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1

### IIT-JEE 1998

Subjective
Prove that a triangle $$ABC$$ is equilateral if and only if $$\tan A + \tan B + \tan C = 3\sqrt 3$$.

Solve it.
2

### IIT-JEE 1998

Subjective
A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose $${60^ \circ }$$ and $${30^ \circ }$$ are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is at the points $$P$$ and $$Q$$ respectively on its path. Let $$\theta$$ be the angle of elevation of the bird when it is a point on the are of the circle exactly midway between $$P$$ and $$Q$$. Find the numerical value of $${\tan ^2}\theta$$. (Assume that the observer is not inside the vertical projection of the path of the bird.)

$${3 \over 5}$$
3

### IIT-JEE 1994

Subjective
Consider the following statements connecting a triangle $$ABC$$

(i) The sides $$a, b, c$$ and area $$\Delta$$ are rational.

(ii) $$a,\tan {B \over 2},\tan {c \over 2}$$ are rational.

(iii) $$a,\sin A,\sin B,\sin C$$ are rational.
Prove that $$\left( i \right) \Rightarrow \left( {ii} \right) \Rightarrow \left( {iii} \right) \Rightarrow \left( i \right)$$

Solve it.
4

### IIT-JEE 1994

Subjective
Let $${A_1},{A_2},........,{A_n}$$ be the vertices of an $$n$$-sided regular polygon such that $${1 \over {{A_1}{A_2}}} = {1 \over {{A_1}{A_3}}} + {1 \over {{A_1}{A_4}}}$$, Find the value of $$n$$.

$$7$$

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