Question

Let \( \alpha, \beta; \, \alpha > \beta \), be the roots of the equation $$ x^2 - \sqrt{2}x - \sqrt{3} = 0. $$ Let \( P_n = \alpha^n - \beta^n, \, n \in \mathbb{N} \). Then $$ \left( 11\sqrt{3} - 10\sqrt{2} \right) P_{10} + \left( 11\sqrt{2} + 10 \right) P_{11} - 11P_{12} $$ is equal to:  

• A. \( 10\sqrt{2}P_9 \)
• B. \( 10\sqrt{3}P_9 \)
• C. \( 11\sqrt{2}P_9 \)
• D. \( 11\sqrt{3}P_9 \)

Answer 1

The Correct Option is B

We are given that α and β are the roots of the quadratic equation:

\(x^2 - \sqrt{2}x - \sqrt{3} = 0\)

Step 1: Find the roots α and β
To find α and β, we use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the given equation, \(a = 1\), \(b = -\sqrt{2}\), and \(c = -\sqrt{3}\). Substituting these values into the quadratic formula:

\[ x = \frac{\sqrt{2} \pm \sqrt{(\sqrt{2})^2 - 4(1)(-\sqrt{3})}}{2(1)} \]

\[ x = \frac{\sqrt{2} \pm \sqrt{2 + 4\sqrt{3}}}{2} \]

Thus, the roots α and β are:

\[ \alpha = \frac{\sqrt{2} + \sqrt{2 + 4\sqrt{3}}}{2}, \quad \beta = \frac{\sqrt{2} - \sqrt{2 + 4\sqrt{3}}}{2} \]

Step 2: Use recurrence relation for \(P_n\)
We are given that \(P_n = \alpha^n - \beta^n\). From the given quadratic equation, we know that:

\[ \alpha + \beta = \sqrt{2}, \quad \alpha\beta = -\sqrt{3} \]

Using this, we can derive a recurrence relation for \(P_n\). The recurrence relation is:

\[ P_n = (\alpha + \beta)P_{n-1} - \alpha\beta P_{n-2} \]

Substituting the values \(\alpha + \beta = \sqrt{2}\) and \(\alpha\beta = -\sqrt{3}\), we get:

\[ P_n = \sqrt{2}P_{n-1} + \sqrt{3}P_{n-2} \]

Step 3: Calculate the required expression
Now, we need to calculate the following expression:

\[ (11\sqrt{3} - 10\sqrt{2})P_{10} + (11\sqrt{2} + 10)P_{11} - 11P_{12} \]

Using the recurrence relation for \(P_n\), we can express each term in terms of \(P_9\):

\[ P_{10} = \sqrt{2}P_9 + \sqrt{3}P_8 \]

\[ P_{11} = \sqrt{2}P_{10} + \sqrt{3}P_9 \]

\[ P_{12} = \sqrt{2}P_{11} + \sqrt{3}P_{10} \]

Substituting these into the original expression, and simplifying, we find that the value of the expression is:

\[ 10\sqrt{3}P_9 \]

Thus, the correct answer is:

\[ 10\sqrt{3}P_9 \]