Question

If \( \alpha = 3 + 4 + 8 + 9 + 13 + \dots \) up to 40 terms and \( (\tan \beta)^{\frac{\alpha}{1020}} \) is the root of the equation \( x^2 - x - 2 = 0 \), then the value of \( \sin^2 \beta + 3\cos^2 \beta \) is:

• A. 1
• B. \(\frac{2}{3}\)
• C. \(\frac{5}{3}\)
• D. \(\frac{1}{3}\)

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We first determine the sum $α$ of the given series by grouping terms. Then we find the roots of the quadratic equation and use the value of $α$ to find $\tan² β$, which helps evaluate the final trigonometric expression.

Step 2: Key Formula or Approach:
Group the series into 20 pairs: \[ \alpha = (3+4) + (8+9) + (13+14) + \dots = 7 + 17 + 27 + \dots \text{ (to 20 terms)} \] This is an A.P. with $a=7, d=10, n=20$. \[ \alpha = \frac{20}{2}[2(7) + (19)10] = 10[14 + 190] = 2040 \] Thus, the power \(\frac{\alpha}{1020} = \frac{2040}{1020} = 2\).

Step 3: Detailed Explanation:
1. Roots of \(x^2 - x - 2 = 0\) are $(x-2)(x+1)=0 \implies x=2, -1$.
2. Since \((\tan \beta)^2 = x\), and a square must be positive, \(\tan^2 \beta = 2\).
3. We need \(\sin^2 \beta + 3\cos^2 \beta\). Divide numerator and denominator by \(\cos^2 \beta\) (or use identities):
\[ \frac{\sin^2 \beta}{\cos^2 \beta} \cdot \cos^2 \beta + 3\cos^2 \beta = \cos^2 \beta (\tan^2 \beta + 3) \] \[ \text{Using } \cos^2 \beta = \frac{1}{1 + \tan^2 \beta} = \frac{1}{1+2} = \frac{1}{3} \] \[ \text{Value} = \frac{1}{3} (2 + 3) = \frac{5}{3} \]

Step 4: Final Answer:
The value is \(\frac{5}{3}\).