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Trigonometric Functions and Equations question

2019 · Shift 2 · Q19
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  2. /JEE Advanced
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  5. /2019 · Shift 2 · Q19

Trigonometric Functions and Equations question

2019 · Shift 2 · Q19

JEE AdvancedMathematicsTrigonometric Functions and EquationsMultiple correct+4 / −1
For non-negative integers n, let f(n)=∑k=0nsin⁡(k+1n+2π)sin⁡(k+2n+2π)∑k=0nsin⁡2(k+1n+2π)f(n) = {{\sum\limits_{k = 0}^n {\sin \left( {{{k + 1} \over {n + 2}}\pi } \right)} \sin \left( {{{k + 2} \over {n + 2}}\pi } \right)} \over {\sum\limits_{k = 0}^n {{{\sin }^2}\left( {{{k + 1} \over {n + 2}}\pi } \right)} }}f(n)=k=0∑n​sin2(n+2k+1​π)k=0∑n​sin(n+2k+1​π)sin(n+2k+2​π)​ Assuming cos −1-1−1 x takes values in [0, π\piπ], which of the following options is/are correct?
  1. A
    If α\alphaα = tan(cos −-− 1 f(6)), then α\alphaα 2 + 2 α−\alpha -α− 1 = 0
  2. B
    f(4)=32f(4) = {{\sqrt 3 } \over 2}f(4)=23​​
  3. C
    sin(7 cos −-− 1 f(5)) = 0
  4. D
    lim⁡n→∞ f(n)=12\mathop {\lim }\limits_{n \to \infty } \,f(n) = {1 \over 2}n→∞lim​f(n)=21​
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