41.

The distance between the focii of the ellipse
x = 3cos$\left(\mathrm{\theta }\right)$, y = 4sin$\left(\mathrm{\theta }\right)$ is

• $2\sqrt{7}$

• $7\sqrt{2}$

• $\sqrt{7}$

• $3\sqrt{7}$

The equations of the latus rectum of the ellipse
9x² + 25y² - 36x + 50y - 164 = 0 are

• x - 4 = 0, x + 2 = 0

• x - 6 = 0, x + 2 = 0

• x + 6 = 0, x - 2 = 0

• x + 4 = 0, x + 5 = 0

The values of m for which the line y = mx + 2
becomes a tangent to the hyperbola 4x² - 9y² = 36 is

• $\pm \frac{2}{3}$

• $\pm \frac{2\sqrt{2}}{3}$

• $\pm \frac{8}{9}$

• $\pm \frac{4\sqrt{2}}{3}$

$$\begin{gathered} \mathrm{The} \mathrm{harmonic} \mathrm{conjugate} \mathrm{of} (2, 3, 4) \mathrm{with} \\ \mathrm{respect} \mathrm{to} \mathrm{the} \mathrm{points} (3, - 2, 2), (6, - 17, - 4) \mathrm{is} \end{gathered}$$

• $\left(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\right)$

• $\left(\frac{18}{5}, - 5, \frac{4}{5}\right)$

• $\left(\frac{- 18}{5}, \frac{5}{4}, \frac{4}{5}\right)$

• $\left(\frac{18}{5}, - 5, \frac{- 4}{5}\right)$

$$\begin{gathered} \mathrm{If} \mathrm{a} \mathrm{line} \mathrm{makes} \mathrm{angles} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{and} \mathrm{\delta } \mathrm{with} \mathrm{the} \mathrm{four} \mathrm{diagonals} \mathrm{of} \mathrm{a} \mathrm{cube}, \mathrm{then} \mathrm{the} \mathrm{value} \\ \mathrm{of} {\sin }^2\left(\mathrm{\alpha }\right) +{\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right) + {\sin }^2\left(\mathrm{\delta }\right) \mathrm{is} \end{gathered}$$

• $\frac{4}{3}$

• $\frac{8}{3}$

• $\frac{7}{3}$

• $\frac{5}{3}$

$\underset{\mathrm{x}\to 0}{\lim }\frac{6^{\mathrm{x}} - 3^{\mathrm{x}} - 2^{\mathrm{x}} + 1}{{\mathrm{x}}^2} = ?$

• $\left({\log }_{\mathrm{e}}\left(2\right)\right){\log }_{\mathrm{e}}\left(3\right)$

• ${\log }_{\mathrm{e}}\left(5\right)$

• ${\log }_{\mathrm{e}}\left(6\right)$

• 0

$$\begin{gathered} \mathrm{Define} \mathrm{f}\left(\mathrm{x}\right) =\left\{ {\mathrm{x}}^2 + \mathrm{bx} + \mathrm{c}, \mathrm{x} < 1 \\ \mathrm{x}, \mathrm{x} \ge 1\right\} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{differentiable} \mathrm{at} \mathrm{x} = 1, \mathrm{then} \\ \left(\mathrm{b} - \mathrm{c}\right) = ? \end{gathered}$$

•  - 2

• 0

• 1

• 2

If x = a is a root of multiplicity two of a polynomial equation f(x) = 0, then

• f'(a) = f''(a) = 0

• f''(a) = f(a) = 0

• $\mathrm{f}\text{'}\left(\mathrm{a}\right) \ne 0 \ne \mathrm{f}\text{'}\text{'}\left(\mathrm{a}\right)$

• $\mathrm{f}\left(\mathrm{a}\right) = \mathrm{f}\text{'}\left(\mathrm{a}\right) = 0, \mathrm{f}\text{'}\text{'}\left(\mathrm{a}\right) \ne 0$

$\mathrm{If} \mathrm{A} > 0, \mathrm{B} > 0 \mathrm{and} \mathrm{A} + \mathrm{B} = \frac{\mathrm{\pi }}{3}, \mathrm{then} \mathrm{the} \mathrm{maximum} \mathrm{value} \mathrm{of} \mathrm{Atan}\left(\mathrm{B}\right) \mathrm{is}$

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\sqrt{3}$

The equation of the common tangent drawn to the curves y = 8x and xy = - 1 is

• y = 2x + 1

• 2y = x + 6

• y = x + 2

• 3y = 8x + 2