1

IIT-JEE 1983

Subjective
The ex-radii $${r_1},{r_2},{r_3}$$ of $$\Delta$$$$ABC$$ are H.P. Show that its sides $$a, b, c$$ are in A.P.

Solve it.
2

IIT-JEE 1982

Subjective
A vertical pole stands at a point $$Q$$ on a horizontal ground. $$A$$ and $$B$$ are points on the ground, $$d$$ meters apart. The pole subtends angles $$\alpha$$ and $$\beta$$ at $$A$$ and $$B$$ respectively. $$AB$$ subtends an angle $$\gamma$$ and $$Q$$. Find the height of the pole.

$${d \over {\sqrt {{{\cot }^2}\alpha + {{\cot }^2}\beta - \cot \alpha \cot \beta \cot \gamma } }}$$
3

IIT-JEE 1981

Subjective
Let the angles $$A, B, C$$ of a triangle $$ABC$$ be in A.P. and let $$b:c = \sqrt 3 :\sqrt 2$$. Find the angle $$A$$.

$${75^ \circ }$$
4

IIT-JEE 1980

Subjective
(i) $$PQ$$ is a vertical tower. $$P$$ is the foot and $$Q$$ is the top of the tower. $$A, B, C$$ are three points in the horizontal plane through $$P$$. The angles of elevation of $$Q$$ from $$A$$, $$B$$, $$C$$ are equal, and each is equal to $$\theta$$. The sides of the triangle $$ABC$$ are $$a, b, c$$; and the area of the triangle $$ABC$$ is $$\Delta$$. Show that the height of the tower is $${{abc\tan \theta } \over {4\Delta }}$$.

(ii) $$AB$$ is vertical pole. The end $$A$$ is on the level ground. $$C$$ is the middle point of $$AB$$. $$P$$ is a point on the level ground. The portion $$CB$$ subtends an angle $$\beta$$ at $$P$$. If $$AP = n\,AB,$$ then show that tan$$\beta$$ $$= {n \over {2{n^2} + 1}}$$

(i) Solve it.

(ii) Solve it.

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