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1

### IIT-JEE 2003 Screening

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
$$f\left( x \right) = \left| x \right|$$
2

### IIT-JEE 2002 Screening

The point(s) in the curve $${y^3} + 3{x^2} = 12y$$ where the tangent is vertical, is (are)
A
$$\left( { \pm {4 \over {\sqrt 3 }}, - 2} \right)$$
B
$$\left( { \pm \sqrt {{{11} \over 3}} ,1} \right)$$
C
$$(0,0)$$
D
$$\left( { \pm {4 \over {\sqrt 3 }}, 2} \right)$$
3

### IIT-JEE 2002 Screening

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
$$\pi$$
4

### IIT-JEE 2001 Screening

Let $$f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1$$ and let $$m(b)$$ be the minimum value of $$f(x)$$. As $$b$$ varies, the range of $$m(b)$$ is
A
$$\left[ {0,1} \right]$$
B
$$\left( {0,\,1/2} \right]$$
C
$$\left[ {1/2,\,1} \right]$$
D
$$\left( {0,\,1} \right]$$

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