Class 12 Mathematics (NCERT)

1. The area between a curve y = f(x) and the x-axis from x = a to x = b is generally given by:

• A. ∫[a to b] f(x) dx (taking the absolute value where the curve dips below the axis)
• B. f(b) - f(a)
• C. f'(b) - f'(a)
• D. Just the value f(a)

2. The area enclosed between two curves y = f(x) and y = g(x) is found by integrating:

• A. [f(x) - g(x)] over the appropriate interval
• B. The product f(x) · g(x)
• C. The sum f(x) + g(x)
• D. Only f(x)

3. To find area using integration, we typically consider:

• A. Vertical or horizontal elementary strips
• B. Only triangles
• C. Isolated points
• D. Only circular arcs

4. The area of a circle x² + y² = r² can be determined using:

• A. Integration techniques with appropriate limits
• B. Differentiation alone
• C. Matrices
• D. Simple algebra without calculus

5. When a portion of a curve lies below the x-axis, the corresponding definite integral gives:

• A. A negative value, so its absolute value must be taken for area
• B. Always a positive value directly
• C. Always zero
• D. An undefined result

6. The area bounded by a curve and the y-axis is obtained by integrating with respect to:

• A. y
• B. x
• C. Both x and y simultaneously
• D. Neither variable

7. When finding the area between two intersecting curves, the first step is generally to find:

• A. The points of intersection by solving the equations simultaneously
• B. The derivative of both curves
• C. The slopes of the curves
• D. The matrix of coefficients

8. The area under a curve, as studied in this chapter, is a direct application of the:

• A. Definite integral
• B. Indefinite integral
• C. Derivative
• D. Limit of a sequence

9. If a region is bounded by a curve and both coordinate axes, its area is computed by:

• A. Integrating over the appropriate sub-intervals
• B. Using differentiation only
• C. Using matrices
• D. Using determinants

10. The area of the region bounded by an ellipse can be computed using:

• A. Integration involving the ellipse's equation
• B. Simple geometry formulas alone
• C. Matrices
• D. Vector algebra

11. To compute area using horizontal strips, integration is carried out with respect to:

• A. y
• B. x
• C. Time
• D. Neither variable

12. The application of integrals, as studied in this chapter, is primarily used to determine:

• A. Areas of regions bounded by curves
• B. Slopes of tangent lines
• C. Only instantaneous rates of change
• D. Limits of sequences