If z = x + iy, w = $\frac{2 - \mathrm{iz}}{2\mathrm{z} - \mathrm{i}} \mathrm{and} \left|\mathrm{w}\right| = 1$, find the locus of z and illustrate it in the Argand plane.

Find the volume of a parallelopiped whose eidges are reprsented by the vectors:

$\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}, \mathrm{and} \overrightarrow{\mathrm{c}} = 3\hat{\mathrm{i}} +\hspace{0.17em}\hat{\mathrm{j}} +\hspace{0.17em}2\hat{\mathrm{k}}$.

Find the equation of the plane passing through the intersection of the planes:

x + y +z + 1 = 0and 2x - 3y + 5z - 2 = 0 and the point (- 1, 2 , 1)

Find the shortest distance between the lines ${\mathrm{r}}_1 = \hat{\mathrm{i}} +\hspace{0.17em}2\hat{\mathrm{j}} +\hspace{0.17em}3\hat{\mathrm{k}} + \mathrm{\lambda }(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} +\hspace{0.17em}4\hat{\mathrm{k}}) \mathrm{and} {\mathrm{r}}_2 = 2\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 5\hat{\mathrm{k}} + \mathrm{\mu }(4\hat{\mathrm{i}} + 6\hat{\mathrm{j}} + 8\hat{\mathrm{k}})$

Mr. Nirav borrowed Rs. 50,000 from the bank for 5 years. The rate of interest is 9 % per annum compounded monthly. Find the payment he makes monthly if he pays back at the beginning of each month.

A dietician wishes to mix two kinds of food X and Y in such a way that the mixture contains atleast 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below:

One kg of food X costs Rs.24 and one kg of food Y costs Rs.36. Using Linear Programming, find the least cost of the total mixture which will contain the required vitamins.

27.

A bill for Rs. 7650 was drawn on 8^th March, 2013, at 7 months. It was discounted on 18^th May, 2013 and the holder of bill received Rs. 7497. What is the rate of interest charged by bank?

The average cost function, AC for a commodity is given by AC = $\mathrm{x} + 5 + \frac{36}{\mathrm{x}}$, in terms of output x. Find:

(i) The total cost, C and marginal cost, MC as a function of x.

(ii) The outputs for which AC increases.

Calculate the index number for the year 2014, with 2010 as the base year by the weighted aggregate method from the following data:

The quarterly profits of a small scale industry (in thousands of rupees) is as follows:

Calculate four quarterly moving averages. Display these and the original figures graphically on the same graph sheet.