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CBSE Class 12 Mathematics Solved Question Paper 2010

Long Answer Type

31.
Using integration, find the area of the following region:
$\{(x, y) : \frac{x^{2}}{9} + \frac{y^{2}}{4} \leq 1 \leq \frac{x}{3} + \frac{y}{2}\}$

Given ellipse

$\Rightarrow \frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \Rightarrow y = \frac{2}{3} \sqrt{9 - x^{2}} Given line \frac{x}{3} + \frac{y}{2} = 1 \Rightarrow y = (2 - \frac{2 x}{3}) Required Area \{(x, y) : \frac{x^{2}}{9} + \frac{y^{2}}{4} \leq 1 \leq \frac{x}{3} + \frac{y}{2}\} is given below$

$Required Area = \int_{0}^{3} (y_{1} - y_{2}) dx = \int_{0}^{3} [\frac{2}{3} \sqrt{9 - x^{2}} - (2 - \frac{2 x}{3})] dx = {[\frac{2}{3} (\frac{x}{2} \sqrt{9 - x^{2}} + \frac{9}{2} \sin^{- 1} \frac{x}{3}) - 2 x + \frac{x^{2}}{3}]}_{0}^{3} = [\frac{2}{3} (\frac{9}{2} \sin^{- 1} 1) - 6 + 3] - 0 = 3 \times \frac{π}{2} - 3 = \frac{3}{2} (π - 2) sq . units$

32.

A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an L.P.P. and solve it graphically

33.

A card form a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn at random and are found to be both clubs. Find the probability of the lost card being of clubs.

34.

From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability distribution of the number of defective bulbs.