1
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
For non-negative integers n, let

$$f(n) = {{\sum\limits_{k = 0}^n {\sin \left( {{{k + 1} \over {n + 2}}\pi } \right)} \sin \left( {{{k + 2} \over {n + 2}}\pi } \right)} \over {\sum\limits_{k = 0}^n {{{\sin }^2}\left( {{{k + 1} \over {n + 2}}\pi } \right)} }}$$

Assuming cos$$-1$$ x takes values in [0, $$\pi$$], which of the following options is/are correct?
A
If $$\alpha$$ = tan(cos$$-$$1 f(6)), then $$\alpha$$2 + 2$$\alpha$$ $$-$$1 = 0
B
$$f(4) = {{\sqrt 3 } \over 2}$$
C
sin(7 cos$$-$$1 f(5)) = 0
D
$$\mathop {\lim }\limits_{n \to \infty } \,f(n) = {1 \over 2}$$
2
JEE Advanced 2018 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
In a $$\Delta$$PQR = 30$$^\circ$$ and the sides PQ and QR have lengths 10$$\sqrt 3$$ and 10, respectively. Then, which of the following statement(s) is(are) TRUE?
A
$$\angle QPR = 45^\circ$$
B
The area of the $$\Delta PQR$$ is $$25\sqrt 3$$ and $$\angle QRP = 120^\circ$$
C
The radius of the incircle of the $$\Delta PQR$$ is $$10\sqrt 3$$ $$-$$ 15
D
The area of the circumcircle of the $$\Delta PQR$$ is 100$$\pi$$
3
JEE Advanced 2017 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
Let $$\alpha$$ and $$\beta$$ be non zero real numbers such that $$2(\cos \beta - \cos \alpha ) + \cos \alpha \cos \beta = 1$$. Then which of the following is/are true?
A
$$\sqrt 3 \tan \left( {{\alpha \over 2}} \right) - \tan \left( {{\beta \over 2}} \right) = 2$$
B
$$\tan \left( {{\alpha \over 2}} \right) - \sqrt 3 \tan \left( {{\beta \over 2}} \right) = 0$$
C
$$\tan \left( {{\alpha \over 2}} \right) + \sqrt 3 \tan \left( {{\beta \over 2}} \right) = 0$$
D
$$\sqrt 3 \tan \left( {{\alpha \over 2}} \right) + \tan \left( {{\beta \over 2}} \right) = 2$$
4
JEE Advanced 2013 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$f\left( x \right) = x\sin \,\pi x,\,x > 0.$$ Then for all natural numbers $$n,\,f'\left( x \right)$$ vanishes at
A
A unique point in the interval $$\left( {n,\,n + {1 \over 2}} \right)$$
B
A unique point in the interval $$\left( {n + {1 \over 2},n + 1} \right)$$
C
A unique point in the interval $$\left( {n,\,n + 1} \right)$$
D
Two points in the interval $$\left( {n,\,n + 1} \right)$$
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