For any positive integer $n$, define $f_{n}:(0, \infty) \rightarrow \mathbb{R}$ as \[ f_{n}(x)=\sum_{j=1}^{n} \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \text { for all } x \in(0, \infty) \text {. } \] (Here, the inverse trigonometric function $\tan ^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.) Then, which of the following statement(s) is (are) TRUE?

(D) For any fixed positive integer $n, \lim _{x \rightarrow \infty} \sec ^{2}\left(f_{n}(x)\right)=1$

(A) $\sum_{j=1}^{5} \tan ^{2}\left(f_{j}(0)\right)=55$

(B) $\sum_{j=1}^{10}\left(1+f_{j}^{\prime}(0)\right) \sec ^{2}\left(f_{j}(0)\right)=10$

(C) For any fixed positive integer $n, \lim _{x \rightarrow \infty} \tan \left(f_{n}(x)\right)=\frac{1}{n}$

JEE Advanced · 2018 · 20 May 2018 · Paper 2 · Inverse Trigonometric Functions

For any positive integer

n

, define

f_{n} : (0, \infty) \to R

f_{n} (x) = j = 1 \sum n tan^{- 1} (\frac{1}{1 + ( x + j ) ( x + j - 1 )}) for all x \in (0, \infty) .

(Here, the inverse trigonometric function

tan^{- 1} x

assumes values in

(- \frac{π}{2}, \frac{π}{2})

.)
Then, which of the following statement(s) is (are) TRUE?

(A) $\sum_{j = 1}^{5} tan^{2} (f_{j} (0)) = 55$
(B) $\sum_{j = 1}^{10} (1 + f_{j}^{'} (0)) sec^{2} (f_{j} (0)) = 10$
(C)For any fixed positive integer $n, lim_{x \to \infty} tan (f_{n} (x)) = \frac{1}{n}$
(D)For any fixed positive integer $n, lim_{x \to \infty} sec^{2} (f_{n} (x)) = 1$ ✓ Correct

Answer: (D)

Using the identity for inverse tangent differences:

f_{n} (x) = j = 1 \sum n [tan^{- 1} (x + j) - tan^{- 1} (x + j - 1)] = tan^{- 1} (x + n) - tan^{- 1} x = tan^{- 1} (\frac{n}{1 + x ( x + n )})

(A) and (B): Zero is not in the domain

(0, \infty)

, so these statements are invalid.

(C):

lim_{x \to \infty} tan (f_{n} (x)) = lim_{x \to \infty} \frac{n}{1 + x ( x + n )} = 0 \neq = \frac{1}{n}

(D):

lim_{x \to \infty} sec^{2} (f_{n} (x)) = lim_{x \to \infty} [1 + tan^{2} (f_{n} (x))] = 1 + 0 = 1

✓

JEE Advanced · 2018 · 20 May 2018 · Paper 2 · One or more correct options