Exercise - 5.1

. Expand (i) x − (ii) x − − x + x + − x . . Compute (i) (ii) (iii) .

. Using binomial theorem, indicate which of the following two number is larger: ( . ) 1000000 , 10000 . .

Find the coefficient of x in x + x . . Find the coefficient of x and the coefficient of x in x − x . .

Find the coefficient of x in the expansion of ( + x ) ( x + x ) . . Find the constant term of x − x . .

Find the last two digits of the number . . If n is a positive integer, using Binomial theorem, show that, n + − n − is always divisible by . .

If n is an odd positive integer, prove that the coefficients of the middle terms in the expansion of ( x + y ) n are equal. . If n is a positive integer and r is a nonnegative integer, prove that the coefficients of x r and x n − r in the expansion of ( + x ) n are equal. .

If a and b are distinct integers, prove that a − b is a factor of a n − b n , whenever n is a positive integer. [Hint: write a n = ( a − b + b ) n and expand] . In the binomial expansion of ( a + b ) n , if the coefficients of the th and th terms are equal then, find n . .

If the binomial coefficients of three consecutive terms in the expansion of ( a + x ) n are in the ratio : : , then find n . . In the binomial expansion of ( + x ) n , the coefficients of the th , th and th terms are in AP. Find all values of n .