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12th Standard English Medium Business Maths Reduced Syllabus Annual Exam Model Question Paper with Answer Key - 2021

12th Standard

    Reg.No. :
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Business Maths

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

        Part-I

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


    20 x 1 = 20
  1. In a transition probability matrix, all the entries are greater than or equal to _______.

    (a)

    2

    (b)

    1

    (c)

    0

    (d)

    3

  2. The rank of an n x n matrix each of whose elements is 2 is __________

    (a)

    1

    (b)

    2

    (c)

    n

    (d)

    n2

  3. The value of \(\int _{ -\frac{\pi}{2}}^{ \frac{\pi}{2}}\) cos x dx is _______.

    (a)

    0

    (b)

    2

    (c)

    1

    (d)

    4

  4. The value of \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ cosx } { e }^{ sinx }dx=\)

    (a)

    1

    (b)

    e-1

    (c)

    0

    (d)

    -1

  5. Area bounded by the curve y = \(\frac{1}{x}\) between the limits 1 and 2 is ________.

    (a)

    log 2 sq.units

    (b)

    log 5 sq.units

    (c)

    log 3 sq.units

    (d)

    log 4 sq.units

  6. The area of the region bounded by the line 2y = -x + 8, X - axis and the lines x = 2 and x = 4 is ________ sq.units.

    (a)

    \(\frac{1}{5}\)

    (b)

    \(\frac{2}{5}\)

    (c)

    5

    (d)

    \(\frac{5}{2}\)

  7. Which of the following is the homogeneous differential equation?

    (a)

    (3x−5)dx = (4y−1)dy

    (b)

    xy dx−(x3+y3)dy = 0

    (c)

    y2dx+(x− xy  − y2)dy = 0

    (d)

    (x2+y)dx = (y2+x)dy

  8. Solution of \(\frac { dx }{ dy } \)+mx = 0 where m<0 is _______

    (a)

    x = cemy

    (b)

    x = ce-my

    (c)

    x = my + c

    (d)

    x = c

  9. If h = 1, then Δ(x2) = _______.

    (a)

    2x

    (b)

    2x −1

    (c)

    2x +1

    (d)

    1

  10. ∇f(x+ 3h) ______________

    (a)

    f{x + 2h)

    (b)

    f(x + 3h) - f(x + 2h)

    (c)

    f(x + 3h)

    (d)

    f(x + 2h) - f(x - 3h)

  11. An expected value of a random variable is equal to it’s ________.

    (a)

    variance

    (b)

    standard deviation

    (c)

    mean

    (d)

    covariance

  12. If X is a discrete random variable then which of the following is correct?

    (a)

    0 ≤ F(x) ≤ 1

    (b)

    F(-∞) = 0,F(∞) ≤ 1

    (c)

    P(X = Xn) = F(Xn)-F(Xn-1)

    (d)

    F(x) is a constant function

  13. An experiment succeeds twice as often as it fails. The chance that in the next six trials, there shall be at least four successes is ________.

    (a)

    240/729

    (b)

    489/729

    (c)

    496/729

    (d)

    251/729

  14. The area under the standard normal curve between Z=-∞ and z=∞ is

    (a)

    0

    (b)

    0.5

    (c)

    1

    (d)

    0.75

  15. In simple random sampling from a population of N units, the probability of drawing any unit at the first draw is  ______.

    (a)

    \(\frac{n}{N}\)

    (b)

    \(\frac{1}{N}\)

    (c)

    \(\frac{N}{n}\)

    (d)

    1

  16. An _______ is a specific observed value of a statistic

    (a)

    Estimation

    (b)

    Estimator

    (c)

    Estimate

    (d)

    Testing of hypothesis

  17. The additive model of the time series with the components T, S, C and I is ________.

    (a)

    y = T + S + C × I

    (b)

    y = T + S × C × I

    (c)

    y = T + S + C + I

    (d)

    y = T + S × C + I

  18. Chance variation does not affect _____ of the product

    (a)

    price

    (b)

    value

    (c)

    quantity

    (d)

    quality

  19. The Penalty in VAM represents difference between the first ________.

    (a)

    Two largest costs

    (b)

    Largest and Smallest costs

    (c)

    Smallest two costs

    (d)

    None of these

  20. _____determines the lowest out comes for each alternative.

    (a)

    Least cost

    (b)

    Minimax criteria

    (c)

    Maximin criteria

    (d)

    Payoff matrix

    1. Part-II

      Answer any seven questions and Question number 30 is compulsory.


    7 x 2 = 14
  21. If A and B are non-singular matrices, prove that AB is non-singular.

  22. Integrate the following with respect to x
    \(\frac { 1 }{ { 9-16x }^{ 2 } } \)

  23. If the marginal revenue for a commodity is MR = 9 - 6x2 + 2x, find the total revenue function.

  24. Find the differential equation of the following
    x2 + y2 = a2

  25. If f(0) = 5, f(1) = 6, f(3) = 50, find f(2) by using Lagrange's formula.

  26. The time to failure in thousands of hours of an important piece of electronic equipment used in a manufactured DVD player has the density function
    \(f(x)= \begin{cases}2 e^{-2 x}, & x>0 \\ 0,& \text { otherwise }\end{cases}\)
    Find the expected life of this piece of equipment.

  27. If 10 coins are tossed, find the probability that exactly 5 heads appears.

  28. State any two demerits of systematic random sampling.

  29. Calculate the cost of living index by aggregate expenditure method

    Commodity Quantity Price(Rs.)
      2000 2000 2003
    A 100 8 12
    B 25 6 7.50
    C 10 5 5.25
    D 20 48 52
    E 65 15 16.50
    F 30 19 27.00
  30. What do you mean by balanced transportation problem?

      1. Part-III

        Answer any seven questions and Question number 40 is compulsory.


    7x 3 = 21
  31. Find the rank of each of the following matrices.
    \(\left( \begin{matrix} -1 & 2 & -2 \\ 4 & -3 & 4 \\ -2 & 4 & -4 \end{matrix} \right) \)

  32. Evaluate  \(\int { \frac { { { x }^{ 4 }+{ x }^{ 4 }+1 } }{ { x }^{ 2 }-x+1 } } \)

  33. If the marginal revenue function for a commodity is MR = 9 − 4x2. Find the demand function.

  34. Form the differential equation for \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \)=1 where a & b are arbitrary constants.

  35. A second degree polynomial passes though the point (1,-1) (2,-1) (3,1) (4,5). Find the polynomial.

  36. The probability distribution of a discrete random variable. X is given by

    X -2 2 5
    P(X=x) \(\frac{1}{4}\) \(\frac{1}{4}\) \(\frac{1}{2}\)

    then find 4E(X2)- Var (2X)

  37. A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of 2 successes.

  38. A random sample of 500 apples was taken from large consignment and 45 of them were found to be bad. Find the limits at which the bad apples lie at 99% confidence level.

  39. 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
  40. 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
      1. Part-IV

        Answer all the questions.


    7x 5 = 35
    1. The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine while others do not. From this mailing list, 45% of those who already subscribe will subscribe again while 30% of those who do not now subscribe will subscribe. On the last letter, it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current letter can be expected to order a subscription?

    2. Evaluate \(\int _{ 1 }^{ 4 }{ f(x) } dx\), where f(x) = \(\begin{cases} 7x+3,if \ 1\le x\le 3 \\ 8x,if \ 3\le x\le 4 \end{cases}\)

    1. Prove that \(\int _{ a }^{ b }{ \frac { f\left( x \right) }{ f\left( x \right) +f(a+b-x) } } dx=\frac { b-a }{ 2 } \)

    2. The marginal cost and marginal revenue with respect to commodity of a firm are given by C'(x) = 8 + 6x and R'(x)= 24. Find the total Profit given that the total cost at zero output is zero.

    1. Solve: x2\(\frac { dy }{ dx } \) = y2+2xy given that y = 1, when x = 1

    2. Using Newton’s formula for interpolation estimate the population for the year 1905 from the table:

      Year 1891 1901 1911 1921 1931
      Population 98.752 1,32,285 1,68,076 1,95,690 2,46,050
    1. A continuous random variable X has p.d.f
      f(x) = 5x4, 0\(\le\)x\(\le\)
      Find a1 and a2 such that
      i) P[X\(\le\)a1] = P[X>a1]   
      ii) P[X>a2] = 0.05

    2. An insurance company has discovered that only about 0.1 per cent of the population is involved in a certain type of accident each year. If its 10,000 policy holders were randomly selected from the population, what is the probability that not more than 5 of its clients are involved in such an accident next year? (e−10=.000045)

    1. If the height of 300 students are normally distributed with mean 64.5 inches and standard deviation 3.3 inches find the height below which 99% of the student lie?

    2. 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?

    1. 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.

    2. Construct the cost of living index number for 2011 on the basis of 2007 from the given data using family budget method.

      Commodities Price Weights
      2007 2011
      A 350 400 40
      B 175 250 35
      C 100 115 15
      D 75 105 20
      E 60 80 25
    1. Consider the problem of assigning five jobs to five persons. The assignment costs are given as follows. Determine the optimum assignment.

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