JEE Mains · Maths · STD 11 - 8. sequence and series
Let \(\alpha = 3+4+8+9+13+14+\ldots\) upto 40 terms. If \((\tan\beta)^{\frac{\alpha}{1020}}\) is a root of the equation \(x^2+x-2=0\), \(\beta \in \left(0, \dfrac{\pi}{2}\right)\), then \(\sin^2\beta + 3\cos^2\beta\) is equal to:
- A \(2\)
- B \(\dfrac{7}{4}\)
- C \(\dfrac{5}{2}\)
- D \(\dfrac{3}{2}\)
Answer & Solution
Correct Answer
(A) \(2\)
Step-by-step Solution
Detailed explanation
The given series is \(\alpha = 3+4+8+9+13+14+\ldots\) up to \(40\) terms. We can split the series into two arithmetic progressions, each containing \(20\) terms: \(S_1 = 3 + 8 + 13 + \ldots\) up to \(20\) terms \(S_2 = 4 + 9 + 14 + \ldots\) up to \(20\) terms Using the sum…
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