GATE 2023 PH – Question 32

Correct answer: (A) $e^{ix}$ and $e^{-ix}$; (B) $x$ and $e^x$; (D) $e^{ix}$ and $\sin x$

Explanation

Two functions are **linearly dependent** if one is a constant multiple of the other for all $x$.

• **A. $e^{ix}$ and $e^{-ix}$:** their ratio is $e^{2ix}$, which is not constant. Independent. ✓
• **B. $x$ and $e^x$:** the ratio $x/e^x$ is not constant. Independent. ✓
• **C. $2^x$ and $2^{x-3}$:** $2^{x-3}=2^x\cdot2^{-3}=\tfrac18\,2^x$. One is a constant multiple of the other, so they are **dependent**. ✗
• **D. $e^{ix}$ and $\sin x$:** $\sin x=\dfrac{e^{ix}-e^{-ix}}{2i}$ is not a multiple of $e^{ix}$ (it contains $e^{-ix}$ as well). Independent. ✓

Answer **A, B and D**.