total number of dissimilar terms in the expansion of ( x _ 1 + x _ 2…
• If (1 + ax) m =1 + 8x + 24x 2 + ..., then the value of a and m are respectively.
• If [1+2 x+3 x^ 2 ]^ 10 =a_ n +a_ 1 x+a_ 2 x^ 2 + +a_ 2 n x^ 2 n , then l _ 1 equals
• Statement I : _ r =0 ^ n ( r +1) ^ n C _ r =( n +2) 2^ n -1 Statement II : _ r =0 ^ n ( r +1)^ n C _ r x ^ r =(1+ x )^…
• The coefficient of x 50 in the binomial expansion of (1 + x) 1000 + x(1 + x) 999 + x 2 (1 + x) 998 + ..... + x 1000 is:
• If n,p N and in the expansion of (1 + x) n the coefficient of p th and (p + 1) th terms are respectively p and q. The p…
• Let 5_ n =1+q+q^ 2 + +q^ n and T_ n =1+ ( q+1 2 )+ ( q+1 2 )^ 2 + + ( q+1 2 )^ n where q is a real number and q ≠ 1. If…
• The value of n C 0 – n C 1 + n C 2 – .... + (–1) n n C n is
• If n N and (1+x) n = 1 + a 1 x + a 2 x 2 + ... + a n x n . If a 1 , a 2 and a 3 are in A.P., then the value of n is
• Using binomial theorem, the value of (0.999) 3 correct to 3 decimal places is
• In the expansion of (1+x)^ 7 (1+y)^ 7 (1+z)^ 7 , the sum of the coefficients of the terms of degree m is :
• The number of terms which are free from radical sign in the expansion of (y^ 1 / 5 +x^ 1 / 16 )^ 55 is :
• The coefficient of x^ 4 in the expansion of (1+2 x+3 x^ 2 + )^ 1 / 2 is :
• If x is positive, the first negative term in the expansion of (1+x)^ 27 5 is :

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