If ` f(x) = {:{(px^(2)-q, x in [0,1)), ( x+1 , x in (1,2]):}`
and `f(1) =2` , then the value of the pair `( p,q)` for which `f(x)` cannot be continuous at `x =1` is:

Options

A. (a) `(2,0)`
B. (b) `(1,-1)`
C. (c) `(4,2)`
D. (d) `(1,1)`

Answer

To solve the problem, we need to analyze the function \( f(x) \) and determine the conditions under which it cannot be continuous at \( x = 1 \).

### Step 1: Define the function
The function is defined as follows:
\[ f(x) = \begin{cases} px^2 - q & \text{for } x \in [0, 1) \\ 2 & \text{for } x = 1 \\ x + 1 & \text{for } x \in (1, 2] \end{cases} \]

### Step 2: Evaluate \( f(1) \)
From the problem, we know that \( f(1) = 2 \).

### Step 3: Find the right-hand limit as \( x \) approaches 1
To find the right-hand limit as \( x \) approaches 1, we evaluate:
\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x + 1) = 1 + 1 = 2 \]

### Step 4: Find the left-hand limit as \( x \) approaches 1
To find the left-hand limit as \( x \) approaches 1, we evaluate:
\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (px^2 - q) = p(1)^2 - q = p - q \]

### Step 5: Set up the condition for discontinuity
For \( f(x) \) to be discontinuous at \( x = 1 \), the left-hand limit must not equal the value of the function at that point:
\[ p - q \neq f(1) = 2 \]
This gives us the inequality:
\[ p - q \neq 2 \]

### Step 6: Analyze the given options
We need to check which pairs \( (p, q) \) from the options lead to \( p - q \neq 2 \):
1. **Option A:** \( (2, 0) \)
\[ 2 - 0 = 2 \quad \text{(not valid)} \]
2. **Option B:** \( (1, -1) \)
\[ 1 - (-1) = 1 + 1 = 2 \quad \text{(not valid)} \]
3. **Option C:** \( (4, 2) \)
\[ 4 - 2 = 2 \quad \text{(not valid)} \]
4. **Option D:** \( (1, 1) \)
\[ 1 - 1 = 0 \quad \text{(valid)} \]

### Conclusion
The only pair \( (p, q) \) for which \( f(x) \) cannot be continuous at \( x = 1 \) is:
\[ \boxed{(1, 1)} \]