Question

Let \( \alpha, \beta \in \mathbb{N} \) be roots of the equation \( x^2 - 70x + \lambda = 0 \), where \( \frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{N} \). If \( \lambda \) assumes the minimum possible value, then \[ \frac{\left( \sqrt{\alpha - 1} + \sqrt{\beta - 1} \right)(\lambda + 35)}{|\alpha - \beta|} \] is equal to____.

Answer 1

Correct Answer: 60

The problem involves finding the minimum possible value of a parameter \( \lambda \) in a quadratic equation whose roots, \( \alpha \) and \( \beta \), are natural numbers, subject to certain divisibility conditions on \( \lambda \). Once these values are found, we need to evaluate a given expression.

Concept Used:

The solution relies on Vieta's formulas for the roots of a quadratic equation and basic number theory principles (divisibility rules).

For a quadratic equation of the form \( ax^2 + bx + c = 0 \) with roots \( \alpha \) and \( \beta \), Vieta's formulas state:

• Sum of roots: \( \alpha + \beta = -\frac{b}{a} \)
• Product of roots: \( \alpha \beta = \frac{c}{a} \)

We will use these relations to connect the roots \( \alpha, \beta \) to the parameter \( \lambda \) and then apply the given constraints to find the minimum possible value for \( \lambda \).

Step-by-Step Solution:

Step 1: Apply Vieta's formulas to the given equation.

The equation is \( x^2 - 70x + \lambda = 0 \). The roots are \( \alpha \) and \( \beta \), which are natural numbers (\( \mathbb{N} \)).

From Vieta's formulas, we have:

\[ \alpha + \beta = -(-70) = 70 \] \[ \alpha \beta = \lambda \]

Step 2: Find the minimum possible value for \( \lambda \) satisfying the given conditions.

The problem states that \( \lambda \) assumes the minimum possible value under the conditions that \( \frac{\lambda}{2} \notin \mathbb{N} \) and \( \frac{\lambda}{3} \notin \mathbb{N} \). This means \( \lambda \) must not be divisible by 2 (i.e., it must be odd) and must not be divisible by 3.

Since \( \alpha, \beta \in \mathbb{N} \) and \( \alpha + \beta = 70 \), we can express \( \lambda \) as a function of one root, say \( \alpha \):

\[ \lambda = \alpha \beta = \alpha(70 - \alpha) \]

To find the minimum value of \( \lambda = 70\alpha - \alpha^2 \), we should test values of \( \alpha \) starting from \( \alpha = 1 \).

• If \( \alpha = 1 \), then \( \beta = 69 \), and \( \lambda = 1 \times 69 = 69 \). Here, \( \frac{69}{3} = 23 \in \mathbb{N} \), so this value is not allowed.
• If \( \alpha = 2 \), then \( \beta = 68 \), and \( \lambda = 2 \times 68 = 136 \). Here, \( \frac{136}{2} = 68 \in \mathbb{N} \), so this value is not allowed.
• If \( \alpha = 3 \), then \( \beta = 67 \), and \( \lambda = 3 \times 67 = 201 \). Here, \( \frac{201}{3} = 67 \in \mathbb{N} \), so this value is not allowed.
• If \( \alpha = 4 \), then \( \beta = 66 \), and \( \lambda = 4 \times 66 = 264 \). This is divisible by 2. Not allowed.
• If \( \alpha = 5 \), then \( \beta = 65 \), and \( \lambda = 5 \times 65 = 325 \).
  • Check divisibility by 2: 325 is odd, so \( \frac{325}{2} \notin \mathbb{N} \). This condition is satisfied.
  • Check divisibility by 3: The sum of the digits is \( 3+2+5=10 \), which is not divisible by 3. So, \( \frac{325}{3} \notin \mathbb{N} \). This condition is also satisfied.

Since we are checking values of \( \alpha \) in increasing order, the first value of \( \lambda \) that satisfies all conditions will be the minimum possible value. Therefore, the minimum value of \( \lambda \) is 325, with corresponding roots \( \alpha = 5 \) and \( \beta = 65 \).

Step 3: Evaluate the given expression.

The expression to evaluate is:

\[ \frac{\left( \sqrt{\alpha - 1} + \sqrt{\beta - 1} \right)(\lambda + 35)}{|\alpha - \beta|} \]

We have \( \alpha = 5 \), \( \beta = 65 \), and \( \lambda = 325 \). Let's calculate each part of the expression:

• Numerator Part 1: \( \sqrt{\alpha - 1} + \sqrt{\beta - 1} = \sqrt{5 - 1} + \sqrt{65 - 1} = \sqrt{4} + \sqrt{64} = 2 + 8 = 10 \).
• Numerator Part 2: \( \lambda + 35 = 325 + 35 = 360 \).
• Denominator: \( |\alpha - \beta| = |5 - 65| = |-60| = 60 \).

Final Computation & Result:

Now, substitute these computed values back into the expression:

\[ \frac{(10)(360)}{60} \] \[ = 10 \times \frac{360}{60} = 10 \times 6 = 60 \]

The value of the expression is 60.

Answer 2

Analyze \( \alpha \) and \( \beta \):

Since \( \alpha + \beta = 70 \), assume possible integer values for \( \alpha \) and \( \beta \) that satisfy \( \alpha \beta = \lambda \) and check divisibility conditions. A suitable pair is \( \alpha = 5 \) and \( \beta = 65 \), giving \( \lambda = 5 \times 65 = 325 \).

Verification:

Check that \( \frac{325}{2} \notin \mathbb{N} \) and \( \frac{325}{3} \notin \mathbb{N} \), satisfying the divisibility conditions.

Calculate the Required Expression:

Substitute \( \alpha = 5 \), \( \beta = 65 \), and \( \lambda = 325 \):

\[ \frac{\sqrt{\alpha - 1} + \sqrt{\beta - 1}(\lambda + 35)}{|\alpha - \beta|} = \frac{\sqrt{5 - 1} + \sqrt{65 - 1} \times (325 + 35)}{|5 - 65|} \]

Simplifying this gives:

\[ = \frac{\sqrt{4 + 8 \times 360}}{60} = \frac{\sqrt{12 \times 360}}{60} = 60 \]