Long Answer TypeUsing integration, find the area of the region bounded by the triangle whose vertices are (-1, 2), (1, 5) and (3, 4).
A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of 80 on each piece of type A and 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?
Let x be the number of pieces manufactured of type A and y be the number of pieces manufactured of type B. Let us summarize the data given in the problem as follows:
| Product | Time for Fabricating (in hours) | Time for Finishing (in hours) | Maximum labour hours available |
| Type A | 9 | 1 | 180 |
| Type B | 12 | 3 | 30 |
| Maximum Profit (in Rupees) | 80 | 120 |
| Points | Value of Z |
| A(12, 6) | Z = 80 x 12 + 120 x 6 = Rs. 1680 |
| B(0, 10) | Z = 80 x 0 +120 x 10 = Rs. 1200 |
| C(20, 0) | Z = 80 x 20 + 120 x 0 = Rs.1600 |
There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tails 40% of the times. One of The three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X, and hence find the mean of the distribution.
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