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ISC Class 12 Mathematics Solved Question Paper 2015

Short Answer Type

21.
If z = x + iy, w = $\frac{2 - iz}{2 z - i} and |w| = 1$ , find the locus of z and illustrate it in the Argand plane.

Given, $\frac{2 - iz}{2 z - i} and |w| = 1$

$\Rightarrow |\frac{2 - iz}{2 z - i}| = 1$

$\Rightarrow |2 - iz| = |2 z - i|$

$\Rightarrow |2 - i (x + iy)| = |2 (x + iy) - i|$

$\Rightarrow |(2 + y) - ix| = |2 x + i (2 y - 1)|$

$\Rightarrow \sqrt{(2 + y)^{2} + x^{2}} = \sqrt{4 x^{2} + (2 y - 1)^{2}}$

Squaring,

4 + 4y + y² + x² = 4x² + 4y² - 4y + 1

$\Rightarrow 3 x^{2} + 3 y^{2} - 8 y - 3 = 0$

$\Rightarrow x^{2} + y^{2} - \frac{8}{3} y - 1 = 0$

Circle, centre (0, $\frac{4}{3}$ ) and r = $\sqrt{\frac{16}{9} + 1}$

= $\frac{5}{3}$

22.

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

$\vec{a} = 2 \hat{i} - 3 \hat{j} - 4 \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - \hat{k}, and \vec{c} = 3 \hat{i} + \hat{j} + 2 \hat{k}$ .

23.

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)

24.

Find the shortest distance between the lines $r_{1} = \hat{i} + 2 \hat{j} + 3 \hat{k} + λ (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) and r_{2} = 2 \hat{i} + 4 \hat{j} + 5 \hat{k} + μ (4 \hat{i} + 6 \hat{j} + 8 \hat{k})$

25.

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.

26.

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:

Food	Vitamin A	Vitamin B	Vitamin C
X	1 unit	2 units	3 units
Y	2 units	2 units	1 unit

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^thMay, 2013 and the holder of bill received Rs. 7497. What is the rate of interest charged by bank?