1

### JEE Advanced 2016 Paper 2 Offline

The value of

$$\sum\limits_{k = 1}^{13} {{1 \over {\sin \left( {{\pi \over 4} + {{\left( {k - 1} \right)\pi } \over 6}} \right)\sin \left( {{\pi \over 4} + {{k\pi } \over 6}} \right)}}}$$ is equal to
A
$$3 - \sqrt 3$$
B
$$2\left( {3 - \sqrt 3 } \right)$$
C
$$2\left( {\sqrt 3 - 1} \right)\,\,\,$$
D
$$2\left( {2 - \sqrt 3 } \right)$$

## Explanation

It is given that,

$$\sum\limits_{k = 1}^{13} {{1 \over {\sin \left( {{\pi \over 4} + {{(k - 1)\pi } \over 6}} \right)\sin \left( {{\pi \over 4} + {{k\pi } \over 6}} \right)}}}$$

Let $$\alpha = {\pi \over 4}$$ and $$\beta = {\pi \over 6}$$. Therefore,

$$\sum\limits_{k = 1}^{13} {{1 \over {\sin (\alpha + k\beta )sin(\alpha + (k - 1)\beta )}}}$$

$$= {1 \over {\sin \beta }}\sum\limits_{k = 1}^{13} {{{\sin ((\alpha + k\beta ) - (\alpha + (k - 1)\beta ))} \over {\sin (\alpha + k\beta )\sin (\alpha + (k - 1)\beta )}}}$$

$$= {1 \over {\sin \beta }}\sum\limits_{k = 1}^{13} {(\cot (\alpha + (k - 1)\beta ) - \cot (\alpha + k\beta ))}$$

$$= {1 \over {\sin \beta }}\{ [\cot (\alpha ) - \cot (\alpha + \beta )] + [\cot (\alpha + \beta ) - \cot (\alpha + 2\beta )] + ...... + [\cot (\alpha + 12\beta ) - \cot (\alpha + 13\beta )]\}$$

$$= {1 \over {\sin \beta }}(\cot \alpha - \cot (\alpha + 13\beta ))$$

$$= {1 \over {\sin (\pi /6)}}\left( {\cot {\pi \over 4} - \cot \left( {{\pi \over 4} + {{13\pi } \over 6}} \right)} \right)$$

$$= 2(1 - 2 + \sqrt 3 ) = 2(\sqrt 3 - 1)$$

2

### JEE Advanced 2016 Paper 1 Offline

Let $$S = \left\{ {x \in \left( { - \pi ,\pi } \right):x \ne 0, \pm {\pi \over 2}} \right\}.$$ The sum of all distinct solutions of the equation $$\sqrt 3 \,\sec x + \cos ec\,x + 2\left( {\tan x - \cot x} \right) = 0$$ in the set S is equal to
A
$$- {{7\pi } \over 9}$$
B
$$- {{2\pi } \over 9}$$
C
0
D
$${{5\pi } \over 9}$$
3

### JEE Advanced 2014 Paper 2 Offline

For $$x \in \left( {0,\pi } \right),$$ the equation $$\sin x + 2\sin 2x - \sin 3x = 3$$ has
A
infinitely many solutions
B
three solutions
C
one solution
D
no solution
4

### JEE Advanced 2013 Paper 1 Offline

The number of points in $$\left( { - \infty \,\infty } \right),$$ for which $${x^2} - x\sin x - \cos x = 0,$$ is
A
6
B
4
C
2
D
0

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