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1.Maximize and Minimize Z=3x+9y subjected to the constraints
x + 3y ≤ 60
x + y ≥ 10
x = y and x ≥ 0, y ≥ 0 graphically
2. A furniture dealer deals in only two items –tables and chairs. He has Rs 50,000 to invest and has storage space of at most 60 pieces. A table costs Rs 2500 and a chair Rs 500. He estimates that from the sale of one table , he can make a profit of Rs 250 and that from the sale of one chair a profit of Rs 75. How many tables and chairs he should buy from the available money so as to maximize his total profit assuming that he can sell all the items which he buys
3. A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs 17.50 per package on nuts and Rs 7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day?
4. One kind of cake requires 200g of flour and 25g of fat and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1kg of fat assuming that there is no shortage of the other ingredient, used in making the cakes.
5. Consider f:R → R defined by f(x)=4x+3. Show that f is invertible. Find the inverse of f.
6. Solve by matrix method: 2x+y+2z=5, x-y-z=0, x+2y+3z=5.
7. A ladder long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of . How fast is its height on the wall decreasing when the foot of the ladder is away from the wall?
8. Using integration find the area bounded by the circle x2 +y2= 16 and the parabola y2=6x.
9. Derive equation of plane perpendicular to a given vector and passing through a given point both in the vector and Cartesian form
10. Five cards are drawn successively with replacement from well shuffled deck of 52 cards. What is the probability that
(i) all the five card are spade. (ii) none is spade.
11. A die is thrown 6 times , if ‘getting an odd number is success‟ What is the probability of
i) 5 success ?
ii) at least 5 success?
iii) at most 5 success?
12. There are 5 % defective items in a large bulk of items. What is the probability that sample of 10 items will include not more than one defective item?
13. The length of a rectangle is decreasing at the rate of 3 cm/minute and the width y is increasing at the of 2 cm/minute. When x = 10 cm and y = 6 cm, find the rate of change of (i) the perimeter and (ii) the area of the rectangle
14. If y= 5 cos(log x)+7sin(logx), show that x2y2 +xy1=0
15. Two groups are competing for the position on the board of directors of a corporation. The probability of I and II groups will win are 0.6 and 0.4 respectively. Further, if I group wins,the probability of introducing a new product if 0.7 and corresponding probability is 0.3 if the II group wins. Find the probability that new product introduced was by the II group.
16.Find the area of the region enclosed by the parabola x2=4y and the line x=4y-2 and the x axis.
17. Find the area of triangle with vertices A(1,1,2), B(2,3,5), C(1,5,5).
18. In a bank, principal “P” increases continuously at the rate of 5% per year. Find the principal interest of time t.
19. Find the area of the circle x2+y2 =4 bounded by the lines x=0 and x=2 which is lying in the first quadrant.
20. Solve the equations x-y+3z=10, x-y-z=-2 and 2x+3y+4z=4 by matrix method.
21. Show that the relation R in the set of all integers Z defined by R= {(a, b): 2 divides a-b} is an equivalence relation.
23. Prove that if the function is differentiable at a point c,then it is also continuous at that point.
24. Verify mean value theorem for the function f(x)=x2-4x-3 in the interval [1,4].
25. Form the differential equation representing the family of curves y = a sin )x =b) where a and b arbitrary constants.
26. Find the area of a triangle having the points A (1, 1, 1 ), B(1,2,3 ) and (2,3,1 ) as its vertices using vector method.
27. Two cards are drawn successively with replacement from a well shuffled deck of 52 cards. Find the probability distribution of the number of aces.
28. Show that the relation R in the set of real numbers R defined as R = {(a, b): a ≤ b2} + is neither reflexive nor symmetric nor transitive
29. Verify Rolle’s theorem for the function f(x)=x2+2x-8,xϵ[-4,2].
30. Find the absolute maximum value and the absolute minimum value of the function f(x)=sinx+cosx, xϵ[0, π]
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