1
IIT-JEE 2004
Subjective
+2
-0
Using Rolle's theorem, prove that there is at least one root
in $$\left( {{{45}^{1/100}},46} \right)$$ of the polynomial
$$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$$.
2
IIT-JEE 2004
Subjective
+4
-0
Prove that for $$x \in \left[ {0,{\pi \over 2}} \right],$$ $$\sin x + 2x \ge {{3x\left( {x + 1} \right)} \over \pi }$$. Explain
the identity if any used in the proof.
3
IIT-JEE 2003
Subjective
+4
-0
Using the relation $$2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0$$ or otherwise,
prove that $$\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]$$
4
IIT-JEE 2003
Subjective
+2
-0
Find a point on the curve $${x^2} + 2{y^2} = 6$$ whose distance from
the line $$x+y=7$$, is minimum.
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