<p>Which of the following statement(s) is/are incorrect?</p><p>(a) The lines \(\frac{x-4}{-3} = \frac{y+6}{-1} = \frac{z+6}{-1}\) and \(\frac{x-1}{-1} = \frac{y-2}{-2} = \frac{z-3}{2}\) are orthogonal</p><p>(b) The planes \(3x - 2y - 4z = 3\) and the plane \(x - y - z = 3\) are orthogonal</p><p>(c) The function \(f(x) = \ln(e^{-2} + e^x)\) is monotonic increasing for all \(x \in \mathbb{R}\)</p><p>(d) If g is the inverse of the function \(f(x) = \ln(e^{-2} + e^x)\) then \(g(x) = \ln(e^x - e^{-2})\)</p>
Step-by-Step Solution
Key Concept: Check orthogonality conditions for lines and planes using dot product of direction/normal vectors, and verify inverse function relationships.
<p><strong>(a)</strong> Direction vectors are \((-3, -1, -1)\) and \((-1, -2, 2)\). Dot product: \(3 + 2 - 2 = 3 \neq 0\), so not orthogonal. Incorrect.</p><p><strong>(b)</strong> Normal vectors are \((3, -2, -4)\) and \((1, -1, -1)\). Dot product: \(3 + 2 + 4 = 9 \neq 0\), so not orthogonal. Incorrect.</p><p><strong>(c)</strong> \(f'(x) = \frac{e^x}{e^{-2} + e^x} > 0\) for all \(x\), so monotonic increasing. Correct.</p><p><strong>(d)</strong> From \(y = \ln(e^{-2} + e^x)\), we get \(e^y = e^{-2} + e^x\), so \(e^x = e^y - e^{-2}\), thus \(g(x) = \ln(e^x - e^{-2})\). Correct.</p>
Correct Answer: a, b