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1

IIT-JEE 1992

Subjective
What normal to the curve $$y = {x^2}$$ forms the shortest chord?

Answer

$$x + \sqrt 2 \,y = \sqrt 2 $$

or

$$x - \sqrt {2\,} y = - \sqrt 2 $$

2

IIT-JEE 1992

Subjective
A cubic $$f(x)$$ vanishes at $$x=2$$ and has relative minimum / maximum at $$x=-1$$ and $$x = {1 \over 3}$$ if $$\int\limits_{ - 1}^1 {f\,\,dx = {{14} \over 3}} $$, find the cubic $$f(x)$$.

Answer

$${x^3} + {x^2} - x + 2$$
3

IIT-JEE 1991

Subjective
A window of perimeter $$P$$ (including the base of the arch) is in the form of a rectangle surmounded by a semi circle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass transmits three times as such light per square meter as the coloured glass does.

What is the ratio for the sides of the rectangle so that the window transmits the maximum light ?

Answer

$${{6 + \pi } \over 6}$$
4

IIT-JEE 1990

Subjective
A point $$P$$ is given on the circumference of a circle of radius $$r$$. Chord $$QR$$ is parallel to the tangent at $$P$$. Determine the maximum possible area of the triangle $$PQR$$.

Answer

$${{3\sqrt 3 } \over 4}\,\,{r^2}$$

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