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

• $\left(\log \left(2\right)\right)\left(\log \left(3\right)\right)$

• $\left(\log \left(18\right)\right)$

• $\left(\log {\left(18\right)}^2\right){\mathrm{y}}^2$

• $\left(\log \left(18\right)\right)\mathrm{y}$

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

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

• 3

• 5

• $\frac{3}{5}$

• 15

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

• 4 m

• 3 m

• 6 m

• 5 m

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

• $\frac{\cos \left(\mathrm{x}\right)}{2\mathrm{y} - 1}$

• $\frac{- \cos \left(\mathrm{x}\right)}{2\mathrm{y} - 1}$

• $\frac{\sin \left(\mathrm{x}\right)}{1 - 2\mathrm{y}}$

• $\frac{- \sin \left(\mathrm{x}\right)}{1 - 2\mathrm{y}}$

Given f(0) = 0 and f(x) = $\frac{1}{\left(1 - {\mathrm{e}}^{- {\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}}\right)}$ for $\mathrm{x} \ne 0$. Then only one of the following statements on f (x) is true. That is f(x), is :

• continuous at x = 0

• not continuous at x = 0

• both continuous and differentiable at x = 0

• not defined at x = 0

The value of x for which the polynomial 2x³ - 9x² + 12x + 4 is a decreasing function of x, is :

• - 1 < x < 1

• 0 < x < 2

• x > 3

• 1 < x < 2

The range of the function $\sin \left({\sin }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{x}\right)\right), \left|\mathrm{x}\right| \le 1$ :

• [- 1, 1]

• [1, - 1]

• {0}

• {1}

Let f : R $\to$ R : f(x) = x² and g : R $\to$ R : g(x) = x + 5, then gof is :

• (x + 5)

• (x + 5²)

• (x² + 5²)

• (x² + 5)