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1

### IIT-JEE 2005 Screening

If $$P(x)$$ is a polynomial of degree less than or equal to $$2$$ and $$S$$ is the set of all such polynomials so that $$P(0)=0$$, $$P(1)=1$$ and $$P'\left( x \right) > 0\,\,\forall x \in \left[ {0,1} \right],$$ then
A
$$S = \phi$$
B
$$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,2} \right)$$
C
$$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,\infty } \right)$$
D
$$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,1} \right)$$
2

### IIT-JEE 2004 Screening

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
$$1/2$$
3

### IIT-JEE 2004 Screening

If $$f\left( x \right) = {x^3} + b{x^2} + cx + d$$ and $$0 < {b^2} < c,$$ then in $$\left( { - \infty ,\infty } \right)$$
A
$$f\left( x \right)$$ is a strictly increasing function
B
$$f\left( x \right)$$ has a local maxima
C
$$f\left( x \right)$$ is a strictly decreasing function
D
$$f\left( x \right)$$ is bounded
4

### IIT-JEE 2003 Screening

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
$$\pi /4$$

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