Exams › JEE Advanced › Maths
The equation 2tan²(theta) - 5sec(theta) = 1 has exactly 7 solutions in the interval [0, n*pi/2]. For the least natural number n satisfying this, find the value of sum(k=1 to n) k / 2^k.
- (1/2¹³)(2¹⁴ - 15)
- 1 - 15/2¹³
- (1/2¹⁵)(2¹⁴ - 14)
- (1/2¹⁴)(2¹⁵ - 15)
Correct answer: (1/2¹³)(2¹⁴ - 15)
Solution
The equation reduces to sec(theta) = 3, giving cos(theta) = 1/3. Each period 2*pi contributes two solutions. Listing all solutions shows exactly 7 fit when n = 13. The finite sum S = sum(k/2^k, k=1 to n) equals 2 - (n+2)/2ⁿ.
Related JEE Advanced Maths questions
- If a, b, and c are in harmonic progression, then e raised to the power of -a, e raised to the power of -b, and e raised to the power of -c will be in which progression?
- If x, y, and z represent the pᵗʰ, qᵗʰ, and rᵗʰ terms of both an arithmetic progression and a geometric progression, what is the value of (xʳ)(yᵖ)(zᵠ)?
- Let ϕ(x) represent a quadratic polynomial. Given that ϕ(1) equals ϕ(−1) and the terms a₁, a₂, a₃ form an arithmetic progression, then the values ϕ(a₁), ϕ(a₂), ϕ(a₃) will be in which sequence?
- Let Sₙ = Σ (k+1)/2 * k². Then Sₙ can take value(s)
- If a, b, and c are positive integers such that b is divisible by a, and they form a geometric sequence, while their arithmetic mean equals b + 2, what is the value of (a² + a - 14)/(a + 1)?
- Let bᵢ > 1 for i = 1, 2,..., 101. Assume that logₑb₁, logₑb₂,..., logₑb₁₀₁ form an arithmetic sequence with a common difference of log₂. Also, let a₁, a₂,..., a₁₀₁ form an arithmetic sequence where a₁ = b₁ and a₅₁ = b₅₁. If t represents the sum b₁ + b₂ +... + b₅₁ and s represents the sum a₁ + a₂ +... + a₅₁, then which of the following is true?
⚔️ Practice JEE Advanced Maths free + battle 1v1 →