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Application of Derivatives

Multiple Choice Questions

241.

The point on the curve y² = x the tangent at which makes an angle 45° with X-axis ts

$(\frac{1}{4}, \frac{1}{2})$
$(\frac{1}{2}, \frac{1}{4})$
$(\frac{1}{2}, \frac{- 1}{2})$
$(\frac{1}{2}, \frac{1}{2})$

242.

The length of the subtangent to the curve x²y² = a⁴ at (- a, a)

$\frac{a}{2}$
2a
a
$\frac{a}{3}$

243.

A stone is thrown vertically upwards and the height x ft reached by the stone in t seconds is given by x = 80 t - 16t² The stone reaches the maximum height in

2 s
2.5 s
3 s
1.5 s

244.
The maximum value of $\frac{\log (x)}{x} in (2, \infty)$ is

1

$\frac{2}{e}$

e

$\frac{1}{e}$

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$\frac{1}{e}$

$Let y = \frac{\log (x)}{x} On differentiating w . r . t . x, we get \frac{dy}{dx} = \frac{x . \frac{1}{x} - \log (x) . 1}{x^{2}} = \frac{1 - \log (x)}{x^{2}} For maxima, put \frac{dy}{dx} = 0 \Rightarrow \frac{1 - \log (x)}{x^{2}} = 0 \Rightarrow \log (x) = 1 \Rightarrow x = e Now, \frac{d^{2} y}{{dx}^{2}} = \frac{x^{2} (- \frac{1}{t}) - (1 - \log (x)) 2 x}{{(x^{2})}^{2}} At x = e, \frac{d^{2} y}{{dx}^{2}} \leq 0, maxima ∴ The maximum value at x = e is y = \frac{\log (e)}{e} = \frac{1}{e}$

245.

The tangent to a given curve y = f(x) is perpendicular to the x-axis, if

$\frac{dy}{dx} - 1$
$\frac{dx}{dy} = 0$
$\frac{dx}{dy} = 1$
$\frac{dy}{dx} = 0$

246.

The minimum value of $27^{\cos (2 x)} 81^{\sin (2 x)}$ is

- 5
$\frac{1}{5}$
$\frac{1}{243}$
$\frac{1}{27}$

247.

A stone is thrown vertically upwards from the top of a tower 64 m high according to the law s = 48t - 16t². The greatest height attained by the stone above ground is