Mathematics & Statistics  – Important Question Bank for Maharashtra HSC Board Exam 2017

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1. The probability that a certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive.
2. Find the approximate value of tan ^{– 1} (1.0001).
3. Find the volume of the solid generated, when the area between ellipse 4x² + 9y² = 36 and the chord AB, with A = (3,0), B = (0,2); is revolved about X-axis.
4. Find the area of the region common to the circle x ² + y ² = 9 and the parabola y² = 8x
5. The surface area of a spherical balloon is increasing at the rate of 2 cm2/sec. At what rate is the volume of the balloon is increasing when the radius of the balloon is 6 cm?
6. A random variable X has the following probability distribution:
   

   X =  x	0	1	2	3	4	5	6
   P(X=x)	K	3k	5 k	7k	9k	11k	13k

   (a) Find k, (b) Find P (0<X<4)

   (c) Obtain cumulative distribution function (c.d.f) of X.

   7. Verify Rolle ’s Theorem for the function: f(x) = x² – 5x + 9 on [1, 4]
   8. If A,B,C,D are (1,1,1), (2,1,3), (3,2,2), and (3,3,4) respectively, then find the volume of the parallelepiped with AB, AC and AD are concurrent edges.
   9. In any ΔABC, with usual notations, prove that b² = c² + a² – 2ca cos B.
   10. A baker man sells 5 types of cakes. Profits due to the sale of each type of cake are respectively Rs 3, Rs 2.5, Rs 2, Rs 1.5, Rs 1. The demands for these cakes are 10%, 5%, 25%, 45% and 15% respectively. What is the expected profit per cake?
   11. A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.
   12. Solve: 3e^x tan ydx + (1 + e^x) sec² ydy = 0, also find the particular solution when x = 0 and y = π
   13. Amit and Rohit started a business by investing Rs 20,000 each. After 3 months Amit withdrew Rs 5,000 and Rohit put in Rs 5,ooo additionally. How should a profit of Rs 12.800 be divided between them at the end of the year?
   14. Find the present value of an annuity immediate Rs 18,000 p.a for 3 years at 9% p.a. compounded annually. [Given (1.09)^-3 = 0.7722]
   15. Using the truth table, examine whether the statement pattern (p →q) ↔ (~ p v q) is an tautology, a contradiction or a contingency.
   16. Discuss extreme value of the function f(x) = x log x
   17. A card is drawn at random and replaced four times from a well shuffled pack of 52 cards. Find the probability that-a) Tow diamond cards are drawn b) At least one diamond card is drawn.
   18. Find the agent’s commission at 15% on the first premium if he places insurance for Rs. 2,000,000 on the life of a person, the premium being at the rate of Rs. 35 per thousand, per annum, paid annually.
   19. A banker’s discount calculated for 1 years is 26 times his gain. Find the rate of interest.
   20. Examine whether the following statement (p ^ q) v ( ~p v ~ q ) is a tautology or contradiction or neither of them.