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

• $\left(\frac{1}{4}, \frac{1}{2}\right)$

• $\left(\frac{1}{2}, \frac{1}{4}\right)$

• $\left(\frac{1}{2}, \frac{- 1}{2}\right)$

• $\left(\frac{1}{2}, \frac{1}{2}\right)$

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

• $\frac{\mathrm{a}}{2}$

• 2a

• a

• $\frac{\mathrm{a}}{3}$

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

The maximum value of $\frac{\log \left(\mathrm{x}\right)}{\mathrm{x}} \mathrm{in} \left(2, \infty \right)$ is

• 1

• $\frac{2}{\mathrm{e}}$

• e

• $\frac{1}{\mathrm{e}}$

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

• $\frac{\mathrm{dy}}{\mathrm{dx}} - 1$

• $\frac{\mathrm{dx}}{\mathrm{dy}} = 0$

• $\frac{\mathrm{dx}}{\mathrm{dy}} = 1$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

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

• - 5

• $\frac{1}{5}$

• $\frac{1}{243}$

• $\frac{1}{27}$

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

• 36 m

• 32 m

• 100  m

• 64 m

The length of the subtangent at t on the curve $\mathrm{x} = \mathrm{a}\left(\mathrm{t} + \sin \left(\mathrm{t}\right)\right)$, y = $\mathrm{a}\left(1 - \cos \left(\mathrm{t}\right)\right)$

• $\mathrm{asin}\left(\mathrm{t}\right)$

• $2\mathrm{asin}\left(\frac{\mathrm{t}}{2}\right)\tan \left(\frac{\mathrm{t}}{2}\right)$

• $2\mathrm{asin}\left(\frac{\mathrm{t}}{2}\right)$

• $2{\mathrm{asin}}^3\left(\frac{\mathrm{t}}{2}\right)\sec \left(\frac{\mathrm{t}}{2}\right)$

A wire of length 20 cm is bent in the form of a sector of a circle. The maximum area that can be enclosed by the wire is

• 20 sq cm

• 25 sq cm

• 10 sq cm

• 30 sq cm

The condition for the line y = mx + c to be a normal to the parabola y² = 4ax is

• c = - 2am - am³

• $\mathrm{c} = - \frac{\mathrm{a}}{\mathrm{m}}$

• $\mathrm{c} = \frac{\mathrm{a}}{\mathrm{m}}$

• c = 2am + am³