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1

IIT-JEE 1993

Subjective
Tagent at a point $${P_1}$$ {other than $$(0, 0)$$} on the curve $$y = {x^3}$$ meets the curve again at $${P_2}$$. The tangent at $${P_2}$$ meets the curve at $${P_3}$$, and so on. Show that the abscissae of $${P_1},\,{P_2},{P_3}......{P_n},$$ form a G.P. Also find the ratio.

[area $$\left( {\Delta {P_1},{P_2},{P_3}} \right)$$]/[area $$\left( {{P_2},{P_3},{P_4}} \right)$$]

Answer

$${1 \over {64}}\,sq.\,unit$$
2

IIT-JEE 1993

Subjective
A line through $$A (-5, -4)$$ meets the line $$x + 3y + 2 = 0,$$ $$2x + y + 4 = 0$$ and $$x - y - 5 = 0$$ at the points $$B, C$$ and $$D$$ respectively. If $${\left( {15/AB} \right)^2} + {\left( {10/AC} \right)^2} = {\left( {6/AD} \right)^2},$$ find the equation of the line.

Answer

$$2x + 3y + 22 = 0$$
3

IIT-JEE 1992

Subjective
Determine all values of $$\alpha $$ for which the point $$\left( {\alpha ,\,{\alpha ^2}} \right)$$ lies insides the triangle formed by the lines $$$\matrix{ {2x + 3y - 1 = 0} \cr {x + 2y - 3 = 0} \cr {5x - 6y - 1 = 0} \cr } $$$

Answer

$$\alpha \in \left( { - {3 \over 2}, - 1} \right) \cup \left( {{1 \over 2},1} \right)$$
4

IIT-JEE 1991

Subjective
Find the equation of the line passing through the point $$(2, 3)$$ and making intercept of length 2 units between the lines $$y + 2x = 3$$ and $$y + 2x = 5$$.

Answer

$$3x + 4y - 18 = 0$$ or $$x - 2 = 0$$

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