# COMMERCIAL MATHEMATICS AND STATISTICS: 2013

2013
H.S. 2nd Year
(COMMERCIAL MATHEMATICS AND STATISTICS)
Full Marks: 100
Time: 3 hours

The figures in the margin indicate full marks for the questions.
1. (a) Simplify:        1
Solution:
Solution:
(b) For what value of x the matrix        1

Will be singular?

Solution:
(c) Evaluate:        1
X
Solution:
(d) If a dice and a coin are thrown simultaneously, write down the sample space.   1
Solution:
(e) If SD of be Ïƒ, what will be        1
SD of
Solution:

(f) Which of the following is affected by extreme values?        1
(i) Arithmetic mean
(ii) Median
(iii) Mode

Solution:
1. AM = Affected
2. Median = Not affected
3. Mode = Not affected

(g) Write down the formula of amount (A) on principal (P) for (n) years @ r% p.a. compounded monthly.        1
Solution:
(h) Whether the above statement is true or false?        1
Solution:
2. Answer the following questions: 2x5=10
(a) If cov (x, y) =0.9, then find the value of r.
Solution:

(b) Find mean deviation about median of the following data: Wt (kg) 3, 4, 8, 10, 12
Solution:
 3 4 8 10 12 5 4 0 2 4
(c) In the determinant given below, what are the minor and cofactor of the element – 2?
Solution:
(d) Find x and y, if
=
Solution:

(e) Find the weighted AM of 1,2,3,4 with corresponding weights 4, 3, 2, and 1 respectively. 2
Solution:
3. Answer any five parts of the following: 3x5=15
(a) If
A = {2, 4, 5, 6}
B = {3, 4, 6, 9, 10}
C = {2, 6, 7, 10}

Find
(i)
(ii)
Solution:
(b) Form a matrix A of order 3x2, where
Solution:
(c) Prove that
=
Solution:

(d) AM and GM of any two positive numbers are 10 and 8 respectively. Find the two numbers.

(e) Two dice are thrown simultaneously. Find the probability of getting an odd number on one dice and an even number on the other.
Solution:
(f) Each side of a square field is of length 100 meters. A cyclist covers the four sides at speed of 50 meters, 20 meters, 10 meters, and 25 meters per hour respectively. Find his average speed.
Solution:
4. Prove that:        5
Solution:
Or
Solve by using Cramer’s rule:
2x – 3y = 5
4y + 5z = 6
6x + 7z =20
Solution:

5. Using mathematical induction, prove that        5
Solution:
6. Find the coefficient of in
Solution:
Or
If in the expansion
The coefficient of is 160, then find the value of K.
Solution:
7. How many numbers less than 700 may be formed by using the digits 3,4,5,6,7,8?       5

8. Find r from the following data:        5
X : 6 2 10 4 8
Y : 9 11 5 8 7
Solution:
 6 2 10 4 8 0 4 2 0 16 16 4 4 9 11 5 8 7 1 3 0 1 9 9 0 1 0 0

9. A man deposited Rs. 400 every year in a fund at 3% p.a. compounded annually. After some years he received Rs. 4,064 as amount from the fund. Find the number of years.        5

10. There are 20 pens out of which 5 are defective, 3 pens are picked at random. Find the probability of selecting:
(i) all defective pens;
(ii) exactly I defective pen.
Solution:
5
Or
2 cards are drawn from a pack of cards. Find the probability of getting 2 cards of  same colour.
Solution:
11. Write down the definition of sinking fund and perpetuity/ The difference between simple and compound interest on a certain sum of money for 3 years at 5% p.a. is Rs. 213.50. Find the principal, simple interest and compound interest.    8

12. (a) Draw the graph of the following linear inequalities. 4+4=8
(i)
Solution:
 x 6 4 2 Y 2 3 4 x 1 0 2 Y 0 1 -1

Diagram

Or
(ii)

Solution:
 x 5 0 10 Y 0 3 -3 x 2 1 3 Y 0 2 -2

(b) Prove that
Solution:
13. (a) From the following data , find and if AM of the following distribution is 72.5 marks :

 Marks No. of Students 30-39 40-49 50-59 60-69 70-79 80-89 90-99 2 3 11 32 7 Total =100

Solution:
 Marks 29.5 – 39.5 39.4 – 49.5 49.5 – 59.5 59.5 – 69.5 69.5 – 79.5 79.5 – 89.5 89.5 – 99.5 2 13 11 32 7 34.5 44.5 54.5 64.5 74.5 84.5 94.5 0 10 20 30 0 1 2 3 0 32 21 100
(b) Write two demerits of mode.
14. (a) If mean deviation for a group of 50 items is 16.2, what will be their SD?         1
Solution:
(b) Write four properties of a good measure of dispersion.         2
(c) From the following data, find standard deviation and CV.         5
 X : 10 11 12 13 14 15 16 Y : 2 7 11 15 10 4 1

Solution:
 10 11 12 13 14 15 16 2 7 11 15 10 4 1 0 1 2 3 0 10 8 3 18 28 11 0 10 16 9

Or
Find quartile deviation from the following data:

 Height (in cm) Frequency 25-35 35-45 45-55 55-65 65-75 2 10 25 16 7

Solution:
 25 – 35 35 – 45 45 – 55 55 – 65 65 – 75 2 10 25 16 7 2 12 37 53 60