Identify the correct set from the following for fundamental particles
`{:(,"List-A",,"List-B",),((I),"Decreasing order of masses",(a),e^(-) gt p gt n,),((II),"Decreasing order of e/m values",(b),p gt e^(-) gt n,),((III),"Decreasing order of de-Broglie",(c),n gt p gt e^(-),),(,"wavelength with same velocities",,,),((IV),"Decreasing order of uncertainity",(d),n gt e^(-) gt p,),(,"in velocity when " Deltax " is same",,,):}`
The correct match is

Options

A. I-C, II-a, III-d, IV-a
B. I-C, II-a, III-a, IV-a
C. I-c, II-d, III-b, IV-a
D. I-c, II-b, III-d, IV-a

Answer

To solve the question regarding the correct set of matches for fundamental particles, we will analyze each option step by step.

### Step 1: Decreasing Order of Masses
We need to identify the masses of the three fundamental particles:
- Electron (e⁻): Mass ≈ \(9.1 \times 10^{-31}\) kg
- Proton (p): Mass ≈ \(1.67 \times 10^{-27}\) kg
- Neutron (n): Mass ≈ \(1.68 \times 10^{-27}\) kg

**Conclusion**: The order from lightest to heaviest is:
\[ e^- < p < n \]
Thus, the decreasing order of masses is:
\[ n > p > e^- \]
This matches with option (c).

### Step 2: Decreasing Order of e/m Values
The e/m ratio is calculated as charge divided by mass. The charge (e) and mass (m) for each particle are:
- Electron: \( e = 1.6 \times 10^{-19} \) C, \( m = 9.1 \times 10^{-31} \) kg
- Proton: \( e = 1.6 \times 10^{-19} \) C, \( m = 1.67 \times 10^{-27} \) kg
- Neutron: \( e = 0 \) (neutral), \( m = 1.68 \times 10^{-27} \) kg

Calculating e/m:
- For electron: \( \frac{e}{m} = \frac{1.6 \times 10^{-19}}{9.1 \times 10^{-31}} \)
- For proton: \( \frac{e}{m} = \frac{1.6 \times 10^{-19}}{1.67 \times 10^{-27}} \)
- For neutron: \( \frac{e}{m} = 0 \)

**Conclusion**: The order from highest to lowest e/m is:
\[ e^- > p > n \]
This matches with option (b).

### Step 3: Decreasing Order of de-Broglie Wavelength with Same Velocities
The de-Broglie wavelength (\( \lambda \)) is given by the formula:
\[ \lambda = \frac{h}{mv} \]
Where \( h \) is Planck's constant and \( v \) is velocity. Since the velocity is the same for all particles, the wavelength is inversely proportional to mass.

**Conclusion**: The order from longest to shortest wavelength is:
\[ e^- > p > n \]
This matches with option (a).

### Step 4: Decreasing Order of Uncertainty in Velocity when Δx is Same
According to Heisenberg's Uncertainty Principle, the uncertainty in position (\( \Delta x \)) and momentum (\( \Delta p \)) is given by:
\[ \Delta x \Delta p \geq \frac{h}{4\pi} \]
Where \( \Delta p = m \Delta v \). Thus, if \( \Delta x \) is constant, then:
\[ \Delta v \propto \frac{1}{m} \]
This means that the particle with the smallest mass has the largest uncertainty in velocity.

**Conclusion**: The order from highest to lowest uncertainty in velocity is:
\[ e^- > p > n \]
This matches with option (d).

### Final Matching
Now we can summarize the matches:
1. Decreasing order of masses: **(I) → (c)**
2. Decreasing order of e/m values: **(II) → (a)**
3. Decreasing order of de-Broglie wavelength: **(III) → (b)**
4. Decreasing order of uncertainty in velocity: **(IV) → (d)**

### Final Answer
The correct matches are:
- (I) → (c)
- (II) → (a)
- (III) → (b)
- (IV) → (d)