Let P(x) = a₀ + a₁x² + a₂x² + a₃x⁶ + ... + a_nx²ⁿ be a polynomial in a real variables with 0 < a₀ < a₁ < a₂ < .... < a_n. The function P(x) has

• neither a maxima nor a minima

• only one maxima

• both maxima and minima

• only one minima

The maximum value of f(x) = $2\sin \left(\mathrm{x}\right) + \cos \left(2\mathrm{x}\right), 0 \le \mathrm{x} \le \frac{\mathrm{\pi }}{2}$ occurs at x is equal to

• 0

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• None of these

The equation of tangent of the curve y = be^-x/a at the point, where the curve meet y-axis is

• bx + ay - ab = 0

• ax + by - ab = 0

• bx - ay - ab = 0

• ax + by - ab = 0

If y = 4x - 5 is a tangent to the curve y² = px³ + q at (2, 3), then

• p = 2, q = - 7

• p = - 2, q = 7

• p = - 2, q = - 7

• p = 2, q = 7

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

• $\left(\frac{1}{2}, \infty \right)$

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

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

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

206.

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