1
IIT-JEE 2005
Subjective
+4
-0
If $$p(x)$$ be a polynomial of degree $$3$$ satisfying $$p(-1)=10, p(1)=-6$$ and $$p(x)$$ has maxima at $$x=-1$$ and $$p'(x)$$ has minima at $$x=1$$. Find the distance between the local maxima and local minima of the curve.
2
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
$$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$$.
3
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.
4
IIT-JEE 2003
Subjective
+4
-0
Using the relation $$2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0$$ or otherwise,
prove that $$\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]$$
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