11.

Write the intercept cut off by the plane 2x + y – z = 5 on x-axis.

Prove the following:

${\cot }^{-1 }\left[ \frac{\sqrt{ 1 + \sin \mathrm{x}} + \sqrt{ 1 - \sin \mathrm{x}}}{\sqrt{ 1 + \sin \mathrm{x}} - \sqrt{ 1 - \sin \mathrm{x}}} \right] = \frac{\mathrm{x}}{2}, \mathrm{x} \in \left( 0, \frac{\mathrm{\pi }}{4} \right)$

Find the value of  ${\tan }^{-1} \left( \frac{\mathrm{x}}{\mathrm{y}} \right) - {\tan }^{-1} \left(\frac{\mathrm{x} - \mathrm{y}}{\mathrm{x} + \mathrm{y}}\right)$

Using properties of determinants, prove that

$\left| \begin{array}{ccc} - {\mathrm{a}}^2& \mathrm{ab} & \mathrm{ac}\\ \mathrm{ba}& -{\mathrm{b}}^2& \mathrm{bc}\\ \mathrm{ca}& \mathrm{cb}& - {\mathrm{c}}^2\end{array} \right| = 4 {\mathrm{a}}^2{\mathrm{b}}^2{\mathrm{c}}^2$

Find the value of ‘a’ for which the function f defined as

$\mathrm{f} ( \mathrm{x} ) = \left\{\begin{array}{l} \mathrm{a} \sin \frac{\mathrm{\pi }}{2} ( \mathrm{x} + 1 ), \mathrm{x} \le 0\\ \frac{\tan \mathrm{x} - \sin \mathrm{x} }{{\mathrm{x}}^3}, \mathrm{x} > 0 \end{array}\right.$

is continuous at x = 0.

Differentiate  $\mathrm{X}{ }^{\mathrm{x} \cos \mathrm{x}} + \frac{{\mathrm{x}}^2 + 1}{{\mathrm{x}}^2 - 1} \mathrm{w}.\mathrm{r}.\mathrm{t}. \mathrm{x}$

If   $\mathrm{x} = \mathrm{a} \left( \mathrm{\theta } - \sin \mathrm{\theta } \right), \mathrm{y} = \left( 1 + \cos \mathrm{\theta } \right), \mathrm{find} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$

Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of t heradius of the base. How fast is the sand cone increasing when the height is 4 cm?

Find the points on the curve  x² + y² – 2x – 3= 0  at  whichthe tangents are parallel to x-axis.

Using matrix method, solve the following system of equations:

$\frac{2}{\mathrm{x}} + \frac{3}{\mathrm{y}} + \frac{10}{\mathrm{z}} = 4, \frac{4}{\mathrm{x}} - \frac{6}{\mathrm{y}} + \frac{5}{\mathrm{z}}, \frac{6}{\mathrm{x}} + \frac{9}{\mathrm{y}} - \frac{20}{\mathrm{z}}; \mathrm{x}, \mathrm{y}, \mathrm{z} \ne 0$