1

IIT-JEE 1985

Subjective
Let $$f\left( x \right) = {\sin ^3}x + \lambda {\sin ^2}x, - {\pi \over 2} < x < {\pi \over 2}.$$ Find the intervals in which $$\lambda $$ should lie in order that $$f(x)$$ has exactly one minimum and exactly one maximum.

Answer

$$\lambda \in \left( { - {3 \over 2},0} \right) \cup \left( {0,{3 \over 2}} \right)$$
2

IIT-JEE 1985

Subjective
Find all the tangents to the curve
$$y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,$$ that are parallel to the line $$x+2y=0$$.

Answer

$$2x + 4y - \pi = 0$$
$$2x + 4y + 3\pi = 0$$
3

IIT-JEE 1983

Subjective
Find the coordinates of the point on the curve $$y = {x \over {1 + {x^2}}}$$
where the tangent to the curve has the greatest slope.

Answer

$$(0. 0)$$
4

IIT-JEE 1983

Subjective
Show that $$1+x$$ $$In\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}} $$ for all $$x \ge 0$$

Answer

Solve it.

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