Here’s a breakdown of the solutions presented in the PDF:

Section A: True/False Questions

1. Answers for each statement with brief explanations:
   • (i) True: By the Intermediate Value Theorem, a cubic equation with real coefficients must have at least one real root.
   • (ii) False: Not all sets A and B fulfill the given condition.
   • (iii) True: The curve in question has an oblique asymptote due to the properties of its degree.
   • (iv) False: The greatest integer function is not continuous across all real numbers.
   • (v) False: A critical point is not necessarily an extremum; it could be a saddle point.
   • (vi) False: The function’s domain excludes 0.
   • (vii) True: Differentiation properties apply.
   • (viii) False: The expression diverges as $x \to \infty$.
   • (ix) True: The property holds under differentiation.
   • (x) True: The function is integrable over finite intervals of $\mathbb{R}$.

Section B: Problems on Integration and Differentiation

2. • Integral: Evaluate $\int \frac{x^2 - 2}{x^2 - 6x + 10} \, dx$.
   • Differentiation:
     • (i) Differentiate $y = [(x^2 + 3)\sin x]^4$ with respect to $x$.
     • (ii) Differentiate an implicit function $4x^4 + 16y^4$.
3. • Integral: Solve $\int \frac{t^2 + 9}{(2t - 1)^2} \, dt$.
   • Differentiation for an implicit function: If $x^y = e^x$, find $\frac{dy}{dx}$.

Section C: Curve Tracing, Limits, and Area Calculations

5. Curve Tracing:

   • Trace the curve $y = \frac{1}{x - 1}$.

6. Limits and Graphing:

   • Find the limit and sketch the graph for $f(x) = \frac{x^2 + 6}{x - 2}$.

7. Area Between Curves:

   • Calculate the area enclosed between the line $y = x - 1$ and the parabola $y = x^2 + 6$.

8. Cardioid Perimeter:

   • Find the perimeter of a cardioid given by $r = 1 + \sin(\theta)$.

9. Equivalence Relation:

   • Verify whether the given relation $R$ has the equivalence property.

10. Taylor Expansion and Rolle’s Theorem:

• Expand $e^{2x}$ in a Taylor series.
• Apply Rolle’s theorem on the interval $[0, 2]$.

If you want a specific problem solved or a more detailed breakdown for any section, let me know!

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Follow-Up Questions:

1. Do you want full step-by-step solutions to any specific question?
2. Would you like a graph or visualization for any of the problems?
3. Is there a particular calculus concept you'd like explained in more detail?
4. Would you like tips on solving these types of calculus problems independently?
5. Do you need further clarification on the True/False statements?

Tip:

When solving calculus problems, always start by identifying whether the problem involves differentiation or integration, as the approach differs significantly.