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JEE Mathematics Solved Question Paper 2004

Multiple Choice Questions

If y = 2^x . 3^{2x - 1}, then $\frac{d^{2} y}{{dx}^{2}}$ is equal to :

$(\log (2)) (\log (3))$
$(\log (18))$
$(\log {(18)}^{2}) y^{2}$
$(\log (18)) y$

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing, is

a constant
proportional to the radius
inversely proportional to the radius
inversely proportional to the surface area

3.
The length of the subtangent to the curve x² + xy + y² = 7 at (1, - 3) is :

3

5

$\frac{3}{5}$

15

Given curve is x² + xy + y² = 7

On differentiating w.r.t. x,

$2 x + x \frac{dy}{dx} + y + 2 y \frac{dy}{dx} = 0 \Rightarrow (x + 2 y) \frac{dy}{dx} = - (2 x + y) \Rightarrow {(\frac{dy}{dx})}_{(1, - 3)} = \frac{- (2 - 3)}{(1 - 6)} = - \frac{1}{5} ∴ Length of subtangent = \frac{y}{\frac{dy}{dx}} = \frac{- 3}{\frac{1}{- 5}} = 15$

If y = x + x² + x³ + ... to $\infty$ where $|x| < 1$ , then for $|y| < 1$ , $\frac{dx}{dy}$ is equal to :

y + y² + y³ + ... to $\infty$
1 - y + y² - y³ + ... to $\infty$
1 - 2y + 3y²- ... to $\infty$
1 + 2y + 3y² + ... to $\infty$

Twenty two metres are available to fence a flower bed in the form of a circular sector. If the flower bed should have the greatest possible surface area, the radius of the circle must be :

If $y = \sqrt{\sin (x) + \sqrt{\sin (x) + \sqrt{\sin (x) + . . . \infty}}}$ , then $\frac{dy}{dx}$ is equal to :

$\frac{\cos (x)}{2 y - 1}$
$\frac{- \cos (x)}{2 y - 1}$
$\frac{\sin (x)}{1 - 2 y}$
$\frac{- \sin (x)}{1 - 2 y}$

Given f(0) = 0 and f(x) = $\frac{1}{(1 - e^{- \frac{1}{x}})}$ for $x \neq 0$ . Then only one of the following statements on f (x) is true. That is f(x), is :