If `x^(2)-2 (4 lamda-1) x+ (15 lamda^(2)-2 lamda -7) gt 0` for all real x, then `lamda` belongs to

Options

A. `(0,2)`
B. `(1,3)`
C. `(2,4)`
D. none of these

Answer

To solve the inequality \( x^2 - 2(4\lambda - 1)x + (15\lambda^2 - 2\lambda - 7) > 0 \) for all real \( x \), we need to analyze the quadratic in the form \( ax^2 + bx + c > 0 \).

### Step 1: Identify coefficients
The quadratic can be expressed as:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -2(4\lambda - 1) = -8\lambda + 2 \)
- \( c = 15\lambda^2 - 2\lambda - 7 \)

### Step 2: Condition for the quadratic to be positive for all \( x \)
For the quadratic \( ax^2 + bx + c > 0 \) to be positive for all real \( x \), the following conditions must be satisfied:
1. \( a > 0 \) (which is true since \( a = 1 \))
2. The discriminant \( D < 0 \)

### Step 3: Calculate the discriminant
The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-8\lambda + 2)^2 - 4 \cdot 1 \cdot (15\lambda^2 - 2\lambda - 7) \]

### Step 4: Expand the discriminant
Calculating \( D \):
\[ D = (64\lambda^2 - 32\lambda + 4) - (60\lambda^2 - 8\lambda - 28) \]
\[ D = 64\lambda^2 - 32\lambda + 4 - 60\lambda^2 + 8\lambda + 28 \]
\[ D = (64\lambda^2 - 60\lambda^2) + (-32\lambda + 8\lambda) + (4 + 28) \]
\[ D = 4\lambda^2 - 24\lambda + 32 \]

### Step 5: Set the discriminant less than zero
Now we set the discriminant less than zero:
\[ 4\lambda^2 - 24\lambda + 32 < 0 \]
Dividing the entire inequality by 4:
\[ \lambda^2 - 6\lambda + 8 < 0 \]

### Step 6: Factor the quadratic
Factoring the quadratic:
\[ (\lambda - 2)(\lambda - 4) < 0 \]

### Step 7: Determine the intervals
To find the intervals where this inequality holds, we analyze the sign of the product:
- The roots are \( \lambda = 2 \) and \( \lambda = 4 \).
- The intervals to test are \( (-\infty, 2) \), \( (2, 4) \), and \( (4, \infty) \).

### Step 8: Test the intervals
1. For \( \lambda < 2 \) (e.g., \( \lambda = 0 \)):
\[ (0 - 2)(0 - 4) = 8 > 0 \]
2. For \( 2 < \lambda < 4 \) (e.g., \( \lambda = 3 \)):
\[ (3 - 2)(3 - 4) = 1 \cdot (-1) = -1 < 0 \]
3. For \( \lambda > 4 \) (e.g., \( \lambda = 5 \)):
\[ (5 - 2)(5 - 4) = 3 \cdot 1 = 3 > 0 \]

### Conclusion
The inequality \( (\lambda - 2)(\lambda - 4) < 0 \) holds for the interval:
\[ \lambda \in (2, 4) \]

Thus, the final answer is:
\[ \lambda \text{ belongs to } (2, 4) \]