Gradually Varied Flow (GVF) profiles in open channels given in Column 1 are to be matched with the water surface slopes in Column 2 in the table below.

$$\begin{gathered} \begin{array}{cc}\text{Column 1} \\ \text{GVF Profile}& \text{Column 2} \\ \text{Water Surface Slope}\\ \text{(P) M1}& \text{(I) Positive}\\ \text{(Q) M2}& \text{(II) Negative}\\ \text{(R) M3}& \text{(III) Zero}\end{array} \end{gathered}$$

Which of the following options is/are NOT correct?

The correct options that are NOT correct are:
1. (P)- (I) ; (Q)- (II) ; (R)- (III)
2. (P)- (II) ; (Q)- (I) ; (R)- (I)
3. (P)- (I) ; (Q)- (III) ; (R)- (II)

Step-by-Step Explanation:

To determine which options are incorrect, we first need to find the correct relationship between the Gradually Varied Flow (GVF) profiles and their corresponding water surface slopes (dy/dx).

The dynamic equation for Gradually Varied Flow in an open channel is given by:

$$\frac{dy}{dx}=S_0\frac{1-{\left(\frac{y_n}{y}\right)}^N}{1-{\left(\frac{y_c}{y}\right)}^M}$$

where:
- y is the depth of flow at any section.
- y_n is the normal depth of flow.
- y_c is the critical depth of flow.
- S₀ is the longitudinal bed slope (positive for mild slope channels).
- dy/dx is the water surface slope relative to the channel bed.
- N and M are positive exponents.

For a mild slope (M), the normal depth is greater than the critical depth:

$y_n>y_c$

Now, let us analyze each of the three zones for a mild slope:

1. M1 Profile (Zone 1):
This zone lies above both the normal depth line and the critical depth line, where:

$y>y_n>y_c$

Since y > y_n, we have:

$1-{\left(\frac{y_n}{y}\right)}^N>0$

Since y > y_c, we have:

$1-{\left(\frac{y_c}{y}\right)}^M>0$

Since both the numerator and the denominator are positive, the slope dy/dx is positive:

$\frac{dy}{dx}>0$

Therefore, (P) matches with (I) Positive.

2. M2 Profile (Zone 2):
This zone lies between the normal depth line and the critical depth line, where:

$y_n>y>y_c$

Since y < y_n, we have:

$1-{\left(\frac{y_n}{y}\right)}^N<0$

Since y > y_c, we have:

$1-{\left(\frac{y_c}{y}\right)}^M>0$

Thus, the numerator is negative and the denominator is positive, meaning the slope dy/dx is negative:

$\frac{dy}{dx}<0$

Therefore, (Q) matches with (II) Negative.

3. M3 Profile (Zone 3):
This zone lies below both the normal depth line and the critical depth line, where:

$y_n>y_c>y$

Since y < y_n, we have:

$1-{\left(\frac{y_n}{y}\right)}^N<0$

Since y < y_c, we have:

$1-{\left(\frac{y_c}{y}\right)}^M<0$

Since both the numerator and the denominator are negative, the ratio becomes positive, meaning the slope dy/dx is positive:

$\frac{dy}{dx}>0$

Therefore, (R) matches with (I) Positive.

Conclusion:
The correct matching sequence is:
(P)- (I) ; (Q)- (II) ; (R)- (I)

Since the question asks for the options that are NOT correct, any option other than the correct match is correct as a choice for the question. Thus, the three incorrect options are:
- (P)- (I) ; (Q)- (II) ; (R)- (III)
- (P)- (II) ; (Q)- (I) ; (R)- (I)
- (P)- (I) ; (Q)- (III) ; (R)- (II)