2014
(COMMERCIAL MATHEMATICS AND STATISTICS)
Full Marks: 100
Pass Marks: 30
Time: 3 hours
The figures in the margin indicate full marks for the questions.
1. Answer the following questions as directed: 1x8=8
(a) If, then find r.
Solution:
(b) Write true or false:
Ans.
(c) Find the total number of terms in the expansion below:
(d) Find the mode: 7, 9, 10, 8, 9, 11, 6, 8, 4, and 9
Solution:
(e) What is the difference between simple interest and compound interest?
(f) Fill in the blank:
(g) Fill in the blank: If AM and HM of two numbers are 25, and 9, then their GM = _____.
Solution:
(h) A student calculated r = 2.62. Do you agree? State with reasons.
Ans. Not possible, r cannot be more than.
2. Answer the following questions: 2x5=10
(a) Write two properties of determinant.
(b) AM and SD of a set of values are 30 and 8 respectively. If 2 is added to each value, then what will be the coefficient of variation of the series? Find it.
Solution:
(c) If A = {2, 4, 5, 6, 7}
B = {4, 6, 8, 9}
Then find
Solution:
(d) If AM of x – 6, x – 3, x + 3 and x + 6 is 10, then find x.
Solution:
(e) If then find
Solution:
3. Answer the following questions: 3x5=15
(a) If
Then find the sets A, B and C.
Solution:
Or
Show that
Solution:
(b) Find the probability of getting 53 Mondays in a leap year.
Solution:
Total no. of days in a leap year is 366 and therefore there are 52 weekend 2 additional days. These 2 additional days may be –
(c) Find the simple interest on Rs. 5,000 from 4th March, 1998 to 28th July 1998 @ 6% p.a.
(d) If
Then find,
Solution:
(e) Write down the second and fifth term of the following expansion:
Solution:
Or
Prove that
Solution: Same as 2012 Paper
4. In a questions paper, there are two groups A and B consisting of 5 questions each. In how many ways a student can answer 6 questions in all if he has to answer not less than 2 questions from each group? 5
Solution:
5. Using mathematical induction, prove that 5
Solution:
Or
Find the term involving in
Solution:
6. If median of the following distribution is 32.5 marks, then find the missing frequency 5
Marks : 1020 2030 3040 4050 5060
No. of students: 3 5 3 1
Solution:
10 – 20
20 – 30
30 – 40
40 – 50
50 – 60

3
5
3
1

3
8

7. Draw the graph of the following inequalities: 5
Solution: Same as 2013
Or
Solution:
x

2

7

3

Y

1

1

3

x

0

4

2

Y

1

1

0

8. A machine depreciates @ 10% p.a. of its value at the beginning of the year. If the original value of the machine Rs. 5,810 and scrap value is Rs. 2,250, find for how many years the machine was used. 5
9. Mr. Hazarika borrows Rs. 20,000 at 4% compound interest p.a. and agrees to pay both the principal and interest in 10 equal installments at the end of each year. Find the value of each installment.
10. A bag contains 4 gold, 3 silver and 3 copper coins. 2 coins are drawn at the ransom from it. Find the probability of drawing: 2+3=5
(i) 1 gold and 1 silver coin
(ii) 2 coins of the same type
Solution:
Or
2 dice are thrown simultaneously. Find the probability of getting –
(i) the number 5 on at least one of the dice;
(ii) a sum equal to at least 10 of the numbers on the two dice.
Solution:
11. (a) If SD of 1, 2, 3,............, n be , then find the value of n. 1
Solution:
(b) Why SD is called the best measure of dispersion? 2
(c) Find the mean deviation about arithmetic mean and its coefficient : 5
X : 15 19 23 27 31
F : 5 4 5 4 2
Solution:
15
19
23
27
31

5
4
5
4
2

75
76
115
108
62

6.8
2.8
1.2
5.2
9.2

34
11.2
6
20.8
18.4

Or
The mean of five observations is 4.4 and variance is 8.24. If three of the observations are 4, 6 and 9, then find the other two.
Solution:
12. (a) What type of correlation is involved in the following cases? Positive correlation / negative correlation / no correlation: 2
(i) Income and expenditure of some families
(ii) Weight and intelligence of a group of persons
(b) Find Karl Pearson’s correlation coefficient : 6
X : 1 2 3 4 5 6 7
Y : 3 5 6 8 10 11 13
Solution:
1
2
3
4
5
6
7

0
1
2
3

9
4
1
0
1
4
9

3
5
6
8
10
11
13

0
2
3
5

25
9
4
0
4
9
25

15
6
2
0
2
6
15

13. (a) Find the middle term of the following expansion : 5
Solution:
Or
Prove that
Solution: Same as 2013 Paper
(b) What types of average should be used in the following cases? 3
(i) Size of readymade shirts in a shop
(ii) Estimation of intelligence of students in a class
(iii) To find the average speed when time of journey is given
14. (a) Fill in the blanks :
(i) SD of 7, 7, 7, 19, 19, and 19 is ____.
Solution:
(ii) Coefficient of variation is expressed in ____.
(b) Prove that
Solution:
Or
Evaluate :
× ×
Solution:
(c) Prove, using Venn diagram : 2
Solution: