2. (i) For which values of a and b does the following pair of linear equations have an infinite number of solutions? 2x + 3y = 7 (a – b) x + (a + b) y = 3a + b – 2 (ii) For which value of k will the following pair of linear equations have no solution? 3x + y = 1 (2k – 1) x + (k – 1) y = 2k + 1

Solution: (i) 3y + 2x -7 =0 (a + b)y + (a-b)y – (3a + b -2) = 0 a1/a2 = 2/(a-b) , b1/b2 = 3/(a+b) , c1/c2 = -7/-(3a + b -2) For infinitely many solutions, a1/a2 = b1/b2 = c1/c2 Thus 2/(a-b) = 7/(3a+b– 2) 6a + 2b – 4 = 7a – 7b a – 9b = -4 ……………………………….(i) 2/(a-b) = 3/(a+b) 2a + 2b = 3a – 3b a – 5b = 0 ……………………………….….(ii) Subtracting (i) from (ii), we get 4b = 4 b =1 Substituting this eq. in (ii), we get a -5 x 1= 0 a = 5 Thus at a = 5 and b = 1 the given equations will have infinite solutions. (ii) 3x + y -1 = 0 (2k -1)x + (k-1)y – 2k -1 = 0 a1/a2 = 3/(2k -1) , b1/b2 = 1/(k-1), c1/c2 = -1/(-2k -1) = 1/( 2k +1) For no solutions a1/a2 = b1/b2 ≠ c1/c2 3/(2k-1) = 1/(k -1) ≠ 1/(2k +1) 3/(2k –1) = 1/(k -1) 3k -3 = 2k -1 k =2 Therefore, for k = 2 the given pair of linear equations will have no solution.