Q592: If \( f(x) = \begin{cases} 1, & \text{if } x \leq 3 \\ ax + ...

Class CBSE Class 12 Mathematics Continuity and Differentiability Q #592

COMPETENCY BASED

APPLY

1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE

If $ f(x) = \begin{cases} 1, & \text{if } x \leq 3 \\ ax + b, & \text{if } 3 < x < 5 \\ 7, & \text{if } x \geq 5 \end{cases} $ is continuous for all real numbers, then find the values of $a$ and $b$:

(A) $a=3$, $b=-8$

(B) $a=3$, $b=8$

(D) $a=-3$, $b=8$

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AI Tutor Explanation

Detailed Solution

Step 1: Understanding Continuity

For $f(x)$ to be continuous at $x=3$ and $x=5$, the left-hand limit (LHL) must equal the right-hand limit (RHL) at these points, and both must equal the function's value at those points.

Step 2: Continuity at x = 3

At $x=3$, we need to ensure that the function is continuous. The left-hand limit (LHL) at $x=3$ is: $$ \lim_{x \to 3^-} f(x) = 1 $$ The right-hand limit (RHL) at $x=3$ is: $$ \lim_{x \to 3^+} f(x) = a(3) + b = 3a + b $$ For continuity at $x=3$, LHL = RHL: $$ 1 = 3a + b $$

Step 3: Continuity at x = 5

At $x=5$, we also need to ensure continuity. The left-hand limit (LHL) at $x=5$ is: $$ \lim_{x \to 5^-} f(x) = a(5) + b = 5a + b $$ The right-hand limit (RHL) at $x=5$ is: $$ \lim_{x \to 5^+} f(x) = 7 $$ For continuity at $x=5$, LHL = RHL: $$ 5a + b = 7 $$

Step 4: Solving the System of Equations

We now have a system of two linear equations: $$ 3a + b = 1 \quad (1) $$ $$ 5a + b = 7 \quad (2) $$ Subtract equation (1) from equation (2): $$ (5a + b) - (3a + b) = 7 - 1 $$ $$ 2a = 6 $$ $$ a = 3 $$ Substitute $a = 3$ into equation (1): $$ 3(3) + b = 1 $$ $$ 9 + b = 1 $$ $$ b = 1 - 9 $$ $$ b = -8 $$

Step 5: Final Answer

Therefore, $a = 3$ and $b = -8$.

Final Answer: a=3, b=-8

AI Suggestion: Option A

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because the student needs to apply the concept of continuity to find the values of $a$ and $b$. They must use the definition of continuity (LHL = RHL = function value) and solve a system of equations.

Knowledge Dimension: CONCEPTUAL

Justification: The question requires understanding the conceptual definition of continuity and applying it to a piecewise function. It involves understanding limits and how they relate to the function's value at a point.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply the concept of continuity to a piecewise function and solve for unknown parameters, which is a competency-based application of the theoretical knowledge.