Half Yearly Model Question Paper 2019

12th Standard EM

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Business Maths

Time : 02:30:00 Hrs
Total Marks : 90

    Part A

    Answer all the questions.

    Choose the most suitable answer from the given four alternatives and write the option code with the corresponding answer.

    20 x 1 = 20
  1. \(\left| { A }_{ n\times n } \right| \)=3 \(\left| adjA \right| \) =243 then the value n is

    (a)

    4

    (b)

    5

    (c)

    6

    (d)

    7

  2. For what value of k, the matrix \(A=\left( \begin{matrix} 2 & k \\ 3 & 5 \end{matrix} \right) \) has no inverse?

    (a)

    \(\cfrac { 3 }{ 10 } \)

    (b)

    \(\cfrac { 10 }{ 3 } \)

    (c)

    3

    (d)

    10

  3. If \(\int _{ 0 }^{ 1 }{ f(x) } dx=1,\int _{ 0 }^{ 1 }{ xf(x) } dx=a\) and \(\int _{ 0 }^{ 1 }{ { x }^{ 2 }f(x) } dx={ a }^{ 2 }\), then \(\int _{ 0 }^{ 1 }{ { (a-x) }^{ 2 } } f(x)\) is

    (a)

    4a2

    (b)

    0

    (c)

    2a2

    (d)

    1

  4. \(\int _{ 0 }^{ \infty }{ { x }^{ 4 }{ e }^{ -x } } \)dx is

    (a)

    12

    (b)

    4

    (c)

    4!

    (d)

    64

  5. If \(\int { \frac { 1 }{ \left( x+2 \right) \left( { x }^{ 2 }+1 \right) } } \) dx = a log \(\left| 1+{ x }^{ 2 } \right| \) +b tan-1 x + \(\frac { 1 }{ 5 } log\left| x+2 \right| \) +c then

    (a)

    \(a=-\frac { 1 }{ 10 } ,b=\frac { -2 }{ 5 } \)

    (b)

    \(a=\frac { 1 }{ 10 } ,b=\frac { -2 }{ 5 } \)

    (c)

    \(a=-\frac { 1 }{ 10 } ,b=\frac { 2 }{ 5 } \)

    (d)

    \(a=\frac { 1 }{ 10 } ,b=\frac { 2 }{ 5 } \)

  6. The area lying above the X-axis and under the parabola y = 4x - x2 is ______ sq. units

    (a)

    \(\frac{16}{3}\)

    (b)

    \(\frac{8}{3}\)

    (c)

    \(\frac{32}{3}\)

    (d)

    \(\frac{64}{3}\)

  7. The solution of \(\frac { dy }{ dx } \) =ex-y is

    (a)

    eyex = c

    (b)

    y=log cex

    (c)

    y=log(ex+c)

    (d)

    ex+y = c

  8. Δ2y0 =

    (a)

    y−2y+ y0

    (b)

    y+ 2y− y0

    (c)

    y2 + 2y1 + y0

    (d)

    y+ y+ 2y0

  9. Lagrange’s interpolation formula can be used for

    (a)

    equal intervals only

    (b)

    unequal intervals only

    (c)

    both equal and unequal intervals

    (d)

    none of these.

  10. Newton's forward interpolation formula is used when the value of y is required near the ______ of the, table

    (a)

    end

    (b)

    beginning

    (c)

    left

    (d)

    right

  11. If X is a discrete random variable and p x ( ) is the probability of X , then the expected value of this random variable is equal to

    (a)

    \(\sum { f(x) } \)

    (b)

    \(\sum { [x+f(x)] } \)

    (c)

    \(\sum { f(x)+x } \)

    (d)

    \(\sum { xp(x) } \)

  12. If the p.d.f of a continuous random variable. X is \(f(x)=\begin{cases} \frac { x }{ 2 } ,0<x<2 \\ 0,\quad elsewhere \end{cases}\) then E(3X2-2X)=

    (a)

    \(\frac{2}{3}\)

    (b)

    \(\frac{4}{3}\)

    (c)

    \(\frac{10}{3}\)

    (d)

    \(\frac{7}{3}\)

  13. If F(x) is the probability distribution function, then F(- ∞) is_______.

    (a)

    1

    (b)

    2

    (c)

    (d)

    0

  14. If X ~N(μ, σ2), the maximum probability at the point of inflexion of normal distribution is

    (a)

    \({ \left( \frac { 1 }{ \sqrt { 2\pi } } \right) }^{ { e }^{ \frac { 1 }{ 2 } } }\)

    (b)

    \({ \left( \frac { 1 }{ \sqrt { 2\pi } } \right) }^{ { e }^{ \left( -\frac { 1 }{ 2 } \right) } }\)

    (c)

    \({ \left( \frac { 1 }{ \sigma \sqrt { 2\pi } } \right) }^{ { e }^{ \left( \frac { 1 }{ 2 } \right) } }\)

    (d)

    \({ \left( \frac { 1 }{ \sqrt { 2\pi } } \right) }\)

  15. The point estimate mean of the following data is __________.
    21.1, 25.0, 20.0, 16.0, 12.0, 10.0, 17.0, 18.0, 13.0,11.0

    (a)

    16.3

    (b)

    13.6

    (c)

    21.21

    (d)

    212:10

  16. Any hypothesis which is complementary to the null hypothesis is _______ hypothesis.

    (a)

    Null

    (b)

    Alternative

    (c)

    Statistical

    (d)

    testing

  17. Least square method of fitting a trend is

    (a)

    Most exact

    (b)

    Least exact

    (c)

    Full of subjectivity

    (d)

    Mathematically unsolved

  18. Cyclic variations in a time series are caused by

    (a)

    Lock out in a factor

    (b)

    war

    (c)

    floods

    (d)

    none of above

  19. Variation due to assignable causes in the product occur due to, _____

    (a)

    faulty process

    (b)

    carelessness of operators

    (c)

    poor quality of raw material

    (d)

    all the above.

  20. The methods of funding feasible solution to a transportation problem

    (a)

    North West Corner Rule

    (b)

    Least Cost Method

    (c)

    Hungarian Method

    (d)

    Vogel's Approximation Method

  21. Part B

    Answer any 7 questions. Question no. 30 is compulsory.

    7 x 2 = 14
  22. Solve: 2x + 3y = 4 and 4x + 6y = 8 using Cramer's rule.

  23. Calculate the producer’s surplus at x = 5 for the supply function p = 7 + x.

  24. Find the area of the region bounded by the parabola x2 = 4y, Y = 2, Y = 4 and the y-axis.

  25. Solve: x dy +y dx = 0

  26. When h = 1, find Δ (x3).

  27. Find the mean for the probability density function \(f(x)=\begin{cases} \frac { 1 }{ 24 } ,-12\le x\le 12 \\ 0,\quad otherwise \end{cases}\)

  28. A car hiring firm has two cars. The demand for cars on each day is distributed as a Poisson variate, with mean 1.5. Calculate the proportion of days on which
    (i) Neither car is used
    (ii) Some demand is refused

  29. A sample of 400 students is found to have mean height of 171.38 cms, Can it reasonable be regarded as a sample from a large population with mean height of 171.17 cms and standard deviation of 3.3 cms (Test at 5% level)

  30. Calculate the 3-yearlymoving averages of the production figures (in tonnes) for the following data.
     

    Year 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987
    Production 15 21 30 36 42 46 50 56 63 70 74 82 90 95 102

     

  31. For the given pay-off matrix, find the optimal decision under the minimax principle.

  32. Part C

    Answer any 7 questions. Question no. 40 is compulsory.

    7 x 3 = 21
  33. Solve: 2x - 3y - 1 = 0, 5x + 2y - 12 = 0 by Cramer's rule.

  34. Find the area under the demand curve xy = 1 bounded by the ordinates x = 3, x = 9 and x-axis

  35. Form the differential equation for y=(A+Bx)e3x where A and B are constants.

  36. If y75 = 2459, y50 = 2018, y85 = 1180, and y90 =402, find y82

    x 75 80 85 90
    y 2459 2018 1180 402
  37. If you toss a fair coin three times, the outcome of an experiment consider as random variable which counts the number of heads on the upturned faces. Find out the probability mass function and check the properties of the probability mass function.

  38. An urn contains 4 white and 6 red balls. Four balls are drawn at random from the urn. Find the probability distribution of the number of white balls.

  39. The standard deviation of a binomial distribution (q +p)16 is 2. Find its mean.

  40. A sample of 100 students is chosen from a large group of students. The average height of these students is 162 cm and standard deviation (S.D) is 8 cm. Obtain the standard error for the average height of large group of students of 160 cm?

  41. Fit a trend line to the following data by graphic  method.

    Year 1978 1979 1980 1981 1982 1983 1984 1985 1986
    Production of steel 20 22 24 21 23 25 23 26 25

     

  42. For the given pay-off matrix, choose the best alternative for the given states of nature under
    (i) Maximin (ii) Minimax princple

    Alternative States of Nature
      Good Fair Bad
    A 100 60 +50
    B 80 50 +10
    C 40 20 +5

     

  43. Part D

    Answer all the questions.

    7 x 5 = 35
  44. A new transit system has just gone into operation in a city. Of those who use the transit system this year, 10% will switch over to using their own car next year and 90% will continue to use the transit system. Of those who use their cars this year, 80% will continue to use their cars next year and 20% will switch over to the transit system. Suppose the population of the city remains constant and that 50% of the commuters use the transit system and 50% of the commuters use their own car this year,
    (i) What percent of commuters will be using the transit system after one year?
    (ii) What percent of commuters will be using the transit system in the long run?

  45. Evaluate \(\int _{ 0 }^{ \infty }{ { e }^{ -\frac { x }{ 2 } } } dx\)

  46. Evaluate ഽ x3 sin (x4) dx

  47. Solve (x2 + 1)\(\frac { dy }{ dx } \) + 2xy = 4x

  48. The net profit p and quantity x satisfy the differential equation \(\frac { dp }{ dx } =\frac { 2{ p }^{ 3 }-{ x }^{ 3 } }{ 3x{ p }^{ 2 } } \). Find the relationship between the net profit and demand given that p = 20, when x = 10.

  49. From the following data, calculate the value of e1.75

    x 1.7 1.8 1.9 2.0 2.1
    ex 5.474 6.050 6.686 7.386 8.166
  50. The following information is the probability distribution of successes.

    No. of Successes 0 1 2
    Probability \(\frac{6}{11}\) \(\frac{9}{22}\) \(\frac{1}{22}\)

    Determine the expected number of success.

  51. The probability distribution of a random variation X is given below.

    X 0 1 2 3 4
    P(X) 0.1 0.25 0.3 0.2 0.15

    Find (i) V(X)
    ii) V\((\frac{X}{2})\)

  52. Assuming one in 80 births is a case of twins, calculate the probability of 2 or more sets of twins on a day when 30 births occur.

  53. Marks in an aptitude test given to 800 students of a school was found to be normally distributed 10% of the students scored below 40 marks and 10% of the students scored above 90 marks. Find the number of students scored between 40 and 90?

  54. A manufacturer of ball pens claims that a certain pen he manufactures has a mean writing life of 400 pages with a standard deviation of 20 pages. A purchasing agent selects a sample of 100 pens and puts them for test. The mean writing life for the sample was 390 pages. Should the purchasing agent reject the manufactures claim at 1% level?

  55. Measurements of the weights of a random sample of 200 ball bearings made by certain machine during one week showed a mean of 0.824 newtons and a S.D. of 0.042 newton's. Find a) 95% and b) 99% confidence limits for the mean weight of all the ball bearings.

  56. Calculate Fisher’s price index number and show that it satisfies both Time Reversal Test and Factor Reversal Test for data given below.

    Commodities Base Year Current Year
    Price Quantity Price Quantity
    Rice 150 5 11 6
    Wheat  12 6 13 4
    Rent 14 8 15 7
    Fuel 16 9 17 8
    Transport 18 7 19 5
    Miscellaneous 20 4 21 3
  57. Solve the following assignment problem.

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