# BINOMIAL EXPANSION PRACTICE WORKSHEET

Binomial expansion for (x + a)n is,

nc0xna+ nc1xn-1a+ nc2xn-2a+ .........+ ncnxn-na0(

If X is a set containing n elements, then we know that nCr is the number of subsets of X having exactly r elements. So by adding nCr for r = 0, 1, 2, . . . , n we get the number of subsets of X. So by using the above identity we see that a set of n elements has 2n subsets.

(1)  Expand (i) [2x2 − (3/x)]

(ii)  (2x2 − 3√1 − x2)4  +  (2x2 + 3√1 − x2)Solution

(2)  Compute (i)  1024   (ii)  994   (iii)  97    Solution

(3) Using binomial theorem, indicate which of the following two number is larger: (1.01)100000010000.

Solution

(4)  Find the coefficient of x15 in  (x2 + (1/x3))10  Solution

(5)  Find the coefficient of x6 and the coefficient of x2 in  (x2 - (1/x3))6   Solution

(6)  Find the coefficient of x4 in the expansion of (1 + x3)50(x2 + 1/x)5.   Solution

(7)  Find the constant term of (2x3 - (1/3x2))Solution

(8)  Find the last two digits of the number 3600    Solution

(9)  If n is a positive integer, show that, 9n+1 − 8n − 9 is always divisible by 64.   Solution

(10)  If n is an odd positive integer, prove that the coefficients of the middle terms in the expansion of (x + y)n are equal.              Solution

(11)  If n is a positive integer and r is a non negative integer, prove that the coefficients of xr and xn−r in the expansion of (1 + x)n are equal             Solution

(12)  If a and b are distinct integers, prove that a − b is a factor of an − bn, whenever n is a positive integer. [Hint: write an = (a − b + b)n and expand]    Solution

(13)  In the binomial expansion of (a + b)n, the coefficients of the 4th and 13th terms are equal to each other, find n.      Solution

(14)  If the binomial coefficients of three consecutive terms in the expansion of (a + x)n are in the ratio 1 : 7 : 42, then find n.              Solution

(15)  In the binomial coefficients of (1 + x)n, the coefficients of the 5th, 6th and 7th terms are in AP. Find all values of n.

(16)

Solution

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