ALGEBRA HOTS





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Extra HOTS in ALGEBRA PRACTICE WELL. 

1. In the following experiment write the sample space S, number of sample points n(S), events P, Q, R using set and n(P), n(Q) and n(R). There are 3 men and 2 women. A 'Gramswachaatta Abhiyan' committee of two is to be formed. P is the event that the committee should contain at least one woman. Q is the event that the committee should contain one man and one woman. R is the event that there is not woman in the committee.

2. There are three boys and two girls. A committee of two is to be formed, find the probability of events that the committee contains at least one girl.

3. Sachin buys fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish. What is the probability that the fish taken out is a male fish?

4. A coin is tossed three times then find the probability of getting head on middle coin.

5. In the following experiment write the sample space A, number of sample point n(S), event A, B, C and n(A), n(B), n(C). A die is thrown. A is the event that prime number comes up, B is the event that the number is divisible by three comes up, C is the event that the perfect square comes up.

6. In the following experiments write the sample space S, number of sample points n(S), events P, Q using set and n(P), n(Q). A coin is tossed and a die is thrown simultaneously: P is the event of getting head and an odd number. Q is the event of getting either H or T and an even number.


7. In the following experiment write the sample space S, number of sample points n(S), events P, Q using set and n(P), n(Q). Form two digit number using the digits, 0, 1, 2, 3, 4, 5 without repeating the digits. P is the event that the number so formed is even. Q is the event that the number so formed is divisible by 3.

8. Two coins are tossed. Find the probability of the events head appears on both the coins.

9. Two digit number are formed from the digits 0, 1, 2, 3, 4 where digits are not repeated. A is the event that the number formed is even. Write S, A , n(S) and n(A).

10. One card is drawn from a well – shuffled pack of 52 cards. Find the probability of getting the jack of hearts.

11. A box contains 3 red, 3 white and 3 green balls, A ball is selected at random. Find the probability that ball picked up is a red ball.

12. In the following experiment, write the sample space S, number of sample point n(S), event A, B, n(A), n(B). Two coins are tossed, A is the event of getting at most one head, B is the event of getting both heads.

13. If two coins are tossed then find the probability of the events. At least one tail turns up.


14. In the following experiment write the sample space S, number of sample points n(S), events P, Q, R using set and n(P) , n(Q) and n(R). A die is thrown: P is the event of getting an odd number. Q is the event of getting an even number. R is the event of getting a prime number.


15. One card is drawn from a well – shuffled deck of 52 cards. Find the probability of getting king of red colour.


16. In each of the following experiments, write the sample space S, number of sample point n (S), events A, B and n(A), n(B). A coin is tossed three times. A is the event that head appears once, B is the event that head appears at the most twice.


17. In the following experiment write the sample space S, number of sample points n(S), events P, Q, R using set and n(P), n(Q) and n(R). There are 3 red, 3white and 3 green balls in a bag. One ball is drawn at random from a bag. P is the event that the ball is red. Q is the event that the ball is not green. R is the event that ball is red or white.

18. In the following experiment write the sample space S, number of sample points n(S), events P, Q, n(P), and n(Q). A die is thrown: P is the event of getting an odd number. Q is the event of getting an even number.

19. Two unbiased dice are rolled once. Find the probability of a sum greater than 8

20. Two unbiased dice are rolled once. Find the probability of getting a doublet.

21. Two unbiased dice are rolled once. Find the probability of getting a sum 8

22. In tossing a fair coin twice, find the probability of getting atleast one head.

23. In tossing a fair coin twice, find the probability of getting two heads.

24. A fair die is rolled. Find the probability of getting a number greater than 4.

25. A fair die is rolled. Find the probability of getting a prime factor of 6

26. A fair die is rolled.Find the probability of getting an even number.

27. A fair die is rolled. Find the probability of getting the number 4.

28. An integer is chosen from the first twenty natural numbers. What is the probability that it is a prime number?

29. Find the probability that a non - leap year selected at random will have 53 Fridays.

30. Find the probability that a leap year selected at random will have only 52 Friday.

31. Find the probability that a leap year selected at random will have 53 Fridays.

32. A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of drawing a red ball, then find the number of blue balls in the bag.

33. There are 20 boys and 15 girls in a class of 35 students. A student is chosen at random. Find the probability that the chosen student is a (i) boy (ii) girl.

34. From a well shuffled pack of 52 playing cards, one card is drawn at random. Find the probability of getting a diamond 10

35. From a well shuffled pack of 52 playing cards, one card is drawn at random. Find the probability of getting a spade card.

36. From a well shuffled pack of 52 playing cards, one card is drawn at random. Find the probability of getting a black king.

37. From a well shuffled pack of 52 playing cards, one card is drawn at random. Find the probability of getting a king.

38. In an arithmetic series, the sum of first 14 terms is -203 and the sum of the next 11 terms is –572. Find the arithmetic series.

39. Three numbers are in the ratio 2 : 5 : 7. If 7 is subtracted from the second, the resulting numbers form an arithmetic sequence. Determine the numbers.

40. If a person joins his work in 2010 with an annual salary of Rs.30,000 and receives an annual increment of `Rs. 600 every year, in which year, will his annual salary be `Rs. 39,000?

41. In a flower garden, there are 23 rose plants in the first row, 21 in the second row, 19 in the third row and so on. There are 5 rose plants in the last row. How many rows are there in the flower garden?

42. Find the smallest positive integer n such that tn of the arithmetic sequence 20, 19 1/4, 18 1/2, .... is negative?

43. Find the first term 'a' and the common difference 'd'

44. Find the first term 'a' and the common difference 'd'.

45. Is the sequence is an A.P. ?

46. Is the sequence is an A.P?


47. A car left 30 minutes later than the scheduled time. in order to reach its destination 150 km away in time, it has to increase its speed by 25 km/hr from its usual speed. Find its usual speed.

48. The base of a triangle is 4 cm longer than its altitude. If the area of the triangle is 48 sq. cm, then find its base and altitude.

49. The sum of a number and its reciprocal is 5 1/5 . Find the numbers.

50. Solve by factorization method : 6x2 – 5x – 25 = 0

51. If α and β are the roots of the equation 2x^2 - 3x - 1 = 0 , find the value of (α+1/β) (1/α+β)

52. If α and β are the toots of the equation 2x^2 - 3x - 1 = 0, find the values of α^2/β+β^2/α

53. If α and β are the roots of the equation 2x^2 - 3x - 1 = 0, find the values of α - β.

54. If α and β are the roots of the equation 2x^2 - 3x - 1 = 0, find the values of (α/β) + (β/α)

55. If α and β are the roots of the equation 2x^2 - 3x - 1 = 0, find the values of a^2 + b^2

56. If the sum and product of the roots of the quadratic equation ax^2 - 5x + c = 0 are both equal to 10, then find the values of a and c.

57. If one of the roots of the equation 3x^2 - 10x + k = 0 is 1/3, then find the other root and also the value of k.

58. Find the values of k so that the equation x^2 - 2x ( 1 + 3k) + 7 (3 + 2x) = 0 has real and equal roots.

59. Determine the nature of roots of the following quadratic equation 2x^2 + 5x + 5 = 0 .

60. Determine the nature of roots of the following quadratic equation 4x^2 - 28x + 49 = 0

61. Determine the nature of roots of the following quadratic equation x^2 - 11x - 10 = 0

62. If α and β are the roots of the equation 3x^2 - 4x + 1 = 0. Form a quadratic equation whose roots are α^2 / β and β^2 /α

63. Form the quadratic equation whose roots are 7 + √3 and 7 - √3.