1
JEE Advanced 2018 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
For any positive integer n, define

$${f_n}:(0,\infty ) \to R$$ as

$${f_n} = \sum\limits_{j = 1}^n {{{\tan }^{ - 1}}} \left( {{1 \over {1 + (x + j)(x + j - 1)}}} \right)$$

for all x$$\in$$(0, $$\infty$$). (Here, the inverse trigonometric function tan$$-$$1 x assumes values in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$). Then, which of the following statement(s) is (are) TRUE?
A
$$\sum\limits_{j = 1}^5 {{{\tan }^2}({f_j}(0)) = 55}$$
B
$$\sum\limits_{j = 1}^{10} {(1 + f{'_j}(0)){{\sec }^2}({f_j}(0)) = 10}$$
C
For any fixed positive integer n, $$\mathop {\lim }\limits_{x \to \infty } \tan ({f_n}(x)) = {1 \over n}$$
D
For any fixed positive integer n, $$\mathop {\lim }\limits_{x \to \infty } {\sec ^2}({f_n}(x)) = 1$$
2
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
If $$\alpha$$ $$= 3{\sin ^{ - 1}}\left( {{6 \over {11}}} \right)$$ and $$\beta = 3{\cos ^{ - 1}}\left( {{4 \over 9}} \right),$$ where the inverse trigonimetric functions take only the principal values, then the correct options(s) is (are)
A
$$cos\beta > 0$$
B
$$\sin \beta < 0$$
C
$$\cos \left( {\alpha + \beta } \right) > 0$$
D
$$\cos \alpha < 0$$
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