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1

### IIT-JEE 1993

Subjective
Find the equation of the normal to the curve
$$y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)$$ at $$x=0$$

$$x+y=1$$
2

### IIT-JEE 1992

Subjective
In this questions there are entries in columns $$I$$ and $$II$$. Each entry in column $$I$$ is related to exactly one entry in column $$II$$. Write the correct letter from column $$II$$ against the entry number in column $$I$$ in your answer book.

Let the functions defined in column $$I$$ have domain $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$

$$\,\,\,\,$$Column $$I$$
(A) $$x + \sin x$$
(B) $$\sec x$$

$$\,\,\,\,$$Column $$II$$
(p) increasing
(q) decreasing
(r) neither increasing nor decreasing

$$\left( A \right) - \left( p \right),\left( B \right) - \left( r \right)$$
3

### IIT-JEE 1992

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

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

or

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

4

### 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 $$f(x)$$.

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

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