1

IIT-JEE 1996

Subjective
Determine the points of maxima and minima of the function
$$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$$ where $$b \ge 0$$ is a constant.

min at $$x = {1 \over 4}\left( {b + \sqrt {{b^2} - 1} } \right)$$
max at $$x = {1 \over 4}\left( {b - \sqrt {{b^2} - 1} } \right)$$
2

IIT-JEE 1996

Subjective
A curve $$y=f(x)$$ passes through the point $$P(1, 1)$$. The normal to the curve at $$P$$ is $$a(y-1)+(x-1)=0$$. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the $$y$$-axis, the curve and the normal to the curve at $$P$$.

$$y = {e^{a\left( {x - 1} \right)}}$$
Area $$=$$ $$1$$ sq. unit.
3

IIT-JEE 1995

Subjective
Let $$(h, k)$$ be a fixed point, where $$h > 0,k > 0.$$. A straight line passing through this point cuts the possitive direction of the coordinate axes at the points $$P$$ and $$Q$$. Find the minimum area of the triangle $$OPQ$$, $$O$$ being the origin.

$$2$$ $$kh$$
4

IIT-JEE 1994

Subjective
The circle $${x^2} + {y^2} = 1$$ cuts the $$x$$-axis at $$P$$ and $$Q$$. Another circle with centre at $$Q$$ and variable radius intersects the first circle at $$R$$ above the $$x$$-axis and the line segment $$PQ$$ at $$S$$. Find the maximum area of the triangle $$QSR$$.

$${{4\sqrt 3 } \over 9}$$ sq. units

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