1
AIEEE 2005
+4
-1
Area of the greatest rectangle that can be inscribed in the
ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$
A
$$2ab$$
B
$$ab$$
C
$$\sqrt {ab}$$
D
$${a \over b}$$
2
AIEEE 2005
+4
-1
The normal to the curve
$$x = a\left( {\cos \theta + \theta \sin \theta } \right),y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ at any point
$$\theta\, '$$ is such that
A
it passes through the origin
B
it makes an angle $${\pi \over 2} + \theta$$ with the $$x$$-axis
C
it passes through $$\left( {a{\pi \over 2}, - a} \right)$$
D
it is at a constant distance from the origin
3
AIEEE 2005
+4
-1
A spherical iron ball $$10$$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $$50$$ cm$$^3$$ /min. When the thickness of ice is $$5$$ cm, then the rate at which the thickness of ice decreases is
A
$${1 \over {36\pi }}$$ cm/min
B
$${1 \over {18\pi }}$$ cm/min
C
$${1 \over {54\pi }}$$ cm/min
D
$${5 \over {6\pi }}$$ cm/min
4
AIEEE 2005
+4
-1
If the equation $${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ........... + {a_1}x = 0$$
$${a_1} \ne 0,n \ge 2,$$ has a positive root $$x = \alpha$$, then the equation
$$n{a_n}{x^{n - 1}} + \left( {n - 1} \right){a_{n - 1}}{x^{n - 2}} + ........... + {a_1} = 0$$ has a positive root, which is
A
greater than $$\alpha$$
B
smaller than $$\alpha$$
C
greater than or equal to smaller than $$\alpha$$
D
equal to smaller than $$\alpha$$
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