Consider the function f x = a 7 x - 12 - x 2 b x 2 - 7 x + 12 , x < 3 2 sin ( x - 3 ) x - [ x ] , x > 3 b , x = 3 ,where [ x ] denotes the greatest integer less than or equal to x . If S denotes the set of all ordered pairs ( a , b ) such that f ( x ) is continuous at x = 3 , then the number of elements in S is :

Step-by-step solution

Given: f x = a 7 x - 12 - x 2 b x 2 - 7 x + 12 , x < 3 2 sin ( x - 3 ) x - [ x ] , x > 3 b , x = 3 ⇒ f 3 - = a 7 x - 12 - x 2 b | x 2 - 7 x + 12 | ⇒ f 3 - = - a ( x - 3 ) ( x - 4 ) b ( x - 3 ) ( x - 4 ) ⇒ f 3 - = - a b . . . i Now, f 3 + = 2 lim x → 3 + sin ( x - 3 ) x - 3 ⇒ f 3 + = 2 . . . ii Also, f ( 3 ) = b . . . iii It is given that f x is continuous at x = 3 . Hence, f ( 3 ) = f ( 3 + ) = f ( 3 - ) ⇒ b = 2 = - a b ⇒ b = 2 , a = - 4 Hence, there is only 1 elements in S , which is ordered pair ( - 4 , 2 ) .