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Sequences and Series question

2021 · Shift 1 · Q34
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  5. /2021 · Shift 1 · Q34

Sequences and Series question

2021 · Shift 1 · Q34

JEE AdvancedMathematicsSequences and SeriesMultiple correct+4 / −2
For any positive integer n, let Sn : (0, ∞\infty∞) →\to→ R be defined by Sn(x)=∑olimitsk=1ncot⁡−1(1+k(k+1)x2x){S_n}(x) = \sum olimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)}Sn​(x)=∑olimitsk=1n​cot−1(x1+k(k+1)x2​), where for any x ∈\in∈ R, cot⁡−1(x)∈(0,π){\cot ^{ - 1}}(x) \in (0,\pi )cot−1(x)∈(0,π) and tan⁡−1(x)∈(−π2,π2){\tan ^{ - 1}}(x) \in \left( { - {\pi \over 2},{\pi \over 2}} \right)tan−1(x)∈(−2π​,2π​). Then which of the following statements is (are) TRUE?
  1. A
    S10(x)=π2−tan⁡−1(1+11x210x){S_{10}}(x) = {\pi \over 2} - {\tan ^{ - 1}}\left( {{{1 + 11{x^2}} \over {10x}}} \right)S10​(x)=2π​−tan−1(10x1+11x2​), for all x > 0
  2. B
    lim⁡n→∞cot⁡(Sn(x))=x\mathop {\lim }\limits_{n \to \infty } \cot ({S_n}(x)) = xn→∞lim​cot(Sn​(x))=x, for all x > 0
  3. C
    The equation S3(x)=π4{S_3}(x) = {\pi \over 4}S3​(x)=4π​ has a root in (0, ∞\infty∞)
  4. D
    tan(Sn(x))≤12tan({S_n}(x)) \le {1 \over 2}tan(Sn​(x))≤21​, for all n ≥\ge≥ 1 and x > 0
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