11.

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question?

Find the magnitude of each of the two vectors $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$, having the same magnitude such that the angle between them is 60^o and their scalar product is 9/2.

A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

If θ is the angle between two vectors $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}} \mathrm{and} 3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ find sin θ.

Find the differential equation representing the family of curves y = ae^{bx + 5} , where a and b are arbitrary constants.

Evaluate: $\int \frac{\cos 2\mathrm{x} + 2 {\sin }^2 \mathrm{x} }{{\cos }^2 \mathrm{x} } \mathrm{dx}$

If y = sin (sin x), prove that $\frac{{\mathrm{d}}^2 \mathrm{y}}{{\mathrm{dx}}^2} + \tan \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} {\cos }^2 \mathrm{x} = 0$

Find the particular solution of the differential equation e^x tan ydx + (2 -e^x) sec² ydy = 0, given that $y = \frac{\pi }{4}$ when x = 0

Find the particular solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + 2 \mathrm{y} \tan \mathrm{x} = \sin \mathrm{x}$, given that y = 0 when $\mathrm{x} = \frac{\mathrm{\pi }}{3}$

Find the shortest distance between the lines.

$\overrightarrow{\mathrm{r}} = (4\hat{\mathrm{i}} -\hat{\mathrm{j}}) + \mathrm{\lambda } (\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}) + \mathrm{\mu } (2\hat{\mathrm{i}} + 4\hat{\mathrm{j}}-5\widehat{\mathrm{k})}$