On which of the following intervals is the function f(x) = 2x² - $\log \left|\mathrm{x}\right|, \mathrm{x} \ne 0$ increasing 2.

• $\left(\frac{1}{2}, \infty \right)$

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

• $\left(- \infty , - \frac{1}{2}\right) \cup \left(0, \frac{1}{2}\right)$

• $\left(- \frac{1}{2}, 0\right) \cup \left(\frac{1}{2}, \infty \right)$

The length of the normal to the curve $\mathrm{x} = \mathrm{a}\left(\mathrm{\theta } + \sin \left(\mathrm{\theta }\right)\right)$, y =  $\mathrm{a}\left(1 - \cos \left(\mathrm{\theta }\right)\right) \mathrm{at} \mathrm{\theta } = \frac{\mathrm{\pi }}{2}$ is

• 2a

• $\frac{\mathrm{a}}{2}$

• $\frac{\mathrm{a}}{\sqrt{2}}$

• $\sqrt{2}\mathrm{a}$

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

• e

• 2e

• $\frac{1}{\mathrm{e}}$

• $\frac{2}{\mathrm{e}}$

The smallest circle with centre on y-axis and passing through the point (7, 3) has radius

• $\sqrt{58}$

• 7

• 3

• 4

If sum of two numbers is 6, the minimum value of the sum of their reciprocals is

• $\frac{6}{5}$

• $\frac{3}{4}$

• $\frac{2}{3}$

• $\frac{1}{2}$

The normal to the curve x = $\mathrm{a}\left(\cos \left(\mathrm{\theta }\right) + \sin \left(\mathrm{\theta }\right)\right)$, y = $\mathrm{a}\left(\sin \left(\mathrm{\theta }\right) - \mathrm{\theta }\cos \left(\mathrm{\theta }\right)\right)$ at any point $\mathrm{\theta }$ is such that

• it makes a constant angle with x-axis

• it passes through origin

• it is at a constant distance from origin

• None of the above

647.

The function f(x) = $\log \left(1 + \mathrm{x}\right) - \frac{2\mathrm{x}}{2 + \mathrm{x}}$ is increasing on

• $\left(- 1, \infty \right)$

• $\left(- \infty , 0\right)$

• $\left(- \infty , \infty \right)$

• None of these

The minimum value of ${\mathrm{x}}^2 + \frac{1}{1 + {\mathrm{x}}^2}$ is at

• x = 0

• x = 1

• x = 4

• x = 3

The maximum value of $3 \cos \left(\mathrm{\theta }\right) + 4 \sin \left(\mathrm{\theta }\right)$ is

• 3

• 4

• 5

• None of these

The function x⁵ - 5x⁴ + 5x³ - 1 is

• neither maximum nor minimum at x = 0

• maximum at x = 0

• maximum at x = 1 and minimum at x = 3

• minimum at x = 0