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Q111. The Fourier transform of a function f(x) is 5sin(5w). Then the Fourier transform of the function f(5x) is:
According to the scaling property of Fourier transforms, F[f(ax)] = (1/|a|) * F(w/a). Here, a = 5. Therefore, the new transform is (1/5) * 5sin(5*(w/5)) = sin(w).
Q112. If L = sediment load, D50 = median sediment size, Q = mean discharge and S = slope, then which one of the following relationships is correct?
According to Lane's Balance equation (Q * S is proportional to L * D50), the stream slope (S) must increase to carry a larger sediment load (L) or larger grain size (D50), and can decrease if water discharge (Q) increases.
Q113. If the Fourier transform of a function f(x) is w^2, the Fourier transform of the function xf(x) is:
According to Fourier transform properties, F[x*f(x)] = i * d/dw(F(w)). Taking the derivative of w^2 gives 2w. Multiplying by i yields 2iw.
Q114. During the year 1997 that witnessed one of the strongest El Ninos of the century, which of the following events negated its effect on the Indian subcontinent?
A strongly positive Indian Ocean Dipole (IOD) occurred in 1997. The warmer waters in the western Indian Ocean enhanced moisture supply and monsoon rainfall, successfully counteracting the classic drought effect of El Nino.
Q115. Which of the following conditions is NOT required for tropical cyclone development?
A non-zero Coriolis force is absolutely required to impart rotation to the rising air mass. This is why tropical cyclones do not form exactly on the equator (where the Coriolis force is zero).