Question

Let \(\alpha = 3 + 4 + 8 + 9 + 13 + 14 + \dots\) up to 40 terms. If \((\tan\beta)^{\pi/1020}\) is a root of the equation \(x^2 + x - 2 = 0\), \(\beta \in (0, \pi/2)\), then \(\sin^2\beta + 3\cos^2\beta\) is equal to:

• A. \(2 \)
• B. \(\frac{7}{4} \)
• C. \(\frac{5}{2} \)
• D. \(\frac{3}{2} \)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We must calculate the sum of the given sequence \( \alpha \). The sequence can be decomposed into two distinct arithmetic progressions. After finding \( \alpha \), we use it to determine the exact value of \( \beta \) by solving the given quadratic equation.
Step 2: Key Formula or Approach:
The sum of an Arithmetic Progression (A.P.) is \( S_n = \frac{n}{2} [2a + (n-1)d] \).
After finding the sum, factor the quadratic equation \( x^2 + x - 2 = 0 \) to identify the correct root for the tangent function.
Step 3: Detailed Explanation:
The series is \( \alpha = 3 + 4 + 8 + 9 + 13 + 14 + \dots \) up to 40 terms.
Let's group the terms into two separate series, picking alternating terms. Each series will have 20 terms:
Series 1: \( 3 + 8 + 13 + \dots \) (20 terms)
Series 2: \( 4 + 9 + 14 + \dots \) (20 terms)
For Series 1: First term \( a_1 = 3 \), common difference \( d_1 = 5 \).
\( S_1 = \frac{20}{2} [2(3) + (20-1)5] = 10 [6 + 95] = 10 \times 101 = 1010 \).
For Series 2: First term \( a_2 = 4 \), common difference \( d_2 = 5 \).
\( S_2 = \frac{20}{2} [2(4) + (20-1)5] = 10 [8 + 95] = 10 \times 103 = 1030 \).
Total sum \( \alpha = S_1 + S_2 = 1010 + 1030 = 2040 \).
Now calculate the exponent given in the problem:
\( \frac{\alpha}{1020} = \frac{2040}{1020} = 2 \).
Thus, the root of the equation is \( (\tan\beta)^2 = \tan^2\beta \).
Now solve the quadratic equation \( x^2 + x - 2 = 0 \):
\( (x+2)(x-1) = 0 \implies x = -2 \) or \( x = 1 \).
Since \( \tan^2\beta \ge 0 \) for all real \( \beta \), we discard \( x = -2 \).
So, \( \tan^2\beta = 1 \).
Given \( \beta \in (0, \pi/2) \), \( \tan\beta \) must be positive.
\( \tan\beta = 1 \implies \beta = \pi/4 \).
Finally, evaluate the required expression:
\( \sin^2\beta + 3\cos^2\beta = \sin^2(\pi/4) + 3\cos^2(\pi/4) \).
\( = \left(\frac{1}{\sqrt{2}}\right)^2 + 3\left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} + 3\left(\frac{1}{2}\right) = \frac{1+3}{2} = \frac{4}{2} = 2 \).
Step 4: Final Answer:
The value of the expression is 2.