1
IIT-JEE 2001 Screening
+2
-0.5
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
$$\left( {0,\,1} \right]$$
2
IIT-JEE 2000 Screening
+2
-0.5
Consider the following statements in $$S$$ and $$R$$
$$S:$$ $$\,\,\,$$\$ Both $$\sin \,\,x$$ and $$\cos \,\,x$$ are decreasing functions in the interval $$\left( {{\pi \over 2},\pi } \right)$$
$$R:$$$$\,\,\,$$ If a differentiable function decreases in an interval $$(a, b)$$, then its derivative also decreases in $$(a, b)$$.
Which of the following is true ?
A
Both $$S$$ and $$R$$ are wrong
B
Both $$S$$ and $$R$$ are correct, but $$R$$ is not the correct explanation of $$S$$
C
$$S$$ is correct and $$R$$ is the correct explanation for $$S$$
D
$$S$$ is correct and $$R$$ is wrong
3
IIT-JEE 2000 Screening
+2
-0.5
Let $$f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx.}$$ Then $$f$$ decreases in the interval
A
$$\left( { - \infty ,2} \right)$$
B
$$\left( { - 2, - 1} \right)$$
C
$$\left( {1,2} \right)$$
D
$$\left( {2, + \infty } \right)$$
4
IIT-JEE 2000 Screening
+2
-0.5
If the normal to the curve $$y = f\left( x \right)$$ and the point $$(3, 4)$$ makes an angle $${{{3\pi } \over 4}}$$ with the positive $$x$$-axis, then $$f'\left( 3 \right) =$$
A
$$-1$$
B
$$- {3 \over 4}$$
C
$${4 \over 3}$$
D
$$1$$
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