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Question #41811

Difficulty: 8/10 | 🟢 Free

The Question

Let $g(x) = \left| \int_{-1}^x (2|t| - 2) \, dt \right|$ . Let $m$ be the number of points in $\mathbb{R}$ where $g(x)$ is non-differentiable. A biased coin with probability of heads $p = \frac{m}{5}$ is tossed 75 times. If $X$ is the number of heads obtained, find the value of $E(X) + \text{Var}(X)$ .

Options

A.
27
B.
48
C.
63
D.
72

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Bernoulli Trials Logic|Non-Differentiable Extrema|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze the differentiability of a modulus function defined via an integral with a sign-dependent integrand.

The primary trap lies in assuming that all roots of the inner function $f(x)$ lead to non-differentiable points for $|f(x)|$ .

Students often find the roots $x = -1$ and $x = 1+\sqrt{2}$ and incorrectly assume $m=2$ , failing to check that $f'(-1) = 0$ , which makes $g(x)$ differentiable at $x=-1$ .

Another common trap is ignoring the modulus on $t$ inside the integral, which also yields two roots and leads to the incorrect parameter $p=2/5$ .

The Bernoulli trial acts as a strict discriminator, amplifying the error in $m$ into an incorrect expectation and variance.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Bernoulli Trials Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free