1. It is known that (r + 1)th term, (Tr+1), in the binomial expansion of (a + b)n is given by .The first three terms of the expansion are given as 729, 7290, and 30375 respectively.Therefore, we obtainDividing (2) by (1), we obtainDividing (3) by (2), we obtainFrom (4) and (5), we obtainSubstituting n = 6 in equation (1), we obtaina6 = 729From (5), we obtainThus, a = 3, b = 5, and n = 6.3 .Using Binomial Theorem, the expressions, (1 + 2x)6 and (1 – x)7, can be expanded asThe complete multiplication of the two brackets is not required to be carried out. Only those terms, which involve x5, are required.The terms containing x5 areThus, the coefficient of x5 in the given product is 171.2 It is known that (r + 1)th term, (Tr+1), in the binomial expansion of (a + b)n is given by .Assuming that x2 occurs in the (r + 1)th term in the expansion of (3 + ax)9, we obtainComparing the indices of x in x2 and in Tr + 1, we obtainr = 2Thus, the coefficient of x2 isAssuming that x3 occurs in the (k + 1)th term in the expansion of (3 + ax)9, we obtainComparing the indices of x in x3 and in Tk+ 1, we obtaink = 3Thus, the coefficient of x3 isIt is given that the coefficients of x2 and x3 are the same.Thus, the required value of a is.4In order to prove that (a – b) is a factor of (an – bn), it has to be proved thatan – bn = k (a – b), where k is some natural numberIt can be written that, a = a – b + bThis shows that (a – b) is a factor of (an – bn), where n is a positive integer5 Firstly, the expression (a + b)6 – (a – b)6 is simplified by using Binomial Theorem.This can be done as6Firstly, the expression (x + y)4 + (x – y)4 is simplified by using Binomial Theorem.This can be done as70.99 = 1 – 0.01Thus, the value of (0.99)5 is approximately 0.951.8In the expansion, ,Fifth term from the beginning Fifth term from the end Therefore, it is evident that in the expansion of, the fifth term from the beginning is and the fifth term from the end is.It is given that the ratio of the fifth term from the beginning to the fifth term from the end is. Therefore, from (1) and (2), we obtainThus, the value of n is 10.9Using Binomial Theorem, the given expression can be expanded asAgain by using Binomial Theorem, we obtainFrom (1), (2), and (3), we obtain10Using Binomial Theorem, the given expression can be expanded asAgain by using Binomial Theorem, we obtainFrom (1) and (2), we obtain