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1

WB JEE 2009

MCQ (Single Correct Answer)

Angle between y2 = x and x2 = y at the origin is

A
$$2{\tan ^{ - 1}}\left( {{3 \over 4}} \right)$$
B
$${\tan ^{ - 1}}\left( {{4 \over 3}} \right)$$
C
$$\pi$$/2
D
$$\pi$$/4

Explanation

Tangent at (0, 0) on the curve y2 = x is y-axis while tangent at (0, 0) on the curve x2 = y is x-axis, so from figure angle between x-axis and y-axis is 90$$^\circ$$.

2

WB JEE 2009

MCQ (Single Correct Answer)

If the rate of increase of the radius of a circle is 5 cm/sec., then the rate of increase of its area, when the radius is 20 cm, will be

A
10$$\pi$$
B
20$$\pi$$
C
200$$\pi$$
D
400$$\pi$$

Explanation

Let radius r $$\therefore$$ $${{dr} \over {dt}} = 5$$ cm/sec (given)

Area of circle $$(A) = \pi {r^2}$$

$$\therefore$$ $${{dA} \over {dt}} = {{d(\pi {r^2})} \over {dt}} = \pi (2r){{dr} \over {dt}}$$

when r = 20 then $${{dA} \over {dt}} = \pi \,.\,2\,.\,20\,.\,5 = 200\pi $$.

3

WB JEE 2009

MCQ (Single Correct Answer)

The distance covered by a particle in t seconds is given by x = 3 + 8t $$-$$ 4t2. After 1 second its velocity will be

A
0 unit/second
B
3 units/second
C
4 units/second
D
7 units/second

Explanation

x = 3 + 8t $$-$$ 4t2

$${{dx} \over {dt}} = 8 - 8t$$

$$\therefore$$ Velocity at $$t = 1 = {\left( {{{dx} \over {dt}}} \right)_{t = 1}} = 8 - 8\,.\,1 = 0$$

4

WB JEE 2009

MCQ (Single Correct Answer)

The Rolle's theorem is applicable in the interval $$-$$1 $$\le$$ x $$\le$$ 1 for the function

A
f(x) = x
B
f(x) = x2
C
f(x) = 2x3 + 3
D
f(x) = |x|

Explanation

(a) f(x) = x

$$f'(x) = {{df(x)} \over {dx}} = 1$$ which is greater than zero

$$\therefore$$ f (x) is strictly increasing in [$$-$$1, 1].

So Rolle's theorem is not applicable.

(b) $$\because$$ f($$-$$1) = f(1) = 1

Also f(x) = x2 is continuous in [$$-$$1, 1] and differentiable in ($$-$$1, 1)

$$\therefore$$ Rolle's theorem is applicable.

(c) f(x) = 2x3 + 3 $$\Rightarrow$$ f'(x) = 6x2 > 0

$$\therefore$$ f(x) is strictly increasing in [$$-$$1, 1].

So, Rolle's theorem is not applicable.

(d) f(x) = |x| = x, x $$\ge$$ 0 and $$-$$x, x < 0

f(1) = f($$-$$1) = 1, also f(x) is continuous but f(x) is not differentiable at x = 0 $$\in$$ ($$-$$1, 1). So all conditions of Rolle's theorem is not satisfied.

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