A speaks truth in 60% of the cases, while B is 40% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact ?

From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find :

(a) The probability distribution of X

(b) Mean of X

(c) Variance of X

Find $\mathrm{\lambda }$ if the scalar projection of $\overrightarrow{\mathrm{a}} = \mathrm{\lambda }\hat{\mathrm{i}} + \hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ on $\overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ is 4 units.

The Cartesian equation of line is : 2x - 3 = 3y + 1 = 5 - 6z. Find the vector equation of a line passing through (7, – 5, 0) and parallel to the given line.

25.

Water is dripping out from a conial funnel of semi-verticle angle $\frac{\mathrm{\pi }}{4}$ at the uniform rate of 2 cm²/sec in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.

Using matrices, solve the following system of equations :

2x - 3y + 5z = 11

3x + 2y - 4z = - 5

x + y - 2z = - 3

Using elementary tranformation, find the inverse of the matrix :

$\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$

A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height.

Evaluate: $\int \frac{\mathrm{x} - 1}{\sqrt{{\mathrm{x}}^2 - \mathrm{x}}}\mathrm{dx}$

Evaluate: ${\int }_0^{\mathrm{\pi }/2}\frac{{\cos }^2\left(\mathrm{x}\right)}{1 + \sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}d\mathrm{x}$