JEE 2024 — Mathematics PYQJEE | Mathematics | 2024Consider the function. f(x)=⎩⎨⎧b∣x∣2−7x+12a(7x−12−x2),2∣x−[x]∣sin(x−3),b,amp;xamp;xamp;x=3lt;3gt;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:Choose the correct answer:A. 1 (Correct Answer)B. 4C. 2D. infinitely manyCorrect Answer: 1ExplanationSince, f(x) is continuous at x=3. ∴ x→3−limf(x)=f(3)=x→3+limf(x) ⇒x→3−limb∣(x−3)(x−4)∣−a(x−4)(x−3)=b=x→3lim2x−3+sin(x−3) ⇒b−a=b=21=2 ⇒b=2 and 2a=2⇒a=−4 ∴ S=(−4,2) i.e. only one pairA1B4C2Dinfinitely manyShow Answer SExplanationSince, f(x) is continuous at x=3. ∴ x→3−limf(x)=f(3)=x→3+limf(x) ⇒x→3−limb∣(x−3)(x−4)∣−a(x−4)(x−3)=b=x→3lim2x−3+sin(x−3) ⇒b−a=b=21=2 ⇒b=2 and 2a=2⇒a=−4 ∴ S=(−4,2) i.e. only one pair
