If `f(x)={{:(,ax^(2)-b,ale xlt 1),(,2,x=1),(,x+1,1 le xle2):}` then the value of the pair (a,b) for which f(x) cannot be continuous at x=1, is

Options

A. (2,0)
B. (1,-1)
C. (4,2)
D. (1,1)

Explanation

To determine the values of \( (a, b) \) for which the function \( f(x) \) is not continuous at \( x = 1 \), we need to analyze the left-hand limit (LHL), right-hand limit (RHL), and the value of the function at that point.

### Step 1: Define the function
The function is defined as:
\[ f(x) = \begin{cases} ax^2 - b & \text{if } x < 1 \\ 2 & \text{if } x = 1 \\ x + 1 & \text{if } 1 < x \leq 2 \end{cases} \]

### Step 2: Find the Left-Hand Limit (LHL) as \( x \) approaches 1
The LHL is calculated using the expression for \( f(x) \) when \( x < 1 \):
\[ \text{LHL} = \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (ax^2 - b) = a(1)^2 - b = a - b \]

### Step 3: Find the Right-Hand Limit (RHL) as \( x \) approaches 1
The RHL is calculated using the expression for \( f(x) \) when \( x > 1 \):
\[ \text{RHL} = \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x + 1) = 1 + 1 = 2 \]

### Step 4: Set up the condition for discontinuity
For \( f(x) \) to be continuous at \( x = 1 \), the following condition must hold:
\[ \text{LHL} = \text{RHL} = f(1) \]
Thus, for discontinuity:
\[ a - b \neq 2 \]

### Step 5: Analyze the pairs \( (a, b) \)
We need to find pairs \( (a, b) \) such that \( a - b \neq 2 \). Let's check the given options:

1. **Option 1:** \( (2, 0) \)
- \( a - b = 2 - 0 = 2 \) (Continuous)

2. **Option 2:** \( (1, -1) \)
- \( a - b = 1 - (-1) = 1 + 1 = 2 \) (Continuous)

3. **Option 3:** \( (4, 2) \)
- \( a - b = 4 - 2 = 2 \) (Continuous)

4. **Option 4:** \( (1, 1) \)
- \( a - b = 1 - 1 = 0 \) (Not continuous)

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