## Friday, March 16, 2018

### COMMERCIAL MATHEMATICS AND STATISTICS: 2015

2015
(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 A = { 1, 2, 3, 5 }, B = { 5, 3, 7, 8 } and C = { 2, 4, 7, 8 }, then find
Ans.
(b) Write True or False : Every diagonal matrix is a scalar matrix.
Ans. False
(c) Fill in the blank :If standard deviation of be , then SD of is .
(d) Fill in the blank : If , then
(e) When two sets A and B are said to be disjoint?
(f) Which decile is equal to median?
Ans.
(g) Write down the basic difference between a matrix and a determinant.
(h) Write True of False : If two sets are equal, then they will be equivalent.
Ans. Equivalent sets are not always equivalent equal set are always equivalent.

2. Answer the following questions : 2x5=10
(a) Evaluate :
Solution:
(b) Write down the subsets of the set given below : { 1, { 3, 5 }, 6 }
(c) Define symmetric matrix and give an example.
Ans. A square matrix A is said to be a symmetric matrix if
E.g.
(d) Find the harmonic mean of the following series :
Solution:
(e) If a and b are constants and x and y are variables and they are related as   then prove that
Solution:
3. If 3
Then show that
Solution:

4. Show that         3
Solution:
LHS RHS
5. There are 50 balls numbered from 1 to 50. One ball is drawn at random from these balls. Find the probability that the number on the ball is multiple of 4 or 6.         3
Solution:
Or
3 Coins are thrown simultaneously. Find the probability of getting at least 2 tails.
Solution:
6. Using properties of determinant, prove that         3
Solution:

7. Calculate mean deviation about median and its coefficient :         3
Income : 40, 48, 32, 64, 52
Solution:
 32 40 48 52 64 16 8 0 4 16

Or
The following are the heights of eleven students (in cm). Calculate quartile deviation : 124,  127, 126, 123, 127, 129, 125, 130, 132, 130, 121
Solution:
8. A machine costs a company Rs. 52,000 and its effective life is estimated to be 25 years. A sinking fund is created for replacing the machine by a new model at the end of its life  time, when its scrap realises a sum of Rs. 2,500 only. The price of the new model is estimated to be 25 % more than the price of the present one. What amount should be retained out of profit at the end of each year for the sinking fund, if it accumulated at  3.5% p.a. CI? 5
9. Find the coefficient of
Solution:
Or
Prove that
Solution: Same as 2012 Paper

10. Using mathematical induction, prove that         5
Solution:
11. Show that         5
Solution: Same as 2013 Paper
Or
If
, and
Then find the value of a and b
Solution:

12. How many different words may be formed by sing the letters of the word DAUGHTER taken all together if the vowels always remain together?         5
Solution:
Or
From 7 teachers and 5 students a committee of 6 is to be formed. In how many ways can this be done if it must include at most 4 teachers?
Solution:
13. Draw the graph of any one of the following inequalities:         5
(a)
Solution: Same as 2012 Paper
(b)
Solution:
 x 4 3 5 Y 0 1 -1 x 3 1 2 Y 1 -1 0

14. The mode of the following distribution is 65 inches. Find           5
Height (in inches) Frequency
60-62 12
62-64
64-66 32
66-68 16
68-70  8
Solution:
 60 – 62 62 – 64 64 – 66 66 – 68 68 – 70 12 32 16 8

15. (a) If, A = Amount
P = Principal
n = Number of years
r = Rate of interest
then write down the formula for finding compound interest if interest is compounded monthly.
(b) The difference between simple and compound interests on a certain sum of money @ 6% p.a. for 2 years is Rs. 13.50. Find the principal, simple interest and  the compound interest.       4+1+1=6

16. (a) What type of correlation exists between the following pairs  of variable – positive / negative / no correlation? 2
(i) Atmospheric temperature and sale of woolen garments
(ii) Colour of sari and intelligence of lady who wears it
(b) Find Karl Pearson’s correlation coefficient :         6
X : 12 9 8 10 11 13 7
Y : 14 8 6 9 11 12 3
Solution:
 12 9 8 10 11 13 7 2 0 1 3 4 1 4 0 1 9 9 14 8 6 9 11 12 3 5 0 2 3 25 1 9 0 4 9 36 10 1 6 0 2 9 18

17. (a) The AM and SD of 20 observations are found to be 20 cm and 5 cm, respectively. On checking, it was found that an item 13 was misread as 30. Find the correct AM and SD. 4
Solution:
Or
The mean and variance of five observations are 4.4 and 8.24 respectively. If three of them are 1, 2 and 6, then find the other two.
Solution:
(b)Find the standard deviation from the data given below :         4
: 15 25 35 45 55
: 18 17 27 23 15
Solution:
 15 25 35 45 55 18 17 27 23 15 0 10 20 0 230 300 7200 1700 0 2300 6000

18. (a) The simple interest on a sum of money is of the principal and the number of  years is double the rate percent of the interest. Find the rate of interest.         2
(b) For any two observations, prove that         2
Solution:
(c) Two cards are drawn from a pack of cards. Find the probability of getting a  diamond and club.     4
Solution:

Or
A pair of unbiased dice is thrown simultaneously at random. Find the probability that sum of the numbers on the two dice is at least equal to 10.
Solution: