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1

### IIT-JEE 1993

Subjective
Let $$f\left( x \right) = \left\{ {\matrix{ { - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2} + 3b + 2} \right)}},} & {0 \le x < 1} \cr {2x - 3} & {1 \le x \le 3} \cr } } \right.$$

Find all possible real values of $$b$$ such that $$f(x)$$ has the smallest value at $$x=1$$.

$$b \in \left( { - 2, - 1} \right) \cup \left( {1,x} \right)$$
2

### 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 $$x=0$$

$$x+y=1$$
3

### 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

$$\left( A \right) - \left( p \right),\left( B \right) - \left( r \right)$$
4

### IIT-JEE 1992

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

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

or

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

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