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Question #41581

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all real numbers $k$ such that the inequality $\frac{2x^2 + 4x + 5}{x^2 - kx + 4} \ge 1$ is satisfied for all $x \in \mathbb{R}$ . If $\alpha = \inf(S)$ and $\beta = \sup(S)$ , then the value of the integral $\int_{\alpha}^{\beta} |x^2 + 4x + 3| \, dx$ is equal to:

Options

A.
2
B.
$\frac{56}{3}$
C.
$\frac{2}{3}$
D.
$\frac{52}{3}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Compound Function Boundaries|Maximum/Minimum of Quadratic Expressions|Cross Multiplication Trap|Concavity Analysis

Why This Question Is Tricky

Common Mistakes

This question tests the 'Cross Multiplication Trap' in rational inequalities.

Students frequently cross-multiply without ensuring the denominator is strictly positive for all $x$ , leading them to incorrectly solve only the numerator's discriminant condition ( $k \in [-6, -2]$ ) and obtain $\alpha = -6$ .

The correct approach requires intersecting this with the denominator's positivity condition ( $k \in (-4, 4)$ ).

Furthermore, the integral evaluates their ability to perform concavity analysis and split the bounds at the roots of the absolute value function, acting as a secondary filter against careless integration.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Compound Function Boundaries
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free