Using integration find the area of the region {(x, y): x^{2} +y^{2} â‰¤2ax, y^{2} â‰¥ ax, x,yâ‰¥ 0}

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

A bag ‘A’ contains 4 black and 6 red balls and bag ‘B’ contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B. If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.

Using integration find the area of the triangle formed by positive x-axis and tangent and normal to the circle x^{2} + y^{2}= 4 at (1, âˆš3).

If A is a square matrix such that A^{2} = I, then find the simplified value of (A -1)^{3} + (A +1)^{3} – 7A.

Find the local maxima and local minima, of the function f(x) = sin x – cos x, 0 < x < 2Ï€. Also find the local maximum and local minimum values.

Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below:

2x + 4y â‰¤ 8

3x + y â‰¤ 6

x + y â‰¤ 4

xâ‰¥ 0, y â‰¤ 0.6

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3. Also find maximum volume in terms of volume of the sphere.

Solve the differential equation: (tan^{-1} y-x)dy = (1 + y^{2}) dx.

Solution. Same as solution Q. 23 (OR) Set 1 (Outside Delhi) up to eq.

x = tan^{-1} y -1 + ce ^{-tan}^{-1} y

OR

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game’ then find the probability that B wins.

A manufacturer produces two products A and 6. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

Show that the equation of normal at any point on the curve x = 3 cost t – cos^{3} t and y = 3 sin t – sin^{3} t is 4(ycos^{3} t- sin^{3}t) = 3 sin 4t.

Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio 1:2:4. The probabilities that A, B and C can introduce changes to improve profits of the , company are 0.8,0.5 and 0.3 respectively. If the change does not take place, find the probability that it is due to the appointment of C.

A dealer in rual area wishes to purchase a number of sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs 360 and a manually operated sewing machine Rs 240. He can sell an electronic sewing machine at a profit of Rs 22 and a manually operated sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit ? Make it as a LPP and solve it graphically.

From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.