1
IIT-JEE 2004 Screening
+2
-0.5
If $$f\left( x \right) = {x^a}\log x$$ and $$f\left( 0 \right) = 0,$$ then the value of $$\alpha$$ for which Rolle's theorem can be applied in $$\left[ {0,1} \right]$$ is
A
$$-2$$
B
$$-1$$
C
$$0$$
D
$$1/2$$
2
IIT-JEE 2003 Screening
+2
-0.5
In $$\left[ {0,1} \right]$$ Languages Mean Value theorem is NOT applicable to
A
$$f\left( x \right) = \left\{ {\matrix{ {{1 \over 2} - x} & {x < {1 \over 2}} \cr {{{\left( {{1 \over 2} - x} \right)}^2}} & {x \ge {1 \over 2}} \cr } } \right.$$
B
$$f\left( x \right) = \left\{ {\matrix{ {\sin x,} & {x \ne 0} \cr {1,} & {x = 0} \cr } } \right.$$
C
$$f\left( x \right) = x\left| x \right|$$
D
$$f\left( x \right) = \left| x \right|$$
3
IIT-JEE 2003 Screening
+2
-0.5
Tangent is drawn to ellipse
$${{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)$$.

Then the value of $$\theta$$ such that sum of intercepts on axes made by this tangent is minimum, is

A
$$\pi /3$$
B
$$\pi /6$$
C
$$\pi /8$$
D
$$\pi /4$$
4
IIT-JEE 2002 Screening
+2
-0.5
The length of a longest interval in which the function $$3\,\sin x - 4{\sin ^3}x$$ is increasing, is
A
$${\pi \over 3}$$
B
$${\pi \over 2}$$
C
$${3\pi \over 2}$$
D
$$\pi$$
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