Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
If $x^a y^b = e^m$, $x^c y^d = e^n$, $P = \begin{pmatrix} m & b \\ n & d \end{pmatrix}$, $Q = \begin{pmatrix} a & m \\ c & n \end{pmatrix}$, $R = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then:
$P \log_e e = Q \log_e y$
$x = e^{P/R}$, $y = e^{Q/R}$
$P \log_e x = Q \log_e y$
$x = e^{R/P}$, $y = e^{R/Q}$
Step-by-Step Solution
Key Concept: Exponential equations can be solved by taking logarithms and dividing corresponding equations to eliminate variables.
Given $x^c y^b = e^m$, we take logarithms and manipulate to find relationships between variables. From $x^c y^d = e^n$, dividing gives $y^{b-d} = e^{m-n}$, so $y = e^Q$ where $Q = \frac{m-n}{b-d}$. Similarly, from $x^{ad} y^{bd} = e^{md}$ and $x^{cd} y^{bd} = e^{nd}$, dividing yields $x^{ad-cd} = e^{md-nd}$, so $x = e^P$ where $P = \frac{m-n}{a-c}$. Taking logarithms gives $R = P\log e$ and $R = Q\log e$.
Correct Answer: 1,2