If H is the harmonic mean between P and Q, then the value of $\frac{\mathrm{H}}{\mathrm{P}} + \frac{\mathrm{H}}{\mathrm{Q}}$ is

• 2

• $\frac{\mathrm{PQ}}{\mathrm{P} +\hspace{0.17em}\mathrm{Q}}$

• $\frac{1}{2}$

• $\frac{\mathrm{P} +\hspace{0.17em}\mathrm{Q}}{\mathrm{PQ}}$

If $\cos \left(\mathrm{x}\right) + {\cos }^2\left(\mathrm{x}\right) = 1$, then the value of ${\sin }^{12}\left(\mathrm{x}\right) + 3{\sin }^{10}\left(\mathrm{x}\right) + 3{\sin }^8\left(\mathrm{x}\right) + {\sin }^6\left(\mathrm{x}\right) - 1$, is equal to :

• 2

• 1

• - 1

• 0

The product of all values of $\left(\cos \left(\mathrm{\alpha }\right) + \mathrm{isin}\left(\mathrm{\alpha }\right)\right)$^3/5 is :

• 1

• $\cos \left(\mathrm{\alpha }\right) + \mathrm{isin}\left(\mathrm{\alpha }\right)$

• $\cos \left(3\mathrm{\alpha }\right) + \mathrm{isin}\left(3\mathrm{\alpha }\right)$

• $\cos \left(5\mathrm{\alpha }\right) + \mathrm{isin}\left(5\mathrm{\alpha }\right)$

The imaginary part of $\frac{{\left(1 +\hspace{0.17em}\mathrm{i}\right)}^2}{\mathrm{i}\left(2\mathrm{i} - 1\right)}$ is :

• $\frac{4}{5}$

• 0

• $\frac{2}{5}$

• $- \frac{4}{5}$

The equation of a directrix of the ellipse $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{25} = 1$ is :

• 3y = 5

• y = 5

• 3y = 25

• y = 3

If the normal at (ap, 2ap) on the parabola y² = 4ax, meets the parabola again at (aq² , 2aq), then

• p² + pq + 2 = 0

• p² - pq + 2 = 0

• q² + pq + 2 = 0

• p² + pq + 1

The length of the straight line x - 3y = 1 intercepted by the hyperbola x² - 4y² = 1 is :

• $\sqrt{10}$

• $\frac{6}{5}$

• $\frac{1}{\sqrt{10}}$

• $\frac{6}{5}\sqrt{10}$

The curve described parametrically by x = t² + 2t - 1, y = 3t + 5 represents :

• an ellipse

• a hyperbola

• a parabola

• a circle

19.

The function f (x) = x² e^{- 2x}, x > 0. Then the maximum value of f (x) is :

• $\frac{1}{\mathrm{e}}$

• $\frac{1}{2\mathrm{e}}$

• $\frac{1}{{\mathrm{e}}^2}$

• $\frac{4}{{\mathrm{e}}^4}$

If (x + y)$\sin \left(\mathrm{u}\right) = {\mathrm{x}}^2{\mathrm{y}}^2, \mathrm{then} \mathrm{x}\frac{\partial \mathrm{u}}{\partial \mathrm{x}} + \mathrm{y}\frac{\partial \mathrm{u}}{\partial \mathrm{y}}$ is equal to

• $\frac{1}{\mathrm{e}}$

• $\frac{1}{2\mathrm{e}}$

• $\frac{1}{{\mathrm{e}}^2}$

• $\frac{4}{{\mathrm{e}}^4}$