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Question #41222

Difficulty: 8/10 | 🟢 Free

The Question

Let $w, x, y, z \in \mathbb{R}$ such that the following matrix equation holds: $\begin{pmatrix} w & x \\ y & z \end{pmatrix} = \begin{pmatrix} \int_{-e}^{-1} t \ln|t| dt & \int_{-2}^{2} \frac{t^3}{\sqrt{4-t^2}} dt \\ \int_{1}^{e} t \ln t dt & \int_{1}^{2} \left( 3t^2 - 2t + \frac{5}{4} \right) dt \end{pmatrix}$ Find the value of $w + x + y + z$ .

Options

A.
$\frac{315}{60}$
B.
$\frac{e^2+1}{2}$
C.
$\frac{21}{4} - \frac{e^2+1}{2}$
D.
\text{Does not exist}

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

Matrix Equality|Integration by Parts (LIATE Rule)|Logarithmic Integration Modulus|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of definite integrals within matrix equality.

The primary trap lies in the element $w$ : forgetting the modulus in $\ln|t|$ for negative bounds leads to complex numbers (e.g., $\ln(-1) = i\pi$ ), which violates the real-field constraint $w \in \mathbb{R}$ and tempts students to choose 'Does not exist'.

A secondary trap is the denominator restriction in $x$ at $t = \pm 2$ ; students must recognize it as a convergent improper integral of an odd function.

Finally, the correct sum $21/4$ is disguised as the unsimplified fraction $315/60$ to test confidence in the derived result.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Matrix Equality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free