Question

1. If () ∶ 2 + 4 + 6 + ⋯ + 2 = ( + 1), then verify that (1), (2) and (3) are true. (3) 2. Find the Variance of the numbers 6, 7, 8, 9, 10. (3) 3. (a) The sum of all deviations of the observations of a data from its A.M. is (1) (i) Zero (ii) Maximum (iii) Minimum (iv) Negative number (b) Find the mean of the observations 4, 7, 8, 9, 10, 12,13, 17. (2) 4. If the mean of 3, 4, , 7, 10 is 6, then find the value of . (3) Each question carries 4 marks 5. Prove by using the principle of Mathematical Induction (4) () ∶ 1 + 3 + 3 2 + ⋯ . . +3 −1 = 3 −1 2 is true for all ∈ . 6. A statement p (n) for a natural number n is given by (4) () ∶ 1 2 + 1 4 + 1 8 … … … + 1 2 = 1 − 1 2 (a) Verify that (1) is true. (b) By assuming that () is true for a natural number k, show that ( + 1) is true. 7. (a) Suppose that the mean of a certain number of observations is 50 and the sum of all the observations is 450. Write down the number of observations. (1) (b) Find mean deviation about the mean for the data 6, 7, 10 , 12, 13, 4, 8, 12. (3) 8. Consider the data 6,8,10,12,14,16,18,20,22,24. Find (a) Variance. (3) (b) Standard deviation. (1) 9. (a) Find the mean of the frequency distribution. (2) 15 25 35 45 55 65 75 85 3 1 1 8 17 38 9 3 (b) Hence find variance.
10. If () ∶ 1

3 + 2

3 + 3

3 + ⋯ +

3 = (

(+1)

2

)

2

(a) Check whether (1) is true. (2)

(b) If () is true, prove that ( + 1) is also true. (3)

(c) Is the statement () true for all ∈ ? Justify your answer. (1)

11.From the following table. Find

Class Frequency

30-40 3

40-50 7

50-60 12

60-70 15

70-80 8

80-90 3

90-100 2

(a) Mean. (3)

(b) Variance. (3)

12.The following data gives marks of 40 students.

(a) Find the arithmetic mean of marks given in the above data. (3)

(b) Find the standard deviation of the marks in the above data. (3)
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Answer 1

Answer:

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