The circle $${x^2} + {y^2} = 1$$ cuts the $$x$$-axis at $$P$$ and $$Q$$. Another circle with centre at $$Q$$ and variable radius intersects the first circle at $$R$$ above the $$x$$-axis and the line segment $$PQ$$ at $$S$$. Find the maximum area of the triangle $$QSR$$.
Answer
$${{4\sqrt 3 } \over 9}$$ sq. units
2
IIT-JEE 1994
Subjective
The curve $$y = a{x^3} + b{x^2} + cx + 5$$, touches the $$x$$-axis at $$P(-2, 0)$$ and cuts the $$y$$ axis at a point $$Q$$, where its gradient is $$3$$. Find $$a, b, c$$.
Answer
$$a = - {1 \over 2},b = - {3 \over 4},c = 3$$
3
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 $$x=1$$.