1

### IIT-JEE 1999

Subjective
Let $$ABC$$ be a triangle having $$O$$ and $$I$$ as its circumcenter and in centre respectively. If $$R$$ and $$r$$ are the circumradius and the inradius, respectively, then prove that $${\left( {IO} \right)^2} = {R^2} - 2{\mathop{\rm Rr}\nolimits}$$. Further show that the triangle BIO is a right-angled triangle if and only if $$b$$ is arithmetic mean of $$a$$ and $$c$$.

Solve it.
2

### 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.
3

### 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.)

$${3 \over 5}$$
4

### IIT-JEE 1994

Subjective
A tower $$AB$$ leans towards west making an angle $$\alpha$$ with the vertical. The angular elevation of $$B$$, the topmost point of the tower is $$\beta$$ as observed from a point $$C$$ due west of $$A$$ at a distance $$d$$ from $$A$$. If the angular elevation of $$B$$ from a point $$D$$ due east of $$C$$ at a distance $$2d$$ from $$C$$ is $$\gamma$$, then prove that $$2$$ tan $$\alpha = - \cot \beta + \cot \gamma$$.

Solve it.

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