For non-negative integers n, let f(n)= _ k=0 ^ n ( k+1 n+2 ) ( k+2 n+2 ) _ k=0 ^ n ^2 ( k+1 n+2 ) Assuming ^ -1 x takes values in [0, ], which of the following options is/are correct?

JEE Advanced 2019 Mathematics question

For non-negative integers , let

f(n)=

Assuming takes values in , which of the following options is/are correct?

Answer: (A),(C),(D)

Let . Using , the numerator is (the cosine sum vanishes over the symmetric angles), and the denominator is , with

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Question ID VK-026900