2012
( COMMERCIAL MATHEMATICS AND STATISTICS )
( COMMERCIAL ARITHMETIC AND ELEMENTS OF STATISTICS )
Full Marks : 100
Time : 3 hours
The figures in the margin indicate full marks for the questions.
( GROUP – A : New Course )
( COMMERCIAL MATHEMATICS AND STATISTICS )
1. (a) Find 
if 
1
if 1
Solution:
(b) Find x if 
1
1
Solution:
(c) Is the set A = {x: x + 5 = 5} null? 1
Ans. No, It is single ton set having the only element 0.
(d) Every scalar matrix is a diagonal matrix. (Write True or False) 1
Ans. True.
(e) Define a symmetric matrix. Give example also. 1
Ans. A square matrix A is said to be a symmetric matrix if
E.g. 
(f) If AM of 
is 
, then AM of 
is 
. (Fill in the blank). 1
is , then AM of is . (Fill in the blank). 1
(g) SD of 1, 2, 3, 4, and 5 is
(i) 2
(ii) 
(iii) 4 ( Choose the correct answer ) 1
Solution:
(h) 
( Fill in the blank ) 1
( Fill in the blank ) 1
(i) If two dice are thrown at random, they can fall together in 36 number of ways. ( Fill in the blank )1
Solution: 
(j) If the two variables x and y are independent, then value of r will be ___0._.( Fill in the blank ) 1
2. Write down two differences between a matrix and a determinant. 2
3. Find the value of x, when 
and 
2
and 2
Solution:
4. Find n if 
2
2
Solution:
5. Are the following two sets disjoint? 2
A = { x : 3x – 8 = 0 }
B = { x : x is a prime number < 5 }
Solution:
6. Arithmetic mean of samples of sizes 50 and 75 are 60 and x respectively. If the arithmetic mean of 125 observations of both the samples taken together be 54, find x. 2
Solution:
7. If 
.Find coefficient of variation. 2
.Find coefficient of variation. 2
Solution:
8. Find the simple interest on Rs. 6,000 from 4th March, 1992 to 28th July, 1992 at the rate of 5% p.a. 3
Solution:
9. Write down the first four terms of the expansion given below : 3
Solution:
10. Let A = { p, q, r }, B = { s }, C = { r, s }, 3
D = { p, q, s }, E = { p, q }
State whether the following statements are true or false :
(i)
(ii) 
(iii)
(iv)
(v)
(vi)
Solution:
11. Solve : 3
Solution:
12. AM of the following distribution is 17 years. Find the value of x : 3
Age (years) : 8 20 26 29
No. of Persons : 3 2 x 1
Solution:
 |  |  |
8
20
26
29
|
3
2
1
|
24
40
26
29
|
13. Prove that 4
Solution:
Or
Given
Find a and b if 2A + 5B = C.
Solution:
14. Find the middle term of expansion : 4
Solution:
Or
Show that
Solution:
15. Prove by mathematical induction that 4
Solution:
16. Mohit borrows Rs. 12,500 at 5% p.a. CI and agrees to replay the loan with interest in 5 equal annual installments, the first payment being made at the end of first year. Find the value of each yearly installment to be paid by him. 5
17. Write down the formula for finding the amount (A) on a principal (P) at r% p.a. compound interest, interest being compounded monthly for a period of n years.
The value of machine at the end of year becomes 90% of its value at the beginning of that year. The machine was bought at Rs. 4,800 and after using it for some years it was sold at Rs. 1,800. For how many years the machine was in use? 2+5=7
18. Draw the graph of the following inequalities : 5
Solution:
x
|
3
|
0
|
6
|
Y
|
0
|
8
|
-8
|
Or
Draw the graph of the following inequalities :
Solution:
x
|
3
|
6
|
0
|
Y
|
0
|
2
|
-2
|
x
|
6
|
3
|
0
|
Y
|
0
|
1
|
2
|
19. Find median and mode of the following frequency distribution : 3+3=6
Height (in inches)
|
Frequency
|
30-40
40-50
50-60
60-70
70-80
80-90
|
18
37
45
27
15
8
|
Solution:
 |  |  |
30 – 40
40 – 50
50 – 60
60 – 70
70 – 80
80 – 90
|
18
37
45
27
15
8
|
18
55
100
127
142
150
|
20. A committee of 6 is to be formed out of 7 gentlemen and 4 ladies. In how many ways can the committee be formed, if:
(a) at least 2 ladies are to be included;
(b) at most 2 ladies are to be included ? 3+3=6
Solution:
(b)
Or
Find the coefficient of 
in
in
Solution:
21. (a) Find the mean and standard deviation of the following frequency distribution : 3+3=6
Marks
|
Frequency
|
60-62
62-64
64-66
66-68
68-70
|
34
27
20
13
6
|
Solution:
 |  |
Min value
|  |  |  |
60 – 62
62 – 64
64 – 66
66 – 68
68 – 70
|
7
7
7
7
6
|
61
63
65
67
69
|  
0
1
2
|  
0
7
12
|
28
7
0
7
24
|
 |  |  |
(b) In which factory is there more variability in the distribution of wages? 3
Factory A
|
Factory B
| |
Mean Income (Rs.)
Standard Deviation (Rs.)
|
80
5
|
90
7
|
Solution:
22. (a) Three coins are thrown simultaneously at random. Write down the sample space. 3
(b) A pair of unbiased dice is thrown at random. If two numbers appearing be different, find the probability that the sum is 8. 4
(c) Find the Karl Pearson’s Correlation coefficient from the data given below : 6
X : 2 4 5 6 8 11
Y : 18 12 10 8 7 5
Solution:
 |  |  |  |  |  |  |
2
4
5
6
8
11
|   
0
2
5
|
16
4
1
0
4
25
|
18
12
10
8
7
5
|
8
2
0
  |
64
4
0
4
9
25
|  
0
0
  |
 |  |  |  |  |
( GROUP – B : Old Course )
( COMMERCIAL ARITHMETIC AND ELEMENTS OF STATISTICS)
23. (a) If 
, find x. 1
, find x. 1
(b) Find x if 
1
1
(c) Is the set A = { x : x + 5 = 5 } null? 1
(d) Banker’s gain = _____ on TD. (Fill in the blank ) 1
(e) Write down the nit matrix 
1
1
(f) 
( Fill in the blank ) 1
( Fill in the blank ) 1
(g) _____ is called the average of position. ( Fill in the blank ) 1
(h) If 
find 
1
find 1
(i) If two dice are thrown at random, they can fall together in _____ number of ways.(Fill in the blank ) 1
(j) What is the income obtained from Rs. 4,500, 4% stock at Rs. 120? 1
24. Prove that 
2
2
25. Write down two differences between a matrix and a determinant. 2
26. Find SD if 
2
2
27. Are the following two sets disjoint? 2
A = { x : 3x – 8 = 0 }
B = { x : x is a prime number < 5 }
28. Arithmetic mean a samples of sizes 50 and 75 are 60 and x respectively. If the arithmetic mean of 125 observations of both the samples taken together be 54, find x.
29. If A = { 1, 2, 3, 4 } 2
B = { 2, 4, 6, 7 }
C = { 3, 4, 5, 7}
Find (i) 
(ii) 
30. Solve : 
3
3
31. Evaluate : 
3
3
32. AM of the following distribution is 17 years. Find the value of x : 3
Age (years) : 8 20 26 29
No. of Persons : 3 2 x 1
33. Define with examples : 3
(a) Row Matrix
(b) Column Matrix
(c) Scalar Matrix
34. Mention three advantages of standard deviation. 3
35. Calculate mean deviation about median of the following numbers : 3
45, 37, 69, 57, 53.
36. Prove That 4
37. (a) A certain sum of money amounts to Rs. 8,820 and Rs. 9,261 at a certain rate of compound interest compounded annually in 2 years and 3 years respectively. Find the sum and the rate of interest. 5
(b) What do you mean by ‘yield’ in case of stock? 1
(c) A person invests equal sums in the 4% stock and 4½% stock and obtains equal incomes. If the 4% stock is at 4% discount, find the market price of 4½% stock. 5
38. A banker discounts a bill 73 days before maturity. If the discounted value of the bill be Rs. 990, find the face value of the bill, rate of interest being 5% p.a. 5
39. Mohit borrows Rs. 12,500 at 5% p.a. compound interest and agrees to pay in 5 equal annual installments, the first payment being made at the end of first year. Find the value of each annual installment to be paid by him. 5
40. (a) Find mean and standard deviation of the following frequency distribution : 4+4=8
Marks
|
Frequency
|
60-62
62-64
64-66
66-68
68-70
|
34
27
20
13
6
|
(b) Give two examples where range is used. 2
(c) The AM and GM of two numbers are 30 and 18 respectively. Find their HM and also the two numbers. 6
41. (a) Three coins are thrown simultaneously at random. Write down the sample space. 3
(b) A pair of unbiased dice is thrown at random. If two numbers appearing be different, find the probability that the sum is 8. 4
42. The mean and SD of a group of 20 items were calculated by a student as 20 cm and 5 cm respectively. But while calculating them, an item 13 was misread as 30. Find the correct mean and standard deviation. 4
43. Simplify : 
3
3
44. If
Prove that
5
Or
If
Find matrix X such that 3A – 4B – 5X = 0