Consider f(x)= a(7x-12-x^2), b x^2-7x+12 for continuity. Find n(S… — Solution | JEE Mathematics

MCQMediumJEE 2024Continuity

Consider $f(x)= a(7x-12-x^2), b \lvert x^2-7x+12 \rvert$ for continuity. Find $n(S)$ .

Answer

Correct answer:D

Standard Method

Given: The function is defined using the expressions $a(7x-12-x^2)$ and $b\lvert x^2-7x+12 \rvert$ , and continuity is required.

Find: $n(S)$ , the number of ordered pairs in the set $S$ .

From the extracted solution, solving the continuity condition at $x=3$ gives the unique solution

(a,b)=(-4,2)

Hence the set $S$ contains exactly one element.

Therefore, the correct option is D and $n(S)=1$ .

Treating continuity at only one point as giving multiple valid pairs. This is wrong because the extracted solution states that the continuity condition leads to a unique pair $(a,b)=(-4,2)$ . Always use the continuity equations to determine whether the solution is unique or not.
Ignoring the modulus expression $\lvert x^2-7x+12 \rvert$ while analysing continuity. This is wrong because modulus-based expressions can change form at the zeros of the quadratic. Check the critical points before concluding the number of valid parameter pairs.

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