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:
 AM = Affected
 Median = Not affected
 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

3039
4049
5059
6069
7079
8089
9099

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

2535
3545
4555
5565
6575

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
