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1

### JEE Advanced 2015 Paper 1 Offline

Numerical
The number of distinct solutions of the equation $${5 \over 4}{\cos ^2}\,2x + {\cos ^4}\,x + {\sin ^4}\,x + {\cos ^6}\,x + {\sin ^6}\,x\, = \,2$$\$ in the interval $$\left[ {0,\,2\pi } \right]$$ is

2

### IIT-JEE 2011 Paper 1 Offline

Numerical
The positive integer value of $$n\, > \,3$$ satisfying the equation $${1 \over {\sin \left( {{\pi \over n}} \right)}} = {1 \over {\sin \left( {{{2\pi } \over n}} \right)}} + {1 \over {\sin \left( {{{3\pi } \over n}} \right)}}$$ is

3

### IIT-JEE 2010 Paper 2 Offline

Numerical
Two parallel chords of a circle of radius 2 are at a distance $$\sqrt 3 + 1$$ apart. If the chords subtend at the center , angles of $${\pi \over k}$$ and $${{2\pi } \over k},$$ where$$k > 0,$$ then the value of $$\left[ k \right]$$ is

[Note :[k] denotes the largest integer less than or equal to k ]

4

### IIT-JEE 2010 Paper 1 Offline

Numerical
The maximum value of the expression $${1 \over {{{\sin }^2}\theta + 3\sin \theta \cos \theta + 5{{\cos }^2}\theta }}$$ is

## Explanation

Let

$$f(\theta ) = {1 \over {{{\sin }^2}\theta + 3\sin \theta \cos \theta + 5{{\cos }^2}\theta }}$$

Again let

$$g(\theta ) = {\sin ^2}\theta + 3\sin \theta \cos \theta + 5{\cos ^2}\theta$$

$$= {{1 - \cos 2\theta } \over 2} + 5\left( {{{1 + \cos 2\theta } \over 2}} \right) + {3 \over 2}\sin 2\theta$$

$$= 3 + 2\cos 2\theta + {3 \over 2}\sin 2\theta$$

$$\therefore$$ $$g{(\theta )_{\min }} = 3 - \sqrt {4 + {9 \over 4}} = 3 - {5 \over 2} = {1 \over 2}$$

$$\therefore$$ $$f(\theta ) = {1 \over {g{{(\theta )}_{\min }}}} = 2$$

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