CSIR NET June 2016 Q87: Quantitative Aptitude
The Fourier transform of x·f(x) is related to the derivative of the transform of f(x) with respect to angular frequency, with a sign/factor depending on the transform convention (here using F(ω)=∫f(x) e^{iωx} dx).
Differentiating F(ω)=∫f(x)e^{iωx}dx with respect to ω gives dF/dω = i∫x f(x) e^{iωx}dx = i·FT[xf(x)](ω), so FT[xf(x)](ω) = (1/i)·dF/dω = -i·dF/dω. Given F(ω)=ω², dF/dω = 2ω, so FT[xf(x)](ω) = -i(2ω) = -2iω. Option 4 is correct.
1) and 2) Incorrect — these omit the required factor of i entirely. 3) Incorrect — this has the correct magnitude and i-factor but the wrong sign for the e^{+iωx} convention used here. 4) Correct — -2iω follows directly from differentiating ω² and applying the correct sign for this Fourier convention.