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1

### IIT-JEE 1993

Fill in the Blanks
If $$K = \sin \left( {\pi /18} \right)\sin \left( {5\pi /18} \right)\sin \left( {7\pi /18} \right),$$ then the numerical value of K is ______.

$${\raise0.5ex\hbox{\scriptstyle 1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle 8}}$$
2

### IIT-JEE 1991

Fill in the Blanks
The value of
$$\sin {\pi \over {14}}\sin {{3\pi } \over {14}}\sin {{5\pi } \over {14}}\sin {{7\pi } \over {14}}\sin {{9\pi } \over {14}}\sin {{11\pi } \over {14}}\sin {{13\pi } \over {14}}$$ is equal to ______.

$${\raise0.5ex\hbox{\scriptstyle 1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle {64}}}$$
3

### IIT-JEE 1987

Fill in the Blanks
The set of all $$x$$ in the interval $$\left[ {0,\,\pi } \right]$$ for which $$2\,{\sin ^2}x - 3$$ $$\sin x + 1 \ge 0,$$ is _____.

$$\,\left[ {0,{\pi \over 6}} \right]U\left\{ {{\pi \over 2}} \right\}U\left[ {{{s\pi } \over 6},\pi } \right]$$
4

### IIT-JEE 1987

Fill in the Blanks
The sides of a triangle inscribed in a given circle subtend angles $$\alpha$$, $$\beta$$ and $$\gamma$$ at the centre. The minimum value of the arithmetic mean of $$cos\left[ {\alpha + {\pi \over 2}} \right],\,\cos \left[ {\beta + {\pi \over 2}} \right]$$ and $$cos\left[ {\gamma + {\pi \over 2}} \right]$$ is equal to _______.

$${{ - \sqrt 3 } \over 2}$$

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Class 12