1
IIT-JEE 2001
Subjective
+4
-0
Let $$P$$ be a point on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,0 < b < a$$. Let the line parallel to $$y$$-axis passing through $$P$$ meet the circle $${x^2} + {y^2} = {a^2}$$ at the point $$Q$$ such that $$P$$ and $$Q$$ are on the same side of $$x$$-axis. For two positive real numbers $$r$$ and $$s$$, find the locus of the point $$R$$ on $$PQ$$ such that $$PR$$ : $$RQ = r: s$$ as $$P$$ varies over the ellipse.
2
IIT-JEE 2001
Subjective
+6
-0
If $$\Delta$$ is the area of a triangle with side lengths $$a, b, c,$$ then show that $$\Delta \le {1 \over 4}\sqrt {\left( {a + b + c} \right)abc}$$. Also show that the equality occurs in the above inequality if and only if $$a=b=c$$.
3
IIT-JEE 2001
Subjective
+5
-0
Let $$- 1 \le p \le 1$$. Show that the equation $$4{x^3} - 3x - p = 0$$
has a unique root in the interval $$\left[ {1/2,\,1} \right]$$ and identify it.
4
IIT-JEE 2001
Subjective
+5
-0
Evaluate $$\int {{{\sin }^{ - 1}}\left( {{{2x + 2} \over {\sqrt {4{x^2} + 8x + 13} }}} \right)} \,dx.$$
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