Class CBSE Class 12 Mathematics Applications of Integrals Q #1713
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
An ant is observed crawling on a sheet of paper along a straight line given by equation $y=2x-4$. Area of the surface covered by the ant bounded by y-axis, x-axis and $x=1$ is :
(A) 1 sq. unit
(B) 3 sq. units
(C) 2 sq. units
(D) 4 sq. units
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Correct Answer: B

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Detailed Solution

Step 1: Identify the region of integration

The problem asks for the area bounded by the line y = 2x - 4, the x-axis, the y-axis (x = 0), and the line x = 1. We need to calculate the definite integral of the absolute value of the function over the interval [0, 1].

Step 2: Set up the integral

The area A is given by: $$A = \int_{0}^{1} |2x - 4| \, dx$$ Since for all x in [0, 1], 2x - 4 is negative (e.g., at x=0, y=-4; at x=1, y=-2), we have |2x - 4| = -(2x - 4) = 4 - 2x.

Step 3: Evaluate the integral

Now, integrate the expression: $$A = \int_{0}^{1} (4 - 2x) \, dx$$ $$A = [4x - x^2]_{0}^{1}$$ $$A = (4(1) - (1)^2) - (4(0) - (0)^2)$$ $$A = 4 - 1 = 3$$

Final Answer: 3 sq. units

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must translate a geometric description into a definite integral and handle the absolute value property correctly.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the application of integration techniques to find the area under a curve, which is a standard procedural skill in Calculus.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. It tests the student's ability to interpret boundaries and apply the concept of definite integrals to geometric problems.

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