Find the number of solutions of the equation `e^(2x) + e^x-2=[{x^2 + 10x + 11}]` is(where, {x} denotes fractional part of x and [x] denotes greatest integer function) (a)0 (b)1 (c)2 (d)3

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( e^{2x} + e^x - 2 = \{x^2 + 10x + 11\} \), where \(\{x\}\) denotes the fractional part of \(x\) and \([x]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Understand the Right-Hand Side
The right-hand side of the equation involves the fractional part of \(x^2 + 10x + 11\). The fractional part \(\{y\}\) of any real number \(y\) is defined as \(y - [y]\), where \([y]\) is the greatest integer less than or equal to \(y\). Therefore, we can express the right-hand side as:
\[ \{x^2 + 10x + 11\} = x^2 + 10x + 11 - [x^2 + 10x + 11] \]
This means that the right-hand side will always be in the range \([0, 1)\).

### Step 2: Analyze the Left-Hand Side
The left-hand side of the equation is \( e^{2x} + e^x - 2 \). We can analyze this function by setting \( t = e^x \). Then, we can rewrite the left-hand side as:
\[ t^2 + t - 2 \]
This is a quadratic equation in \(t\).

### Step 3: Solve the Quadratic Equation
We can factor the quadratic:
\[ t^2 + t - 2 = (t + 2)(t - 1) = 0 \]
Setting each factor to zero gives us:
\[ t + 2 = 0 \quad \Rightarrow \quad t = -2 \quad (\text{not valid since } t = e^x > 0) \]
\[ t - 1 = 0 \quad \Rightarrow \quad t = 1 \quad \Rightarrow \quad e^x = 1 \quad \Rightarrow \quad x = 0 \]

### Step 4: Check the Validity of the Solution
Now, we need to check if \(x = 0\) satisfies the original equation. We substitute \(x = 0\) into the right-hand side:
\[ \{0^2 + 10 \cdot 0 + 11\} = \{11\} = 0 \]
Now, substituting \(x = 0\) into the left-hand side:
\[ e^{2 \cdot 0} + e^0 - 2 = 1 + 1 - 2 = 0 \]
Both sides are equal, confirming that \(x = 0\) is indeed a solution.

### Step 5: Conclusion
Since the left-hand side \(e^{2x} + e^x - 2\) can only equal \(0\) at \(x = 0\) and the right-hand side is constrained between \(0\) and \(1\), we conclude that there is only one solution to the equation.

Thus, the number of solutions is:
\[ \boxed{1} \]