12th Business Maths Model Exam -2

St. Britto Hr. Sec. School - Madurai

12th Business Maths Model Exam -2-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

The starting annual salaries of newly qualified chartered accountants (CA’s) in South Africa follow a normal distribution with a mean of Rs.180,000 and a standard deviation of Rs. 10,000. What is the probability that a randomly selected newly qualified CA will earn between Rs.165,000 and Rs.175,000 per annum?

0.819
0.242
0.286
0.533

Laspeyre’s index = 110, Paasche’s index = 108, then Fisher’s Ideal index is equal to:

For a normal distribution if the mean is m, mode is n and median is m, then

m > m₁ > m₀
m < m₁ < m₀
m \(\neq\) m₁ x m₀
m = m₁ = m₀

In 8 throws of a die, 1 or 3 is considered a success. Then the mean number of successes is

\(\frac83\)
\(\frac43\)
\(\frac23\)
\(\frac53\)

The starting annual salaries of newly qualified chartered accountants (CA's) in South Africa follow a normal distribution with a mean of Rs.1,80,000 and a standard deviation of Rs.10,000. What is the probability that a randomly selected newly qualified CA will earn between Rs.1,65,000 and Rs.1,75,000 per annum?

0.819
0.242
0.286
0.533

Cape town is estimated to have 21% of homes whose owners subscribe to the satellite service, DSTV. If a random sample of your home is taken, what is the probability that all four home subscribe to DSTV?

0.2100
0.5000
0.8791
0.0019

In a non – degenerate solution number of allocations is

Equal to m+n–1
Equal to m+n+1
Not equal to m+n–1
Not equal to m+n+1

A statistical analysis of long-distance telephone calls indicates that the length of these calls is normally distributed with a mean of 240 seconds and a standard deviation of 40 seconds. What proportion of calls lasts less than 180 seconds?

0.214
0.094
0933
0.067

The probability density function p(x) cannot exceed

zero
one
mean
infinity

A die is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of number of success is

\(\frac83\)
\(\frac38\)
\(\frac45\)
\(\frac54\)

The standard normal distribution is represented by

N(0, 0)
N(1, 0)
N(0, 1)
N(1, 1)

Δf(x) =

f(x+ h)
f(x) − f(x+h)
f(x + h) − f(x)
f (x) − f(x−h)

Value which is obtained by multiplying possible values of random variable with probability of occurrence and is equal to weighted average is called

Discrete value
Weighted value
Expected value
Cumulative value

If n > 0, then \(\Gamma \)(n) is
\(\int _{ 0 }^{ 1 }{ { e }^{ -x }x^{ n-1 } } \) dx

\(\int _{ 0 }^{ 1 }{ { e }^{ -x }{ x }^{ n } } dx\)

\(\int _{ 0 }^{ \infty }{ { e }^{ x }{ x }^{ -n } } \ dx\)

\(\int _{ 0 }^{ \infty }{ { e }^{ -x }{ x }^{ n-1 } } \ dx\)
The solution of the differential equation \(\frac { dy }{ dx } \) + Py = Q where P and Q are the function of x is
y = ഽQe^ഽPdxdx + c

y = ഽQe^-^ഽPdxdx + c

ye^ഽPdx= ഽQe^ഽPdxdx + c

yeഽPdx=ഽQe^-ഽPdxdx + c
The P.I of (3D²+ D − 14)y = 13e^2x is
\(\frac {x}{2}\)e^2x

xe^2x

\(\frac {x^2}{2}\)e^2x

13xe^2x

A discrete probability distribution may be represented by

table
graph
mathematical equation
all of these

\(\overset { \infty }{ \underset { -\infty }{ \int } } \)f(x)dx is always equal to

zero
one
E(X)
f(x) +1

Which of the following statements is/are true regarding the normal distribution curve?

it is symmetrical and bell shaped curve
it is asymptotic in that each end approaches the horizontal axis but never reaches it
its mean, median and mode are located at the same point
all of the above statements are true.

E[X-E(X)] is equal to

E(X)
V(X)
0
E(X)-X

Part - B

Generic 2 - Answer All Question

The price elasticity of demand for a commodity is \(\frac { p }{ { x }^{ 3 } } \). Find the demand function if the quantity of demand is 3, when the price is Rs. 2

Evaluate Δ\(\left[ \frac { 1 }{ (x+1)(x+2) } \right] \) by taking ‘1’ as the interval of differencing

A sample of 1000 students whose mean weight is 119 lbs(pounds) from a school in Tamil Nadu State was taken and their average weight was found to be 120 lbs with a standard deviation of 30 lbs. Calculate standard error of mean.

X is normally distributed with mean 12 and sd 4. Find P(X \(\le\) 20) and P(0 \(\le\) X \(\le\) 12)

A random variable X has the probability mass function

X -2 3 1

P (X = x) \(\frac { k }{ 6 }\) \(\frac { k }{ 4 }\) \(\frac { k }{ 12 }\)

then find k

In an entrance examination a student has to answer all the 120 questions. Each question has four options and only one option is correct. A student gets 1 mark for a correct answer and loses \(\frac{1}{2}\) mark for a wrong answer. What is the expectation of the mark scored by a student if he chooses the answer to each question at
random?

What is statistic?

Define Bernoulli trials.
Find the order and degree of the following differential equations.
\({ \left( \frac { dy }{ dx } \right) }^{ 3 }+y=x-\frac { dx }{ dy } \)

Solve the following equation by using Cramer’s rule
2x + y −z = 3, x + y + z =1, x− 2y− 3z = 4

Solve yx²dx + e^−xdy = 0

Two types of soaps A and B are in the market. Their present market shares are 15% for A and 85% for B. Of those who bought A the previous year, 65% continue to buy it again while 35% switch over to B. Of those who bought B the previous year, 55% buy it again and 45% switch over to A. Find their market shares after one year and when is the equilibrium reached?

State the uses of Index Number.

Define random variable.

X is a normally normally distributed variable with mean μ = 30 and standard deviation σ = 4. Find
(a) P(x < 40)
(b) P(x > 21)
(c) P(30 < x < 35)

Part - C

Generic 3 - Answer 3 Questions

If a random variable. X has the probability distribution

X 0 1 2 3 4 5

P (X = x) a 2a 3a 4a 5a 6a

then find F(4)

You are given below the values of sample mean ( X ) and the range ( R ) for ten samples of size 5 each. Draw mean chart and comment on the state of control of the process.

Sample number 1 2 3 4 5 6 7 8 9 10

\(\overset{-}{X}\) 43 49 37 44 45 37 51 46 43 47

R 5 6 5 7 7 4 8 6 4 6

Given the following control chart constraint for :n=5, A₂=0.58, D₃=0 and D₄=2.115

If X is a normal variate with mean 30 and SD 5. Find the probabilities that
(i) 26≤X≤40
(ii) X > 45

Using the following random number table (Kendall-Babington Smith)

23 15 75 48 59 01 83 72 59 93 76 24 97 08 86 95 23 03 67 44

05 54 55 50 43 10 53 74 35 08 90 61 18 37 44 10 96 22 13 43

14 87 16 03 50 32 40 43 62 23 50 05 10 03 22 11 54 36 08 34

38 97 67 49 51 94 05 17 58 53 78 80 59 01 94 32 42 87 16 95

97 31 26 17 18 99 75 53 08 70 94 25 12 58 41 54 88 21 05 13

Draw a random sample of 10 four- figure numbers startingfrom 1550 to 8000.

The following data gives readings of 10 samples of size 6 each in the production of a certain product. Draw control chart for mean and range with its control limits.

Sample 1 2 3 4 5 6 7 8 9 10

Mean 383 508 505 582 557 337 514 614 707 753

Range 95 128 100 91 68 65 148 28 37 80

Part - D

Generic 5 - Answer All Question

The mean weight of 500 male students in a certain college is 151 pounds and the S.D is 15 pounds. Assuming the weights are normally distributed, find how many students weight
(i) between 120 and 155 pounds
(ii) more than 185 pounds.
The probability distribution of the discrete random variables X and Y are given below.

X 0 1 2 3

P(X) \(\frac { 1 }{ 5 } \) \(\frac { 2 }{ 5 } \) \(\frac { 1 }{ 5 } \) \(\frac { 1 }{ 5 } \)

Y 0 1 2 3

P(Y) \(\frac { 1 }{ 5 } \) \(\frac{3}{10}\) \(\frac { 2 }{ 5 } \) \(\frac{1}{10}\)

Prove that E(Y²) = 2 E(X).

A sample of 100 measurements at breaking strength of cotton thread gave a mean of 7.4 and a standard deviation of 1.2 gms. Find 95% confidence limits for the mean breaking strength of cotton thread.

the Laspeyre’s, Paasche’s and Fisher’s price index number for the following data. Interpret on the data.

Commodities Price Quandity

2000 2010 2000 2010

Rice 38 35 6 7

Wheat 12 18 7 10

Rent 10 15 10 15

Fuel 25 30 12 16

Miscellaneous 30 33 8 10

Four coins are tossed simultaneously. What is the probability of getting
a) atleast 2 heads b) atmost 2 heads.

Consider a random variable X with probability density function

f(x)={\({ 4x }^{ 2 },\quad if\quad 0<x<1\\ 0,\quad otherwise\)
Find E(X) and V(X)

Determine the mean and variance of the random variable Xhaving the following probability distribution.

X=x 1 2 3 4 5 6 7 8 9 10

P(x) 0.15 0.10 0.10 0.01 0.08 0.01 0.05 0.02 0.28 0.20

An auto company decided to introduce a new six cylinder car whose mean petrol consumption is claimed to be lower than that of the existing auto engine. It was found that the mean petrol consumption for the 50 cars was 10 km per litre with a standard deviation of 3.5 km per litre. Test at 5% level of significance, whether the claim of the new car petrol consumption is 9.5 km per litre on the average is acceptable.

The probability density function of a random variable X is
f(x)=ke^-|x|, -∞<x< ∞
Find the value of k and also find mean and variance for the random variable.

The mean life time of a sample of 169 light bulbs manufactured by a company is found to be 1350 hours with a standard deviation of 100 hours. Establish 90% confidence limits within which the mean life time of light bulbs is expected to lie.

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X	-2	3	1
P (X = x)	\(\frac { k }{ 6 }\)	\(\frac { k }{ 4 }\)	\(\frac { k }{ 12 }\)

Sample number	1	2	3	4	5	6	7	8	9	10
\(\overset{-}{X}\)	43	49	37	44	45	37	51	46	43	47
R	5	6	5	7	7	4	8	6	4	6

23	15	75	48	59	01	83	72	59	93	76	24	97	08	86	95	23	03	67	44
05	54	55	50	43	10	53	74	35	08	90	61	18	37	44	10	96	22	13	43
14	87	16	03	50	32	40	43	62	23	50	05	10	03	22	11	54	36	08	34
38	97	67	49	51	94	05	17	58	53	78	80	59	01	94	32	42	87	16	95
97	31	26	17	18	99	75	53	08	70	94	25	12	58	41	54	88	21	05	13

Sample	1	2	3	4	5	6	7	8	9	10
Mean	383	508	505	582	557	337	514	614	707	753
Range	95	128	100	91	68	65	148	28	37	80

X	0	1	2	3
P(X)	\(\frac { 1 }{ 5 } \)	\(\frac { 2 }{ 5 } \)	\(\frac { 1 }{ 5 } \)	\(\frac { 1 }{ 5 } \)

Y	0	1	2	3
P(Y)	\(\frac { 1 }{ 5 } \)	\(\frac{3}{10}\)	\(\frac { 2 }{ 5 } \)	\(\frac{1}{10}\)

Commodities	Price		Quandity
Commodities	2000	2010	2000	2010
Rice	38	35	6	7
Wheat	12	18	7	10
Rent	10	15	10	15
Fuel	25	30	12	16
Miscellaneous	30	33	8	10

23	15	75	48	59	01	83	72	59	93	76	24	97	08	86	95	23	03	67	44
05	54	55	50	43	10	53	74	35	08	90	61	18	37	44	10	96	22	13	43
14	87	16	03	50	32	40	43	62	23	50	05	10	03	22	11	54	36	08	34
38	97	67	49	51	94	05	17	58	53	78	80	59	01	94	32	42	87	16	95
97	31	26	17	18	99	75	53	08	70	94	25	12	58	41	54	88	21	05	13

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12th Standard English Medium

12th Standard Tamil Medium

11th Standard English Medium

11th Standard Tamil Medium

10th Standard English Medium

9th Standard English Medium

8th Standard English Medium

7th Standard English Medium

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11Th Standard

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9Th Standard

12th Business Maths Model Exam -2

St. Britto Hr. Sec. School - Madurai

12th Business Maths Model Exam -2-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

Laspeyre’s index = 110, Paasche’s index = 108, then Fisher’s Ideal index is equal to:

For a normal distribution if the mean is m, mode is n and median is m, then

In 8 throws of a die, 1 or 3 is considered a success. Then the mean number of successes is

Cape town is estimated to have 21% of homes whose owners subscribe to the satellite service, DSTV. If a random sample of your home is taken, what is the probability that all four home subscribe to DSTV?

In a non – degenerate solution number of allocations is

A statistical analysis of long-distance telephone calls indicates that the length of these calls is normally distributed with a mean of 240 seconds and a standard deviation of 40 seconds. What proportion of calls lasts less than 180 seconds?

The probability density function p(x) cannot exceed

A die is tossed 5 times. Getting an odd number is considered a success. Then the variance of distribution of number of success is

The standard normal distribution is represented by

Δf(x) =

Value which is obtained by multiplying possible values of random variable with probability of occurrence and is equal to weighted average is called

If n > 0, then \(\Gamma \)(n) is

The solution of the differential equation \(\frac { dy }{ dx } \) + Py = Q where P and Q are the function of x is

The P.I of (3D2 + D − 14)y = 13e2x is

A discrete probability distribution may be represented by

\(\overset { \infty }{ \underset { -\infty }{ \int } } \)f(x)dx is always equal to

Which of the following statements is/are true regarding the normal distribution curve?

E[X-E(X)] is equal to

Part - B

Generic 2 - Answer All Question

The price elasticity of demand for a commodity is \(\frac { p }{ { x }^{ 3 } } \). Find the demand function if the quantity of demand is 3, when the price is Rs. 2

Evaluate Δ\(\left[ \frac { 1 }{ (x+1)(x+2) } \right] \) by taking ‘1’ as the interval of differencing

A sample of 1000 students whose mean weight is 119 lbs(pounds) from a school in Tamil Nadu State was taken and their average weight was found to be 120 lbs with a standard deviation of 30 lbs. Calculate standard error of mean.

X is normally distributed with mean 12 and sd 4. Find P(X \(\le\) 20) and P(0 \(\le\) X \(\le\) 12)

A random variable X has the probability mass function X -2 3 1 P (X = x) \(\frac { k }{ 6 }\) \(\frac { k }{ 4 }\) \(\frac { k }{ 12 }\) then find k

What is statistic?

Define Bernoulli trials.

Find the order and degree of the following differential equations. \({ \left( \frac { dy }{ dx } \right) }^{ 3 }+y=x-\frac { dx }{ dy } \)

Solve the following equation by using Cramer’s rule 2x + y −z = 3, x + y + z =1, x− 2y− 3z = 4

Solve yx2dx + e−xdy = 0

State the uses of Index Number.

Define random variable.

X is a normally normally distributed variable with mean μ = 30 and standard deviation σ = 4. Find (a) P(x < 40) (b) P(x > 21) (c) P(30 < x < 35)

Part - C

Generic 3 - Answer 3 Questions

If a random variable. X has the probability distribution X 0 1 2 3 4 5 P (X = x) a 2a 3a 4a 5a 6a then find F(4)

If X is a normal variate with mean 30 and SD 5. Find the probabilities that (i) 26≤X≤40 (ii) X > 45

The following data gives readings of 10 samples of size 6 each in the production of a certain product. Draw control chart for mean and range with its control limits. Sample 1 2 3 4 5 6 7 8 9 10 Mean 383 508 505 582 557 337 514 614 707 753 Range 95 128 100 91 68 65 148 28 37 80

Part - D

Generic 5 - Answer All Question

The mean weight of 500 male students in a certain college is 151 pounds and the S.D is 15 pounds. Assuming the weights are normally distributed, find how many students weight (i) between 120 and 155 pounds (ii) more than 185 pounds.

A sample of 100 measurements at breaking strength of cotton thread gave a mean of 7.4 and a standard deviation of 1.2 gms. Find 95% confidence limits for the mean breaking strength of cotton thread.

the Laspeyre’s, Paasche’s and Fisher’s price index number for the following data. Interpret on the data. Commodities Price Quandity 2000 2010 2000 2010 Rice 38 35 6 7 Wheat 12 18 7 10 Rent 10 15 10 15 Fuel 25 30 12 16 Miscellaneous 30 33 8 10

Four coins are tossed simultaneously. What is the probability of getting a) atleast 2 heads b) atmost 2 heads.

Consider a random variable X with probability density function f(x)={\({ 4x }^{ 2 },\quad if\quad 0<x<1\\ 0,\quad otherwise\) Find E(X) and V(X)

Determine the mean and variance of the random variable Xhaving the following probability distribution. X=x 1 2 3 4 5 6 7 8 9 10 P(x) 0.15 0.10 0.10 0.01 0.08 0.01 0.05 0.02 0.28 0.20

The probability density function of a random variable X is f(x)=ke-|x|, -∞<x< ∞ Find the value of k and also find mean and variance for the random variable.

The mean life time of a sample of 169 light bulbs manufactured by a company is found to be 1350 hours with a standard deviation of 100 hours. Establish 90% confidence limits within which the mean life time of light bulbs is expected to lie.

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The P.I of (3D²+ D − 14)y = 13e^2x is

A random variable X has the probability mass function

X -2 3 1

P (X = x) \(\frac { k }{ 6 }\) \(\frac { k }{ 4 }\) \(\frac { k }{ 12 }\)

then find k

Find the order and degree of the following differential equations.
\({ \left( \frac { dy }{ dx } \right) }^{ 3 }+y=x-\frac { dx }{ dy } \)

Solve the following equation by using Cramer’s rule
2x + y −z = 3, x + y + z =1, x− 2y− 3z = 4

Solve yx²dx + e^−xdy = 0

X is a normally normally distributed variable with mean μ = 30 and standard deviation σ = 4. Find
(a) P(x < 40)
(b) P(x > 21)
(c) P(30 < x < 35)

If a random variable. X has the probability distribution

X 0 1 2 3 4 5

P (X = x) a 2a 3a 4a 5a 6a

then find F(4)

If X is a normal variate with mean 30 and SD 5. Find the probabilities that
(i) 26≤X≤40
(ii) X > 45

The following data gives readings of 10 samples of size 6 each in the production of a certain product. Draw control chart for mean and range with its control limits.

Sample 1 2 3 4 5 6 7 8 9 10

Mean 383 508 505 582 557 337 514 614 707 753

Range 95 128 100 91 68 65 148 28 37 80

The mean weight of 500 male students in a certain college is 151 pounds and the S.D is 15 pounds. Assuming the weights are normally distributed, find how many students weight
(i) between 120 and 155 pounds
(ii) more than 185 pounds.

the Laspeyre’s, Paasche’s and Fisher’s price index number for the following data. Interpret on the data.

Commodities Price Quandity

2000 2010 2000 2010

Rice 38 35 6 7

Wheat 12 18 7 10

Rent 10 15 10 15

Fuel 25 30 12 16

Miscellaneous 30 33 8 10

Four coins are tossed simultaneously. What is the probability of getting
a) atleast 2 heads b) atmost 2 heads.

Consider a random variable X with probability density function

f(x)={\({ 4x }^{ 2 },\quad if\quad 0<x<1\\ 0,\quad otherwise\)
Find E(X) and V(X)

Determine the mean and variance of the random variable Xhaving the following probability distribution.

X=x 1 2 3 4 5 6 7 8 9 10

P(x) 0.15 0.10 0.10 0.01 0.08 0.01 0.05 0.02 0.28 0.20

The probability density function of a random variable X is
f(x)=ke^-|x|, -∞<x< ∞
Find the value of k and also find mean and variance for the random variable.

23	15	75	48	59	01	83	72	59	93	76	24	97	08	86	95	23	03	67	44
05	54	55	50	43	10	53	74	35	08	90	61	18	37	44	10	96	22	13	43
14	87	16	03	50	32	40	43	62	23	50	05	10	03	22	11	54	36	08	34
38	97	67	49	51	94	05	17	58	53	78	80	59	01	94	32	42	87	16	95
97	31	26	17	18	99	75	53	08	70	94	25	12	58	41	54	88	21	05	13