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Question #40771

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \int_0^x (t^2 - 5t + 6) dt$ . According to Lagrange's Mean Value Theorem applied to $f(x)$ on the interval $[0, 3]$ , there exists at least one $c \in (0, 3)$ such that the tangent to the curve $y = f(x)$ at $x = c$ is parallel to the secant line joining the points $(0, f(0))$ and $(3, f(3))$ . A line from the family of lines given by $(8y - 4) + ([\lambda]^2 - 5[\lambda])(x - y + 1) = 0$ is parallel to this tangent, where $[\cdot]$ denotes the greatest integer function. If the set of all possible values of $\lambda$ is $S = [a, b) \cup [c, d)$ with $a < b < c < d$ , find the value of $a^2 + b^2 + c^2 + d^2$ .

Options

A.
158
B.
138
C.
62
D.
148

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

Family of Lines|LMVT Tangent Parallelism|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

The question tests the synthesis of Lagrange's Mean Value Theorem (LMVT) with the concept of a family of lines, while embedding a Greatest Integer Function (GIF) trap.

Students must first correctly apply LMVT to an integral function to find the specific tangent slope.

The core trap lies in equating this slope to the parameterized slope of the family of lines, $m(\lambda)$ .

Because the parameter involves $[\lambda]$ , the slope equality $m([\lambda]) = f'(c)$ transforms a continuous algebraic equation into a discrete condition.

Students often incorrectly assume $\lambda$ takes a single discrete value or mishandle the inequality bounds of the GIF (e.g., $[\lambda] = n \implies \lambda \in [n, n+1)$ ), leading to incorrect interval endpoints.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Family of Lines
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free