AHSEC HS 2nd Year Question Paper 2019 Commercial Mathematics and Statistics

Total number of pages—12
29T CMST

2019

COMMERCIAL MATHEMATICS
AND STATISTICS

Full Marks : 100

Pass Marks : 30

Time : Three hours

The figures in the margin indicate full marks
for the questions.

Q. No. 1 1 mark each 1×8 = 8

Q. No. 2 2 marks each 2×5 = 10

Q. Nos. 3 – 7 3 marks each 3×5 = 15

Q. Nos. 8 – 14 5 marks each 5×7 = 35

Q. Nos. 15–18 8 marks each 8×4 = 32

Total =100

Contd.

1. Answer the following questions as directed : 1×8=8

Ó¬˘1 √õ∂ùüÀfl¬˝◊Ȭ±1 øÚÀ«˙ ’Ú≈¸±À1 Î◊¬M1 ø˚˛± :

(a) Is the set A   x x 1, x 1 a null set ?

A   x x 1, x 1 ¤˝◊ ¸—˝øÓ¬ÀȬ± ø1Mê ¸—˝øÓ¬ ˝˚˛ÀÚ∑

(b) Find the co-factor of –1 in the following determinant.

Ó¬˘1 øÚÌ«±˚˛fl¬Ó¬ –1 1 ¸˝1±ø˙ øÚÌ«˚˛ fl¬1±º

2 3 5
5 2 7
4 2 1

(c) What is the difference between simple interest and compound
interest ?

¸1˘ ¸≈Ó¬ ’±1n∏ ‰¬Sê¬ı‘øX√√ ¸≈Ó¬1 ¬Û±Ô«fl¬… øfl¬∑

(d) Write True or False :
qX√√ ŒÚ ’qX√√ ø˘‡± –
‘Every null matrix is a square matrix’

ë¸fl¬À˘± ˙”Ú… Œ˜Ã˘fl¬é¬ ¤È¬± ¬ı·« Œ˜Ã˘fl¬é¬í

(e) Fill in the gap :

‡±˘œ ͬ±˝◊ ¬Û≈1Ì fl¬1± –
A.M. × H.M. = ——————

¸˜±ôL1 ˜±Ò… × ˝1±Rfl¬ ˜±Ò… = ——————

29T CMST [2]

(f) What is the arithmetic mean of first n natural numbers ?

√õ∂Ô˜ n Ȭ± ¶§±ˆ¬±øªfl¬ ¸—‡…±1 ¸˜±ôL1 ˜±Ò… øfl¬˜±Ú∑

(g) What is the minimum value that the probability of an event
can take ?

¤È¬± ‚Ȭڱ1 ¸y±øªÓ¬±1 ˜±Ú Ú”…ÚÓ¬˜ øfl¬ ˝¬ı ¬Û±À1∑

(h) Fill in the gap :

‡±˘œ ͬ±˝◊ ¬Û≈1Ì fl¬1± –

Standard deviation is always ————— .

˜±Úfl¬ ø¬ı‰¬˘Ú ¸±˚˛ ————— ˝˚˛º

2. Answer the following questions in brief : 2×5=10

‰¬˜≈Õfl¬ Ó¬˘1 √õ∂ùüÀfl¬˝◊Ȭ±1 Î◊¬M1 ø˚˛± –

(a) What is the difference between  and    ?

 ’±1n∏    1 ˜±Ê1 ¬Û±Ô«fl¬… øfl¬∑

For what value of x the matrix A  ⎡⎢
5 x⎤
(b) will be singular ?
⎣  2 4 ⎥⎦

x 1 øfl¬ ˜±Ú1 ¬ı±À¬ı A  ⎡⎢
5 x⎤
¤È¬± √õ∂øÓ¬˜ Œ˜Ã˘fl¬é¬ ˝í¬ı∑
⎣  2 4 ⎥⎦

29T CMST [3] Contd.

(c) What is perpetual annuity ?

ø‰¬1¶ö±˚˛œ ¬ı±ø¯∏«fl¬œ øfl¬∑

(d) If n P6  30.n P4 , find n.

˚ø n P6  30  n P4 ˝˚˛ ŒÓ¬ÀôL n 1 ˜±Ú øÚÌ«˚˛ fl¬1±º

(e) If y  a  bx , show that y  a  bx , a and b are constants.

˚ø y  a  bx , ŒÓ¬ÀôL Œ‡≈›ª± Œ˚ y  a  bx , a ’±1n∏ b ÒËnªfl¬º

3. Find the simple interest on Rs. 6000 from 4th March, 2017 to 28th
July, 2017 @ 5% p.a. 3

6000 Ȭfl¬±1 4 ˜±‰«¬ 2017 1 ¬Û1± 28 Ê≈˘±˝◊ 2017 Δ˘Àfl ¬ıÂø1 5% ˝±1Ó¬ ¸1˘ ¸≈Ó¬ øÚÌ«˚˛
fl¬1±º

4. If n P4 : n 1P4  5 : 9 , find n. 3

˚ø n P4 : n 1P4  5 : 9 ˝˚˛ ŒÓ¬ÀôL n øÚÌ«˚˛ fl¬1±º

Or / Ú±˝◊¬ı±

If n Pr  132 and n Cr  66 , find n and r. 3

¬˚ø n Pr  132 ’±1n∏ n Cr  66 ˝˚˛ ŒÓ¬ÀôL n ’±1n∏ r øÚÌ«˚˛ fl¬1±º

29T CMST [4]

5. Without expanding show that 3
¬ø¬ıô¶‘øÓ¬ Úfl¬1±Õfl¬ Œ‡≈›ª± Œ˚

0 c b
c 0 a 0
b a 0

Or / Ú±˝◊¬ı±

If 3
˚ø

⎡3 5⎤ ⎡ 2 5⎤ ⎡1 0⎤
⎢⎣ 1 y ⎥⎦ ⎢⎣  1 x ⎥⎦  ⎢⎣ 0 1 ⎥⎦ ,

what is the value of x and y ?

x ’±1n∏ y 1 ˜±Ú øfl¬∑

6. Find the Mean Deviation from median from the given marks of 7
students. 3
7 ÊÚ Â±S1 ø˚˛± Ú•§11 ¬Û1± ˜Ò…˜±1 ¬Û1± ·Î¬ˇ ø¬ı‰¬˘Ú øÚÌ«˚˛ fl¬1±º

18, 26, 15, 20, 17, 12, 25

7. Write any two algebraic properties of Arithmetic Mean. 3
¬¸±˜ôL1 ˜±Ò…1 ø˚Àfl¬±ÀÚ± ≈Ȭ± ¬ıœÊ·øÌÓ¬œ˚˛ Ò˜« ø˘‡±º

Or / Ú±˝◊¬ı±

Calculate the Harmonic Mean of 4, 8, and 12. 3

4, 8, ’±1n∏ 121 ˝1±Rfl¬ ˜±Ò… øÚÌ«˚˛ fl¬1±º

29T CMST [5] Contd.

8. The value of a machine at the end of a year becomes 90% of its
value at the beginning of that year. The machine was bought at Rs.
4800 and after using it for some years it was sold at Rs. 1800. How
many years was the machine in use ? 5

¤È¬± Œ˜ø‰¬Ú1 ˜”˘… ¬ıÂ1ÀȬ±1 Œ˙¯∏Ó¬ ’ªé¬˚˛1 ø¬ÛÂÓ¬ ¬ıÂ11 ’±1yøÌ1 ˜”˘…1 90% ˝˚˛Õ·º
Œ˜ø‰¬ÚÀȬ± 4800 Ȭfl¬±Ó¬ øfl¬øÚ øfl¬Â≈∏ ¬ıÂ1 ¬ı…ª˝±11 ø¬ÛÂÓ¬ 1800 Ȭfl¬±Ó¬ ø¬ıSêœ fl¬ø1 ø˚˛± ˝í˘º
Œ˜ø‰¬ÚÀȬ± øfl¬˜±Ú ¬ıÂ1 ¬ı…ª˝±1 fl¬1± Δ˝ø¢∑

9. 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 at least 2 ladies
are to be included ? 5
7 ÊÚ ˆ¬^À˘±fl¬ ’±1n∏ 4 ·1±fl¬œ ˜ø˝˘±1 ¬Û1± 6 ÊÚœ˚˛± fl¬ø˜øȬ ¤È¬± ·Í¬Ú fl¬ø1¬ı ˘±À·º ˚ø
fl¬ø˜øȬӬ fl¬À˜› 2 ·1±fl¬œ ˜ø˝˘± ’ôLˆ«≈¬Mê fl¬ø1¬ı ˘±À·, ŒÓ¬ÀôL fl¬ø˜øȬ‡Ú øfl¬˜±Ú Ò1ÀÌ ·Í¬Ú
fl¬ø1¬ı ¬Û1± ˚±¬ı∑

10. Mr. Roy borrows Rs. 20,000 at 4% compound interest and agrees
to pay both principal and interest in 10 equal annual instalments
at the end of each year. Find the amount of each instalment.
Given  1  04  10  0  6761 5

ø˜. 1À˚˛ ¬ıÂø1 4% ‰¬Sê¬ı‘øX ¸≈Ó¬1 ˝±1Ó¬ 20,000 Ȭfl¬± Ò±1Õ˘ ˘íÀ˘º ŒÓ¬›“ 10 Ȭ± ¸˜±Ú
¬ıÂÀ1fl¬œ˚˛± øfl¬øô¶Ó¬ ˜”˘ÒÚ ’±√√1n∏ ¸≈Ó¬ ¬Ûø1À˙±Ò fl¬ø1¬ıÕ˘ ø¬ı‰¬±ø1À˘º ˚ø ŒÓ¬›“ √õ∂øÓ¬ÀȬ± øfl¬øô¶
¬ıÂ11 Œ˙¯∏Ó¬, ¬Ûø1À˙±Ò fl¬À1, ŒÓ¬ÀÚ˝íÀ˘ √õ∂øÓ¬ÀȬ± øfl¬øô¶1 ¬Ûø1˜±Ì øÚÌ«˚˛ fl¬1±º
ø˚˛± ’±À  1  04 10  0  6761

11. Show that there will be no term containing x 9 in the expansion of
20
⎛ 2x 2  1 ⎞
⎜ ⎟ . 5
⎝ x ⎠
20
Œ‡≈›ª± Œ˚ ⎛⎜ 2x 2  ⎞⎟
1
1 ø¬ıô¶‘øÓ¬Ó¬ x 9 Ôfl¬± Œfl¬±ÀÚ± ¬Û Ú±Ô±Àfl¬º
⎝ x ⎠

29T CMST [6]

Or / Ú±˝◊¬ı±

If the coefficients of x 2 and x 3 in the expansion of  3  kx 9 are
equal, find the value of k. 5

˚ø  3  kx 9 1 ø¬ıô¶‘øÓ¬Ó¬ x 2 ’±1n∏ x 3 1 ¸˝· ¤Àfl¬ ˝˚˛ ŒÓ¬ÀôL , k 1 ˜±Ú øÚÌ«˚˛ fl¬1±º

12. Prove by mathematical induction that the sum of first n odd

natural numbers is n 2 . 5

·±øÌøÓ¬fl¬ ’±À¬ı˙ Ó¬N1 Z±1± √õ∂˜±Ì fl¬1± Œ˚ √õ∂Ô˜ n Ȭ± ’˚≈¢¨ ¶§±ˆ¬±øªfl¬ ¸—‡…±1 Œ˚±·Ù¬˘ n 2 .

Or / Ú±˝◊¬ı±

Using mathematical induction prove that 5

·±øÌøÓ¬fl¬ ’±À¬ı˙ Ó¬N ¬ı…ª˝±1 fl¬ø1 √õ∂˜±Ì fl¬1± Œ˚

n n  1 2n  1 
12  22  32  ....  n 2 
6

for all n  N
¸fl¬À˘± ¶§±ˆ¬±øªfl¬ ¸—‡…± n 1 ¬ı±À¬ıº

13. Draw the graph of : (any one) 5

Œ˘‡ ’—fl¬Ú fl¬1± – [ø˚Àfl¬±ÀÚ± ¤È¬±]

(i) x  2y  3, 3x  4y  12, x  0, y  1

(ii) 5x  4y  20, x  1, y  2

29T CMST [7] Contd.

14. Calculate the missing frequency, you are given that arithmetic
mean is 50  9 . 5

˘≈5 ¬ı±1—¬ı±1Ó¬±ÀȬ± øÚÌ«˚˛ fl¬1±, ø˚˛± ’±À Œ˚ ¸±˜ôL1 ˜±Ò… 50  9 .

Marks
: 0  20 20  40 40  60 60  80 80  100
Ú•§1
Frequency
: 5  30 12 8
¬ı±1—¬ı±1Ó¬±

Or / Ú±˝◊¬ı±

If f  x   x 2  5x  6 , find f  A  . 5

˚ø f  x   x 2  5x  6 ˝˚˛, ŒÓ¬ÀôL f  A  øÚÌ«˚˛ fl¬1±º

⎡ 2 0 1⎤
A  ⎢2 1 3⎥
⎢ ⎥
⎣ 1 1 0 ⎦

15. (a) Prove that 4

√õ∂˜±Ì fl¬1± Œ˚
n
Pr  n 1Pr  r . n 1Pr 1

(b) Solve : 4

¸˜±Ò±Ú fl¬1± –

a a x
m m m 0
b x b

29T CMST [8]

Or / Ú±˝◊¬ı±

If A   a , b , B   2, 3  and C  1, 2  , show that 4

˚ø A   a , b , B   2, 3  ’±1n∏ C  1, 2  ˝˚˛, Œ‡≈›ª± Œ˚
A   B  C    A  B    A C 

16. (a) Calculate Standard Deviation from the following data :
4

Ó¬˘Ó¬ ø˚˛± Ó¬Ô…1 ˜±Úfl¬ ø¬ı‰¬˘Ú øÚÌ«˚˛ fl¬1± –

Class
Œ|Ìœ : 10  19 20  29 30  39 40  49 50  59 60  69 70  79
Frequency
: 3 61 223 137 53 19 4
¬ı±1—¬ı±1Ó¬±

(b) Calculate coefficient of variation from the data given in
Question 16. (a). 4

¬1√6. (a) õ∂ùü1 Ó¬Ô…1 ¬Û1± ø¬ı‰¬1Ì &̱—fl¬ øÚÌ«˚˛ fl¬1±º

17. (a) Two dice are thrown simultaneously. Find the probability of
getting even number on both the dice. 4

≈Ȭ± ˘≈Î≈¬&øȬ ¤Àfl¬˘À· ø˘›ª± ˝í˘º ≈À˚˛±È¬± ˘≈Î≈¬&øȬӬ ˚≈¢¨ ¸—‡…± Œ¬Û±ª±1 ¸y±øªÓ¬±
øfl¬˜±Ú∑

29T CMST [9] Contd.

(b) A bag contains 4 red, 3 blue and 3 white balls. 2 balls are
drawn at random from it. Find the probability of getting

(i) 2 red and 1 blue ball

(ii) 2 balls of the same colour. 4

¤È¬± Œ˜±Ú±Ó¬ 4 Ȭ± 1„±, 3 Ȭ± Úœ˘± ’±1n∏ 3 Ȭ± ¬ı·± ¬ı˘ ’±Àº ˚±‘ø2Âfl¬ ˆ¬±À¬ı 2 Ȭ± ¬ı˘
Ȭڱ ˝˘, Ó¬˘Ó¬ ø˚˛± Ò1ÀÌ ¬ı˘ Œ¬Û±ª±1 ¸y±øªÓ¬± øÚÌ«˚˛ fl¬1±º
(i) 2 Ȭ± 1„± ’±1n∏ 1Ȭ± Úœ˘±

(ii) ¤Àfl¬ ¬ı1Ì1 ¬ı˘ 2 Ȭ±º

Or / Ú±˝◊¬ı±

Define and give one example of each : 2+2=4

¬¸—:± ø˘‡± ’±1n∏ √õ∂øÓ¬ÀȬ±1 ¤È¬± Î◊¬±˝1Ì ø˚˛± –
(i) Mutually exclusive events

¬Û1¶Û1 ¬ıø˝ˆ«”¬Ó¬ ‚Ȭڱ
(ii) Positive correlation

ÒÚ±Rfl¬ ¸˝¸•§g¬

18. (a) Find out Karl Pearson’s coefficient of correlation. 6

fl¬±˘« ø¬ÛÀ˚˛1‰¬Ú1 ¸˝¸•§g¬&̱—fl¬ øÚÌ«˚˛ fl¬1± º

x: 2 2 4 5 5
y: 6 3 2 6 4

29T CMST [ 10 ]

(b) Karl Pearson’s coefficient of correlation between two variables

x and y is 0  28 and their covariance is 7  6 . If the variance
of x is 9, find the standard deviation of y. 2

fl¬±˘« ø¬ÛÀ˚˛1‰¬Ú1 ¸˝¸•§g¬&̱—fl¬ ≈Ȭ± ‰¬˘fl¬ x ’±1n∏ y 1 ˜±Ê1 Œ¬Û±ª± ·˘ 0  28 ’±1n∏
ø¸“˝Ó¬1 ¸˝ø¬ı‰¬±˘Ú 7  6 º ˚ø x 1 √õ∂¸1Ì 9 ˝˚˛, ŒÓ¬ÀôL y 1 ˜±Úfl¬ ø¬ı‰¬˘Ú øfl¬˜±Ú∑

——— ×———

29T CMST [ 11 ]

29T CMST [ 12 ] 8+