On a multiple choice examination with three possible answers (out of which only one is correct) for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?

Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, -1, 2) and parallel to the line  $\frac{\mathrm{x} - 4}{1} = \frac{\mathrm{y} + 3}{-4} = \frac{\mathrm{z} + 1}{7}$

23.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  $\left( 2\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} \right) \mathrm{and} \left( \overrightarrow{\mathrm{a}} - 3\overrightarrow{\mathrm{b}} \right)$ respectively, externally in the ratio 1:2. Also, show that P is the midpoint of the line segment R.

Evaluate: ${\int }_0^{\mathrm{\pi }} \frac{\mathrm{x}}{1 + \mathrm{sinx}} \mathrm{dx}$

Evaluate: $\int {\mathrm{e}}^{\mathrm{x}} \left( \frac{\sin 4\mathrm{x} - 4}{1 - \cos 4\mathrm{x}}\right) \mathrm{dx}$

Evaluate: $\int \frac{1 - {\mathrm{x}}^2}{\mathrm{x} \left( 1 - 2\mathrm{x} \right)} \mathrm{dx}$

Find the particular solution of the differential equation satisfying the given conditions: x² dy + (xy + y² )dx = 0; y = 1 when x = 1.

Find the general solution of the differential equation,

$\mathrm{x} \log \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = \frac{2}{\mathrm{x}} \log \mathrm{x}$

Find the particular solution of the differential equation satisfying the given conditions:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y} \tan \mathrm{x}$,    given that   y = 1  when   x= 0.

Evaluate ${\int }_1^3 \left( 3 {\mathrm{x}}^2 + 2 \mathrm{x} \right) \mathrm{dx}$  as limit of sums.