## Tuesday, July 14, 2020 2019
COMMERCIAL MATHEMATICS AND STATISTICS
Full Marks: 100
Pass Marks: 30
Time: Three hours
The figures in the margin indicate full marks for the questions
1. Answer the following questions as directed: 1x8=8
1. Is the set a null set?
2. Find the co-factor of in the following determinant. 1. What is the difference between simple interest and compound interest?
2. Write True or False: ‘Every null matrix is a square matrix’
3. Fill in the gap: A.M. x H.M. = _____
4. What is the arithmetic mean of first n natural numbers?
5. What is the minimum value that the probability of an event can take?
6. Fill in the gap: Standard deviation is always _____.
2. Answer the following questions in brief: 2x5=10
1. What is the difference between and ?
2. For what value of x the matrix will be singular?
3. What is perpetual annuity?
4. If find n.
5. If , show that , a and b are constants.
3. Find the simple interest on Rs. 6000 from 4th March, 2017 to 28th July, 2017 @ 5% p.a. 3
4. If , find n. 3
Or
If and , find n and r.
5. Without expanding show that: 3 Or
If 3
what is the value of x and y?
6. Find the Mean Deviation from median from the given marks of 7 students. 3
18, 26, 15, 20, 17, 12, 25
7. Write any two algebraic properties of Arithmetic Mean. 3
Or
Calculate the Harmonic Mean of 4, 8 and 12. 3
8. The value of a machine at the end of a year becomes 90% of its value at the beginning of that year. The machine was bought at Rs. 4800 and after using it for some years it was sold at Rs. 1800. How many years was the machine in use?     5
9. A committee of 6 is to be formed out of 7 gentleman and 4 ladies. In how many ways can the committee be formed, if at least 2 ladies are to be included? 5
10. Mr. Roy borrows Rs. 20,000 at 4% compound interest and agrees to pay both principal and interest in 10 equal annual installment at the end of each year. Find the amount of each installment.
Given 11. Show that there will be no term containing in the expansion of 5
Or
If the coefficients of and in the expansion of are equal, find the value of k. 5
12. Prove by mathematical induction that the sum of first n odd natural numbers is Or
Using mathematical induction prove that 5 for all 13. Draw the graph of: (any one) 5
1. 2. 14. Calculate the missing frequency, you are given that arithmetic mean is 50.9.
 Marks: 0-20 20-40 40-60 60-80 80-100 Frequency: 5 - 30 12 8

Or
If find 4 15. (a) Prove that: 4
(b) Solve: 4
Or
If and , show that 4 16. (a) Calculate Standard Deviation from the following data: 4
 Class: 10-19 20-29 30-39 40-49 50-59 60-69 70-79 Frequency: 3 61 223 137 53 19 4

(b) Calculate coefficient of variation from the data given in Question 16. (a). 4
17. (a) Two dice are thrown simultaneously. Find the probability of getting even number on both the dice. 4
(b) A bag contains 4 red, 3 blue and 3 white balls. 2 balls are drawn at random from it. Find the probability of getting
1. 2 red and 1 blue ball.
2. 2 balls of the same colour. 4
Or
Define and give one example of each: 2+2=4
1. Mutually exclusive events.
2. Positive correlation.
18. (a) Find out Karl Person’s coefficient of correlation. 6
 X: 2 2 4 5 5 Y: 6 3 2 6 4

(b) Karl Person’s coefficient of correlation between two variables x and y is 0.28 and their covariance is 7.6. If the variance of x is 9, find the standard deviation of y. 2