If t is real and `lambda = (t^2-3t+4)/(t^2+3t+4)` then find number of the solution of the systems of equation `3x-y+4z =0,x+2y-3z=-2.6x+5y+lambda z=-3` for a particular value of `lambda`.

Options

A. one
B. two
C. zero
D. infinite

Answer

To solve the given problem, we need to find the number of solutions of the system of equations for a particular value of \(\lambda\). The equations are:

1. \(3x - y + 4z = 0\)
2. \(x + 2y - 3z = -2\)
3. \(6x + 5y + \lambda z = -3\)

Given that \(\lambda = \frac{t^2 - 3t + 4}{t^2 + 3t + 4}\), we will first determine the range of \(\lambda\) and then analyze the system of equations.

### Step 1: Finding the range of \(\lambda\)

We start with the expression for \(\lambda\):

\[ \lambda = \frac{t^2 - 3t + 4}{t^2 + 3t + 4} \]

Cross-multiplying gives us:

\[ \lambda(t^2 + 3t + 4) = t^2 - 3t + 4 \]

Rearranging this, we have:

\[ (\lambda - 1)t^2 + (3\lambda + 3)t + (4\lambda - 4) = 0 \]

This is a quadratic equation in \(t\). For \(t\) to be real, the discriminant must be non-negative:

\[ D = b^2 - 4ac \geq 0 \]

Where \(a = \lambda - 1\), \(b = 3\lambda + 3\), and \(c = 4\lambda - 4\).

Calculating the discriminant:

\[ D = (3\lambda + 3)^2 - 4(\lambda - 1)(4\lambda - 4) \]

Expanding this:

\[ D = 9\lambda^2 + 18\lambda + 9 - 4(\lambda - 1)(4\lambda - 4) \]

Calculating \(4(\lambda - 1)(4\lambda - 4)\):

\[ = 4(4\lambda^2 - 4\lambda - 4\lambda + 4) = 16\lambda^2 - 16\lambda \]

So, substituting back:

\[ D = 9\lambda^2 + 18\lambda + 9 - (16\lambda^2 - 16\lambda) \]

Combining like terms:

\[ D = -7\lambda^2 + 34\lambda + 9 \geq 0 \]

### Step 2: Solving the quadratic inequality

To find the values of \(\lambda\), we will find the roots of the equation:

\[ -7\lambda^2 + 34\lambda + 9 = 0 \]

Using the quadratic formula:

\[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-34 \pm \sqrt{34^2 - 4 \cdot (-7) \cdot 9}}{2 \cdot (-7)} \]

Calculating the discriminant:

\[ 34^2 - 4 \cdot (-7) \cdot 9 = 1156 + 252 = 1408 \]

Now substituting back:

\[ \lambda = \frac{34 \pm \sqrt{1408}}{-14} \]

Calculating \(\sqrt{1408} \approx 37.5\):

\[ \lambda_1 \approx \frac{34 + 37.5}{-14}, \quad \lambda_2 \approx \frac{34 - 37.5}{-14} \]

Calculating the roots gives us the range of \(\lambda\).

### Step 3: Analyzing the system of equations

We need to find the determinant of the coefficient matrix \(D\):

\[ D = \begin{vmatrix} 3 & -1 & 4 \\ 1 & 2 & -3 \\ 6 & 5 & \lambda \end{vmatrix} \]

Calculating \(D\):

\[ D = 3(2\lambda + 12) + 1(-18 - 24) + 4(5 - 6) = 6\lambda + 36 - 42 - 4 \]

Simplifying gives:

\[ D = 6\lambda - 10 \]

For the system to have a unique solution, \(D \neq 0\).

### Conclusion

Since \(D\) is a linear function of \(\lambda\) and does not equal zero for the integer values of \(\lambda\) derived from the range, we conclude that for each integer value of \(\lambda\) in the range, there will be exactly one unique solution.

### Final Answer

The number of solutions for a particular value of \(\lambda\) is **1**.