# Sample Papers

### Sample Papers

#### Q 3.

Using integration find the area of the region {(x, y): x2 +y2 â‰¤2ax, y2 â‰¥ ax, x,yâ‰¥ 0}

#### Q 5.

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 ?

#### Q 8.

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.

#### Q 9.

Using integration find the area of the triangle formed by positive x-axis and tangent and normal to the circle x2 + y2= 4 at (1, âˆš3).

#### Q 15.

If A is a square matrix such that A2 = I, then find the simplified value of (A -1)3 + (A +1)3 – 7A.

#### Q 16.

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.

#### Q 17.

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

#### Q 20.

If sin [cot-1 (x +1)] = cos(tan-1 x), then find x.

#### Q 21.

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.

#### Q 32.

Solve the differential equation: (tan-1 y-x)dy = (1 + y2) 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

#### Q 34.

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.

Prove that:

#### Q 38.

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.

#### Q 43.

Show that the equation of normal at any point on the curve x = 3 cost t – cos3 t and y = 3 sin t – sin3 t is 4(ycos3 t- sin3t) = 3 sin 4t.

#### Q 45.

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.

#### Q 47.

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.

#### Q 48.

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.