A physical quantity x can dimensionally represented in terms of M, L and T that is `x=M^(a) L^(b) T^(c)`. The quantity time-

Options

A. May be dimensionally represented in terms of x, M and L if `c ne 0`
B. May be dimensionally represented in terms of x, M and L if `c=0`
C. May be dimensionally represented in terms of x, M and L, irrespective of value of c
D. Can never be dimensionally represented in terms of x, M and L

Answer

To solve the problem of representing the physical quantity \( x \) in terms of dimensions \( M \), \( L \), and \( T \), we will follow these steps:

### Step 1: Write the dimensional formula for the physical quantity \( x \)
We start with the dimensional representation of the physical quantity \( x \):
\[ x = M^a L^b T^c \]
where \( a \), \( b \), and \( c \) are the powers corresponding to the dimensions of mass (M), length (L), and time (T), respectively.

### Step 2: Express time \( T \) in terms of \( x \), \( M \), and \( L \)
To express time \( T \) in terms of \( x \), \( M \), and \( L \), we can rearrange the equation:
\[ T^c = \frac{x}{M^a L^b} \]
Taking the \( c \)-th root on both sides gives us:
\[ T = \left( \frac{x}{M^a L^b} \right)^{\frac{1}{c}} \]

### Step 3: Simplify the expression for \( T \)
Now we can simplify the expression:
\[ T = x^{\frac{1}{c}} M^{-\frac{a}{c}} L^{-\frac{b}{c}} \]

### Step 4: Analyze the conditions for representing time \( T \)
From the expression derived, we can see that:
- If \( c \neq 0 \), we can express \( T \) in terms of \( x \), \( M \), and \( L \).
- If \( c = 0 \), then \( x = M^a L^b \) and \( T \) cannot be expressed in terms of \( x \) since \( T^0 = 1 \).

### Step 5: Evaluate the options given in the question
- **Option A**: Quantity time may be dimensionally represented in terms of \( x \), \( M \), and \( L \) if \( c \neq 0 \). **True**.
- **Option B**: Quantity time may be dimensionally represented in terms of \( x \), \( M \), and \( L \) if \( c = 0 \). **False**.
- **Option C**: Quantity time may be dimensionally represented in terms of \( x \), \( M \), and \( L \) irrespective of the value of \( c \). **False**.
- **Option D**: Quantity time can never be dimensionally represented in terms of \( x \), \( M \), and \( L \). **False**.

### Conclusion
The correct option is **Option A**.

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