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1

IIT-JEE 1998

Subjective
Suppose $$f(x)$$ is a function satisfying the following conditions
(a) $$f(0)=2,f(1)=1$$,
(b) $$f$$has a minimum value at $$x=5/2$$, and
(c) for all $$x$$, $$f'\left( x \right) = \matrix{ {2ax} & {2ax - 1} & {2ax + b + 1} \cr b & {b + 1} & { - 1} \cr {2\left( {ax + b} \right)} & {2ax + 2b + 1} & {2ax + b} \cr }$$\$
where $$a,b$$ are some constants. Determine the constants $$a, b$$ and the function $$f(x)$$.

$$a = {1 \over 4},\,\,b = - {5 \over 4}$$
$$f\left( x \right) = {1 \over 4}{x^2} - {5 \over 4}x + 2$$
2

IIT-JEE 1998

Subjective
A curve $$C$$ has the property that if the tangent drawn at any point $$P$$ on $$C$$ meets the co-ordinate axes at $$A$$ and $$B$$, then $$P$$ is the mid-point of $$AB$$. The curve passes through the point $$(1, 1)$$. Determine the equation of the curve.

$$xy=1$$
3

IIT-JEE 1997

Subjective
Let $$a+b=4$$, where $$a<2,$$ and let $$g(x)$$ be a differentiable function.

If $${{dg} \over {dx}} > 0$$ for all $$x$$, prove that $$\int_0^a {g\left( x \right)dx + \int_0^b {g\left( x \right)dx} }$$
increases as $$(b-a)$$ increases.

Solve it.
4

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

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